Composite Materials Research Progress

COMPOSITE MATERIALS
RESEARCH PROGRESS

COMPOSITE MATERIALS
RESEARCH PROGRESS
LUCAS P. DURAND
EDITOR
Nova Science Publishers, Inc.
New York

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Composite materials research progress / Lucas P. Durand, Editor.
p. cm.
Includes index.
ISBN-13: 978-1-60692-496-9
1. Composite materials. I. Durand, Lucas P.
TA418.9.C6C594 2008
620.1'18--dc22 2007034054
Published by Nova Science Publishers, Inc. � New York

CONTENTS
Preface vii
Chapter 1 Multi-scale Analysis of Fiber-Reinforced Composite Parts
Submitted to Environmental and Mechanical Loads
Jacquemin Frédéric and Fréour Sylvain
Chapter 2 Optimization of Laminated Composite Structures: Problems,
Solution Procedures and Applications
Michaël Bruyneel
Chapter 3 Major Trends in Polymeric Composites Technology 109
W.H. Zhong, R.G. Maguire, S.S. Sangari and P.H. Wu
Chapter 4 An Experimental and Analytical Study of Unidirectional
Carbon Fiber Reinforced Epoxy Modified
by SiC Nanoparticle
Yuanxin Zhou, Hassan Mahfuz, Vijaya Rangari
and Shaik Jeelani
Chapter 5 Damage Evaluation and Residual Strength Prediction of CFRP
Laminates by Means of Acoustic Emission Techniques
Giangiacomo Minak and Andrea Zucchelli
Chapter 6 Research Directions in the Fatigue Testing of Polymer
Composites
W. Van Paepegem, I. De Baere, E. Lamkanfi,
G. Luyckx and J. Degrieck
Chapter 7 Damage Variables in Impact Testing
of Composite Laminates
Maria Pia Cavatorta and Davide Salvatore Paolino
Chapter 8 Electromechanical Field Concentrations and Polarization
Switching by Electrodes in Piezoelectric Composites
Yasuhide Shindo and Fumio Narita
1
51
129
165
209
237
257

vi
Contents
Chapter 9 Recent Advances in Discontinuously Reinforced Aluminum
Based Metal Matrix Nanocomposites
S.C. Tjong
Index 297
275

PREFACE
Composite materials are engineered materials made from two or more constituent
materials with significantly different physical or chemical properties and which remain
separate and distinct on a macroscopic level within the finished structure. Fiber Reinforced
Polymers or FRPs include Wood comprising (cellulose fibers in a lignin and hemicellulose
matrix), Carbon-fiber reinforced plastic or CFRP, Glass-fiber reinforced plastic or GFRP
(also GRP). If classified by matrix then there are Thermoplastic Composites, short fiber
thermoplastics, long fiber thermoplastics or long fiber reinforced thermoplastics There are
numerous thermoset composites, but advanced systems usually incorporate aramid fibre and
carbon fibre in an epoxy resin matrix.
Composites can also utilise metal fibres reinforcing other metals, as in Metal matrix
composites or MMC. Ceramic matrix composites include Bone (hydroxyapatite reinforced
with collagen fibers), Cermet (ceramic and metal) and Concrete. Organic matrix/ceramic
aggregate composites include Asphalt concrete, Mastic asphalt, Mastic roller hybrid, Dental
composite, Syntactic foam and Mother of Pearl. Chobham armour is a special composite used
in military applications. Engineered wood includes a wide variety of different products such
as Plywood, Oriented strand board, Wood plastic composite (recycled wood fiber in
polyethylene matrix), Pykrete (sawdust in ice matrix), Plastic-impregnated or laminated paper
or textiles, Arborite, Formica (plastic) and Micarta.
Composite materials have gained popularity (despite their generally high cost) in highperformance
products such as aerospace components (tails, wings , fuselages, propellors),
boat and scull hulls, and racing car bodies. More mundane uses include fishing rods and
storage tanks.
This new book presents the latest research from around the world.
The purpose of Chapter 1 is to present various application of statistical scale transition
models to the analysis of polymer-matrix composites submitted to thermo-hygro-mechanical
loads. In order to achieve such a goal, two approaches, classically used in the field of
modelling heterogeneous material are studied: Eshelby-Kröner self-consistent model on the
one hand and Mori-Tanaka approximate, on the second hand. Both models manage to handle
the question of the homogenization of the microscopic properties of the constituents (matrix
and reinforcements) in order to express the effective macroscopic coefficients of moisture
expansion, coefficients of thermal expansion and elastic stiffness of a uni-directionally
reinforced single ply. Inversion scale transition relations are provided also, in order to identify
the effective unknown behaviour of a constituent. The proposed method entails to inverse

viii
Lucas P. Durand
scale transition models usually employed in order to predict the homogenised macroscopic
elastic/hygroscopic/thermal properties of the composite ply from those of the constituents.
The identification procedure involves the coupling of the inverse scale transition models to
macroscopic input data obtained through either experiments or in the already published
literature. Applications of the proposed approach to practical cases are provided: in particular,
a very satisfactory agreement between the fitted elastic constants and the corresponding
properties expected in practice for the reinforcing fiber of typical composite plies is achieved.
Another part of this work is devoted to the extensive analysis of macroscopic mechanical
states concentration within the constituents of the plies of a composite structure submitted to
thermo-hygro-elastic loads. Both numerical and a fully explicit version of Eshelby-Kröner
model are detailed. The two approaches are applied in the viewpoint of predicting the
mechanical states in both the fiber and the matrix of composites structures submitted to a
transient hygro-elastic load. For this purpose, rigorous continuum mechanics formalisms are
used for the determination of the required time and space dependent macroscopic stresses.
The reliability of the new analytical approach is checked through a comparison between the
local stress states calculated in both the resin and fiber according to the new closed form
solutions and the equivalent numerical model: a very good agreement between the two
models was obtained.
The purpose of the final part of this work consists in the determination of microscopic
(local) quadratic failure criterion (in stress space) in the matrix of a composite structure
submitted to purely mechanical load. The local failure criterion of the pure matrix is deduced
from the macroscopic strength of the composite ply (available from experiments), using an
appropriate inverse model involving the explicit scale transition relations previously obtained
for the macroscopic stress concentration at microscopic level. Convenient analytical forms are
provided as often as possible, else procedures required to achieve numerical calculations are
extensively explained. Applications of this model are achieved for two typical carbon-fiber
reinforced epoxies: the previously unknown microscopic strength coefficients and ultimate
strength of the considered epoxies are identified and compared to typical expected values
published in the literature.
In Chapter 2 the optimal design of laminated composite structures is considered. A
review of the literature is proposed. It aims at giving a general overview of the problems that
a designer must face when he works with laminated composite structures and the specific
solutions that have been derived. Based on it and on the industrial needs an optimization
method specially devoted to composite structures is developed and presented. The related
solution procedure is general and reliable. It is based on fibers orientations and ply
thicknesses as design variables. It is used daily in an (European) industrial context for the
design of composite aircraft box structures located in the wings, the center wing box, and the
vertical and horizontal tail plane. This approach is based on sequential convex programming
and consists in replacing the original optimization problem by a sequence of approximated
sub-problems. A very general and self adaptive approximation scheme is used. It can consider
the particular structure of the mechanical responses of composites, which can be of a different
nature when both fiber orientations and plies thickness are design variables. Several
numerical applications illustrate the efficiency of the proposed approach.
As explained in Chapter 3, composites have been growing exponentially in technology
and applications for decades. The world of aerospace has been one of the earliest and
strongest proponents of advanced composites and the culmination of the recent advances in

Preface ix
composite technology are realized in the Boeing Model 787 with over 50% by weight of
composites, bringing the application of composites in large structures into a new age. This
mostly-composite Boeing 787 has been credited with putting an end to the era of the all-metal
airplane on new designs, and it is perhaps the most visible manifestation of the fact that
composites are having a profound and growing effect on all sectors of society.
It is generally well-known that composite materials are made of reinforcement fibers and
matrix materials, and light weight and high mechanical properties are the primary benefits of
a composite structure. Accordingly, the development trends in composite technology lie in 1)
new material technology specifically for developing novel fibers and matrices, enhancing
interfacial adhesion between fiber and matrix, hybridization and multi-functionalization, and
2) more reliable, high quality, rapid and low cost manufacturing technology.
New reinforcement fiber technology including next generation carbon fibers and organic
fibers with improved mechanical and physical properties, such as Spectra®, Dyneema®, and
Zylon®, have been developing continuously. More significantly, various nanotechnology
based novel fiber reinforcements have conspicuously and rapidly appeared in recent years.
Matrix materials have become as complex as the fibers, satisfying increasing demands for
impact resistant and damage tolerant structure. Various means of accomplishing this have
ranged from elastomeric/thermoplastic minor phases to discrete layers of toughened
materials. Nano-modified polymeric matrices are mostly involved in the development trends
for matrix polymer materials. Technology for enhancing the interfacial adhesion properties
between the reinforcement and matrix for a composite to provide high stress-transfer ability is
more critically demanded and the science of the interface is expanding. Fiber/matrix
interfacial adhesion is vital for the application of the newly developed advanced
reinforcement materials. Effective approaches to improving new and non-traditional
treatment methods for better adhesion have just started to receive sufficient attention. Multifunctionality
is also an important trend for advanced composites, in particular, utilizing
nanotechnology developments in recent years to provide greater opportunities for forcing
materials to play a more comprehensive role in the designs of the future.
More reliable and low cost manufacturing technology has been pursued by industry and
academic researchers and the traditional material forms are being replaced by those which
support the growing need for high quality, rapid production rates and lower recurring costs.
Major trends include the recognition of the value of resin infusion methods, automated
thermoplastic processing which takes advantage of the unique advantages of that material
class, and the value of moving away from dependence on the large and expensive autoclaves.
In Chapter 4, an innovative manufacturing process was developed to fabricate
nanophased carbon prepregs used in the manufacturing of unidirectional composite laminates.
In this technique, prepregs were manufactured using solution impregnation and filament
winding methods and subsequently consolidated into laminates. Spherical silicon carbide
nanoparticles (β-SiC) were first infused in a high temperature epoxy through an ultrasonic
cavitation process. The loading of nanoparticles was 1.5% by weight of the resin. After
infusion, the nano-phased resin was used to impregnate a continuous strand of dry carbon
fiber tows in a filament winding set-up. In the next step, these nanophased prepregs were
wrapped over a cylindrical foam mandrel especially built for this purpose using a filament
winder. Once the desired thickness was achieved, the stacked prepregs were cut along the
length of the cylindrical mandrel, removed from the mandrel, and laid out open to form a
rectangular panel. The panel was then consolidated in a regular compression molding

x
Lucas P. Durand
machine. In parallel, control panels were also fabricated following similar routes without any
nanoparticle infusion. Extensive thermal and mechanical characterizations were performed to
evaluate the performances of the neat and nano-phased systems. Thermo Gravimetric
Analysis (TGA) results indicate that there is an increase in the degradation temperature (about
7 0 C) of the nano-phased composites. Similar results from Differential Scanning Calorimetry
(DSC) and Dynamic Mechanical Analysis (DMA) tests were obtained. An improvement of
about 5 0 C in glass transition temperature (Tg) of nano-phased systems were also seen.
Mechanical tests on the laminates indicated improvement in flexural strength and stiffness by
about 32% and 20% respectively whereas in tensile properties there was a nominal
improvement between 7-10%. Finally, micro numerical constitutive model and damage
constitutive equations were derived and an analytical approach combining the modified shearlag
model and Monte Carlo simulation technique to simulate the tensile failure process of
unidirectional layered composites were also established to describe stress-strain relationships.
A new approach that integrates acoustic emission (AE) and the mechanical behaviour of
composite materials is presented in Chapter 5. Usually AE information is used to evaluate
qualitatively the damage progression in order to assess the structural integrity of a wide
variety of mechanical elements such as pressure vessels. From the other side, the mechanical
information, e.g. the stress-strain curve, is used to obtain a quantitative description of the
material behaviour. In order to perform a deeper analysis, a function that combines AE and
mechanical information is introduced. In particular, this function depends on the strain energy
and on the AE events energy, and it was used to study the behaviour of CFRP composite
laminates in different applications: (i) to describe the damage progression in tensile and
transversal load testing; (ii) to predict residual tensile strength of transversally loaded
laminates (condition that simulates a low velocity impact).
For a long time, fatigue testing of composites was only focused on providing the S-N
fatigue life data. No efforts were made to gather additional data from the same test by using
more advanced instrumentation methods. The development of methods such as digital image
correlation (strain mapping) and optical fibre sensing allows for much better instrumentation,
combined with traditional equipment such as extensometers, thermocouples and resistance
measurement. In addition, validation with finite element simulations of the realistic boundary
conditions and loading conditions in the experimental set-up must maximize the generated
data from one single fatigue test.
This research paper presents a survey of the authors’ recent research activities on fatigue
in polymer composites. For almost ten years now, combined fatigue testing and modelling has
been done on glass and carbon polymer composites with different lay-ups and textile
architectures. Chapter 6 wants to prove that a synergetic approach between instrumented
testing, detailed damage inspection and advanced numerical modelling can provide an answer
to the major challenges that are still present in the research on fatigue of composites.
Chapter 7 presents an overview of the damage variables proposed in the literature over
the years, including a new variable recently introduced by the Authors to specifically address
the problem of thick laminates subject to repeated impacts. Numerous impact data are used as
a basis to address and comment potentials and limitations of the different variables. Impact
data refer to single impact events as well as repeated impact tests performed on laminates
with different fiber and matrix combinations and various lay-ups. Laminates of different
thickness are considered, ranging from tenths to tens of millimeters.

Preface xi
The analysis shows that some of the variables can indeed be used for assessing the
damage tolerance of the laminate. In single impact tests, results point out the existence of an
energy threshold at about 40-50% of the penetration energy, below which the damage threat
is quite negligible. Other variables are not directly related to the amount of damage induced in
the laminate but rather give an indication of the laminate efficiency of energy absorption.
The electromechanical field concentrations due to electrodes in piezoelectric composites
are investigated through numerical and experimental characterization. Chapter 8 consists of
two parts. In the first part, a nonlinear finite element analysis is carried out to discuss the
electromechanical fields in rectangular piezoelectric composite actuators with partial
electrodes, by introducing models for polarization switching in local areas of the field
concentrations. Two criteria based on the work done by electromechanical loads and the
internal energy density are used. Strain measurements are also presented for a four layered
piezoelectric actuator, and a comparison of the predictions with experimental data is
conducted. In the second part, the electromechanical fields in the neighborhood of circular
electrodes in piezoelectric disk composites are reported. Nonlinear disk device behavior
induced by localized polarization switching is discussed.
Aluminum-based alloys reinforced with ceramic microparticles are attractive materials
for many structural applications. However, large ceramic microparticles often act as stress
concentrators in the composites during mechanical loading, giving rise to failure of materials
via particle cracking. In recent years, increasing demand for high performance materials has
led to the development of aluminum-based nanocomposites having functions and properties
that are not achievable with monolithic materials and microcomposites. The incorporation of
very low volume contents of ceramic reinforcements on a nanometer scale into aluminumbased
alloys yields remarkable mechanical properties such as high tensile stiffness and
strength as well as excellent creep resistance. However, agglomeration of nanoparticles
occurs readily during the composite fabrication, leading to inferior mechanical performance
of nanocomposites with higher filler content. Cryomilling and severe plastic deformation
processes have emerged as the two important processes to form ultrafine grained composites
with homogeneous dispersion of reinforcing particles. In Chapter 9, recent development in the
processing, structure and mechanical properties of the aluminum-based nanocomposites are
addressed and discussed.

In: Composite Materials Research Progress ISBN: 1-60021-994-2
Editor: Lucas P. Durand, pp. 1-50 © 2008 Nova Science Publishers, Inc.
Chapter 1
MULTI-SCALE ANALYSIS OF FIBER-REINFORCED
COMPOSITE PARTS SUBMITTED
TO ENVIRONMENTAL AND MECHANICAL LOADS
Jacquemin Frédéric and Fréour Sylvain *
GeM -Institut de Recherche en Génie Civil et Mécanique, Université de Nantes-Ecole
Centrale de Nantes-CNRS UMR 6183, 37 Boulevard de l’Université, BP 406,
44 602 Saint-Nazaire, France
Abstract
The purpose of this work is to present various application of statistical scale transition
models to the analysis of polymer-matrix composites submitted to thermo-hygro-mechanical
loads. In order to achieve such a goal, two approaches, classically used in the field of
modelling heterogeneous material are studied: Eshelby-Kröner self-consistent model on the
one hand and Mori-Tanaka approximate, on the second hand. Both models manage to handle
the question of the homogenization of the microscopic properties of the constituents (matrix
and reinforcements) in order to express the effective macroscopic coefficients of moisture
expansion, coefficients of thermal expansion and elastic stiffness of a uni-directionally
reinforced single ply. Inversion scale transition relations are provided also, in order to identify
the effective unknown behaviour of a constituent. The proposed method entails to inverse
scale transition models usually employed in order to predict the homogenised macroscopic
elastic/hygroscopic/thermal properties of the composite ply from those of the constituents.
The identification procedure involves the coupling of the inverse scale transition models to
macroscopic input data obtained through either experiments or in the already published
literature. Applications of the proposed approach to practical cases are provided: in particular,
a very satisfactory agreement between the fitted elastic constants and the corresponding
properties expected in practice for the reinforcing fiber of typical composite plies is achieved.
Another part of this work is devoted to the extensive analysis of macroscopic mechanical
states concentration within the constituents of the plies of a composite structure submitted to
thermo-hygro-elastic loads. Both numerical and a fully explicit version of Eshelby-Kröner
model are detailed. The two approaches are applied in the viewpoint of predicting the
mechanical states in both the fiber and the matrix of composites structures submitted to a
* E-mail address: sylvain.freour@univ-nantes.fr. Fax number : +33240172618. (Corresponding author)

2
Jacquemin Frédéric and Fréour Sylvain
transient hygro-elastic load. For this purpose, rigorous continuum mechanics formalisms are
used for the determination of the required time and space dependent macroscopic stresses. The
reliability of the new analytical approach is checked through a comparison between the local
stress states calculated in both the resin and fiber according to the new closed form solutions
and the equivalent numerical model: a very good agreement between the two models was
obtained.
The purpose of the final part of this work consists in the determination of microscopic
(local) quadratic failure criterion (in stress space) in the matrix of a composite structure
submitted to purely mechanical load. The local failure criterion of the pure matrix is deduced
from the macroscopic strength of the composite ply (available from experiments), using an
appropriate inverse model involving the explicit scale transition relations previously obtained
for the macroscopic stress concentration at microscopic level. Convenient analytical forms are
provided as often as possible, else procedures required to achieve numerical calculations are
extensively explained. Applications of this model are achieved for two typical carbon-fiber
reinforced epoxies: the previously unknown microscopic strength coefficients and ultimate
strength of the considered epoxies are identified and compared to typical expected values
published in the literature.
Keywords: scale transition modelling, homogenization, identification, polymer-matrix
composites.
1. Introduction
Carbon-reinforced epoxy based composites offer design, processing, performance and cost
advantages compared to metals for manufacturing structural parts. Among the advantages,
provided by carbon-reinforced epoxies over metals and ceramics, that have been recognised
for years, improved fracture toughness, impact resistance, strength to weight ratio as well as
high resistance to corrosion and enhanced fatigue properties have often been put in good use
for practical applications (Karakuzu et al., 2001).
Now, the accurate design and sizing of any structure requires the knowledge of the
mechanical states experienced by the material for the possibly various loads, expected to
occur during service life. Since high performance composites are being increasingly used in
aerospace and marine structural applications, where they are exposed to severe environmental
conditions, these composites experience hygrothermal loads as well as more classical
mechanical loads. Now, unlike metallic or ceramic materials, composites are susceptible to
both temperature and moisture when exposed to such working environments. These
environmental conditions are known to possibly induce sometimes critical stresses
distributions within the plies of the composite structures or even within their very constituents
(i.e. the reinforcements on the one hand and the matrix on the second hand). Actually,
carbon/epoxy composites can absorb significant amount of water and exhibit heterogeneous
Coefficients of Moisture Expansion (CME) and Coefficients of Thermal Expansion (CTE)
(i.e. the CME/CTE of the epoxy matrix are strongly different from the CME/CTE of the
carbon fibers, as shown in: Tsai, 1987; Agbossou and Pastor, 1997; Soden et al., 1998),
moreover, the diffusion of moisture in such materials is a rather slow process, resulting in the
occurrence of moisture concentration gradients within their depth, during at least the transient
stage (Crank, 1975). As a consequence, local stresses take place from hygro-thermal loading
of composite structures which closely depends on the experienced environmental conditions,
on the local intrinsic properties of the constituents and on its microstructure (the morphology

Multi-scale Analysis of Fiber-Reinforced Composite Parts… 3
of the constituents, the lay-up configuration, ... fall in this last category of factors). Now, the
knowledge of internal stresses is necessary to predict a possible damage occurrence in the
material during its manufacturing process or service life. Thus, the study of the development
of internal stresses due to thermo-hygro-elastic loads in composites is very important in
regard to any engineering application. Numerous papers, available in the literature, deal with
this question, using Finite Element Analysis or Continuum Mechanics-based formalisms.
These methods allow the calculation of the macroscopic stresses in each ply constituting the
composite (Jacquemin and Vautrin, 2002). But, they do not provide information on the local
mechanical states, in the fibers and matrix of a given ply, and, consequently, do not allow to
explain the phenomenon of matrix cracking and damage development in composite structures,
which originate at the microscopic level. The present work is precisely focused on the study
of the internal stresses in the constituents of the ply. In order to reach this goal, scale
transition models are required.
The present work underlines the potential of scale-transition models, as predictive tools,
complementary to continuum mechanics in order to address: i) the estimation of the effective
hygro-thermo-elastic properties of a composite ply from those of its constituents (section 2),
ii) the identification of the hygro-thermo-elastic properties of one constituent of a composite
ply (section 3), iii) the estimation of the local mechanical states experienced in each
constituent of a composite structure (section 4), iv) the identification of the local strength of
the constitutive matrix (section 5).
Section 6 of this paper is mainly dedicated to conclusions about the above listed sections
the whereas section 7 is devoted to the introducing some scientifically appealing perspectives
of research in the field of composites materials which are highly considered for further
investigation in the forthcoming years.
2. Scale-Transition Model for Predicting the Macroscopic
Thermo-Hygro-Elastic Properties of a Composite Ply
2.1. Introduction
Scale transition models are based on a multi-scale representation of materials. In the case of
composite materials, for instance, a two-scale model is sufficient:
- The properties and mechanical states of either the resin or its reinforcements are
respectively indicated by the superscripts m and r . These constituents define the socalled
“pseudo-macroscopic” scale of the material (Sprauel and Castex, 1991).
- Homogenisation operations performed over its aforementioned constituents are
assumed to provide the effective behaviour of the composite ply, which defines the
macroscopic scale of the model. It is denoted by the superscript I . This definition
also enables to consider an uni-directional reinforcement at macroscopic scale, which
is a satisfactorily realistic statement, compared to the present design of composite
structures (except for the particular case of woven-composites that will be
specifically discussed in section 7.1).

4
Jacquemin Frédéric and Fréour Sylvain
As for the composite structure, it is actually constituted by an assembly of the above
described composite plies, each of them possibly having the principal axis of their
reinforcements differently oriented from one to another. This approach enables to treat the
case of multi-directional laminates, as shown, for example, in (Fréour et al., 2005a).
2.2. The Classical Practical Strategy for the Direct Application of
Homogenisation Procedures
Within scale transition modeling, the local properties of the i−superscripted constituents are
usually considered to be known (i.e. the pseudo-macroscopic stiffnesses, L i , coefficients of
thermal expansion M i and coefficients of moisture expansion β i ), whereas the corresponding
effective macroscopic properties of the composite structure (respectively, L I , M I and β I ) are a
priori unknown and results from (often numerical) computations.
Among the numerous, available in the literature scale transition models, able to handle
such a problem, most involve rough-and-ready theoretical frameworks: Voigt (Voigt, 1928),
Reuss, (Reuss, 1929), Neerfeld-Hill (Neerfeld, 1942; Hill, 1952), Tsai-Hahn (Tsai and Hahn,
1980) and Mori-Tanaka (Mori and Tanaka, 1973; Tanaka and Mori, 1970) approximates fall
in this category. This is not satisfying, since such a model does not properly depict the real
physical conditions experienced in practice by the material. In spite of this lack of physical
realism, some of the aforementioned models do nevertheless provide a numerically satisfying
estimation of the effective properties of a composite ply, by comparison with the
experimental values or others, more rigorous models. Both Tsai-Hahn and Mori-Tanaka
models fulfil this interesting condition (Jacquemin et al., 2005; Fréour et al., 2006a).
Nevertheless, in the field of scale transition modelling, the best candidate remains Kröner-
Eshelby self-consistent model, because only this model takes into account a rigorous
treatment of the thermo-hygro-elastic interactions between the homogeneous macroscopic
medium and its heterogeneous constituents, as well as this model enables handling the
microstructure (i.e. the particular morphology of the constituents, especially that of the
reinforcements).
2.3. Estimating the Effective Properties of a Composite Ply through Eshelby-
Kröner Self-consistent Model
Self-consistent models based on the mathematical formalism proposed by Kröner (Kröner,
1958) constitute a reliable method to predict the micromechanical behavior of heterogeneous
materials. The method was initially introduced to treat the case of polycrystalline materials,
i.e. duplex steels, aluminium alloys, etc., submitted to purely elastic loads.
Estimations of homogenized elastic properties and related problems have been given in
several works (François, 1991; Mabelly, 1996; Kocks et al., 1998). The model was thereafter
extended to thermoelastic loads and gave satisfactory results on either single-phase (Turner
and Tome, 1994; Gloaguen et al., 2002) or two-phases (Fréour et al., 2003a and 2003b)
materials. More recently, this classical model was improved in order to take into account
stresses and strains due to moisture in carbon fiber-reinforced polymer–matrix composites.

Multi-scale Analysis of Fiber-Reinforced Composite Parts… 5
Therefore, the formalism was extent so that homogenisation relations were established for
estimating the macroscopic CME from those of the constituents (Jacquemin et al., 2005).
Many previously published documents have been dedicated to the determination of (at
least some of) the effective thermo-hygro-elastic properties of heterogeneous materials
through Kröner-Eshelby self-consistent approach (Kocks et al., 1998; Gloaguen et al., 2002;
Fréour et al., 2003a-b; Jacquemin et al., 2005). The main involved equations are:
( I + E
I
: [ L
i
− L
I ] )
L
I
= L
i
:
−1
(1)
i=
r, m
1
−1
i I I − I
i I I
( + L : R ) : L : ( L + L : R )
I 1
−1
i i i
β = L : L : β ΔC (2)
I
ΔC
i=
r, m
i=
r, m
1
−1
i I I − I
i I I
[ L + L : R ] : L : [ L + L : R ]
I
M =
−1
i i
: L : M
(3)
i=
r, m
i=
r, m
Where ΔC i is the moisture content of the studied i element of the composite structure.
The superscripts r and m are considered as replacement rule for the general superscript i, in
the cases that the properties of the reinforcements or those of the matrix have to be
considered, respectively. Actually, the pseudo-macroscopic moisture contents ΔC r and ΔC m
can be expressed as a function of the macroscopic hygroscopic load ΔC I (Loos and Springer,
1981), so that the hygro-mechanical states cancels in relation (2) that can finally be rewritten
as a function of the materials properties only, but at the exclusion of the ΔC i that are
unexpected to appear in such an expression (Jacquemin et al., 2005). Relation (2), that is
provided in the present work for predicting the macroscopic CME, is given for its enhanced
readability, compared to the more rigorous state exclusive relation.
In relations (1-3), the brackets < > stand for volume weighted averages (that in fact
replace volume integrals that would require Finite Elements Methods instead). Empirically, as
stated by Hill (Hill, 1952), arithmetic or geometric averages suggest themselves as good
approximations. On the one hand, the geometric mean of a set of positive data is defined as
the n th root of the product of all the members of the set, where n is the number of members.
On the other hand, in mathematics and statistics, the arithmetic mean (or simply the mean) of
a list of numbers is the sum of all the members of the list divided by the number of items in
the list. For Young’s modulus, as an example, the Geometric Average Y GA of the moduli
GA
according to the Reuss (YR) and Voigt (YV) models is defined as Y = YR
YV
,
whereas the corresponding Arithmetic Average Y AA AA Y Y
is: Y R +
= V
.
2

6
Jacquemin Frédéric and Fréour Sylvain
In statistics, given a set of data, X = { x1, x2, ..., xn} and corresponding weights, W = {
w1, w2, ..., wn}, the weighted geometric (respectively, arithmetic) mean
i
X
GA
i=
1,2,..., n
(respectively,
AA
i
X ) is calculated as:
i=
1,2,..., n
⎛ n ⎞
1/
⎜ ∑ w ⎟
GA ⎛ n ⎞ ⎜ i ⎟
i ⎜ w
X = ⎟ ⎝i=
1 ⎠
= ⎜∏
x i
i 1,2,..., n
i ⎟
⎝ i=
1 ⎠
∑
∑ =
i
X
AA 1 n
= xi
wi
i=
1,2,..., n n
w i 1
i
i=
1
(5)
Both averages have been extensively used in the field of materials science, in order to
achieve various scale transition modelling over a wide range of materials. The interested
reader can refer to: (Morawiec, 1989; Matthies and Humbert, 1993; Matthies et al., 1994) that
can be considered as typical illustrations of works taking advantage of the geometric average
for estimating the properties and mechanical states of polycrystals (nevertheless, Eshelby-
Kröner self-consistent model was not involved in any of these articles), whereas the
previously cited references (Kocks et al., 1998; Gloaguen et al., 2002; Fréour et al., 2003;
Jacquemin et al., 2005) show applications of arithmetic averages for studying of polycrystals
or composite structures.
According to equations (4) and (5), the explicit writing of a volume weighted average
directly depend on the averaging method chosen to perform this operation. Since the present
work aims to express analytical forms involving such volume averages, it is necessary to
select one average type in order to ensure a better understanding for the reader. Usually, in
this field of research, the arithmetic and not the geometric volume weighted average is used.
Moreover, in a recent work, the alternative geometric averages were also used for estimating
the effective properties of carbon-epoxy composites (Fréour et al., to be published).
Nevertheless, the obtained results were not found as satisfactory than in the previously
studied cases of metallic polycrystals or metal ceramic assemblies. Actually, the very strongly
heterogeneous properties presented by the constituents of carbon reinforced polymer matrix
composites yields a strong underestimation of the effective properties of the composite ply
predicted according to Eshelby-Kröner model involving the geometric average, by
comparison to the expected (measured) properties. Thus, the geometric average should not be
considered as a reliable alternate solution to the classical arithmetic average for achieving
scale transition modelling of composite structures. Consequently, arithmetic averages
satisfying to relation (4) only will be used in the following of this manuscript.
(4)

Multi-scale Analysis of Fiber-Reinforced Composite Parts… 7
Now, in the present case, where the macroscopic behaviour is described by two separate
heterogeneous inclusions only (i.e. one for the matrix and one for the reinforcements),
convenient simplifications of equation (5) do occur.
Actually, introducing v r and v m as the volume fractions of the ply constituents, and
taking into account the classical relation on the summation over the volume fractions (i.e. v r +
v m =1), equation (5) applied to the volume average of any tensor A writes:
AA
i
A =
i=
r, m
i
r r m m
A = v A + v A
i=
r, m
In the following of the present work, the superscript AA denoting the selected volume
average type will be omitted.
According to equations (1-3), the effective properties expressed within Eshelby-Kröner
self-consistent model involve a still undefined tensor, R I . This term is the so-called “reaction
tensor” (Kocks et al., 1998). It satisfies:
I
I I −1
−1
−1
( ) : ⎜
⎛ I I
−
= − ⎟
⎞ I
I S S L E : E
R = esh esh
(7)
⎝ ⎠
In the very preceding equation, I stands for the fourth order identity tensor. Hill’s tensor
E I , also known as Morris tensor (Morris, 1970), expresses the dependence of the reaction
tensor on the morphology assumed for the matrix and its reinforcements (Hill, 1965). It can
I
I I I
−1
be expressed as a function of Eshelby’s tensor S esh , through E = Sesh
: L . It has to be
underlined that both Hill’s and Eshelby’s tensor components are functions of the macroscopic
stiffness L I (some examples are given in Kocks et al., 1998; Mura, 1982).
In the case, when ellipsoidal-shaped inclusions have to be taken into account, the
following general form enables the calculation of the components of this tensor (see the
works of Asaro and Barnett, 1975 or Kocks et al. 1998):
⎧
⎪E
⎪
⎨
⎪
⎪⎩
γ
I
ijkl
ikjl
=
=
1
4π
π
∫
0
I [ Kik
() ξ ]
sinθ dθ
In the case of an orthotropic macroscopic symmetry, the components Kjp(ξ) were given in
(Kröner, 1953):
I
K
⎡ I
L
⎢
11
= ⎢
⎢
⎢⎣
−1
ξ
( ) ( )
( ) ( )
( ) ( ) ⎥ ⎥⎥⎥
2 I 2 I 2 I I I I
ξ + +
+
+
⎤
1 L66ξ
2 L55ξ3
L12
L66
ξ1ξ
2 L13
L55
ξ1ξ
2
I I I 2 I 2 I 2 I I
L12
+ L66
ξ1ξ
2 L66ξ1
+ L22ξ
2 + L44ξ3
L23
+ L44
ξ 2ξ3
I I I I I 2 I 2 I 2
L13
+ L55
ξ1ξ
2 L23
+ L44
ξ2ξ
3 L55ξ1
+ L44ξ
2 + L33ξ3
2π
j
∫
0
ξ
γ
l
ikjl
dφ
⎦
(6)
(8)
(9)

8
with
Jacquemin Frédéric and Fréour Sylvain
sinθ cosφ sinθ sinφ cosθ
ξ = ,ξ = ,ξ = (10)
1 2 3
a1 a2 a3
where 2 a1, 2 a2, 2 a3 are the lengths of the principal axes of the ellipsoid (representing the
considered inclusion) assumed to be respectively parallel to the longitudinal, transverse and
normal directions of the sample reference frame.
According to equations (2-3, 7), the determination of both the macroscopic coefficients of
thermal and moisture expansion are somewhat straightforward, while the effective stiffness is
known, because the involved expressions are explicit. On the contrary, the estimation of the
macroscopic stiffness of the composite ply through (1) cannot be as easily handled.
Expression (1) is implicit because it involves L I tensor in both its right and left members.
Moreover, calculating the right member of equation (1) entails evaluating the reaction tensor
(7) which also depends on the researched elastic stiffness, at least because of the occurrence
of Hill’s tensor (or Eshelby’s tensor, if that notation is preferred) in relation (1). As a
consequence, the effective elastic properties of a composite ply satisfying to Eshelby-Kröner
self-consistent model constitutive relations are estimated at the end of an iterative numerical
procedure. This is the main drawback of the self-consistent procedure preventing from
achieving an analytical determination of the effective macroscopic thermo-hygro-elastic
properties of a composite ply, in the case where this scale transition model is employed.
Therefore, managing to express explicit solutions for estimating the macroscopic properties
(or at least the macroscopic stiffness) requires focusing on a less intricate, less rigorous model
but still providing realistic numerical values. Mori-Tanaka approach suggest itself as an
appropriate candidate, for reasons that will be comprehensively explained in the next
subsection.
2.4. Introducing Mori-Tanaka Model as a Possible Alternate Solution to
Eshelby-Kröner Model
As Eshelby-Kröner self-consistent approach, Mori Tanaka estimate is a scale transition model
derived from the pioneering mathematical work of Eshelby (Eshelby, 1957). Mori and Tanaka
actually investigated the opportunity of extending Eshelby’s single-inclusion model (which is
sometimes presented as an “infinitely dilute solution model”) to the case where the volume
fraction of the ellipsoidal heterogeneous inclusion embedded in the matrix is not tending
towards zero anymore, but admits a finite numerical value (Mori and Tanaka, 1973; Tanaka
and Mori, 1970). Calculations show that, in many cases, the effective homogenised
macroscopic properties deduced from Mori-Tanaka approximate are close to their
counterparts, estimated from the previously described Eshelby-Kröner self-consistent
procedure (Baptiste, 1996, Fréour et al., 2006a). Exceptions to this statement occur
nevertheless in the cases where extreme heterogeneities in the constituents properties have to
be accounted for. For example, handling a significant porosities volume fraction yields Mori-
Tanaka estimations deviating considerably from the self-consistent corresponding

Multi-scale Analysis of Fiber-Reinforced Composite Parts… 9
calculations, according to (Benveniste, 1987). However, Mori-Tanaka approach is reported to
remain reliable for treating cases similar to those aimed by the present work.
It has previously been demonstrated that the effective macroscopic homogenised thermohygro-elastic
properties exhibited by a composite ply, according to Mori and Tanaka
approximation satisfy the following relations (Baptiste, 1996; Fréour et al., 2006a):
−1
−1
−1
I i i i
i i
i
L = T : L : T
= L : T : T
(11)
i=
r, m
i=
r, m i=
r, m
i=
r, m
T
I 1 ⎛
−1
⎞
⎜ i i i i
i
: : : ⎟ i
β = T L L T : β ΔC
(12)
I
ΔC
⎜
⎟
⎝
i=
r, m ⎠
i=
r, m
T
I ⎛
−1
⎞
⎜ i i i i
: : : ⎟ i
M = T L L T : M
(13)
⎜
⎟
⎝
i=
r, m ⎠
i=
r, m
The superscript T appearing in relations (12-13) denotes transposition operation.
The same remarks as indicated in the preceding subsection holds for the determination of
the effective macroscopic CME using relation (12). This equation can be rewritten as a
function of the materials properties only, thus excluding the moisture contents.
In equations (11-13), T i is the elastic strain localisation tensor, expressed for the isuperscripted
phase that is considered to interact with the embedding phase (denoted by the
superscript e). Actually, Mori-Tanaka model is based on a two-step scale-transition
procedure. In this theory, contrary to the case of Eshelby-Kröner self-consistent model, the
inclusions are not considered to be directly embedded in the effective material having the
behaviour of the composite structure (and thus interacting with it). In Mori and Tanaka
approximation, the n constituents of a n-phase composite ply are separated in two subclasses:
one of them is designed as the embedding constituent, whereas the n-1 others are considered
as inclusions of the first one. The inclusion particles are embedded in the matrix phase, itself
being loaded at the infinite by the hygro-mechanical conditions applied on the composite
structure. In consequence, the inclusion phase does not experience any interaction with the
macroscopic scale, but with the matrix only. In consequence, Mori and Tanaka model
corresponds to the direct extension of Eshelby’s single inclusion model (Eshelby, 1957) to the
case that the volume fraction of inclusions does not remain infinitesimal anymore. Within
Mori and Tanaka approach, this localisation tensor T i writes as follows:
[ ( ) ] 1 −
i i e
I + E : L − L
i
T =
(14)
Contrary to the case of Eshelby-Kröner scale-transition model (refer to subsection 2.3.
above), the localisation involved within Mori-Tanaka approximate does not explicitly involve

10
Jacquemin Frédéric and Fréour Sylvain
the macroscopic stiffness. Nevertheless, according to the already cited same subsection, the
reaction tensor involved in Eshelby-Kröner model was also implicitely depending on the
macroscopic stiffness through the calculation procedure entailed for estimating Hill’s tensor.
Within Mori-Tanaka procedure (Benveniste, 1987; Baptiste 1996; Fréour et al., 2006a),
Hill’s tensor E i expresses the dependence of the strain localization tensor on the morphology
assumed for the embedding phase and the particulates it surrounds (Hill, 1965). It can be
i
expressed as a function of Eshelby’s tensor S esh , through:
i i e
−1
E = Sesh
: L
(15)
In practice, the calculation of Hill’s tensor for the embedded inclusions phase only would
be necessary, since obvious simplifications of (14), leading to T I
e = , occur in the case that
the embedding constituent localisation tensor is considered. According to relations (14-15),
the strain localization tensor T i does not involve the macroscopic stiffness tensor (or any other
macroscopic property). As a consequence, contrary to Eshelby-Kröner self-consistent
procedure, Mori-Tanaka approximation provides explicit relations (actually, the
homogenization equations (11-13)) for estimating the researched macroscopic effective
properties of a composite ply.
2.5. Example of Homogenization: The Case of T300-N5208 Composites
The present subsection is focused on the application of the theoretical frameworks described
in the above 2.3 and 2.4 sections to the numerical simulation of the effective properties of a
typical, high-strength, fiber-reinforced composite made up of T300 carbon fibers and N5208
epoxy resin. The choice of such a material is justified because of the strong heterogeneities of
the hygro-thermo-elastic properties of its constituents (actually, the numerical deviation
occurring among the macroscopic properties of composites determined through various scale
transition relations rises with this factor, see Jacquemin et al, 2005; Herakovich, 1998). Table
1 accounts for the pseudo-macroscopic properties reported in the literature for these
constituents. The comparison between the results obtained through the two, considered in the
present work alternate scale transition framework of Mori-Tanaka model are displayed on
figure 1, for:
- the longitudinal and transverse Young’s moduli
- Coulomb’s moduli
I
G 12 ,
I
G 23 ,
I I
- the coefficients of thermal expansion M 11,
M 22
I I
- the coefficients of moisture expansion β 11,
β 22 .
I
Y 11 ,
I
Y 22 ,

T300 fibers
(Soden et al.,
1998;
Agbossou and
Pastor, 1997)
N5208 epoxy
matrix (Tsai,
1987;
Agbossou and
Pastor, 1997)
Multi-scale Analysis of Fiber-Reinforced Composite Parts… 11
Table 1. Hygro-thermo-mechanical properties of T300/5208 constituents.
ρ
[g/cm 3 ]
Y 1
[GPa]
Y 2, Y3
[GPa]
ν12
ν
13
G23
[GPa]
G 12
[GPa]
M11
[10 -6 /K]
M22, M 33
[10 -6 /K]
11 22 β , β
1200 230 15 0.2 7 15 -1.5 27 0
1867 4.5 4.5 0.4 6.4 6.4 60 60 0.6
The calculations were achieved assuming that the reinforcements exhibit fiber-like
morphology with an infinite length axis parallel to the longitudinal direction of the ply. For
the determination of the CME, a perfect adhesion between the carbon fibers and the resin was
assumed. Moreover, it also was assumed that the fibers do not absorb any moisture. Thus, the
ratio between the pseudo-macroscopic and the macroscopic moisture contents is deduced
from the expression given in (Loos and Springer, 1981):
m
I
I
ρ
m m
ΔC
= (16)
ΔC v ρ
where ρ stands for the densities. The macroscopic density can be deduced form the classical
rule of mixture:
I m m r r
= v ρ v ρ
(17)
ρ +
The equations required for achieving Mori-Tanaka estimations involve relations (8-17).
For the purpose of the strain localization, the embedding constituent was considered to be the
epoxy matrix, whatever the considered volume fraction of reinforcements (thus, the
e m
transformation rule L = L was considered to be valid in any case). Figure 1 also reports
the numerical results obtained through Kröner-Eshelby Self-Consistent model (1-3, 6-10, 16-
17), in the same conditions (identical inclusion morphology and constituents properties as for
Mori-Tanaka computations).
β
33

12
Macroscopic stiffness
component [GPa]
CME component
1,4
1,2
0,8
0,6
0,4
0,2
0, 25
0,2
0, 15
0,1
0, 05
250
200
150
100
1
50
0
Jacquemin Frédéric and Fréour Sylvain
0 0,25 0,5 0,75 1
matrix volume fraction
0
0 0,25 0,5 0,75 1
0
I
Y22
I
Y11
I
β11
I
β22
epoxy volume fraction
Macroscopic stiffness component
[GPa]
CTE component [10 -6 K -1 ]
80
60
40
20
-20
16
14
12
10
8
6
4
2
0
0 0,25 0,5 0,75 1
epoxy volume fraction
0
0 0,25 0,5 0,75 1
epoxy volume fraction
Longitudinal (KESC) Transverse (KESC)
Longitudinal (Mori-Tanaka) Transverse (Mori-Tanaka)
0 0,25 0,5 0,75 1
matrix volume fraction
Figure 1. Macroscopic effective hygro-thermo-mechanical properties of T300/N5208 plies, estimated as
a function of the epoxy volume fraction, through scale transition homogenisation procedures.
Comparison between Mori-Tanaka approximate and Kröner-Eshelby self-consistent model.
Figure 1 shows the following interesting results:
1) In pure elasticity, both the investigated scale transition methods manage to reproduce
the expected mechanical behaviour of the composite ply: the material is stiffer in the
longitudinal direction than in the transverse direction. Moreover, the bounds are satisfying:
the properties of the single constituents are correctly obtained for those of the composite ply
in the cases where the epoxy volume fraction is either taken equal to v m =0 (transversely
isotropic elastic properties of T300 fibers) or v m =1 (isotropic elastic properties of N5208
resin).
2) The curves drawn for each checked elastic constant are almost superposed, except for
I
Coulomb’s modulus G 12 . Thus Mori-Tanaka model constitutes a rather reliable alternate
homogenization procedure to Eshelby-Kröner rigorous solution for estimating the
macroscopic elastic properties of typical carbon-epoxies.
I
G 23
I
M11
I
G12
I
M22

Multi-scale Analysis of Fiber-Reinforced Composite Parts… 13
3) Kröner-Eshelby self-consistent model and Mori-Tanaka approach both also do manage
to achieve a realistic prediction of the macroscopic coefficients of thermal expansion.
Especially, the expected boundary values are attained when the conditions v m =1 (isotropic
CTE of N5208 resin) or v m =0 (transversely isotropic thermal properties of T300 fibers) are
taken into account.
4) Mori-Tanaka approximate correctly reproduces the expected macroscopic coefficients
of moisture expansion in the longitudinal direction. In the transverse direction, however,
Mori-Tanaka model properly follows Eshelby Kröner model estimates while the epoxy
m
volume fraction is higher than 0.5. In the range 0 ≤ v ≤ 0.5,
discrepancies occur between
two considered scale transition models. In the case that the considered strain localization
assumes the epoxy as the embedding constituent within Mori-Tanaka approximate, the
relative error on I β 22 induced by this localization procedure, compared to Kröner-Eshelby
reference values remains weaker than 9%, and falls below 6% in the range of epoxy volume
fraction that is typical for designing composites structures for engineering applications
m
(0.3 ≤ v ≤ 0.7) .
5) In the range of the epoxy volume fraction, that is typical for designing composites
m
structures for engineering applications ( i.e. 0.3 ≤ v ≤ 0.7)
, according to the above
discussed results 3) and 4), Mori-Tanaka model can be employed as an alternative to Eshelby-
Kröner self-consistent model for estimating the effective macroscopic hygro-thermomechanical
properties of composite plies.
The above listed elements 1) to 5) finally indicate that the effective macroscopic thermohygro-elastic
properties of composite plies can be estimated in a reliable fashion using Mori-
Tanaka approximate, assuming the epoxy as the embedding constituent, instead of the more
rigorous Kröner-Eshelby model. This statement is true while the epoxy volume fraction
remains higher than 40%. Beyond this boundary value, some significant relative error (less
than 10%) may be expected to occur in the estimated transverse CME.
The results, obtained in the present section, will be used in the following as input
parameters for estimating the mechanical states experienced at macroscopic but at
microscopic scale also in composite structures submitted to various loads (the interested
reader should refer to section 4 for details).
3. Inverse Scale Transition Modelling for the Identification of the
Hygro-Thermo-Elastic Properties of One Constituent of a
Composite Ply
3.1. Introduction
The precise knowledge of the pseudo-macroscopic properties of each constituent of a
composite structure is required in order to achieve the prediction of its behavior (and
especially its mechanical states) through scale transition models. Nevertheless, the pseudomacroscopic
stiffness, coefficients of thermal expansion and moisture expansion of the matrix

14
Jacquemin Frédéric and Fréour Sylvain
and its reinforcements are not always fully available in the already published literature. The
practical determination of the hygro-thermo-mechanical properties of composite materials are
most of the time achieved on unidirectionnaly reinforced composites and unreinforced
matrices (Bowles et al., 1981; Dyer et al., 1992; Ferreira et al., 2006a; Ferreira et al., 2006b;
Herakovich, 1998; Sims et al., 1977). In spite of the existence of several articles dedicated to
the characterization of the properties of the isolated reinforcements (Tsai and Daniel, 1994;
DiCarlo, 1986; Tsai and Chiang, 2000), the practical achieving of this task remains difficult
to handle, and the available published data for typical reinforcing particulates employed in
composite design are still very limited. As a consequence, the properties of the single
reinforcements exhibiting extreme morphologies (such as fibers), are not often known from
direct experiment, but more usually they are deduced from the knowledge of the properties of
the pure matrices and those of the composite ply (which both are easier to determine), through
appropriate calculation procedures. The question of determining the properties of some
constituents of heterogeneous materials has been extensively addressed in the field of
materials science, especially for studying complex polycrystalline metallic alloys (like
titanium alloys, cf. Fréour et al., 2002 ; 2005b ; 2006b) or metal matrix composites (typically
Aluminum-Silicon Carbide composites cf. Fréour et al., 2003a ; 2003b or iron oxides from
the inner core of the Earth, cf. Matthies et al., 2001, for instance). The required calculation
methods involved in order to achieve such a goal are either based on Finite Element Analysis
(Han et al., 1995) or on the inversion of scale transition homogenization procedures similar to
those already presented in section 2 of the present paper. It was shown in previous works that
it was actually possible to identify the properties of one constituent of a heterogeneous
material from available (measured) macroscopic quantities through inverse scale transition
models. Such identification methods were successfully used in the field of metal-matrix
composites for the determination of the average elastic (Freour et al., 2002) and thermal
(Freour et al., 2006b) properties of the β-phase of (α+β) titanium alloys. The procedure was
recently extended to the study of the anisotropic elastic properties of the single-crystal of the
β-phase of (α+β) titanium alloys on the basis of the interpretation of X-Ray Diffraction strain
measurements performed on heterogeneous polycrystalline samples in (Freour et al., 2005b).
The question of determining the temperature dependent coefficients of thermal expansion of
silicon carbide was handled using a similar approach from measurements performed on
aluminum – silicon carbide metal matrix composites in (Freour et al., 2003a; Freour et al.,
2003b).Numerical inversion of both Mori-Tanaka and Eshelby-Kröner self-consistent models
will be developed and discussed here.
3.2. Estimating Constituents Properties from Eshelby-Kröner Self-consistent
or Mori-Tanaka Inverse Scale Transition Models
3.2.1. Application of Eshelby-Kröner Self-consistent Framework to the
Identification of the Pseudo-macroscopic Properties of one Constituent
Embedded in a Two-Constituents Composite Material
The pseudomacroscopic stiffness tensor of the reinforcements can be deduced from the
inversion of the Eshelby-Kröner main homogenization form over the constituents elastic
properties (1) as follows :

1 I
L = L :
r
v
Multi-scale Analysis of Fiber-Reinforced Composite Parts… 15
m
I r I v m I m I −1
I r I
[ E : ( L − L ) + I]
− L : [ E : ( L − L ) + I]
: [ E : ( L − L ) + I]
r
v
The application of this equation implies that both the macroscopic stiffness and the
pseudomacroscopic mechanical behaviour of the matrix is perfectly determined. The elastic
stiffness of the matrix constituting the composite ply will be assumed to be identical to the
elastic stiffness of the pure single matrix, deduced in practice from measurements performed
on bulk samples made up of pure matrix. It was demonstrated in (Fréour et al., 2002) that this
assumption was not leading to significant errors in the case that polycrystalline multi-phase
samples were considered. The similarities existing between multi-phase polycrystals and
polymer based composites suggest that this assumption should be suitable in the present
context, at least when scale factors do not occur. Nevertheless, in the case that significant
edge effects, due for instance to a reduced thickness of the matrix layer constituting the
composite ply, might be expected to occur, the identification of the ply embedded matrix
elastic properties to those of the corresponding bulk material would not systematically be
appropriate. Consequently, the application of inverse form (18) given above could lead to an
erroneous estimation of the reinforcements elastic stiffness. Moreover, identification based on
such inverse homogenization methods are sensitive to both the precise knowledge of the
constituents volume fractions (i.e. v m and v r ) and to the presence of porosities (which lowers
the effective stiffness L I of the composite ply).
An expression, analogous to above-relation (18) can be found for the elastic stiffness of
the matrix, through the following replacement rules over the superscripts/subscripts:
m → r, r → m . Nevertheless, the situation, where the properties of the reinforcements are
known, when those of the matrix are unknown is highly improbable.
The pseudomacroscopic coefficients of moisture expansion of the matrix can be deduced
from the inversion of the homogenization form (2) as follows :
where G m writes :
( ) m I I m
L + L : R G
(18)
m
m 1 −1
β = L :
:
(19)
m m
v ΔC
i I I −1
I I r r I I −1
r r r
( L + L : R ) : L : β − v ( L + L : R ) : L : ΔC
m I
G = ΔC
β (20)
i=
r, m
An expression, analogous to above-relation (19) can also be found for the coefficients of
moisture expansion of a permeable reinforcement type, through the following replacement
rules over the superscripts/subscripts: m → r, r → m .
In the particular case, where impermeable reinforcements are present in the composite
structure, the coefficients of moisture expansion of the matrix simplifies as follows (an
extensive study of this very question was achieved in Jacquemin et al., 2005):

16
Jacquemin Frédéric and Fréour Sylvain
m I I i I I −1
I I
( L + L : R ) : ( L + L : R ) : L
I
m ΔC m−1
β = L :
: β (21)
m m
v ΔC
i=
r, m
The pseudomacroscopic coefficients of thermal expansion of the matrix can be deduced
from the inversion of the homogenization form (3) as follows:
( ) ( ) ( ) ⎥ ⎥
⎡
⎤
m I I
−
−
+ ⎢ i I I 1 I
I r r I I 1 r
L L : R : L + L : R : L : M − v L + L : R : L M
m m−1
r (22)
M = L :
:
⎢
⎣
i=
r, m
⎦
Form (22) can be easily rewritten for expressing the coefficients of thermal expansion of
the reinforcements, using the same replacement rules over the superscripts/subscripts:
m → r, r → m , than for the previous cases.
3.2.2. Application of Mori-Tanaka Estimates to the Identification of the Pseudomacroscopic
Properties of one Constituent Embedded in a Two-Constituents
Composite Material
3.2.2.1. Inverse Mori-Tanaka Elastic Model
In the present work, it is be considered, that the reinforcements are surrounded by the matrix,
thus, T m =I and (11) develops as follows:
I
( ) ( ) 1
m m r r r m r r
−
v L + v L : T : v I + v : T
L =
(23)
Thus, from (11) two alternate equations are obtained for identifying the pseudomacroscopic
stiffness of the composite ply constituents:
• On the first hand, the elastic properties of the matrix satisfies
I r r
( L - L ) : T
m
m I 1-
v
L = L +
(25)
m
v
Equation (25) is an implicit equation since both its left and right hand sides involve the
researched stiffness tensor L m .
• whereas, on the second hand, the elastic stiffness of the reinforcements respects
I m r
−1
( L - L ) : T
m
r I v
L = L +
(26)
m
1-
v

Multi-scale Analysis of Fiber-Reinforced Composite Parts… 17
For the same reasons as above (i.e. comments about equation (25)), expression (26) is an
implicit relation. As a consequence, the need of an inverse modelling for achieving the
identification of the elastic properties exhibited by any one constituent of a composite ply
through Mori-Tanaka scale-transition approximate yields the loss of the main advantage of
this very model over the more rigorous Eshelby-Kröner self-consistent approach: the
opportunity to express analytical explicit relations instead of having to perform successive
numerical calculations for solving implicit equations. Moreover, the general remarks about
the sensitivity of identification methods to certain factors, expressed in subsection 3.2.1 are
valid in the present context also.
3.2.2.2. Inverse Mori-Tanaka Model for Identifying Coefficients of Moisture of Thermal
Expansion
Following the same line of reasoning as above, in the purely elastic case, one can inverse
relation (12) in order to express the coefficients of moisture expansion of a constituent
embedded in a composite ply according to Mori-Tanaka estimates, or its coefficients of
thermal expansion, from the homogenization relation (13). In the case of the pure matrix, one
gets:
m 1 m
−1
⎡ i i
I r r r
M = L : ⎢ L : T : M − v L : T : M
m
v ⎢⎣
i=
r, m
m 1 m
−1
⎡ i i
I I r r r r r ⎤
β = L : ⎢ L : T : β ΔC − v L : T : β ΔC ⎥ (28)
m m
v ΔC ⎢⎣
i=
r, m
⎥⎦
This last relation (valid for the general case of a possibly permeable reinforcement type)
yields to the following simplified form if impermeable reinforcements are considered:
r
⎤
⎥
⎥⎦
(27)
m 1 m
−1
i i
I I
β = L : L : T : β ΔC
(29)
m m
v ΔC
i=
r, m
Due to the localization procedure which does not treat in an equivalent way the
embedding matrix and the embedded inclusions (reinforcements) in the point of view of
Mori-Tanaka scale-transition approach, the inverse forms satisfied by the coefficients of
thermal expansion and coefficients of moisture expansion of the reinforcements are not
anymore deduced from the above-relations established for the matrix through simple
replacement rules. Actually, unlike the inverse forms obtained according to Eshelby-Kröner
self-consistent model, Mori-Tanaka model yields non-equivalent inverse forms for the matrix
one the one hand and for the reinforcements, on the second hand. The expressions, required
for identifying the thermal or hygroscopic properties of reinforcements within Mori-Tanaka
model are:

18
Jacquemin Frédéric and Fréour Sylvain
r 1 r 1 r
−1
⎡ i i
I m m
M = T : L : ⎢ L : T : M − v L : M
r
v
⎢⎣
i=
r, m
− m
r 1 r −1 r
−1
⎡ i i
I I m m m m ⎤
β = T : L : ⎢ L : T : β ΔC − v L : β ΔC ⎥ (31)
r r
v ΔC
⎢⎣
i=
r, m
⎥⎦
3.3. Examples of Properties Identification in Composite Structures Using
Inverse Scale Transition Methods
3.3.1. Determination of Reinforcing Fibers Elastic Properties
The literature often provides elastic properties of carbon-fiber reinforced epoxies (see for
instance Sai Ram and Sinha, 1991), that can be used in order to apply inverse scale transition
model and thus identify the properties of the reinforcing fibers, as an example. Table 2 of the
present work summarizes the previously published data for an unidirectional composite
designed for aeronautic applications, containing a volume fraction v r =0.60 of reinforcing
fibers. In order to achieve the calculations, according to relations (18) or (26) depending on
whether Eshelby-Kröner model or Mori-Tanaka approximation, input values are required for
the pseudo-macroscopic properties of the epoxy matrix constituting the composite ply. The
elastic constants considered for this purpose are listed in Table 3 (from Herakovich, 1998).
Both the above-cited inverse scale transition methods have been applied. The obtained results
are provided in Table 4, where they are compared to typical values, reported in the literature,
for high-strength reinforcing fibers (Herakovich, 1998). It is shown that a very good
agreement between the two inverse models is obtained. Moreover, the calculated values are
similar to those expected for typical reinforcements according to the literature. Nevertheless,
some discrepancies between the identified moduli do exist, especially for
r G 12 (that
r
corresponds to L 55 stiffness component). Actually, the value deduced for this component
through Mori-Tanaka inverse model deviates from both the expected properties and the
estimations of Eshelby-Kröner model. This deviation, occurring for this very component, is
Table 2. Macroscopic elastic moduli (from the literature) and stiffness tensor
components (calculated) considered for the composite ply at ΔC 0
I = % and T I = 300
K, according to (Sai Ram and Sinha, 1991).
Elastic moduli
Stiffness tensor
components
I
1
I
Y 2 [GPa]
I
ν 12 [1]
I
G 12 [GPa]
I
23
130 9.5 0.3 6.0 3.0
Y [GPa]
⎤
⎥
⎥⎦
G [GPa]
I
11
I
L 22 [GPa]
I
L 12 [GPa]
I
L 44 [GPa]
I
L 55 [GPa]
134.2 14.8 7.1 6.0 3.0
L [GPa]
(30)

Multi-scale Analysis of Fiber-Reinforced Composite Parts… 19
obviously directly related to the discrepancies previously underlined in subsection 2.5 where
the question of comparing the homogenization relations of the two scale transition methods
presented in this paper, was investigated.
Table 3. Pseudomacroscopic elastic moduli and stiffness tensor components assumed for
the epoxy matrix of the composite plies at ΔC 0
I = % and T I = 300 K, according to
(Herakovich, 1998).
Elastic moduli
Stiffness tensor
components
m
1
m
Y 2 [GPa]
m
ν 12 [1]
m
G 12 [GPa]
m
23
5.35 5.35 0.350 1.98 1.98
Y [GPa]
G [GPa]
m
11
m
L 22 [GPa]
m
L 12 [GPa]
m
L 44 [GPa]
m
L 55 [GPa]
8.62 8.62 4.66 1.98 1.98
L [GPa]
Table 4. Pseudomacroscopic elastic moduli and stiffness tensor components identified
for the carbon fiber reinforcing the composite plies at ΔC 0
I = % and T I = 300 K,
according to either Mori-Tanaka estimates, or Eshelby-Kröner self-consistent model.
Comparison with the corresponding properties exhibited in practice by typical highstrength
carbon fibers, according to (Herakovich, 1998).
Elastic moduli
r
Y 1 [GPa]
r
Y 2 [GPa]
r
ν 12 [1]
r
G 23 [GPa]
r
G 12 [GPa]
Mori-Tanaka estimate 213.1 13.7 0.27 4.1 22.7
Eshelby-Kröner model 213.2 13.3 0.27 4.0 12.1
Typical expected
properties
Stiffness tensor
components
232 15 0.279 5.0 15
r
L 11[GPa]
r
L 22 [GPa]
r
L 12 [GPa]
r
L 44 [GPa]
r
L 55 [GPa]
Mori-Tanaka estimate 219.2 24.9 11.2 4.1 22.7
Eshelby-Kröner model 219.2 23.9 10.8 4.0 12.1
Typical expected
properties
236.7 20.1 8.4 5.02 15
3.3.2. Determination of AS4/3501-6 Matrix Coefficients of Moisture Expansion
Macroscopic values of the Coefficients of Moisture Expansion are sometimes available,
contrary to the corresponding pure epoxy resin CME. Simulations were performed in the cas
e of an AS4/3501-6 composite, with a reinforcing fiber volume fraction v r =0.60. The
calculations were achieved using the elastic properties given in Table 5, and the macroscopic
coefficients of moisture expansion listed in Table 6. The same table summarizes the results
obtained with both inverse Eshelby-Kröner self-consistent model (21) and Mori-Tanaka

20
Jacquemin Frédéric and Fréour Sylvain
estimates (29) assuming a moisture content
ΔC
= 3.
125 (the ratio between composite and
I
ΔC
resin densities being 1.25 in this material, the moisture content ratio assumed in the present
study corresponds to the maximum expected value), in the case that impermeable
reinforcements are considered. According to Table 6, a very good agreement is obtained
between the two inverse models. This result is compatible with the homogenisation
calculation previously achieved in subsection 2.5: for such a volume fraction of
reinforcements, Eshelby-Kröner and Mori-Tanaka models provide identical macroscopic
coefficients of moisture expansion from the pseudomacroscopic data. As a consequence, the
corresponding inverse forms (21) and (29) yields the same estimation for the
pseudomacroscopic CME of the matrix constituting the composite ply.
Table 5. Macroscopic and pseudo-macroscopic mechanical elastic properties of
AS4/3501-6 constituents.
m
E 1 [GPa] E 2, E3
[GPa] ν 12 , ν13
ν23 G 12 [GPa]
AS4 fibers
(Soden et al., 1998) 225 15 0.2 0.40 15
3501-6 epoxy matrix (Soden et
al., 1998)
AS4/3501-6
(KESC homogenisation)
4.2 4.2 0.34 0.34 1.567
135.2 9.2 0.25 0.36 5.2
Table 6. Macroscopic and pseudomacroscopic (3501-6 matrix only) coefficients of
moisture expansion of AS4/3501-6 composite. The pseudomacroscopic values results
from the two inverse scale transition models described in the present work.
Moisture expansion coefficient β 11 β 22, β33
AS4/3501-6 (Daniel and Ishai, 1994) 0.01 0.2
3501-6 epoxy from Eshelby-Kröner self-consistent inverse model 0.148 0.148
3501-6 epoxy from Mori-Tanaka inverse model 0.148 0.148
4. From the Numerical Model to Analytical Solutions for
Estimating the Pseudo-macroscopic Mechanical States
4.1. Introduction
It was extensively discussed in previously published works (the interested reader can, for
instance refer to Benveniste, 1987 and Fréour et al., 2006a, where the question is addressed),
that Mori and Tanaka constitutive assumptions were not suitable for a reliable estimation of
the localization of the macroscopic mechanical states within the constituents of typical

Multi-scale Analysis of Fiber-Reinforced Composite Parts… 21
composites conceived for engineering applications, which often present a significant volume
fraction of reinforcements. As a consequence, only Eshelby-Kröner approach will be
considered in the present section.
4.2. Numerical SC Model Extended to a Thermo-Hygro-Elastic Load
Within Kröner and Eshelby self-consistent framework, the hygrothermal dilatation generated
by a moisture content increment ΔC i is treated as a transformation strain exactly like the
thermal dilatation occurring after a temperature increment ΔT i (that last case was extensively
discussed in the literature, see for example Kocks et al., 1998). Thus, the pseudo-macroscopic
stresses σ i in the considered constituent (i.e. i=r or i=m) are given by:
i
i
i i i i i
( ε − M ΔT ΔC )
σ = L :
− β
(32)
Where, ε stands for the strain tensor. In general case, the moisture content differs at
macroscopic scale and pseudo-macroscopic scale, contrary to the temperature. Actually, the
reinforcements generally do not absorb moisture. In consequence, the mass of water
contained by the composite is: either found in the matrix, locally trapped in porosities
(Mensitieri et al., 1995) or located where fiber debonding occurs.
Replacing the superscripts i by I in (32) leads to the stress-strain relation that holds at
macroscopic scale.
I
I
I I I I I
( ε − M ΔT ΔC )
σ = L :
− β
(33)
The so-called “scale-transition relation” enabling to determine the local stresses and
strains from the macroscopic mechanical states was demonstrated in a fundamental work,
starting from the assumption that the elementary inclusions (here the matrix and the fiber)
have ellipsoidal shapes (Eshelby, 1957):
i
I
i I ( ε )
I I
: R
(34)
σ − σ = −L
: − ε
Actually, (34) is not very useful, because both the unknown pseudo-macroscopic stresses
and strains appear. Nevertheless, combining (32-34) enables to find the following expression
for the pseudo-macroscopic strain (the demonstration is available in Jacquemin et al., 2005
and Fréour et al., 2003b):
i I I
−1
I I I I i i I I i i i I I I
( L + L : R ) : [ ( L + L : R ) .. ε + ( L : M − L : M ) ΔT + L : β ΔC L : β ΔC ]
i
ε =
−
In relation (35), the classical replacement rule ΔT i =ΔT Ι =ΔT was introduced (i.e. the
temperature field is considered to be uniform within the considered ply).
(35)

22
Jacquemin Frédéric and Fréour Sylvain
Moreover, it was established in (Hill, 1967), that the self-consistent model was
compatible with the following volume averages on both pseudo-macroscopic stresses and
strains:
σ
ε
i
i
i=
r, m
i=
r, m
= σ
For a given applied macroscopic thermo-hygro-elastic load {σ I , ΔC I, ΔT} one can easily
determine ε I through (33), provided that the effective elastic behaviour L I of the ply has been
calculated using either the homogenization procedure corresponding to Eshelby-Kröner
model or the corresponding Mori-Tanaka alternate solution (see previous developments
provided in section 2 above). Then, the pseudo-macroscopic strains are determined through
(35).
4.3. Analytical Expression for Calculating the Mechanical States Experienced
by the Constituents of Fiber-Reinforced Composites According to
Eshelby-Kröner Model
The main impediment requiring to be overcome in order to achieve closed-forms from
relation (35) is the determination of Morris’ tensor E I . Actually, according to the integrals
appearing in relation (8), this tensor will admit only numerical solutions in most cases.
However, some analytical forms for Morris’ tensor are actually available in the literature;
the interested reader can for instance refer to (of Mura, 1982; Kocks et al., 1998; or Qiu and
Weng 1991). Nevertheless, these forms were established considering either spherical, discshaped
of fiber-shaped inclusions embedded in an ideally isotropic macroscopic medium, that
is incompatible with the strong elastic anisotropy exhibited by fiber-reinforced composites at
macroscopic scale (Tsai and Hahn, 1987).
In the case of carbon-epoxy composites, a transversely isotropic macroscopic behaviour
being coherent with fiber shape is actually expected (and predicted by the numerical
computations). Assuming that the longitudinal (subscripted 1) axis is parallel to fiber axis, one
obtains the following conditions for the semi-lengths of the microstructure representative
ellipsoid: a1→∞, a2=a3. Moreover, the macroscopic elastic stiffness should satisfy :
I I I I I I
L11 ≠ L12
≠ L22
≠ L23
≠ L44
≠ L55
. Now, it is obvious, that these additional
hypotheses lead to drastic simplifications of Morris’ tensor (8), in the case that fiber
morphology is considered for the reinforcements. The line of reasoning required to achieve
the writing of analytical expressions for Morris’ tensor is extensively presented in (Welzel et
= ε
al., 2005; Fréour et al., 2005). Actually, one obtains (in contracted notation i.e,
components are given here):
I
I
(36)
I
E ij

I
E
⎡0
⎢
⎢0
⎢
⎢
⎢
⎢0
⎢
= ⎢
⎢0
⎢
⎢
⎢0
⎢
⎢
⎢0
⎢
⎣
Multi-scale Analysis of Fiber-Reinforced Composite Parts… 23
0
3 1
+
I I I
8L22
4L22
− 4L23
I I
L22
+ L23
2
I I I
8L22L
23 − 8L22
0
0
0
0
I I
L22
+ L23
2
I I I
8L22L
23 − 8L22
3 1
+
I I I
8L22
4L22
− 4L23
0
0
0
0
0
0
1 1
+
I I I
8L22
4L22
− 4L23
0
0
0
0
0
0
1
I
8L55
0
0 ⎤
⎥
0 ⎥
⎥
⎥
⎥
0 ⎥
⎥
⎥
0 ⎥
⎥
⎥
0 ⎥
⎥
⎥
1 ⎥
I
8L ⎥
55 ⎦
In fact, the epoxy matrix is usually isotropic, so that three components only have to be
considered for its elastic constants: m m m
L 11,
L12
and L44
. One moisture expansion coefficient is
sufficient to describe the hygroscopic behaviour of the matrix: m β 11.
In the case of the carbon fibers, a transverse isotropy is generally observed. Thus, the
corresponding elasticity constants depend on the following components:
r r r r r r
L 11,
L12,
L22,
L23,
L44,
and L55
. Moreover, since the carbon fiber does not absorb water,
r r
its CME β 11 and β22
will not be involved in the mechanical states determination.
Introducing these additional assumptions in (35), and taking into account the form (37)
obtained for Morris’ tensor, one can deduce the following strain tensors for both the matrix
and the fibers:
⎡ i i i
ε
⎤
⎢
11 ε12
ε13
i i i i ⎥
ε = ⎢ε12
ε22
ε23⎥
(38)
⎢ i i i ⎥
⎢
ε
⎣ 13 ε23
ε33
⎥⎦
where, in the case of the matrix,
⎧ m I
ε
⎪
11 = ε11
⎪ I I
m 2 L55
ε12
⎪ε12
=
I m
⎪ L55
+ L44
⎪
I I
⎪ m 2 L55
ε13
⎪ε13
=
I m
⎪ L55
+ L44
⎪ m m m m m
⎨ m N1
+ N2
+ N3
+ N4
+ N
ε
5
=
⎪ 22
m
D
⎪
1
⎪
⎪ m
ε23
=
⎪
2
I I
⎪ 2 L22
+ L23
⎪
⎪ m m I22
ε = −
⎪ 33 ε22
4 L
⎪⎩
L
I I I I
2 L22
( L22
− L23
) ε23
I m I m I m ( L44
− L44
) + L22
( 3 L44
− 2 L23
− 3 L44
)
I I I I
( L22
− L23
)( ε22
− ε33
)
2
I I m m I m I m
22 + 3 L22
( L11
− L12
) - L23(
L11
+ L23
- L12
)
(37)
(39)

24
Jacquemin Frédéric and Fréour Sylvain
m m m m m
( L11
+ 2L12
)( β11
ΔC + M11
ΔT)
I I I I I I I I I
−L12
( β11ΔC
+ M11ΔT)
− ( L22
+ L23
)( β22ΔC
+ M33ΔT)
I m I
( L12
− L12
) ε11
I m m I I m m I
I22 L22
( 5 L11
− L12
+ 3 L22
) − L23(
3 L11
+ L12
+ 4 L22
)
L
2 2
I I m m I I ( 3 L22
− L23
)( L11
− L12
) + L22
− L23
I m m I I m m I
I22 L22
( L11
− 5 L12
− L22
) + L23(
L11
+ 3 L12
+ 4 L22
) −
L
2 2
I I m m I I ( 3 L22
− L23
)( L11
− L12
) + L22
− L23
⎧ m
N
⎪
1 =
⎪ m
N2
=
⎪
⎪ m
N3
=
⎪
⎪
⎪ m
N =
⎨ 4
⎪
⎪
⎪
⎪ m
N5
=
⎪
⎪
⎪ m m m I I
⎩D1
= L11
+ L12
+ L22
− L23
I
2
+ L23
I
ε22
2
I
3 L23
I
ε33
The pseudo-macroscopic stress tensors are deduced from the strains using (32). Thus, in
the matrix, one will have:
with
⎧ m m
σ
⎪
11 = L11
m m
⎨σ22
= L11
⎪ m m
⎪σ
=
⎩ 33 L11
(40)
⎡ m m m m m
σ
⎤
⎢
11 2 L44ε12
2 L44ε13
m m m m m m ⎥
σ = ⎢2
L44ε12
σ22
2 L44ε
23 ⎥
(41)
⎢ m m m m m ⎥
⎢
2 L
⎣ 44ε13
2 L44ε
23 σ33
⎥⎦
m m m m m m m m m ( ε11
− M11
ΔT)
+ L12
( ε22
+ ε33
− 2 M11
ΔT)
− β11(
L11
+ 2 L12
)
m m m m m m m m m ( ε22
− M11
ΔT)
+ L12
( ε11
+ ε33
− 2 M11
ΔT)
− β11(
L11
+ 2 L12
)
m m m m m m m m m ( ε33
− M11
ΔT)
+ L12
( ε11
+ ε22
− 2 M11
ΔT)
− β11(
L11
+ 2 L12
)
m
ΔC
m
ΔC
m
ΔC
The local mechanical states in the fiber are provided by Hill’s strains and stresses average
laws (36):
ε
σ
(42)
m
r 1 I v m
= ε − ε
(43)
r r
v
v
m
r 1 I v m
= σ − σ
(44)
r r
v
v

Multi-scale Analysis of Fiber-Reinforced Composite Parts… 25
4.4. Examples of Multi-scale Stresses Estimations in Composite Structures:
T300-N5208 Composite Pipe Submitted to Environmental Conditions
4.4.1. Macroscopic Analysis
4.4.1.1. Moisture Concentration
Consider an initially dry, thin uni-directionally reinforced composite pipe, whose inner and
outer radii are a and b respectively, and let the laminate be exposed to an ambient fluid with
boundary concentration c0. The macroscopic moisture concentration, c I (r,t), is solution of the
following system with Fick's equation (45), where D I is the transverse diffusion coefficient of
the composite. Boundary and initial conditions are described in (46):
I
∂c
∂t
⎪⎧
c
⎨
⎪⎩ c
I
I
= D
⎡ 2
∂ c
⎢ 2
⎣ ∂r
I
( a,
t)
( r,
0)
= c
= 0
0
I
+
I
1 ∂c
r ∂r
and c
I
⎤
⎥
⎦
( b,
t)
, a

26
Jacquemin Frédéric and Fréour Sylvain
plane tensors of hygroscopic expansion coefficients and moduli. Those tensors are assumed to
be material constants.
with,
I
⎧ I
σ ⎫ ⎡ I
L
11
⎪ ⎪ ⎢
11
I
⎪σ
⎪ ⎢ I
22 L12
⎨ ⎬ = ⎢
I
⎪σ33
⎪ ⎢ I
L12
⎪ I ⎪ ⎢
⎩τ12
⎭ ⎢
⎣0
⎪⎧
τ
⎨
⎪⎩ τ
I
32
I
13
I
L12
I
L22
I
L23
0
⎪⎫
⎡L
⎬ = ⎢
⎪⎭ ⎢⎣
0
I
L ⎤
12 0
⎥
I
L ⎥
23 0
⎥
I
L ⎥
22 0
⎥
I
0 L ⎥
55 ⎦
I
44
0
L
I
55
⎤
⎥
⎥⎦
⎧ε
⎪
⎪ε
⎨
⎪ε
⎪
⎩γ
⎪⎧
γ
⎨
⎪⎩ γ
I
11
I
13
I
22
I
33
I
12
I
32
⎪⎫
⎬
⎪⎭
− β
− β
− β
− β
I
11
I
22
I
12
I
ΔC
⎫
⎪
I
ΔC
⎪
⎬ I
ΔC
⎪
I ⎪
ΔC
⎭
I c
Δ C = . c
I
ρ
I and ρ I are respectively the macroscopic moisture concentration and the
mass density of the dry material.
To solve the hygromechanical problem, it is necessary to express the strains versus the
displacements along with the compatibility and equilibrium equations.
Introducing a characteristic modulus L 0 , we introduce the following dimensionless
variables:
σ
Ι
= σ
Ι I I I I I I I I
/ L0
, L = L /L0
, ( w , u , v ) = ( w , u , v ) / b.
Displacements with respect to longitudinal and circumferencial directions, respectively
u ( x,
r)
I
and v ( x,
r)
I
are then deduced:
⎧ I
u ( x,
r)
= R1x
⎪
I
⎨v
( x,
r)
= R 2xr
⎪
R1,
R 2 are constants.
⎩
It is worth noticing that the displacements ( x,
r)
and ( x,
r)
do not depend on the
moisture concentration field. Finally, to obtain the through-thickness or radial component of
the displacement
concentration (47).
I
w , we shall consider in the following the analytical transient
I
The radial component of the displacement field w satisfies the following equation:
u I
v I
I
22
(48)
(49)
(50)

Multi-scale Analysis of Fiber-Reinforced Composite Parts… 27
r
2
∂
2
w
∂r
2
I
∂w
+ r
∂r
I
− w
I
2 ∂ΔC
K1r
=
∂r
L
I I I I I I
with, K1=
L12
β11
+ L23
β22
+ L22
β22
It is shown that the general solution of equation (51) writes as the sum of a solution of the
homogeneous equation and of a particular solution (Jacquemin et Vautrin, 2002).
w
I
( r)
R 4
= R 3r
+ −
r
∞
∑
m=
1
∞
∑
k=
0
2exp(- ωmτ)
K
ω Δ′ ( ω I ) L
m
u
m
22
k 1 2k+
1
(-1) ( ) ( ωm
)
2
k!
( k + 1)!
k 1 2k+
1 2
(-1) ( ) ( ω )
∞
m
2
2
∑
k=
0 k!
( k + 1)!
2
1
k 1
( −1)
( )
I
22
∞
[ A
2
i
∑
+ {
i
k=
0
2k+
2
1
[ 2ln(
ω
2
k+
2
[
ln( r)
r
(( 2k
k!
( k + 1)!
m
+ 3)
2k+
1
( ω
) − ψ(
k + 1)
− ψ(
k + 2)]
2k+
3
2
−
−1)
2k+
2
m )
2(
2k
(( 2k
I
(( 2k
+ 3)
+ 3)
r
+ 3)
2
r
2k+
3
2k+
3
−1)
2
−1)
(( 2k
2
] −
r
B
π
( 2k+
3)
+ 3)
2
r ln( r)
+
−1)
Finally, the displacement field depends on four constants to be determined : Ri for i=1..4.
These four constants result from the following conditions :
• global force balance of the cylinder;
• nullity of the normal stress on the two lateral surfaces.
4.4.2. Numerical Simulations of Internal Stresses in T300/5208 Composite
Laminated Pipes
4.4.2.1. Introduction
Thin laminated composite pipes, with thickness 4 mm, initially dry then exposed to an
ambient fluid, made up of T300/5208 carbon-epoxy plies, with a fiber volume fraction v r =0.6,
were considered for the determination of both macroscopic stresses and moisture content as a
function of time and space. The closed-form formalism used in order to determine the
mechanical stresses and strains in each ply of the structure is described in subsection 4.4.1.
This model ensures the calculation of the macroscopic moisture content, too.
When the equilibrium state is reached, the maximum moisture content of the neat resin
may be estimated from the maximum moisture content of the composite. By assuming that
the fibers do not absorb any moisture, ΔC I and ΔC m are related by expression (16) given by
(Loos and Springer, 1981). In the case of T300/5208, since the ratio between composite and
resin densities is 1.33 (due to the constituents properties listed in table 1), the maximum
moisture content ratio given by (16) is about 3.33.
} ]
(51)

28
Jacquemin Frédéric and Fréour Sylvain
Figure 2 shows the time-dependent concentration profiles, resulting from the application
of a boundary concentration c0, as a function of the normalized radial distance from the inner
radius rdim. At the beginning of the diffusion process important concentration gradients occur
near the external surfaces. The permanent concentration (noticed perm in the caption) holds
with a constant value because of the symmetrical hygroscopic loading. The macroscopic
mechanical states were calculated for two types of composites structures: a) a unidirectionnaly
reinforced cylinder, and b) a [55°/-55°]S laminated cylinder.
c (%)
1,5
1,4
1,3
1,2
1,1
1
0,9
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
r dim
0.5 month
1 month
1.5 months
2 months
2.5 months
3 months
6 months
Figure 2. Time dependent concentration profiles in T300/5208 as a function of the normalised radial
distance from the inner radius r dim.
Starting with the macroscopic stresses deduced from continuum mechanics, the local
stresses in both the fiber and matrix were calculated either with the new analytical forms or
the fully numerical model. The comparison between the two approaches is plotted on
figures 3 and 4. These figures show the very good agreement between the numerical
approach and the corresponding closed-forms solutions. The slight differences appearing
are due to the small deviations on the components of Morris’ tensor calculated using the
two approaches. Actually, it is not possible to assume the quasi-infinite length of the fiber
along the longitudinal axis in the case of the numerical approach, because the numerical
computation of Morris’ tensor is highly time-consuming. Thus, the numerical version of
Eshelby-Kröner self-consistent model constitutes only an approximation of the real
microstructure of the composite. In consequence, it seems that the new analytical forms,
that are able to take into account the proper microstructure for the fibers, are not only more
convenient, but also more reliable than the initially proposed numerical approach.
4.4.2.2. Interpretation of the Simulations
The highest level of macroscopic tensile stress is reached for the uni-directional composite, in
the transverse direction and in the central ply of the structure (figure 3). The transverse
stresses exceed probably the macroscopic tensile strength in this direction. The choice of a
perm

Multi-scale Analysis of Fiber-Reinforced Composite Parts… 29
[+55°/-55°]S laminated allows to reduce the macroscopic stress in the transverse direction.
Nevertheless, a high shear stress rises along the time in the fibers of the central ply of such a
structure (figure 3).
[MPa]
[MPa]
σ 11
100
50
-50
-100
-150
50
-50
-100
-150
-200
200
0
0
0
0,5 1 1,5 2 2,5 3 6 perm
0,5 1 1,5 2 2,5 3 6 perm
b)
a)
month
month
[MPa]
[MPa]
100
-100
-150
40
35
30
25
20
15
10
-50
5
0
50
0
AS4 50_1_1 ΔC m / ΔC I = 2
1 2 3 4 5 6 7 8
-200
-400
cas
0,5 1 1,5 2 2,5 3 6 perm
a)
b)
month
0,5 1 1,5 2 2,5 3 6 perm
month
composite (CMF) matrix (numerical) fiber (numerical)
matrix (analytical) fiber (analytical)
Figure 3. Local stresses in T300/5208 composite for the central ply, in the case of a) the unidirectionaly
reinforced composite and b) the [+55°/-55°] S symmetric laminate. CMF stands for
Continuum Mechanics Formalisms.
Moreover, the figure 4 shows that the micro-mechanical model always predict a very
high compressive stress in the matrix of the inner ply whatever the laminate studied (the
macroscopic stress is negligible in the radial direction because thin structures are considered).
These local stresses could help to explain damage occurrence in the surface of composite
structures in fatigue.

30
, , [MPa]
, , [MPa]
σ 11
150
100
50
-50
-100
-150
-200
-250
50
0
-50
-100
-150
-200
-250
200
0
0
a)
Jacquemin Frédéric and Fréour Sylvain
0,5 1 1,5 2 2,5 3 6 perm
0,5 1 1,5 2 2,5 3 6 perm
b)
month
month
, , [MPa]
, , [MPa]
150
100
50
0
-50
-100
-150
-200
0
-5
-10
-15
-20
-25
-30
-35
-40
AS4 50_1_1 ΔC m / ΔC I = 2
1 2 3 4 5 6 7 8
-200
-400
cas
0,5 1 1,5 2 2,5 3 6 perm
a)
month
0,5 1 1,5 2 2,5 3 6 perm
b)
month
composite (CMF) matrix (numerical) fiber (numerical)
matrix (analytical) fiber (analytical)
Figure 4. Local stresses in T300/5208 composite for the inner ply, in the case of a) the uni-directionaly
reinforced composite and b) the [+55°/-55°] S symmetric laminate. CMF stands for Continuum
Mechanics Formalisms.
This work demonstrates the complementarities of continuum mechanics and micromechanical
models for the prediction of a possible damage in composite structures submitted
to hygro-elastic loads.
In the following section, the analytical expressions presented here for the localization of
the macroscopic mechanical states within the plies constituents, will be inversed in order to
achieve the identification of the strength of the constitutive matrix of a composite ply.
5. Identification of the Local Strength of the Constitutive Matrix
of a Composite Ply
5.1. Introduction
Damage predictions are important for design and for guiding materials improvement for
engineering applications. Composite structures encountered in engineering applications

Multi-scale Analysis of Fiber-Reinforced Composite Parts… 31
are designed to endure combined mechanical, thermal and hygroscopic loads during their
service life. Besides, composite structures usually benefit from improved properties
granted by a multidirectional arrangement of their plies. The multiplicity of both possible
loads and ply arrangements is not compatible with an extensive experimental
investigation of composite structures damage. As a result, only uniaxial and pure shear
test data of unidirectional composites are usually available in the literature. By
consequence, the estimation of damage occurrence in composite structures requires
introducing adapted failure criteria extending the available data to the combined loads
and composite laminates considered for one particular application. Many published
papers have dealt with this problem: see for instance (Tsai, 1987; Cuntze, 2003).
Nevertheless, it is established for a long time, that in composite structures the damage
initiates at microscopic scale, either (and most of time) in the matrix or (sometimes) in
the fibers. The failure of a ply is thus closely related and explained by the failure of its
microscopic constituents (Tsai, 1987; Cuntze, 2003; Fleck and Jelf, 1995; Kaddour et al.,
2003; Khashaba, 2004). As a consequence, the reliable prediction of a possible damage
occurrence of multi-directionnal laminates submitted to complex loading requires the
knowledge of the microscopic failure criteria of the epoxy matrix and carbon fibers
constituting the plies. Nevertheless, previous published works have emphasised the
following remarkable result: the strength of the pure constituents (i.e. pure epoxy resin)
strongly depends on the size of the sample, and especially on its thickness (Fiedler et al.,
2001). Besides, the thickness of a ply in thin laminates has the magnitude of 150 microns,
that is generally strongly weaker than the thickness of the samples tested for the
experimental determination of the strength of the pure constituents. As a consequence,
the experimental strengths of pure carbon fibers and epoxy matrices, determined on bulk
specimen can hardly be directly used to properly estimate microscopic failure criteria in
real structures. In particular, as shown for instance in (Garett and Bailey, 1977;
Christensen and Rinde, 1979), the effect of the matrix on transverse failure of composite
structures is of interest. The strain to failure of the pure matrix in uniaxial tension varied
from 1.5 to 70 % whereas transverse strains to failure of corresponding fiber reinforced
composites were dramatically smaller and varied only in the range 0.2 to 0.9%.
In the present study, an innovative method, dedicated to the determination of the
microscopic stress/strain failure criteria of the epoxy matrix embedded in a composite
structure is described. This method is based on the inversion of the analytical expressions
presented in section 4.3. The present work describes developments relating the
macroscopic failure envelopes to the microscopic ones. The conditions, indicated in
already published literature, when the macroscopic failure can exclusively be attributed
to matrix failure modes are taken into account as fundamental hypotheses of the present
approach. The model enables the identification of both the strength coefficients and
ultimate strength, so that the microscopic stress/strain failure envelopes can also be
drawn. Applications to the case of two typical carbon/epoxy composites (T300/5208 and
AS4/3501) are achieved: the failure conditions of the N5208 and 3501-6 epoxy resins
will be determined and compared.

32
Jacquemin Frédéric and Fréour Sylvain
5.2. Determination of the Local Failure Criterion of the Matrix from the
Macroscopic Strength Data of the Composite Ply
5.2.1. Introduction – Choice of a Failure Criterion
In this paper, failure is taken in the general sense previously defined in the literature,
including fracture, but also yield, etc. Since this works aims applications to multidirectional
structures submitted to triaxial stresses, general failure criteria are necessary to the description
of the strength in both stress and strain spaces. Failure criteria serve important functions in the
design and sizing of composite laminates. They should provide a convenient framework or
model for mathematical operations. The framework should be the same for different
definitions of failures, such as the ultimate strength, endurance limit, or a working stress
based on design or reliability considerations. However, the criteria are not intended to explain
the mechanisms of failure, that can occur concurrently or sequentially. The quadratic criterion
will be used in the present study: it includes interactions among the stress or strain
components analogous to the Von Mises criterion for isotropic materials, and is compatible
with the existence of strength having the properties, often met in the case that composite
structures are considered, to be anisotropic and also possibly different in tension or
compression. The criterion, expressed in stress space writes as follows :
F
i
mnop
σ
i mn
σ
i op
i
mn
i mn
+ F σ = 1
(52)
where F stands for the strength parameters respectively expressed in stress space. The
superscript i represents the scale considered for failure prediction (macroscopic: i=Ι or
pseudomacroscopic: i=m or i=r).
In order to use the failure criteria (52) presented above, it is necessary to identify the
i
i
quadratic ( F mnop ) and linear ( F mn ) strength parameters involved in the equation.
In the present work, for helping fixing the ideas, the simplified case of three-dimensional
stresses and strains (for both macroscopic and microscopic scales), with a single shear
component, usually met in multi-directional composite laminates submitted to mechanical
i i
loads (see examples given in Tsai, 1987) will be assumed to hold (i.e. σ13
= σ23
= 0 MPa ,
i i
ε13
= ε23
= 0 , where the subscripts 1, 2 and 3 respectively denotes the directions parallel
to the fiber axis, the transverse direction and the normal direction, in the orthogonal frame of
reference of the considered ply). Besides, the strength should be unaffected by the direction or
i
sign of the shear stress component σ 12 : if shear stress is reversed, the strength should be kept
i
i
constant. However, sign reversal for the longitudinal ( σ 11)
and transverse ( σ 22 ) stresses
components from tension to compression is expected to have a significant effect on both the
macroscopic and microscopic strength of the composite. As a consequence, terms of equation
(52) containing first-degree shear stress should be null. Finally, taking into account the
definition chosen for the reference frame, and the properties of (at least) transverse isotropy

Multi-scale Analysis of Fiber-Reinforced Composite Parts… 33
exhibited at any (i.e. macroscopic or microscopic) scale in one ply, the strength parameters
have to satisfy the following relations:
⎧ i i
F
⎪
2222 = F3333,
⎪ i i
⎨F1122
= F1133,
⎪ i i
⎪F
= F .
⎩ 22 33
Taking into account the above listed simplifications, equation (52) can be rewritten:
i i ( σ22
+ σ33
)
⎧ i i
2
i i
2
i ⎛ i
2
i
2 ⎞ i i
i i i
⎪1
= F1111σ11
+ 2 F1212σ12
+ F2222
⎜
⎜σ
22 + σ33
⎟ + 2 F1122σ11
+ 2 F2233σ22σ33
+ (54)
⎨
⎝ ⎠
⎪ i i i i i
⎪⎩
F11σ11
+ F22
( σ22
+ σ33
)
5.2.2. Direct Identification of the Macroscopic Strength Parameters
Most of the unknown macroscopic strength parameters in stress space, appearing in equation
(54) can be identified using information deduced from simple mechanical tests (uniaxial
tension, compression or longitudinal shear tests Tsai, 1987):
/
I
I 1 I 1 I 1 1 I 1 1 I 1
F = , F = , F = - , F = - , F =
1111
I I / 2222
I I
/ 11 I
I
/ 22 I
I
/ 1212
X X Y Y X X Y Y 2 S
Where X I and Y I are respectively the longitudinal and transverse tensile stress strength,
/
I
Y the longitudinal and transverse compressive stress strength, whereas S I is the
X and
longitudinal shear stress.
The two unknown remaining terms, I
F 1122 and I
F 2233 are related to the interaction
between two orthogonal stress components. The practical determination of these interaction
terms requires performing biaxial tests, which are not as easy to achieve than uniaxial tests.
As a consequence, the required data are often not available in the literature. There are,
however, geometric and physical conditions fixing the mathematical form of the failure
criterion (54): for instance, the failure envelope has to be closed so that the material cannot
present infinite strength when submitted to any load. Let us introduce a dimensionless
interaction term:
i*
mmnn
i
Fmmnn
i i
mmmmFnnnn
I 2
(53)
(55)
F = (56)
F

34
Jacquemin Frédéric and Fréour Sylvain
i*
For closed envelopes, the condition − 1 ≤ Fmmnn
≤ 1 has to be satisfied. But a more
detailed theoretical study (see Liu and Tsai, 1998) reduces the admissible range to the domain
1
[-1,0]. The same reference (Liu and Tsai, 1998) advises the choice of F -
2
* I
mmnn = for the
macroscopic interaction term (which corresponds to the generalised Von Mises model), since
this value is reasonable for a wide range of laminates. Taking into account this additional
I I
F = F ensures the
assumption in equation (56), the knowledge of I
F 1111 and
I
determination of the last two missing interaction terms 1122
F and
2222 3333
I
F 2233 , in stress space.
One similar method could be applied in order to determine the macroscopic strength
parameters expressed in strain space from the ultimate strains. Nevertheless, this method is
not useful in practice since uniaxial strains are difficult to apply to a sample. Thus, the
ultimate strains are generally deduced from the ultimate stresses: to reach this goal, one has to
introduce the macroscopic properties, i.e. the stiffness tensor L I , in order to relate both failure
criteria through Hooke’s law (33) expressed at macroscopic scale assuming a purely elastic
load.
5.2.3. Identification of the Microscopic Strength Parameter (of the Matrix Only)
Using an Inverse Method
From the standpoint of the structural designer, it is desirable to have failure criteria which are
applicable at the level of the lamina, the laminate, and the structural component.
Nevertheless, failure at macroscopic scale is often the consequence of an accumulation of
micro-level failure events (Tsai, 1987; Liu and Tsai, 1998). Laminated materials typically
exhibit many local failures prior to rupture. Thus, it is important to build up tools enabling to
enhance the understanding of micro-level failure mechanisms in order to develop higherstrength
materials. The ultimate goal is to have a failure theory that the designer can use with
confidence under the most general structural configuration and loading conditions and that the
developer of materials can use to design and fabricate new products to meet specific needs. In
order to reach this goal, the estimation of microscopic strength criteria would be of a valuable
help.
Since the epoxy resins involved in composite structures generally exhibit an isotropic
hygro-mechanical behaviour, the microscopic strength criterion expressed in terms of stresses
(54) simplifies as follows:
⎧
⎪1
⎨
⎪
⎩
= F
m
1111
+ F
m
11
⎛
⎜σ
⎝
m
2
11
+ σ
+ σ
m m m
( σ + σ + σ )
11
22
m
2
22
33
m
2
33
⎞
⎟ + 2F
⎠
m
1212
σ
m
2
12
+ 2 F
m
1122
m m m m m
[ σ ( σ + σ ) + σ σ ]
Thus, only four strength parameters have to be determined in order to enable failure
m m m m
predictions at microscopic scale:
F1111
, F1212
, F1122
, F11
.
Hypotheses being compatible
with the experimental observations are necessary to build an inverse model enabling the
11
22
33
22
33
(57)

Multi-scale Analysis of Fiber-Reinforced Composite Parts… 35
determination of these four parameters from the corresponding, available from practical
mechanical tests, macroscopic strength stress failure criterion.
The present work is focused on the development of modelling tools for the prediction of a
possible damage occurrence in fiber-reinforced epoxy laminates submitted to mechanical
loads. Actually, fibrous composite materials fail in a variety of mechanisms at the
fiber/matrix microscopic scale. Besides, according to the literature, i) fiber-dominated failures
usually occur when the plies are loaded in planes perpendicular to the fibers axis (longitudinal
tension and compression), whereas ii) matrix-dominated failures often occur in the cases that
the plies are loaded along the transverse and normal directions in tension and compression or
when shear stresses are applied to the considered ply (Tsai, 1987; Liu and Tsai, 1998). Thus,
matrix-dominated failure modes often occur in practice. As a consequence, the above listed i)
and ii) statements will be used in order to identify microscopic strength parameters in stress
and strain spaces for the matrix.
According to the developments of section 4, it is possible to derive the pioneering
numerical self-consistent model of Kröner and Eshelby in order to find the relation between
the macroscopic mechanical states and the researched corresponding microscopic stresses and
strains existing in the matrix of a composite material.
In the present work, the strength parameters in either the matrix or the ply will be
considered to remain independent from the magnitude of the applied mechanical load. Since
the damage envelope has been defined as the strain or stress threshold beyond which nonlinearity
occurs in the behaviour of the material at the scale concerned by damage, and in the
case that a purely mechanical load is taken into account, the material is assumed to behave
elastically until failure occurs. Now, in these conditions, both stress and strain ultimate
strength are simultaneously reached, and satisfy either macroscopic elastic Hooke’s law (33)
or the corresponding microscopic relations that are deduced from (38-42), assuming
I m
I m
ΔC = ΔC = 0 and ΔT = ΔT = 0 K .
It will be assumed that macroscopic failure occurring in the transverse and normal
directions, for a longitudinal stress σ 0 MPa
I
11 = , is governed by local failure of the matrix.
Various macroscopic stress states, compatible with that last hypothesis, are taken on the
macroscopic strength envelope (54), expressed in stress space and, finally implemented in the
scale transition relations (38-42). This leads to the determination of microscopic mechanical
stresses and strains states in the matrix, that are, according to our hypotheses, responsible for
macroscopic damage governed by matrix failure. As a consequence, these local mechanical
states should be compatible with the microscopic failure envelopes of the matrix as written in
equations (57).
According to this relation, four, non equivalent, macroscopic stress states suffice to find
m m m m
the eight researched coefficients involved in (57): F 1111,
F1212
, F1122
, F11
. The whole method
required to perform such estimation is described on table 7. Actually, four macroscopic
loading states taken on the stress failure envelope (defined on table 7) σ , σ , σ and
are required for the determination of the four coefficients of the failure envelopes since
numerical tests shows that equation (56) rewritten at microscopic scale for the epoxy matrix
does not provide an additional relation between m
F 1111 and m
F 1122 :
I
a
I
b
I
c
I
σ d

36
Jacquemin Frédéric and Fréour Sylvain
m*
1122
F
m
F1122
1
= ≠ −
(58)
m
F 2
1111
Moreover, according to (38-42) an uniaxial macroscopic tension or compression along
the transverse (or normal) direction induces local mechanical states in the matrix generally
exhibiting no zero strain and stress on-diagonal components (see for instance the cases of the
macroscopic loads
I
σ a and
I
σ b on Table 7). As a consequence, only the strength coefficient
m
F 1212 can be determined independently from the three others, from the single macroscopic
I
m m m
load σ d . Concerning the calculation of F 1111,
F1122
, F11
, one has to solve numerically the
system (60) (cf. Table 7).
Finally, the uniaxial microscopic ultimate stresses of the epoxy matrix embedded in the
composite structure can be deduced from the set of equations (55) expressed at microscopic
scale (i.e. replacing the subscripts I by the subscript m ), provided that the coefficients of the
local failure envelope are already known:
⎧
⎪X
⎪
⎪
⎪
⎨X
⎪
⎪
⎪S
⎪
⎩
m
/
m
m
= Y
=
m
= Y
m
/
1
2 F
= Z
= Z
m
1212
m
1
=
2 F
m
/
m
1111
1
=
2 F
⎛
⎜
⎝
m
1111
⎛
⎜
⎝
m
2
11
+ 4 F
m
2
11
F
m
1111
+ 4 F
The method, developed in the present paragraph, enables the determination of a) the
coefficients of the microscopic failure envelope of the epoxy matrix in stress and/or strain
space from the macroscopic failure envelope of the ply and scale transition relations
linking macroscopic loads to the corresponding local microscopic mechanical states
experienced by the matrix, only thereafter, b) the local maximum strength of the matrix
embedding the carbon fibers which can be evaluated from the classical formalism relating
the strength to the coefficients of the failure envelope. This inverse method provides an
alternative to the classical direct approach leading to the determination of the failure
envelope from the maximum strength measured on pure epoxies, in the cases that the
required data is not available or when the behaviour of the matrix embedded in the
composite structure is expected to be significantly different from the behaviour of the
pure matrix, as shown for example, in references (Garett and Bailey, 1977; Christensen
and Rinde, 1979).
F
− F
m
1111
m
11
⎞
⎟
⎠
+ F
m
11
⎞
⎟
⎠
(62)

Applied
macroscopic load
Corresponding
macroscopic strain
Corresponding
microscopic stress
according to (15-17)
Corresponding
conditions for
finding the
microscopic strength
coefficients in stress
space from (10, 19)
I
σ
a
I
ε
i
Table 7. One possible set of trials enabling the determination of the microscopic strength
coefficients of the matrix expressed in stress space.
⎡0
⎢
=
⎢
0
⎢
⎣0
=
⎡
⎢
ε
⎢
⎢
⎢
⎢
⎣
i = a, b
0
0
0
I
Y
I
11i
0
0⎤
⎥
0 , I
⎥
σ
b
0⎥
⎦
0
I
ε
22i
0
I
ε
33i
⎡0
⎢
= ⎢0
⎢
⎣
0
0
0
⎤
⎥
⎥
⎥
⎥
⎥
⎦
0
I
/
Y
0
0⎤
⎥
0⎥
0⎥
⎦
I I I
ε11i
= S12
σ22i
,
I I I
ε22i
= S22
σ22i
,
I I I
ε33i
= S23
σ22i
⎡0
0 0 ⎤
I ⎢ I ⎥
σ c =
⎢
0 σ22c
0
⎥ ,
⎢
I
⎣0
0 σ ⎥
33c⎦
⎧ I ⎛ I2
I2
⎞
⎪F2222
⎜
⎜σ
22c + σ33c
⎟ +
⎪ ⎝ ⎠
⎨
⎪ I I I I ⎛ I2
I2
⎞
⎪
2 F2233
σ22c
σ33c
+ F22
⎜
⎜σ22c
+ σ33c
⎟ −1
= 0
⎩
⎝ ⎠
⎡ I
ε
⎢ 11c
I
ε = ⎢
c 0
⎢
⎢
0
⎢⎣
0
I
ε 22c
0
0 ⎤
⎥
0 ⎥
⎥
I
ε
⎥
33c ⎥⎦
I I ( σ + σ )
I I
ε11c
= S12
22c 33c ,
I I I I I
ε22c
= S22
σ22c
+ S23
σ33c
,
I I I I I
ε33c
= S23
σ22c
+ S22
σ33c
⎡ m
σ
⎢ 11i
m
σ ⎢
i = 0
⎢
⎢
0
⎢⎣
0
m
σ 22i
0
0 ⎤
⎥
0 ⎥ , i = a, b, c
⎥
m
σ
⎥
33i ⎥⎦
(59)
⎧
F
m
+
m
+
m
⎪ 1111
Ai
F
1122
Bi
F
11
Ci
−1
= 0
⎪ 2 2 2
⎪A
=
m
+
m
+
m
⎪ i
σ
11i
σ
22i
σ
33i
, i = a, b, c
⎨ ⎡ ⎛
⎞ ⎤
⎪B
= 2 ⎢σ
m ⎜σ
m
+ σ
m ⎟ + σ
m
σ
m ⎥
⎪ i
⎢
11i ⎜ 22i 33i ⎟ 22i 33i
⎥
⎪ ⎣ ⎝
⎠ ⎦
⎪
⎩
C = σ
m
+ σ
m
+ σ
m
i 11i 22i 33i
(60)
I
σ d
⎡ 0
⎢ I
= ⎢S
⎢
⎢
0
⎣
S
⎡ 0
⎢
I ⎢ I
ε d = ε
⎢ 12d
⎢ 0
⎣
I
12d
ε = S
m
σ d
I
66
⎡ 0
⎢
= ⎢S
⎢
⎢
0
⎣
m
S
I
0
0
I
0⎤
⎥ ⎥⎥⎥
0
0
⎦
I
ε12d
S
m
0
0
0
0
0⎤
⎥
0⎥
0
⎥
⎥⎦
0⎤
⎥
0⎥
⎥
0⎥
⎦
2
2 F
m
S
m
1212
-1
= 0 (61)

38
Jacquemin Frédéric and Fréour Sylvain
5.3. Numerical Applications and Examples
5.3.1. Identification of the Microscopic Failure Criteria of Two Typical Epoxies
from the Knowledge of the Macroscopic Failure Envelope of AS4/3501-6 and
T300/N5208 Composite Plies
In the present paper, two types of high strength carbon fiber reinforced epoxies are
considered: a) AS4/3501-6 and b) T300/N5208 composites having identical fiber volume
fraction: v f =0.6. These two materials constitute good candidates for the present work, since
the microscopic strength of their respective matrix is not yet available (at our knowledge) in
the already published literature, in spite of they are quite often considered for illustrating
scientific works in this field of research (Tsai, 1987).
Strengths [MPa] X I
Table 8. Macroscopic strength data.
X I ´ Y I , Z I
Y I ´, Z I ´ S I
T300/5208 (Tsai, 1987) 1500 1500 40 246 68
AS4/3501-6 (Liu and Tsai, 1998) 1950 1480 48 200 79
Table 9. Quadratic macroscopic stress failure criteria deduced from the strength data.
Quadratic ijkl subscripted coefficients [MPa -2 ] and linear ij subscripted coefficients
[MPa -1 ].
Strength
parameters
I
F 1111
T300/5208 4.44 10 -7
I
F , 2222
I
F 3333
1.02 10 -4
I
F 1212
I
F , 1122
I
F 1133
I
F 2233
I
F 11
I
F ,
22
I
F 33
1.08 10 -4 -3.36 10 -6 -5.08 10 -5 0 0.0209
AS4/3501-6 3.46 10 -7 1.04 10 -4 8.01 10 -5 -3.00 10 -6 -5.02 10 -5 -0.0002 0.0158
The macroscopic strength of single plies are given in Table 8. The coefficients of the
corresponding quadratic macroscopic stress failure criteria, deduced from the classical direct
method, through equation (55) are listed in Table 9.
Table 10. Macroscopic stiffness components [GPa] of 60% volume uni-directionally
fiber reinforced plies. Fiber axis is parallel to longitudinal direction.
Stiffness
components
I
L 11
T300/5208 142.72
AS4/3501-6 137.27
I
L , 22
13.92
I
L 33
I
L , 12
I
L 13
I
L 23
I
L 44
I
L ,
55
5.79 7.19 3.34 7.00
11.60 4.20 5.22 3.68 6.45
I
L 66

Multi-scale Analysis of Fiber-Reinforced Composite Parts… 39
Table 11. Quadratic local stress failure criteria in N5208 and 3501-6 epoxy matrices
respectively deduced from the macroscopic failure envelopes of T300/5208 and
AS4/3501-6 plies, taking into account the microscopic elastic properties given on tables 1
and 5. Quadratic ijkl subscripted coefficients [MPa -2 ] and linear ij subscripted
coefficients [MPa -1 ].
Strength
parameters
m
F 1111 , m
F 2222 , m
F 3333
N5208 2.18 10 -4
3501-6 2.15 10 -4
m
F 1212
m
F 1122 , m
F , 1133
m
F 2233
m
F 11 , m
F ,
22
m
F 33
7.82 10 -4 -8.77 10 -5 0.0162
5.04 10 -4 -8.07 10 -5 0.0143
Table 12. Local (matrix embedded in a composite ply) strength data deduced from the
local quadratic stress failure criteria of a N5208 and 3501-6 epoxy matrices respectively.
Strengths [MPa] X m , Y m , Z m X m ´, Y m ´, Z m ´ S m
N5208 40.1 114.7 25.3
3501-6 42.6 108.9 30.9
In order to achieve the identification of the coefficients of the quadratic microscopic
failure criteria of the pure epoxies (3501-6 and N5208, respectively), the method
previously explained in subsection 5.2.3 was applied. The macroscopic stiffnesses
considered for the simulation are provided in Table 10, whereas the elastic constants of
the elastically isotropic resins, required for localising the macroscopic stress/strain states
at the microscopic scale in the matrices, according to equations (38-42), were previously
given in tables 1 and 5. In order to find the microscopic strength coefficients, four
independent macroscopic stress states
I
σ a ,
I
σ b ,
I
σ c ,
I
σ d located on the macroscopic
failure envelope according to the conditions described on the first raw of Table 7. Table
11 shows the strength coefficients found for the quadratic microscopic failure criterion in
stress space of both epoxies by solving equations (60-61). Besides, the microscopic
ultimate uniaxial stresses of the two studied epoxies have been determined by introducing
in equation (62) the results of the previous identification of the strength coefficients of
their respective quadratic failure criterion in stress space (still Table 11). The
corresponding results have been listed in Table 12.
Finally, instances of the microscopic failure envelopes have been drawn and
superimposed to the corresponding macroscopic failure envelopes. Pictures of Figure 5
compare the results obtained in stress space for each couple epoxy/composite.

40
&22 [MPa]
& & [MPa]
& & [MPa]
-4000 -3000 -2000 -1000-50 0 1000 2000
T300/5208
N5208
Jacquemin Frédéric and Fréour Sylvain
50
-150
-250
-350
& 11 [MPa]
-500 -400 -300 -200 -100 0 100
-100
T300/5208
N5208
100
50
-250 -200 -150 -100 -50 0 50
T300/5208
N5208
σ 22 [MPa]
& 22 [MPa]
200
100
0
-200
-300
-400
-500
0
-50
-100
& & &[MPa]
σ 33 [MPa]
& & [MPa]
-4000 -3000 -2000 -1000-50 0 1000 2000
AS4/3501
3501-6
50
-150
-250
-350
σ 11 [MPa]
200
-500 -400 -300 -200 -100 0 100
AS4/3501
3501-6
σ 22 [MPa]
0
-200
-400
100
50
-250 -200 -150 -100 -50 0 50
AS4/3501
3501-6
σ 22 [MPa]
Figure 5. Examples of macroscopic and local (matrix only) stress failure envelopes of T300/5208 and
AS4/3501-6 plies.
5.3.2. Observations on Predicted Results and Discussion
According to the identification procedure described in subsection 5.2.3, an infinite number of
I I I I
macroscopic stress states sets { σ a , σ b , σ c , σ d } can be considered for the determination
I
of the researched microscopic failure envelope strength coefficients. Actually, σ c only may
I
a
I
b
I
d
vary whereas σ , σ and σ are fixed by the macroscopic ultimate stresses
of the considered composite structure (see the first raw of Table 7). Several tests were
0
-50
-100
I
Y ,
/
I
Y ,
I
S

Multi-scale Analysis of Fiber-Reinforced Composite Parts… 41
performed, introducing various numerical stress states (compatible with the constitutive
I
hypotheses of the present work) for σ c . The tests showed that the microscopic strength
coefficients are, as expected, independent from the choice of the initial macroscopic stress
I
state σ c : one set of coefficients only is found as the unique solution of system (60). This
demonstrates that the inverse model presented here is reliable from a numerical point of view.
The obtained results for the ultimate uniaxial stresses of 3501-6 and N5208 epoxies are
close together (Table 12), whereas the macroscopic strength present significant discrepancies
(Table 8). As an example, the relative deviation between the macroscopic longitudinal tensile
ultimate stress of the two composites reaches around 25% when the relative deviation
between the longitudinal tensile ultimate stress of the two epoxies is limited to 6%. Moreover,
the representation of the microscopic failure envelopes are rather similar for the two
considered resins, (Figure 5), whereas the macroscopic failure envelopes differ from one
composite to the other (Figure 5, also). This could be interpreted as follows: for the
considered composites, the observed deviation in the macroscopic failure envelopes comes
from the choice of the reinforcing fibers and not from the choice of the resin. This is
remarkable, since the considered epoxies exhibit a very different elastic mechanical behaviour
(see Tables 1 and 5).
Moreover, the predicted microscopic ultimate uniaxial stresses are coherent with
experimental results measured on plain resins. For instance, reference (Fiedler et al., 2001)
reports a strength value of 117 MPa in compression, and elastic limits reaching respectively
29 MPa in tension and 31 MPa in torsion for small specimen of plain unreinforced Bisphenol-
A type resin (i.e. “small” denotes a significantly reduced sized in normal and transverse
directions compared to “bulk” specimen). These measured strength are of the same order of
magnitude than the strength, calculated in the present work, for 3501-6 and N5208 epoxies.
At the opposite, the strengths determined on bulk specimens of 5208 and 3501-6 plain
epoxies are approximately two times higher than the values obtained in the present work, for
the strength of the corresponding epoxies embedded in thin composite plies. This last result is
also compatible with both the experimental comparison achieved in reference (Fiedler at al.,
2001) on various sized pure epoxies and the practical comparisons of the failure mechanisms
exhibited by composites structures and their constitutive epoxy resin (see Garett and Bailey,
1977; Christensen and Rinde, 1979). The present work allows to represent the scale effects
observed in practice on the composite constituents strengths, because the composite ply
strengths involved in the calculations do actually depend on both the constituents properties
and microstructure.
6. Conclusions
The present work dealt with the question of scale transition modelling of polymer matrix
composites and its application to several fields of investigation. Therefore, Mori-Tanaka and
Eshelby-Kröner self-consistent models, taking advantage of arithmetic averages, were both
considered for achieving the determinaiton of the homogenized properties of composite ply as
a function of the properties of its constituents (on the one hand, the matrix , and on the second
hand, the reinforcements).

42
Jacquemin Frédéric and Fréour Sylvain
The theoretical models properly take into account the specific microstructure of such
materials. Especially the extreme morphology of the reinforcements can be considered, while
the morphology and orientation of the reinforcing inclusions are kept constant in a single ply.
As a consequence, the models manage to reproduce realistically the strong macroscopic
anisotropy observed in practice on uni-directionally fiber-reinforced epoxies. The obtained
results have shown that the two approaches, presented here, yield close together estimations
of the macroscopic coefficients of thermal expansion, coefficients of moisture expansion and
elastic moduli, in the range of the epoxy volume fraction, that is typical for designing
m
composites structures for engineering applications ( i.e. 0.3 ≤ v ≤ 0.7)
. Nevertheless, an
I
exception to this statement occurs for Coulomb modulus G 12 , that is strongly
underestimated in the case that the calculations are performed according to Mori-Tanaka
approximation, in the same range of epoxy volume fractions.
Moreover, realistic inverse scale transition procedures based on Kröner-Eshelby selfconsistent
model and Mori-Tanaka estimates were also provided for achieving the numerical
determination of the mechanical, hygroscopic or thermal properties of one constituent of an
uni-directionally reinforced composite ply. Both models were used in order to estimate the
elastic stiffness of reinforcing fibers embedded in a composite ply, from the knowledge of the
macroscopic properties and those of the matrix. The obtained numerical results were
successfully compared with expected practical results. A similar study was achieved in the
standpoint of estimating the coefficients of moisture expansion of the matrix constituting a
composite ply. In both cases the proposed theoretical approaches led to similar results, which
is satisfying. Thus, the two inverse models described in the present work can be equally used
in order to achieve such an identification.
Another section of this article was devoted to the analysis of the macroscopic mechanical
states localization within the constituents of a composite ply. Since it was previously
demonstrated in the literature, that Mori-Tanaka approximation was not reliable for handling
such a task, only Eshelby-Kröner model was considered. A numerical model, valid for any
morphology of the reinforcing inclusions, was provided. Moreover a rigorous fully analytical
treatment of the classical Kröner and Eshelby Self-Consistent model including morphology
effects was achieved also. Especially, the determination of Morris’ tensor was performed in a
satisfactory agreement with the transverse macroscopic elastic anisotropy expected for the
fiber shape that should be taken into account in order to satisfactory represent the specific
microstructure of carbon-fiber reinforced composites. The new closed-form solutions
obtained for the components of Morris’ tensor were introduced in the classical hygro-thermoelastic
scale transition relation in order to express analytically the internal strains and stresses
in both the fiber and the resin of a ply submitted to a hygro-thermo-elastic load. The closedform
solution demonstrated in the present work was compared to the fully numerical selfconsistent
model for various geometrical arrangements of the fibers: uni-directional or
laminated composites. A very good agreement was obtained between the two models for any
component of the local stress tensors. It was also demonstrated that continuum mechanics and
micro-mechanical models give complementary information about the occurrence of a possible
damage during the loading of the structure.
In a last part, the present study explained a procedure enabling to achieve the
identification of one single set of strength parameters defining completely the microscopic

Multi-scale Analysis of Fiber-Reinforced Composite Parts… 43
failure envelope of the matrix entering in the composition of a composite structure, in the
cases that a pure mechanical load is applied. The identification method was built around an
inverse scale transition method which requires the knowledge of the macroscopic strengths,
and both the macroscopic and microscopic elastic stiffnesses. Besides, it was necessary to
consider some hypotheses in order to proceed to the identification of the coefficients of the
microscopic quadratic failure criteria. In the present work, it was assumed that the
macroscopic failure of a uni-directionally reinforced ply is dominated by the local failure of
the matrix when the external load is applied in planes perpendicular to the fiber axis.
Numerical applications of the proposed inverse method were made considering the cases
of two high-strength composites structures: AS4/3501-6 and T300/N5208. The determination
of the microscopic quadratic failure criterion of the pure epoxies (3501-6 and N5208,
respectively) was achieved. The obtained results are close together and present a good
agreement with ultimate strengths measured on reduced sized plain resins (available from
already published literature). This demonstrates the reliability of the present predictive
method for estimating the local failure behaviour of epoxies whose experimental failure
criterion has not yet been determined.
In further works, the proposed approach will be extended to the more general case of
hygro-thermo-mechanical loads. This will imply to take into account the stress free strains in
order to keep consistency between the failure envelopes expressed in stress and strain spaces.
Besides, the rigorous treatment of the hygro-thermo-mechanical load requires to consider the
dependence on the temperature and moisture content of a) the elastic stiffness, coefficients of
thermal expansion and coefficients of moisture expansion and b) the ultimate strength (and in
general, the coefficients of the considered failure criterion), at both macroscopic and
microscopic scales. Others perspectives of research are proposed in the following section
below.
7. Perspectives
Scale transition modelling based theoretical analysis of composite structures constitutes an
overexpanding field of research, due to multiple factors. Among them, the emergence of new
materials exhibiting a specific, more advanced microstructure, the ambition to account for
additional, sometimes only recently discovered, physical phenomena and the relentless
research for building faster, more convenient but still reliable models stand for the three
essential motivations for achieving further developments in the incoming years.
7.1. Emergence of New Materials
The present development stage of Eshelby’s single inclusion theory involved in the
mechanical modeling of composites is not intended for a rigorous treatment of the
morphology presented by the reinforcements used for manufacturing woven-composites. As a
consequence, answering to the question of a theoretical study, through scale transition
models, of mechanical parts made of such composites will require a specific and still missing
solution.

44
Jacquemin Frédéric and Fréour Sylvain
Since the recent discovery of carbon nanotubes in the 90’s, researchers worldwide have
engaged in fundamental studies of this novel material (Treacy et al., 1996). The pioneering
works have underlined the characteristics of carbon nanotubes such as an extraordinarily high
stiffness (Salvetat et al., 1999) coupled to a high tensile strength (Demczyk et al., 2002)., high
aspect ratio and an especially low density. Actually, for instance, the experimental direct
mechanical measurement of the elastic properties of carbon nanotubes provided Young’s
moduli in the range of 1 TPa, which considerably exceeds the corresponding modulus of any
currently available fiber material (Salvetat et al., 1999; Demczyk et al., 2002).
In consequence, the technological applications of carbon nanotubes as reinforcements for
elastomers (Frogley et al., 2003) or polymer-based composites (Liu and Wagner, 2005;
Breton et al., 2004; Xiao et al., 2006) was very recently investigated. Furthermore, multimaterials
made up of polymer matrix, carbon fibers and carbon nanotubes are considered also
for achieving a new generation of engineering composites.
7.2. Accounting for Additional Physical Factors
The present work is focused on the theoretical prediction of the mechanical behaviour of
composite structures submitted also to environmental conditions. However, every aspect of
the consequences of environmental loading on the constituents of composite materials have
not always been considered in this paper, for the sake of simplicity. Nevertheless, accounting
for some additional physical factors would improve the realism and the reliability of the
predictions obtained through the scale-transition models.
For instance, the moisture diffusion process was assumed, in the present work, to follow
the linear, classical, established for a long time, Fickian model. Nevertheless, some valuable
experimental results, already reported in (Gillat and Broutman, 1978), have shown that
certain anomalies in the moisture sorption process, (i.e. discrepancies from the expected
Fickian behaviour) could be explained from basic principles of irreversible thermodynamics,
by a strong coupling between the moisture transport in polymers and the local stress state
(Weitsman, 1990a, Weitsman, 1990b).
The present work yields several perspectives of research concerning the application of
scale transition model to the identification of composite materials properties. Moisture and
temperature are not the only parameters leading to an evolution of the mechanical properties
of epoxies. According to the literature, thermo-oxidation is reported to enhance the stiffness
of the epoxies (Decelle et al., 2003 ; Ho and al., 2006). The inverse methods presented here
could for instance be directly applied to the estimation of the epoxy stiffening from the
knowledge of the macroscopic elastic properties evolution as a function of the mass loss
during the thermo-oxidation process. Furthermore, extensions of the inverse models could be
achieved in order to account for the variation of the coefficients of thermal and/or moisture
expansion of the constituents of a composite ply, enabling to identify them and their
evolutions as a function of the environmental conditions. Finally, a similar approach could be
developed in order to identify the damage induced evolution of the mechanical behaviour of
the constituents of composite plies from the inelastic part of macroscopic stress/strain curves.
The experimental data required for achieving such analysis is already available in the
literature (Soden et al., 1998). Nevertheless, local and macroscopic damage have still to be

Multi-scale Analysis of Fiber-Reinforced Composite Parts… 45
implemented in the theoretical laws. The above-listed perspectives of research will be
successively considered in further works.
7.3. Improving the Calculation Time While Ensuring the Most Reliable
Predictions
The present work underlines the sometime existing opportunity to replace purely numerical
mathematical solutions by analytical forms enabling to significantly reduce both the time
required for designing the software and the time necessary for achieving one simulation. It
was demonstrated in this paper that Eshelby-Kröner could be, at least partially, presented as
an analytical model, while it was used for predicting mechanical states. Nevertheless, the
estimation of the macroscopic properties (elastic stiffness, coefficients of thermal expansion
and coefficients of moisture expansion) through the homogenization relations deduced from
this very model do still involve an implicit iterative procedure. It was already shown in the
literature by Welzel and his co-authors, that under specific conditions, it was possible to build
a model, numerically equivalent to Eshelby-Kröner model, from the combination of two
(separately less successful) other models (Welzel, 2002 ; Welzel et al. 2003). The concept is
similar to the idea based on empirical comparisons, historically proposed by Neerfeld
(Neerfeld, 1942) and Hill (Hill, 1952) to average Reuss and Voigt rough hypotheses in order
to get a numerically acceptable theoretical solution. In the field of micro-mechanical
modelling of composite materials, a combination of the two possible localization procedures
considered for Mori-Tanaka model in the present work would enable to numerically
reproduce the homogenized properties obtained from Eshelby-Kröner model. Building an
effective model from the two main ways of writing Mori-Tanaka model would mainly enable
to obtain closed-form solutions for the elastic stiffness tensor, instead of having to
numerically solve the iterative procedure involved in Eshelby-Kröner self-consistent model.
Thus, a coupling of this numerically effective solution for predicting realistic hygrothermomechanical
macroscopic properties to the already proposed in this very article analytical
forms for the local mechanical states would yield to a faster but still extremely reliable
innovative scale-transition approach for studying composite materials. The analytical forms
required for achieving the effective Mori-Tanaka model should be derived and published in
the near future.
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In: Composite Materials Research Progress ISBN: 1-60021-994-2
Editor: Lucas P. Durand, pp. 51-107 © 2008 Nova Science Publishers, Inc.
Chapter 2
OPTIMIZATION OF LAMINATED COMPOSITE
STRUCTURES: PROBLEMS, SOLUTION PROCEDURES
AND APPLICATIONS
Michaël Bruyneel
SAMTECH s.a., Liège Science Park
Rue des Chasseurs-ardennais 8, 4031 Angleur, Belgium
Abstract
In this chapter the optimal design of laminated composite structures is considered. A
review of the literature is proposed. It aims at giving a general overview of the problems that a
designer must face when he works with laminated composite structures and the specific
solutions that have been derived. Based on it and on the industrial needs an optimization
method specially devoted to composite structures is developed and presented. The related
solution procedure is general and reliable. It is based on fibers orientations and ply thicknesses
as design variables. It is used daily in an (European) industrial context for the design of
composite aircraft box structures located in the wings, the center wing box, and the vertical
and horizontal tail plane. This approach is based on sequential convex programming and
consists in replacing the original optimization problem by a sequence of approximated subproblems.
A very general and self adaptive approximation scheme is used. It can consider the
particular structure of the mechanical responses of composites, which can be of a different
nature when both fiber orientations and plies thickness are design variables. Several numerical
applications illustrate the efficiency of the proposed approach.
1. Introduction
According to their high stiffness and strength-to-weight ratios, composite materials are well
suited for high-tech aeronautics applications. A large amount of parameters is needed to
qualify a composite construction, e.g. the stacking sequence, the plies thickness and the fibers
orientations. It results that the use of optimization techniques is necessary, especially to tailor
the material to specific structural needs. The chapter will cover this subject and is divided in
three main parts.

52
Michaël Bruyneel
After recalling the goal of optimization, the different laminates parameterizations will be
presented with their limitations (the pros and the cons) in the frame of the optimal design of
composite structures. The issues linked to the modeling of structures made of such materials
and the problems solved in the literature will be reviewed. The key role of fibers orientations
in the resulting laminate properties will be discussed. Finally the outlines of a pragmatic
solution procedure for industrial applications will be drawn. Throughout this section, a
profuse and state-of-the-art review of the literature will be provided.
Secondly, a general solution procedure used daily in industrial problems including fibers
reinforced composite materials will be described. The related optimization algorithm is based
on sequential convex programming and has proven to be very reliable. This algorithm is
presented in detail and validated by comparing its performances to other optimization
methods of the literature.
Finally, it will be shown how this optimization algorithm can efficiently solve several
kinds of composite structure design problems: amongst others, solutions for topology
optimization with orthotropic materials will be presented, important considerations about the
optimal design of composites including buckling criteria will be discussed, optimization with
respect to damage tolerance will be considered (crack delamination in a laminated structure).
On top of that, some key points of the solution procedure based on this optimization
algorithm applied to the pre-sizing of (European) industrial composite aircraft box structures
will be presented.
2. The Optimal Design Problem and Available Optimization
Methods
The goal of optimization is to reach the best solution of a problem under some restrictions. Its
mathematical formulation is given in (2.1), where g0(x) is the objective function to be
minimized, gj(x) are the constraints to be satisfied at the solution, and x={xi, i=1,…,n} is the
set of design variables. The value of those design variables change during the optimization
process but are limited by an upper and a lower bound when they are continuous, which will
be the case in the sequel.
min g ( x )
0
max
g j ( x ) ≤ g j j = 1,...,
m
(2.1)
xi ≤ xi
≤ xi
i
= 1,...,
The problem (2.1) is illustrated in Figure 2.1, where 2 design variables x1 and x2 are
considered. The isovalues of the objective function are drawn, as well as the limiting values
of the constraints. The solution is found via an iterative process. x k is the vector of design
variables at the current iteration k, and x k+1 is the estimation of the solution at the iteration
k+1. Typically a local solution xlocal will be reached when a gradient based optimization
method is used. The best solution xglobal can only be found when all the design space is looked
over: this last can be accessed with specific optimization methods that include a non
deterministic procedure, as the genetic algorithms.
n

Optimization of Laminated Composite Structures… 53
x2
* global
X
a
(k)
S
no
X
( k+
1)
b
(k)
X
α
Initial design
Structural analysis
Optimization
New design
Optimal design ?
yes
End
* local
X
g j (X)
Figure 2.1. Illustration of an optimization problem and its solution.
In structural optimization, the design functions can be global as the weight, the stiffness,
the vibration frequencies, the buckling loads, or local as strength constraints, strains and
failure criteria. When the design variables are linked to the transverse properties of the
structural members (e.g. the cross-section area of a bar in a truss), the related optimization
problem is called optimal sizing (Figure 2.2a). The value of some geometric items (e.g. a
radius of an ellipse) can also be variable: in this case, we are talking about shape optimization
(Figure 2.2b). Topology optimization aims at spreading a given amount of material in the
structure for a maximum stiffness. Here, holes can be automatically created during the
optimization process (Figure 2.2c). Finally, the optimization of the material can be addressed,
e.g. the local design of laminated composite structure with respect to fibers orientations, ply
thickness and stacking sequence (Figure 2.2d).
x1

54
Initial
designs
Final designs
a) Optimal sizing b) Shape
optimization
Michaël Bruyneel
c) Topology
optimization
Figure 2.2. The structural optimization problems.
d) Material
optimization
The structural optimization problems are non linear and non convex, and several local
minimum exist. It is usually accepted that a local solution xlocal gives satisfaction. The global
solution xglobal can only be determined with very large computational resources. In some cases
when the problem includes a very large amount of constraints, a feasible solution is
acceptable.
A lot of methods exist to solve the problem (2.1). Morris (1982), Vanderplaats (1984),
and Haftka and Gurdal (1992) present techniques based on the mathematical programming
approach used in structural optimization. Most of them are compared by Barthelemy and
Haftka (1993), and Schittkowski et al. (1994). Non deterministic methods, such as the genetic
algorithm (Goldberg, 1989), are studied by Potgieter and Stander (1998), and Arora et al.
(1995). Those authors also present a review of the methods used in global optimization.
Optimality criteria for the specific solution of fibers optimal orientations in membrane
(Pedersen 1989) and in plates (Krog 1996) must be mentioned as well. Finally the response
surfaces methods are also used for optimizing laminated structures (Harrison et al. 1995, Liu
et al. 2000, Rikards et al. 2006, Lanzi and Giavotto 2006).
The approximation concepts approach, also called Sequential Convex Programming,
developed in the seventies by Fleury (1973), Schmit and Farschi (1974), and Schmit and
Fleury (1980) has allowed to efficiently solve several structural optimization problems: the
optimal sizing of trusses, shape optimization (Braibant and Fleury, 1985), topology
optimization (Duysinx, 1996, 1997, and Duysinx and Bendsøe, 1998), composite structures
optimization (Bruyneel and Fleury 2002, Bruyneel 2006), as well as multidisciplinary
optimization problems (Zhang et al., 1995 and Sigmund, 2001). In sizing and shape
optimization the solution is usually reached within 10 iterations. For topology optimization,
since a very large number of design variables are included in the problem, a larger number of
design cycles is needed for converging with respect to stabilized design variables values over
2 iterations.
Those approximation methods consist in replacing the solution of the initial optimization
problem (2.1) by the solution of a sequence of approximated optimization problems, as
illustrated in Figure 2.3.

Optimization of Laminated Composite Structures… 55
x2
(k)
*
X
* global
X
no
(k)
X
Initial design
End
yes
* local
X
g j (X)
Approximated optimization problem
Solution of the approximated problem
Optimal design ?
~ ( k)
g j ( X)
Figure 2.3. Definition of an approximated optimization problem based on the information at the current
design point x (k) . The corresponding feasible domain is defined by the constraints of (2.2).
Each function entering the problem (2.1) is replaced by a convex approximation
~ ( k)
g j ( X)
based on a Taylor series expansion in terms of the direct design variables x i or
intermediate ones as for example the inverse design variables 1 xi
. For a current design x (k)
at iteration k, the approximated optimization problem writes:
~
~ ( k)
min g0
( x)
( k)
max
g j ( x ) ≤ g j
j = 1,...,
m
(2.2)
( k)
( k)
xi ≤ xi
≤ xi
i = 1,...,
n
where the symbol ~ is related to an approximated function. The explicit and convex
optimization problem (2.2) is itself solved by dedicated methods of mathematical
x1

56
Michaël Bruyneel
programming (see Section 7). Building an approximated problem requires to carry out a
structural and a sensitivity analyses (via the finite elements method). Solving the related
explicit problem does no longer necessitate a finite element analysis (expensive in CPU for
large scale problems).
The solution obtained with this approach doesn’t correspond to the global optimum, but
to a local one, since gradients and deterministic information are used. Nevertheless this local
solution is found very quickly and several initial designs could be used to try to find a better
solution, as proposed by Cheng (1986). Finally it must be noted that when a very large
number of constraints is considered in the optimal design problem (say more than 10 5 ) the
user is often satisfied with a feasible solution.
3. Parameterizations of Laminated Composite Structures
Before presenting the several possible parameterizations of laminates, with their advantages
and their disadvantages, the classical lamination theory is briefly recalled in order to
introduce the notation that will be used throughout the chapter. See Tsai and Hahn (1980),
Gay (1991) and Berthelot (1992) for details.
3.1. The Classical Lamination Theory
3.1.1. Constitutive Relations for a Ply
Fibers reinforced composite materials are orthotropic along the fibers direction, that is in the
local material axes (x,y,z) illustrated in Figure 3.1. Homogeneous macroscopic properties are
assumed at the ply and at the laminate levels.
y
z,3
2
x
θ
1
Material axes
(orthotropy)
Figure 3.1. The unidirectional ply with its material and structural axes.
For a linear elastic behaviour, the stress-strain relations in the material axes are given by
the Hook’s law Qε
σ = where ε and σ are the strain and stress tensors, respectively, while
Q is the matrix collecting the stiffness coefficients in the orthotropic axes. For a plane stress
assumption, it comes that

⎧σ
x ⎫ ⎡ mE
⎪ ⎪ ⎢
x
⎨σ
y ⎬ = ⎢mν
xy E y
⎪ ⎪ ⎢
⎩
σ xy ⎭ ⎣
0
Optimization of Laminated Composite Structures… 57
mν
yxE
x
mEy
0
0 ⎤⎧
ε x ⎫ ⎡Qxx
⎥⎪
⎪
0 =
⎢
⎥⎨
ε y ⎬ ⎢
Qyx
G ⎥⎪
⎪
xy ⎢
⎦⎩
γ xy ⎭ ⎣ 0
Qxy
Qyy
0
0 ⎤⎧
ε x ⎫
⎪ ⎪
0
⎥
⎥⎨
ε ⎬ ,
1
y m =
Q ⎥⎪
⎪ 1 −ν
xyν
ss ⎦⎩
γ xy ⎭
yx
(3.1)
The stresses and strains can be written in the structural coordinates (1,2,3) as in (3.2) and
(3.3) where θ is the angle between the local and structural axes, defined in Figure 3.1.
⎡ 2
⎧ε1
⎫ cos θ
⎪ ⎪ ⎢ 2
⎨ε
2 ⎬ = ⎢ sin θ
⎪ ⎪ ⎢
⎩ε
6 ⎭ ⎢
2 cosθ
sinθ
⎣
⎡ 2
⎧σ1
⎫ cos θ
⎪ ⎪ ⎢ 2
⎨σ
2 ⎬ = ⎢ sin θ
⎪ ⎪ ⎢
⎩σ
6 ⎭ ⎢
cosθ
sinθ
⎣
sin
cos
2
2
θ
θ
− 2 cosθ
sinθ
sin
cos
2
2
θ
θ
− cosθ
sinθ
− cosθ
sinθ
⎤ ⎧ ε x ⎫
⎥ ⎪ ⎪
cosθ
sinθ
⎥ ⎨ ε y ⎬
2 2
cos θ − sin θ ⎥ ⎪ ⎪
⎥⎦
⎩
γ xy ⎭
− 2cosθ
sinθ
⎤ ⎧σ
x ⎫
⎥ ⎪ ⎪
2 cosθ
sinθ
⎥ ⎨σ
y ⎬
2 2
cos θ − sin θ ⎥ ⎪ ⎪
⎥⎦
⎩
σ xy ⎭
(3.2)
(3.3)
For a ply with an orientation θ with respect to the structural axes, the constitutive
relations write:
⎧σ1
⎫ ⎡Q
⎪ ⎪
=
⎢
⎨σ
2 ⎬ ⎢
Q
⎪ ⎪
⎩σ
6 ⎭ ⎢⎣
Q
11
12
16
Q
Q
Q
12
22
26
Q
Q
Q
16
26
66
⎤⎧ε1
⎫
⎥⎪
⎪
⎥⎨ε
2 ⎬
⎥⎪
⎪
⎦⎩ε
6 ⎭
where the matrix of the stiffness coefficients in the structural axes takes the form:
⎡
(3.4)
⎧Q
4 4 2 2
2 2
11 ⎫ c s 2c
s 4c
s
⎪
Q
⎪ ⎢ 4 4 2 2
2 2 ⎥
⎪ 22 ⎪ ⎢ s c 2c
s 4c
s ⎥ ⎧Qxx
⎫
⎪Q
2 2 2 2 4 4
2 2
12 ⎪ ⎢
⎪
4 Q
⎪
c s c s c + s − c s ⎥ ⎪ yy ⎪
Q ( 1,
2,
3)
= ⎨ ⎬ = ⎢
⎥
2 2 2 2 2 2 2 2 2 ⎨ ⎬ (3.5)
⎪Q66
⎪ ⎢c
s c s − 2c
s ( c − s ) ⎥ ⎪Qxy
⎪
⎪Q
⎪ ⎢ 3 3 3 3 3 3 ⎥
16 c s cs cs c s 2(
cs c s)
⎪
⎩Q
⎪
⎢
− −
−
⎥ ss ⎭
⎪ ⎪
( x,
y,
z)
⎪Q
3 3 3 3 3 3
⎩ 26 ⎪⎭
⎢
( 1,
2,
3)
⎣ cs − c s ( c s − cs ) 2(
c s − cs ) ⎥⎦
with
c = cos θ s = sinθ
The variation of the Q’s with respect to the angle θ is plotted in Figure 3.2. It is observed
that the stiffness coefficients are highly non linear in terms of the fibers orientation.
⎤

58
Michaël Bruyneel
Figure 3.2. Stiffness coefficients in N/mm² in the structural axes for several values of the fibers
orientation in a carbon/epoxy material T300/5208 (after Tsai and Hahn, 1980).
Based on the fact that the trigonometric functions entering the matrix in (3.5) can be
written in the following way:
4 1
cos θ = ( 3 + 4cos
2θ
+ cos 4θ
)
8
3 1
cos θ sinθ
= ( 2sin
2θ
+ sin 4θ
)
8
2 2 1
cos θ sin θ = ( 1 − cos 4θ
)
8
3 1
cosθ
sin θ = ( 2sin
2θ
− sin 4θ
)
8
4 1
sin θ = ( 3 − 4 cos 2θ
+ cos 4θ
)
8
(3.6)
Tsai and Pagano (1968) derived an alternative expression for the Q’s coefficients in the
structural axes given in (3.7):
⎡Q11
Q12
Q16
⎤
Q ( 1,
2,
3)
=
⎢
Q22
Q
⎥
26 = γ0
+ γ1
cos2θ + γ2
cos4θ
+ γ3
sin 2θ
+ γ4
sin 4θ
⎢
⎥
⎢⎣
sym Q66⎥⎦
where the parameters γ are functions of the lamina invariants U1-U5:
(3.7)

and
γ
2
γ
0
Optimization of Laminated Composite Structures… 59
⎡ U1
=
⎢
⎢
⎢⎣
sym
⎡ U3
=
⎢
⎢
⎢⎣
sym
U
U
−U
U
3
3
4
1
0 ⎤
0
⎥
⎥
U5⎥⎦
0 ⎤
0
⎥
⎥
−U
3⎥⎦
1
U1
= ( 3Qxx
+ 3Q
yy + 2Qxy
+ 4Qss
)
8
1
U2
= ( Qxx
− Qyy
)
2
1
U3
= ( Qxx
+ Qyy
− 2Qxy
− 4Qss
)
8
⎡
⎢
0
⎢
γ 3 = ⎢
⎢
⎢sym
⎢⎣
3.1.2. Constitutive Relations for a Laminate
⎡ U2
0 0⎤
γ
⎢
⎥
1 = −U
0
⎢ 2
(3.8)
⎥
⎢⎣
sym 0⎥⎦
0
0
U2
⎤
2 ⎥
U ⎥
2
⎥
2 ⎥
0 ⎥
⎥⎦
⎡ 0
γ
⎢
4 =
⎢
⎢⎣
sym
0
0
U3
⎤
−U
⎥
3⎥
0 ⎥⎦
1
U4
= ( Qxx
+ Qyy
+ 6Qxy
− 4Qss
)
8
1
U5
= ( Qxx
+ Qyy
− 2Qxy
+ 4Qss
)
8
Composite structures are thin membranes, plates or shells made of n unidirectional
orthotropic plies stacked on the top of each other. Such structures can support in and out-of
plane loadings. In the following the constitutive relations for a laminate made of several
individual plies are derived. The notations are defined in Figure 3.3. In the case of plane
stress, i.e. the effects of transverse shear is neglected, in-plane normal and shear loads N, as
well as the flexural and torsional moments M are applied to the laminate. Those loadings are
computed by considering the stress state in each ply with the relations (3.9):
⎧N1
⎫ ⎧σ
⎫
h / 2 1
⎧M1
⎫ ⎧σ
⎫
h / 2 1
⎪ ⎪ ⎪ ⎪
⎪ ⎪ ⎪ ⎪
N = ⎨N
2 ⎬ = ∫ ⎨σ
2 ⎬dz
M = ⎨M
2 ⎬ = ∫ ⎨σ
2 ⎬zdz
(3.9)
⎪N
⎪ −h / 2 ⎪ ⎪
⎩ 6 ⎭ ⎩σ
⎪
6 ⎭
M ⎪ −h / 2 ⎪ ⎪
⎩ 6 ⎭ ⎩σ
6 ⎭
For a first order cinematic theory, where the displacement through the laminate’s
thickness is linear in the z coordinate measured with respect to the mid-plane of the plate/shell
(Figure 3.3), the vector of laminate’s strains εl is linked to the in-plane strains and the
curvatures via the relation εl = ε + zκ
0
. With this definition it turns that the constitutive
relations for a laminate are given by (3.10) where A, B and D are the in-plane, coupling and
bending stiffness matrices of the laminate.

60
⎧ N ⎫ ⎡A
⎨ ⎬ = ⎢
⎩M⎭
⎣B
z k
3
n
k
2
1
⎧ N1
⎫ ⎡ A11
⎪ ⎪ ⎢
⎪
N2
A
⎪ ⎢ 12
B⎤⎪⎧
0
ε ⎪⎫
⎪ N6
⎪ ⎢A16
⎥⎨
⎬ ⇔ ⎨ ⎬ = ⎢
D⎦⎪⎩
κ ⎪⎭ ⎪M1
⎪ ⎢
B11
⎪M
2 ⎪ ⎢B12
⎪ ⎪ ⎢
⎩M
6 ⎭ ⎣B16
Michaël Bruyneel
A12
A22
A26
B12
B22
B26
A16
A26
A66
B16
B26
B66
B11
B12
B16
D11
D12
D16
B12
B22
B26
D12
D22
D26
(a) A laminate with its structural axes. h is the total thickness
(b) Several unidirectional plies stacked on top of each other.
Material axes related to the k th ply .
t k
B ⎤⎧
0
16 ε ⎫
1
⎪ ⎪
B
⎥ 0
26 ⎥⎪ε
2 ⎪
B ⎥⎪
0 ⎪
66 ε6
⎥⎨
⎬
D16
⎥⎪κ1
⎪
D26⎥⎪κ
⎪
2
⎥⎪
⎪
D66⎦⎪⎩
κ6
⎪⎭
(c). Definition of the plies location through the laminate’s thickness.
hk and hk-1 are used to locate the k th ply of the stacking sequence
h k-1
h 2
h 1
h k
h 0
(3.10)
Figure 3.3. A laminate with n layers (a) Structural axes (b) Material axes of ply k (c) Position of each
ply in the stacking sequence.
1

Optimization of Laminated Composite Structures… 61
3.2. The Possible Parameterizations of Laminates
There exist several parameterizations for the laminates depending on the way the coefficients
of the stiffness matrices in (3.10) are computed and depending on the definition of the design
variables. The advantages and disadvantages of those different parameterizations are
compared in the perspective of the optimal design of the laminated composite structures.
3.2.1. Parameterization with Respect to Thickness and Orientation
When the ply thickness and the related fibers orientation are chosen to describe the laminate,
the coefficients of the stiffness matrices can be written as follows:
= ∑
=
− −
n
Aij
Qij
k hk
k 1
hk
1 )
)]( ( [ θ ⇔ = ∑
=
n
Aij
[ Qij
( θ k )] tk
k 1
1 n
2 2
Bij
= ∑ [ Qij
( θ k )]( hk
− hk−1
) ⇔
2 k=
1
= ∑
=
n
Bij
[ Qij
( θ k )] tk
zk
k 1
(3.11)
1 n
3 3
Dij
= ∑ [ Qij
( θk )]( hk
− hk−1
) ⇔
3 k=
1
= ∑ +
=
n
3
2 tk
Dij
[ Qij
( θ k )]( tk
zk
) , i , j = 1,
2,
6
k 1
12
where zk and hk define the position of the k th ply in the stacking sequence. tk and k
θ are the
ply thickness and the fibers orientation, respectively (Figure 3.3).
With such a parameterization the local values (e.g. the stresses in each ply of the
laminate) are available via the relations (3.1) and (3.4). On top of that the design problem is
written in terms of the physical parameters used for the manufacturing of the laminated
structures. Finally several different materials can be considered in the laminate when the
parameterization (3.11) is used.
However when fibers orientations are allowed to change during the structural design
process the resulting mechanical properties are generally strongly non linear (see Figure 3.2)
and non convex, and local minima appear in the optimization problem. This is also illustrated
in Figure 3.4 that draws the variation of the strain energy density in a laminate over 2 fibers
orientations. In Figure 3.5 it is shown that the structural responses entirely differ when either
ply thickness or ply orientation is considered in the design, resulting in mixed monotonousnon
monotonous structural behaviors. It turns that the optimal design task is more
complicated since the optimization method should be able to efficiently take into account
simultaneously both different behaviors.
Additionally using such a parameterization increases the number of design variables that
may appear in the optimal design problem since the thickness and fibers orientation of each
ply are possible variables. Finally optimizing with respect to the fibers orientations is known
to be very difficult and few publications are available on the subject. For a sake of
completion, the sensitivity analysis of the structural responses of composites with respect to
those variables can be found in Mateus et al. (1991), Geier and Zimmerman (1994), and
Dems (1996).

62
Strain energy density
(N/mm)
Michaël Bruyneel
θ2 θ1
Figure 3.4. Variation of the strain energy density in a [θ 1/θ 2] S laminate with respect to the fibers
orientations θ 1 and θ 2.
Strain nenergy
density (N/mm)
θ
1.2
1.4
1.6
1.8
Figure 3.5. Variation of the strain energy density in an unidirectional ply with respect to its thickness t
and its fibers orientation θ.
3.2.2. Parameterization with Sub-laminates
The design parameters are no longer defined based on single unidirectional plies but instead
on predefined sub-laminates. Each sub-laminate is itself made of several single unidirectional
plies. The design parameters are assigned to the sub-laminates and no longer to each
individual ply. Examples of sub-laminates may be [0/45/-45/90], [0/60/-60] or [0/90]. This
parameterization allows to decrease the number of design variables. However the control at
the ply level is lost. The previously presented parameterization in terms of ply thickness and
orientation is a limiting case.
2
t

Optimization of Laminated Composite Structures… 63
3
Sub-laminate 2
[0/45/-45/90]
2
Sub-laminate 1
[30/-30]
Figure 3.6. Parameterization with sub-laminates. Here the symmetric laminate is made of 2 sublaminates.
3.2.3. The Lamination Parameters
The stiffness matrix in (3.10) can be expressed with the lamina invariants defined in (3.8)
together with the lamination parameters. For a given base material identical for each ply of
the laminate the lamination parameters are given by (3.12) in the structural axes:
ξ
h / 2
A,
B,
D 0,
1,
2
[ ] = ∫ z [ cos 2θ
( z),
cos 4θ
( z),
sin 2θ
( z),
sin 4θ
( z)
]
1,
2,
3,
4
−h
/ 2
1
dz
(3.12)
The lamination parameters are the zero, first and second order moments relative to the
plate mid-plane of the trigonometric functions (3.6) entering the rotation formulae for the ply
stiffness coefficients (3.5). With this definition the stiffness matrices A, B and D in (3.10)
write:
A = hγ
B = γ
h
D =
12
+ γ
0
B
1ξ1
3
γ 0
+
A
1ξ1
B
γ 2ξ2
+ γ
D
1 1
+
A
2ξ2
B
γ3ξ
3
+ γ ξ + γ ξ
2
+ γ
D
2
+
A
3ξ3
B
γ 4ξ4
+ γ
+ γ ξ
D
3ξ3
4
A
4
+ γ ξ
4
D
4
(3.13)
Twelve lamination parameters exist in total and characterize the global stiffness of the
laminate. This number is independent of the number of plies that contains the laminate. In
most applications the lamination parameters are normalized with respect to the total thickness
of the laminate (Grenestedt, 1992, and Hammer, 1997). In the case of symmetric laminates
the 4 lamination parameters B
ξ defining the coupling stiffness B vanish. Moreover when the
structure is either subjected to in-plane loads or to out-of-plane loads only the 4 lamination
parameters related to the in-plane stiffness A
ξ or the out-of-plane stiffness D
ξ must be
considered, respectively. In the case of composite membrane or plates presenting orthotropic
material properties 2 lamination parameters are sufficient to characterize the problem.
Lamination parameters are not independent variables. Feasible regions of the lamination
parameters exist which provide realizable laminates. Grenestedt and Gudmundson (1993)

64
Michaël Bruyneel
demontrated that the set of the 12 lamination parameters is convex. It is also observed from
(3.13) that the constitutive matrices A, B and D are linear with respect to the lamination
parameters. This means that the optimization problem is convex if it includes functions
related to the global stiffness of the laminate, as for example the structural stiffness, vibration
frequencies and buckling loads (Foldager, 1999).
Feasible regions were determined for specific laminate configurations (e.g. Miki, 1982
and Grenestedt, 1992), but the region for the 12 lamination parameters has not yet been
determined. Recently the relations between the lamination parameters were derived for ply
angles restricted to 0, 90, 45 and -45 degrees by Liu et al. (2004) for membrane and bending
effects, and by Diaconu and Sekine (2004) for membrane, coupling and bending effects.
One of the feasible regions of lamination parameters is illustrated in Figure 3.7 in the
case of a symmetric and orthotropic laminated plate subjected to bending. As the plate is
assumed orthotropic in bending D ξ 1 and D ξ 2 are enough to identify the stiffness of such a
problem. Those two lamination parameters take their values on the outline delimited by the
points A, B, C, and in the dashed zone. Any combination of the lamination parameters that is
outside of this region will produce a laminate which is not realizable. When this plate is
simply supported and subjected to a uniform pressure, the vertical displacement is a function
of D ξ 1 and D ξ 2 . The iso-values of this structural response are the parallel lines illustrated in
Figure 3.7. According to Grenestedt (1990), the plate stiffness increases in the direction of the
arrow. The stiffest plate is then characterized by the point D in Figure 3.7, which corresponds
to a [(±θ)n]S laminate, defined by a single parameter θ.
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
C
D
D
ξ2
E
D
ξ1
-1
-1 -0.5 0.5 1
B
Figure 3.7. Feasible domain (outline plus dashed zone) of the lamination parameters for a symmetric
and orthotropic laminated plate subjected to a uniform pressure (after Grenestedt, 1990). The points A,
B, C correspond to [0], [(±45) n] S and [90] laminates, respectively. The point D defines a [(±θ) n] S
laminate. The point E is a combination of laminates defined on the outline. The laminate of maximum
stiffness is located on the outline (point D)
A

Optimization of Laminated Composite Structures… 65
This kind of parameterization has allowed to show that optimal solutions – in terms of the
stiffness – are often related to simple laminates with few different ply orientations. For
example only one orientation is necessary for characterizing the optimal laminate in a flexural
problem (Figure 3.7), and at most 3 different ply orientations are sufficient to define the
optimal stacking sequence in the case of a membrane of maximum stiffness (Lipton, 1994).
Table 3.1 summarizes some of those important results.
When using such a parameterization the number of design variables is very small (12 in
the most general case) irrespective to the number of plies that contains the laminate. As
seen in Figure 3.7 the design space is convex, and only one set of lamination parameters
characterizes the optimal solution. However acording to the relations (3.8) and (3.13) only
one kind of material can be used in the laminate: defining a different material for the core
of a sandwich panel is for example not allowed (Tsai and Hahn, 1980). Additionally the
local structural responses (e.g. the stresses in each ply) can not be expressed in terms of the
lamination parameters since those last are defined at the global (laminate) level and are
linked to the structural stiffness. However the global strains of the laminate (but not in each
ply) can be computed with relation (3.10) and used in the optimization, as is done by
Herencia et al. (2006). The feasible regions of the 12 lamination parameters is not yet
determined. As said before those regions are only known for specific laminate
configurations. This strongly limit their use in the frame of the optimal design of composite
structures. Finally when the optimal values of the lamination parameters are known,
coming back to corresponding thicknesses and orientations is a difficult problem and the
solution is non unique (Hammer, 1997). Foldager et al. (1998) proposed a technique based
on a mathematical programming approach while Autio (2000) used a genetic algorithm to
find this solution when the number of layers is limited or for prescribed standardized ply
angles.
Table 3.1. Summary of some important results obtained with the lamination parameters
Kind of
structure
Laminate configuration Criteria Optimal sequence Reference
Grenestedt (1990),
Miki and Sugiyama
(1993)
Plate Symmetric/orthotropic
Symmetric
Stiffness
Vibration
Buckling
Buckling
[ ( ± θ ) n ] S
[ ( ± θ ) n ] S
[ ( ± θ ) n ] S
[ θ ] S
Grenestedt (1991)
Membrane Symmetric
General
Stiffness
Stiffness
[ ( / 90 α)
n ] S
[ θ ] , [ α / 90 + α]
(1993)
Hammer (1997)
Symmetric/orthotropic Buckling [ ( ± θ ) n ] S , [ 0 / 90]
S ,
Cylindrical
shell
3.2.4. Combined Parameterization
α + Fukunaga and Sekine
quasi-isotropic
Fukunaga and
Vanderplaats (1991b)
As shown by Foldager et al. (1998) and Foldager (1999), composite structures can be
designed by combining two parameterizations: the lamination parameters on one hand, and
the plies thickness and fibers orientations on the other hand. The benefit of the approach

66
Michaël Bruyneel
relies on using a convex design space with respect to the lamination parameters, while
keeping in the problem’s definition the physical variables in terms of thickness and
orientation. This iterative procedure – between both design spaces – consists in determining
a first (local) solution in terms of thicknesses and orientations. A new search direction
towards the global optimum is then computed by evaluating the first order derivative of the
objective function at the local solution with respect to the lamination parameters. The
global optimum is reached when this sensitivity is close to zero. Otherwise a new design
point is calculated in the space of the fibers orientations, and the process continues, usually
by adding new plies in the laminate. As seen in Figure 3.8, the structural response is not
convex with respect to θ while it is convex in terms of the lamination parameter ξ. With
this technique the knowledge of the feasible regions of the lamination parameters is not
mandatory.
Although efficient, this solution procedure can only be used for global structural
responses like the stiffness, the vibration frequencies and the buckling load.
f
1
2
3
4
Figure 3.8. Illustration of the optimization process after Foldager et al. (1998) in both spaces of the
lamination parameters ξ and the fibers orientation θ.
3.2.5. Alternative Parameterization
In order to decrease the non linearities introduced by the fibers orientation variables,
Fukunaga and Vanderplaats (1991a) proposed to parameterize the laminated composite
membranes with the following intermediate variables:
i = sin 2 i or xi = cos2θ
i
x θ
based on the relation (3.12) and (3.13). This formulation was tested by Vermaut et al. (1998)
for the optimal design of laminates with respect to strength and weight restrictions. As in the
previous section, the main difficulty is to compute the orientations corresponding to the
optimal intermediate variables values xi.
f(θ)
θ, ξ
f(ξ)

Optimization of Laminated Composite Structures… 67
4. Specific Problems in the Optimal Design of Composite
Structures
For designing laminated composite structures a very large number of data must be considered
(material properties, plies thickness and fibers orientation, stacking sequence) and complex
geometries must be modelled (aircraft wings, car bodies). Therefore the finite element method
is used for the computation of the structural mechanical responses. Usually mass, structural
stiffness, ply strength and strain, as well as buckling loads are the functions used in the
optimization problem. The design variables are classically the parameters defining the
laminate: fibers orientations, plies thickness, and indirectly the number of plies and the
stacking sequence. Some specific problems appear in the formulation of the optimization
problem for laminated structures. They are reported hereafter.
Large number of design variables. Even for a parameterization in terms of the lamination
parameters, the number of design variables can easily reach a large value when the plies
thickness and fibers orientations are allowed to change over the structure, leading to non
homogenenous plies (Figure 4.1) and curvilinear fibers formats (Hyer and Charette 1991,
Hyer and Lee 1991, Duvaut et al. 2000). In industrial applications (Krog et al. 2007),
thicknesses related to specific orientations (0°, ±45°, 90°) are used and several independent
regions are defined throughout the composite structure, what increases the number of design
variables.
Large number of design functions. Not only global structural responses related to the
stiffness are relevant in a composite structure optimization, but also the local strength of each
ply. Damage tolerance and local buckling restrictions are important as well. For an aircraft
wing, it is usual to include about 300000 constraints in the optimization problem (Krog et al.
2007).
Non homogeneous ply
Homogeneous ply
Figure 4.1. Homogeneous and non homogeneous ply in a laminate.
Problems related to the topology optimization of composite structures. In topology
optimization one is looking for the optimal distribution of a given amount of material in a
predefined design space that maximizes the structural stiffness (Figure 4.2).

68
Domain where the
material is
distributed
Solid
Michaël Bruyneel
Figure 4.2. Illustration of a topology optimization problem (after Bruyneel, 2002)
For composite structures, and due to the stratification of the material, it results that 2
topology optimization problems must be defined and solved simultaneously: the optimal
distribution of plies at a given altitude in the laminate (Figure 4.3) and the transverse topology
optimization where the optimal local stacking sequence is looked for (Figure 4.4). Continuity
conditions between adjacent laminates should also be imposed.
Figure 4.3. Topology optimization at a given altitude in the non homogeneous laminate.
Figure 4.4. Transverse topology optimization in a composite structure.
Void

Optimization of Laminated Composite Structures… 69
Specific non linear behaviors of laminated structures. In order to improve the accuracy in
the model, non linear effects, and especially the design with respect to the limit load, should
be considered in the formulation of the optimization of composite structures. This
dramatically increases the computational time of the finite element analysis, and can only be
used for studying small structural parts such as super-stringers, i.e. some stiffeners and the
panel (Colson et al., 2007). Although simple fracture mechanics criteria have been considered
(Papila et al. 2001), damage tolerance and propagation of the cracks (delamination) should be
taken into account in the same way.
Uncertainties on the mechanical properties of composites. There is a larger dispersion in
the mechanical properties of the fibers reinforced composite materials than for metals.
Moreover, some uncertainties concerning the orientations and the plies thickness exist.
Robust optimization should be used in these cases (Mahadevan and Liu, 1998, Chao et al.,
1993, Chao, 1996, and Kristindottir et al., 1996).
Strong link with the manufacturing process. Contrary to the design with metals, there is
a strong link between the material design, the structural design and the manufacturing
process when dealing with composite materials. The constraints linked to manufacturing
can strongly influence the design and the structural performances (Henderson et al., 1999,
Fine and Springer, 1997, Manne and Tsai, 1998) and should be taken into account to
formulate in a rational way the design problem (Karandikar and Mistree, 1992).
Singular optima in laminates design problems. When strength constraints are
considered in the design problem, and if the lower bounds on the plies thickness is set close
to 0 (i.e. some plies can disappear at the solution from the initial stacking sequence), it can
be seen (Schmit and Farschi, 1973, Bruyneel and Fleury, 2001) that the design space can
become degenerated. In this case the optimal design can not be reached with gradient based
optimization methods. Such a degenerated design space is illustrated in Figure 4.5. It is
divided into a feasible and an infeasible region according to the limiting value of the Tsai-
Wu criteria. In this example a [0/90]S laminate’s weight is to be minimized under an inplane
load N1. The optimal solution is a [0] laminate. Unfortunately this optimal laminate
configuration can not be reached with a gradient based method since the 90 degree plies are
still present in the problem even if their thickness is close to zero, and the related Tsai-Wu
criterion penalizes the optimization process. A first solution consists in using the ε-relaxed
approach (Cheng and Guo 1997), which slightly modifies the design space in the
neighborhood of the solution and allows the optimization method to reach the true optimum
[0] * . Alternatively (Bruyneel and Fleury, 2001, and Bruyneel and Duysinx, 2006) when
fibers orientations are design variables the shape of the design space changes, the gap
between the true optimal solution and the one constrained by plies with a vanishing
thickness [0/x] * decreases and the real optimal solution becomes attainable (Figure 4.5).
Optimizing over the fibers orientations allows to circumvent the singularity of the design
space.

70
* represents the obtained solutions, optimum or not
Michaël Bruyneel
Figure 4.5. Design space for [0/90] S and [0/10] S laminates.
Importance of the fibers orientations in the laminate design. Besides their efficiency in
avoiding the singularity in the optimization process as just explained before fibers
orientations play a key role in the design of composite structures. Modifying their value
allows for great weight savings, as illustrated in Figure 4.6. Let’s consider that the initial
laminate design corresponds to fibers orientation and ply thickness at point A. A first way to
obtain a feasible design with respect to strength restrictions is to increase the ply thickness
and go to B, which penalizes the structural weight. Another solution consists in modifying the
fibers orientation, here at constant thickness (point C). A better solution is to simultaneously
optimize with respect to both kinds of design variables (point D). However taking into
account such variables in the optimization problem is a real issue, and providing a reliable
solution procedure is a challenge.
Figure 4.6. Design space for an unidirectional laminate subjected to either N 1 or N 6. Iso-values of the
Tsai-Wu criterion. The ply thickness and fibers orientations are the design variables.
The optimal stacking sequence. A large part of the research effort on composites has been
dedicated to the solution of the optimal stacking sequence problem. As it is a combinatorial

Optimization of Laminated Composite Structures… 71
problem including integer variables, genetic algorithms have been used (Haftka and Gurdal,
1992, Le Riche and Haftka, 1993). The topology optimization formulation of Figure 4.4 was
used by Beckers (1999) and (Stegmann and Lund, 2005) to solve this problem with discrete
and continuous design variables, respectively. Another approach, still based on the discrete
character of the problem, is proposed by Carpentier et al. (2006). It consists in using a lay-up
table defined based on buckling, geometric and industrial rules considerations. This table,
which satisfies the ply drop-off continuity restrictions is determined numerically. Once it is
obtained a given laminate total thickness corresponds to a stacking sequence (via a column of
the table). The optimization process then consists in optimizing the local thickness of a set of
contiguous laminates defining the structure. Each laminate has equivalent homogenized
properties with 0, ±45 and 90° plies. Based on the lay-up table, the stacking sequence is
therefore known everywhere in the structure for different local optimal thicknesses and the
composite material can be drapped.
Figure 4.7. Illustration of a lay-up table for 0, ±45 and 90° plies.
5. Problems Solved in the Literature
5.1. Structural Responses
When designing laminated composite structures the functions entering the optimization
problem (2.1) are classically the stiffness, the vibration frequencies, the structural stability
and the plies’ strength. (see Abrate, 1994, for a detailed review of the literature). It is
interesting to note that for orthotropic laminates maximizing the stiffness, the frequency or
the first buckling load will provide the same solution (Pedersen, 1987 and Grenestedt, 1990).
On top of that, it should be noted that optimizing a laminated structure against plies strength
or stiffness will result in different designs. It results that the local (stress) effects are very
important in the optimal design of composite structures (Tauchert and Adibhatla, 1985,
Fukunaga and Sekine, 1993, and Hammer, 1997).

72
Michaël Bruyneel
5.2. Optimal Design with Respect to Fibers Orientations
Determining the optimal fibers orientation is a very difficult problem since the structural
responses in terms of such variables are highly non linear, non monotonous and non convex.
However it has just been show in the previous section that the design of laminated composite
structures is very sensitive with respect to those variables. As explained by the editors of
commercial optimization software (Thomas et al., 2000) there is a need for an efficient
treatment of such parameters.
A small amount of work has been dedicated to the optimal design of laminated structures
with respect to the fibers orientations. Several kinds of approaches have been investigated and
are reported in the literature:
• Approach by optimality criteria
Optimal orientations of orthotropic materials that maximize the stiffness in membrane
structures were obtained by Pedersen (1989, 1990 and 1991), and by Diaz and Bendsøe
(1992) for multiple load cases. When the unidirectional ply is only subjected to in-plane
loads, Pedersen (1989) proposed to place the fibers in the direction of the principal stresses.
The resulting optimality criterion was used in topology optimization including rank-2
materials (Bendsøe, 1995). This technique was used by Thomsen (1991) in the optimal design
of non homogeneous composite disks. This criterion was extended by Krog (1996) to Mindlin
plates and shells.
• Approach based on the mathematical programming
As soon as 1971, Kicher and Chao solved the problem with a gradients based method.
Hirano (1979a and 1979b) used the zero order method of Powell (conjugate directions) for
buckling optimization of laminated structures. Tauchert and Adibhatla (1984 and 1985) used
a quasi-Newton technique (DFP) able to take into account linear constraints for minimizing
the strain energy of a laminate for a given weight. Cheng (1986) minimized the compliance of
plates in bending and determined the optimal orientations with an approach based on the
steepest descent method.
Martin (1987) found the minimum weight of a sandwich panel subjected to stiffness and
strength restrictions with a method based on the Sequential Convex Programming
(Vanderplaats, 1984). Watkins and Morris (1987) used a similar procedure with a robust
move-limits strategy (see also Hammer 1997).
In Foldager (1999), the method used for determining the optimal fibers orientations is not
cited but belongs according to the author to the family of mathematical programming
methods.
SQP, the feasible directions method and the quasi-Newton BFGS were used by
Mahadevan and Liu (1998), Fukunaga and Vanderplaats (1991a), and Mota Soares et al.
(1993, 1995 and 1997), respectively. Those mathematical programming methods are reported
and explained in Bonnans et al. (2003).
• Approach with non deterministic methods

Optimization of Laminated Composite Structures… 73
Genetic algorithms have been employed by several authors for determining the optimal
stacking sequence of laminated structures (Le Riche and Haftka, 1993, Kogiso et al., 1994
and Potgieter and Stander, 1998) or in the treatment of fibers orientations (Upadhyay and
Kalyanarama, 2000).
5.3. Formulations of the Optimization Problem
Thickness and orientation variables were treated in several ways in the literature. They have
been considered either simultaneously as in Pedersen (1991), and Fukunaga and Vanderplaats
(1991a), or separately (Mota Soares et al. 1993, 1995 and 1997, and Franco Correia et al.
1997).
Weight, stiffness and strength criteria have been separately introduced in the design
problem and taken into account in a bi-level approach by (Mota Soares et al., 1993, 1995,
1997 and Franco Correia et al., 1997): at the first level the weight is kept constant and the
stiffness is optimized over fibers orientations ; at the second level the ply thicknesses are the
only variables in an optimization problem that aims at minimizing the weight with respect to
strength and/or displacements restrictions. A similar approach can be found in Kam and Lai
(1989), and Soeiro et al. (1994). Fukunaga and Sekine (1993) also used a bi-level approach
for determining laminates with maximal stiffness and strength in non homogeneous
composite structures (Figure 4.2) subjected to in-plane loads. In Hammer (1997), both
problems are separately solved and the initial configuration for optimizing with respect to
strength is the laminate previously obtained with a maximal stiffness consideration.
6. Optimal Design of Composites for Industrial Applications
Based on the several possible laminate parameterizations and on the previous discussion it
was concluded in Bruyneel (2002, 2006) that an industrial solution procedure for the design
of laminated composite structures should preferably be based on fibers orientations and ply
thicknesses, instead of intermediate non physical design variables such as the lamination
parameters. Using those variables allows optimizing very general structures (membranes,
shells, volumes, subjected to in- and out-of-plane loads, symmetric or not) and provides a
solution that is directly interpretable by the user.
On the other hand, an optimization procedure used for industrial applications should be
able to consider a large number of design variables and constraints, and find the solution (or
at least a feasible design) in a small number of design cycles. Additionally, the optimization
formulation should be as much general as possible, and not only limited to specific cases (e.g.
not only thicknesses, not only membrane structures, not only orthotropic configurations,…).
For those reasons, a solution procedure based on the approximation concepts approach seems
to be inevitable. Interesting local solutions can be found by resorting to other optimization
methods (e.g. response surfaces coupled with a genetic algorithm) but on structures of limited
size. For the pre-design of large composite structures like a full wing or a fuselage, or when
non linear responses are defined in the analysis (post-buckling, non linear material behavior),
the approximation concepts approach proved to be a fast method not expensive in CPU time
for solving industrial problems (Krog and al, 2007, Colson et al., 2007).

74
Michaël Bruyneel
It results that robust approximation schemes must be available to efficiently optimize
laminated structures. The characteristics of such a reliable approximation are explained in the
following, and tests are carried out to show the efficiency and the applicability of the method.
7. Optimization Algorithm for Industrial Applications
7.1. The Approximation Concepts Approach
In the approximation concepts approach, the solution of the primary optimization problem
(2.1) is replaced with a sequence of explicit approximated problems generated through first
order Taylor series expansion of the structural functions in terms of specific intermediate
variables (e.g. direct xi or inverse 1/xi variables). The generated structural approximations
built from the information known at least at the current design point (via a finite element
analysis), are convex and separable. As will be explained latter a dual formulation can then be
used in a very efficient way for solving each explicit approximated problem.
According to section 2, it is apparent that the approximation concepts approach is well
adapted to structural optimization including sizing, shape and topology optimization
problems. However, the use of the existing schemes (section 7.2) can sometimes lead to bad
approximations of the structural responses and slow convergence (or no convergence at all)
can occur (Figure 7.1).
x2
* global
X
(k ) *
X
x2
(k )
X
* global
X
* local
X
(k ) *
X
x1
x2
* global
X
a. A too conservative approximation b. A too few conservative approximation and unfeasible intermediate
solutions c. An approximation not adapted to the problem, leading to zigzagging
* local
X
Figure 7.1. Difficulties appearing in the approximation of highly non linear structural responses.
x1
(k ) *
X
(k )
X
* local
X
x1

Optimization of Laminated Composite Structures… 75
Such difficulties are met for laminates optimization: their structural responses are mixed,
i.e. monotonous with regard to plies thickness and non monotonous when fibers orientations
are considered (Figure 3.5). Additionally, the non monotonous structural behaviors in terms
of orientations are difficult to manage (Figure 3.4). It results that the selection of a right
approximation scheme is a real challenge. In the next section a generalized approximation
scheme is presented that is able to effectively treat those kinds of problems. This optimization
algorithm will identify the structural behavior (monotonous or not) according to the involved
design variable (orientation or thickness), and will automatically generate the most reliable
approximation for each structural function included in the optimization problem. In section 8
numerical tests will compare the efficiency of the proposed approximation scheme and the
existing ones for laminates optimization including both thickness and orientation variables.
7.2. Selection of an Accurate Approximation Scheme
7.2.1. Monotonous Approximations
Based on the first order derivatives of the structural responses included in the optimization
problem, linear approximations can be built at the current design point x k . It is a first order
Taylor series expansion in terms of the direct design variables xi (7.1).
k
k ∂g
j
k
g ~ ( ) ( ) ( x ) ( )
j ( x ) =g j ( x ) + ∑ ( xi
− xi
)
(7.1)
∂x
i
As it is very simple this approximation is most of the time not efficient for structural
optimization but can anyway be used with some specific move-limits rules (Watkins and
Morris, 1987) that prevent the intermediate design point to go too far from the current one
and to generate large oscillations during the optimization process (Figures 7.1b and 7.1c).
Since the stresses vary as 1/xi in isostatic trusses where xi is the cross section area of the
bars, a linear approximation in terms of the inverse design variables is more reliable for the
optimal sizing of thin structures. The resulting reciprocal approximation is given in (7.2).
k
g
~ ( )
j
( k)
i
( k)
⎛ ⎞
( k)
( k)
2 ∂g
j ( x )
−
⎜ 1 1
( x ) =g
⎟
j ( x ) ∑ ( xi
)
−
(7.2)
∂ ⎜ k ⎟
i
xi
x ( )
⎝ i xi
⎠
The Conlin scheme developed by Fleury and Braibant (1986) is a convex approximation
based on (7.1) and (7.2). It is reported in (7.3) and illustrated in Figure 7.2.
g~
( k)
j
( k)
⎛ ⎞
( k)
∂g
j ( x ) ( k)
( k)
2
∂g
j ( x )
+
⎜ 1 1
( x ) =g
⎟
j ( x ) ∑ ( xi
− xi
) − ∑ ( xi
)
− (7.3)
∂
⎜ ⎟
−
∂
( k)
+ xi
xi
x
⎝ i xi
⎠
( k)

76
Michaël Bruyneel
The symbols ∑ ( +) and ∑ ( −) in (7.3) denote the summations over terms having positive
and negative first order derivatives. When the first order derivative of the considered
structural response is positive a linear approximation in terms of the direct variables is built,
while a reciprocal approximation is used on the contrary.
145
140
135
130
125
120
115
110
105
100
g(x
)
~ ( )
g k
l
( x)
Strain energy density
(N/mm)
~ ( )
g k
r
( x)
45 90 (k )
(k )
180
xr
xl
Figure 7.2. The Conlin approximation.
Conlin can only work with positive design variables since an asymptote is imposed at
xi=0. On top of that, the curvature of this approximation is imposed by the derivative at the
current design point and can not be adapted to better fit the problem.
The Method of Moving Asymptotes or MMA (Svanberg 1987) generalizes Conlin by
introducing two sets of new parameters, the lower and upper asymptotes, Li and Ui, that can
take positive or negative values, in order to adjust the convexity of the approximation in
accordance with the problem under consideration. The asymptotes are updated following
some rules provided by Svanberg (1987). The parameters pij and qij are built with the first
order derivatives.
Strain energy
145 density (N/mm)
140
135
130
125
120
115
110
105
g(x
)
~ ( )
g ( x)
k
100
(k )
L
100
(k )
U
45 90 (k ) 135 (k )* 180 45 90 (k )*
(k )
180
x
x
Strain energy
145 density (N/mm)
140
135
130
125
120
115
110
105
g(x
)
Figure 7.3. The MMA approximation.
x
~ ( )
g ( x)
k
x

Optimization of Laminated Composite Structures… 77
⎛
⎞ ⎛
⎞
~ ( k)
( k)
( k)
⎜ 1
1 ⎟ ( k)
⎜ 1 1
g
⎟
j ( x ) =g j ( x ) + ∑ pij
−
+ ∑q−(7.4)
⎜ ( k)
( k)
( k)
⎟ ij ⎜ ( k)
( k)
( k)
⎟
+ ⎝U
i − xi
U i − xi
⎠ − ⎝ xi
− Li
xi
− Li
⎠
As it will be seen later those monotonous schemes are not efficient for optimizing
structural functions presenting non monotonous behaviors, as in Figure 3.4.
7.2.2. Non Monotonous Approximations
Based on MMA, Svanberg (1995) developed the Globally Convergent MMA approximation
(GCMMA). As illustrated in Figure 7.4 it is non monotonous and still only based on the
information at the current design point (functions values, first order derivatives, asymptotes
values). Here both Ui and Li are used simultaneously. It was not the case in (7.4).
k
g
~ ( )
j
⎛
⎞ ⎛
⎞
( k)
( k)
∑
⎜ 1 1 ⎟ ( k)
+ ∑
⎜ 1 1
( x ) =g +
−
− ⎟
j ( x ) pij
q
(7.5)
⎜ ( k)
( k)
( k)
⎟ ij ⎜ ( k)
( k)
( k)
⎟
i ⎝U
i − xi
Ui
− xi
⎠ i ⎝ xi
− Li
xi
− Li
⎠
Using this method can lead to slow convergence given that it can generated too
conservative approximations of the design functions (Figure 7.1a).
145
140
135
130
125
120
115
110
105
100
Strain energy
density (N/mm)
(k )
L
g(x
)
(k )
U
45 90 (k ) 135 (k )* 180
x
~ ( )
g ( x)
k
Figure 7.4. The GCMMA approximation.
In order to improve the quality of this approximation it was proposed in Bruyneel and
Fleury (2002) and Bruyneel et al. (2002) to use the gradients at the previous iteration to
improve the quality of the approximation, leading to the definition of the Gradient Based
MMA approximations (GBMMA). In those methods the pij and qij parameters of (7.5) are
computed based on the function value and gradient at the current design point and on the
gradient at the previous iteration. The rules defined by Svanberg (1995) for updating the
asymptotes are used.
x

78
Michaël Bruyneel
7.2.3. Mixed Approximation of the MMA Family
When dealing with structural optimization problems including design variables of two
different natures, for example in problems mixing ply thickness and orientation variables, one
is faced to a difficult task because of the simultaneous presence of monotonous and nonmonotonous
behaviors with respect to the set of design variables. In these conditions, most of
the usual approximation schemes presented before have poor convergence properties or even
fail to solve these kinds of problems. Knowing that the MMA approximation is very reliable
for approximating monotonous design functions and based on the GBMMA approximations,
a mixed monotonous – non monotonous scheme is presented in Bruyneel and Fleury (2002)
and Bruyneel et al. (2002), which will automatically adapt itself to the problem to be
approximated (7.6).
⎛
k
k
k
g
~ ( )
( ) ( )
∑
⎜ 1
j ( x)
= g j ( x ) + pij
⎜ ( k)
i∈A⎝Ui−xi 1
⎞ ⎛
⎟ ( k)
+ ∑
⎜ 1
−
q
( k)
( k)
−
⎟ ij ⎜ ( k)
Ui
xi
⎠ i∈A
⎝ xi
− Li
1
⎞
− ⎟
( k)
( k)
x −
⎟
i Li
⎠
⎛
( k)
+ ∑
⎜ 1
pij
⎜ ( k)
+ , i∈B⎝Ui−xi 1 ⎞ ⎛
−
⎟ ( k)
+ ∑ q ⎜ 1
( k)
( k)
−
⎟ ij ⎜ ( k)
Ui
xi
⎠ −,
i∈B
⎝ xi
− Li
1 ⎞
− ⎟
( k)
( k)
x −
⎟
i Li
⎠
In (7.6) the symbols ∑ ( + , i) and ∑( − , i ) designate the summations over terms having
positive and negative first order derivatives, respectively. A and B are the sets of design
variables leading to a non monotonous and a monotonous behavior respectively, in the
considered structural response. At a given stage k of the iterative optimization process, a
monotonous, non monotonous or linear approximation is automatically selected, based on the
tests (7.7), (7.8) and (7.9) computed for given structural response g (X)
and design variable
x i .
j
(7.6)
k
k −1
∂g
j ( x ) ∂g
j ( x )
× > 0 ⇒ MMA (monotonous)
∂xi
∂xi
(7.7)
k
k −1
∂g
j ( x ) ∂g
j ( x )
× < 0 ⇒ GBMMA (non monotonous)
∂xi
∂xi
(7.8)
k
k −1
∂g
j ( x ) ∂g
j ( x )
− = 0 ⇒ linear expansion
∂x
∂x
(7.9)
i
i
The selection of a right approximation is illustrated in Figure 7.5: when a monotonous
approximation is used for approximating a non monotonous function, oscillations can appear,
while a non monotonous approximation is too conservative when the function is monotnous.
The best approximation is therefore selected based on tests (7.7) to (7.9). This strategy proved
to be reliable for simple laminates design (Bruyneel and Fleury 2002) and for general
laminated composite structures design problems (Bruyneel 2006, Bruyneel et al. 2007, Krog
et al. 2007), for truss sizing and configuration (Bruyneel et al. 2002), for topology

Optimization of Laminated Composite Structures… 79
optimization which includes a large amount of design variables (Bruyneel and Duysinx
2005). It has been made available in the BOSS Quattro optimization toolbox (Radovcic and
Remouchamps, 2002). In the following this solution procedure based on a mixed
approximation scheme is called Self Adaptive Method (SAM). Based on this approximation
scheme, it is possible to resort to the other ones (GBMMA, MMA, Conlin and the linear
approximation) by setting specific values to the asymptotes and by limiting the
approximations to the sets A or B in (7.6).
145
140
135
130
125
120
115
110
105
100
Strain energy
density
(N/mm)
(k )
L
g(θ
)
θ
(k ) *
MMA
g GCMMA
~
g MMA
~
(k )
U
45 90 (k ) 135 (k ) * 180
θ θGCMMA
400
350
300
250
200
150
g MMA
~
g(t
)
Strain energy
density
(N/mm)
1.2 1.3 1.4 1.5 (k ) 1.7
Figure 7.5. The mixed SAM approximation.
t
g GCMMA
~
(k )*
GCMMA
A summary of the approximations that will be compared in the following is presented in
Table 7.1.
Table 7.1. Summary of the approximations that will be compared in the numerical tests
Approximation Author Behavior
MMA Svanberg (1987) Monotonous
GCMMA Svanberg (1995) Non monotonous
SAM Bruyneel (2006) Mixed monotonous/non monotonous
7.3. Solution Procedure for Mono and Multi-objective Optimizations
Since the approximations are convex and separable the solution of each optimization subproblem
(Figure 2.3) is achieved by using a dual approach. Based on the theory of the duality,
solving the problem (2.2) in the space of the primal variables xi is equivalent to maximize a
function (7.10) that depends on the Lagrangian multipliers λ j , also called dual variables:
max minL
( x, λ)
λ
x
t
(k ) *
MMA
0 0,...,
( 0 1)
=
= ≥ λ
λ
m j j (7.10)
t

80
Michaël Bruyneel
Solving the primal problem (2.2) requires the manipulation of one design function, m
structural restrictions and 2 × n side constraints (for mono-objective problems). When the
dual formulation is used, the resulting quasi-unconstrained problem (7.10) includes one
design function and m side constraints, if the side constraints in the primal problem are treated
separately. In relation (7.10), L( x,
λ)
is the Lagrangian function of the optimization problem,
which can be written
pij
qij
L ( x,
λ)
= ∑λj( c j + ∑ + ∑ )
(7.11)
k
U − x x − L
j i i i i
according to the general definition of the involved approximations g
~
j ( X ) of the functions.
The parameter λj is the dual variable associated to each approximated function g
~
j ( X ) . Given
that the approximations are separable, the Lagrangian function is separable too. It turns that:
and the Lagrangian problem of (7.10)
can be split in n one dimensional problems
= ∑ x ) , ( ) ( λ
λ x, L
L
i
i
k
minL ( x, λ)
x
i
i
k
minL ( x , λ)
(7.12)
x
i
The primal-dual relations are obtained by solving (7.12) for each primal variable xi:
∂Li
( xi
, λ)
= 0
∂xi
i
i
⇒
xi
= xi
( λ)
k i
(7.13)
Relation (7.13) asserts the stationnarity conditions of the Lagrangian function over the
primal variables xi. Once the primal-dual relations (7.13) are known, (7.10) can be replaced
by
maxl ( λ)
⇔ maxL(
x(
λ)
, λ)
(7.14)
λ
λ
λ j ≥ 0 j = 1,...,
m
Solving problem (2.2) is then equivalent to maximize the dual function l (λ)
with non
negativity constraints on the dual variables (7.14). As it is explained by Fleury (1993), the

Optimization of Laminated Composite Structures… 81
maximization (7.14) is replaced by a sequence of quadratic sub-problems. Each sub-problem
is itself partially solved by a first order maximization algorithm in the dual space.
In the case of a multi-objective formulation the optimization problem writes :
min max gl0
( x)
X l = 1,...,
nc
g j (x ) ≤ g j j = 1,...,
m
(7.15)
where nc is the number of load cases. Using the bound formulation (Olhoff, 1989) the
problem (7.15) can be written as:
g l0
( x)
≤ β
1 2
min β
2
l = 1,...,
nc
(7.16)
g j (x ) ≤ g j
j = 1,...,
m
where β is the multiobjective factor, that is an additional design variable in the optimization
problem. Instead of solving (7.16) problem (7.17) is considered where a new variable δ is
introduced for the possible relaxation of the set of constraints.
( ) ( ) 2
1 2 1 2 C ( k −1)
min β + δ + p + ∑ xi
− xi
2 2 2 i
g j0
( x ) ≤ β j g j0
j = 1,...,
nobj
(7.17)
g j ( x ) ≤ g j ( 1+
δ )
j = 1,...,
m
g j0
are target values on the objective functions. The dual approach described for monoobjective
optimisation problems is then applied to (7.17).
8. Applications of the Optimization Solution Procedure
In the following examples (except the simple laminate designs and the topology optimization
problem), the structural and semi-analytical sensitivity analyses are carried out with
SAMCEF (http://www.samcef.com). The Boss Quattro optimisation tool box
(http://www.samcef.com) is used for defining and solving the optimisation problem
(Radovcic and Remouchamps 2002).
8.1. Laminate Subjected to in- and out-of-plane Loadings
A symmetric 4 plies laminate made of carbon/epoxy is considered. The load case and the
initial configuration are provided in Table 8.1. The fibers orientations of each ply are the

82
Michaël Bruyneel
design variables, while plies thicknesses are kept constant. The optimization consists in
minimizing the laminate’s strain energy density, i.e. maximizing its stiffness. The evolution
of this objective function with respect to the 2 angles θ1 and θ2 is reported in Figure 8.1, with
the initial and optimal design points. A restriction is imposed on the relative variation of the 2
design variables. The optimization problem writes :
1 T T 1 T
min ε 0 Aε 0+
κ Dκ
θ 2 2
θ 2 − θ1
≤ 45
(8.1)
0. 001 θ ≤180
i = 1,
2
≤ i
where the stiffness matrices A, B and D, and the laminate’s strain and curvature were
previously defined in Section 3.
Strain energy
density (N/mm)
θ2
θ1
Initial design
Optimal design
Solution
Figure 8.1. Variation of the strain energy density in the symmetric laminate subjected to the load case
of Table 8.1.
In-plane load case
( N 1,
N 2 , N6
)
in N/mm
Table 8.1. Problem’s definition: load case and initial design
Out-of-plane load case
( M 1,
M 2 , M 6 )
in N
Initial orientations
θ = ( θ1,
θ 2 )
in degrees
23.3°
22.3°
Initial thicknesses
t = ( t1,
t2
)
in mm
(2000,0,1000) (0,500,0) (45,135) (1,2)
In this application the laminate is subjected not only to in-plane but also to out-of-plane
loadings. Since the plies thicknesses are not identical (Table 8.1) the objective function is not
symmetric with regards to the axis θ 1 = θ 2 (Figure 8.2).

Optimization of Laminated Composite Structures… 83
180
160
140
120
100
θ2
80
60
40
20
Strain energy density
(N/mm)
θ init
θ opt
θopt unconstrained
0
0 20 40 60 80 100 120 140 160 180
θ1 Figure 8.2. Illustration of the design space. Staring point, unconstrained and constrained optimum.
The iteration histories for the 3 approximation schemes are illustrated in the Figure 8.3.
The convergence of the optimization process is controlled by the relative variation of the
design variables at 2 successive iterations. The MMA approximation converges in 41
iterations. 29 iterations are enough for GCMMA. When the SAM approximation is used the
solution is reached in a very small number of iterations.
25
20
15
10
5
150
100
Objective function (N/mm)
0
0 20 40 60
50
Evolution of angles (deg.)
0
0 20 40 60
20
15
10
5
150
100
Objective function (N/mm)
0
0 10 20 30
50
Evolution of angles (deg.)
0
0 10 20 30
20
15
10
5
150
100
Objective function (N/mm)
0
0 5 10 15
50
Evolution of angles (deg.)
0
0 5 10 15
MMA GCMMA SAM
Figure 8.3. Iteration history for the 3 approximation methods.

84
8.2. Non Homogeneous Laminate
Michaël Bruyneel
In this application a non homogeneous composite membrane divided in regions of constant
thickness and fibre orientations is studied. Each region is defined with an unidirectional
laminate made of a glass/epoxy material. The design over stiffness is only considered here.
The solution with respect to strength and stiffness is provided in Bruyneel (2006).
2
2
1
1
2
P
1
P
Figure 8.4. Initial configurations with 45 and -45 degrees plies orientations.
The quasi-unconstrained optimization problem (8.2) consists in finding the optimal
values of the plies thickness and fibers orientations in each region of the laminated composite
structure that maximize the overall stiffness (i.e. that minimize the compliance – the potential
energy of the applied loads). The vectors of the design variables are given by
θ = { θi
, i = 1,...,
n}
and t = { ti , i = 1,...,
n}
where n is the number of regions according to
Figure 8.4. The initial thicknesses are of 1 mm.
2
2
1
P
1
min Compliance
θ,t
0° ≤θ
i ≤180°
i = 1,...,
n
(8.2)
0. 01mm
≤ ti
≤ 5mm
In this problem the optimal values of the thickness is 5 mm, that is their upper bound.
Anyway this application illustrates the difficulties encountered when both kinds of design
variables appear in the design problem. The optimal values of the compliances are reported in
Figure 8.5 as a function of the number of regions. As already noticed by Foldager (1999) an
increase of the number of regions of different orientations improves the overall optimal
structural stiffness (i.e. it decreases the compliance).
The optimal fibers orientations are illustrated in Figure 8.6, for the several membrane
configurations of Figure 8.4. The iteration histories are reported in Figure 8.7. When the SAM
method is used, about 10 iterations are enough for reaching a stationary solution with respect
to a small relative variation of the objective at 2 successive iterations. The GCMMA
approximation finds this solution in a larger number of design cycles. It is observed that when
P
P

Optimization of Laminated Composite Structures… 85
the SAM method is used, the structural responses in terms of both the fibers orientations and
the thicknesses are well approximated, while using GCMMA, the approximation in terms of
the thicknesses is too conservative, what slows down the overall convergence speed of the
optimization process.
1
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55
Relative compliances
0.5
1 4 8 12 20
Number of regions : n
Figure 8.5. Evolution of the compliances in the problem (8.2) for the structures illustrated in Figure 8.4.
The compliance of the one region structure is the reference (n = 1)
1 region
4 regions
Figure 8.6. Continued on next page.

86
Michaël Bruyneel
8 regions
12 regions
20 regions
Figure 8.6. Illustration of the optimal fibers orientations for the different composite membranes
illustrated in Figure 7.9.
Pli 19
Pli 5
Figure 8.7. Continued on next page.

4
x 10
5
4
3
2
1
Compliance (Nmm)
0
0 20 40 60 80
4
x 10
5
4
3
2
1
Compliance (Nmm)
0
0 5 10 15
Optimization of Laminated Composite Structures… 87
Mass (kg) and thickness of ply19 (mm)
8
7
6
5
4
3
2
1
0 20 40 60 80
7
6
5
4
3
2
GCMMA
SAM
Total mass
Thickness of ply 19
Mass (kg) and thickness of ply19 (mm)
8
Total mass
Thickness of ply 19
1
0 5 10 15
140
120
100
80
60
Orientation of ply 5 (deg.)
40
0 20 40 60 80
140
120
100
80
60
Orientation of ply 5 (deg.)
40
0 5 10 15
Figure 8.7. Convergence history for GCMMA and SAM for the membrane divided in 20 regions.
Evolution of the thickness and the orientations of the plies number 5 and 19.
2
1.9
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1
Vertical displacement δ max under the load (mm)
1
0 20 40 60 80 100 120 140 160 180
Fibers orientation (deg.)
O 320 finite elements
+ 80 finite elements
* 20 finite elements
Figure 8.8. Evolution of the vertical displacement under the applied load for several discretizations of
the homogeneous composite membrane (Figure 8.4, n=1).

88
Michaël Bruyneel
In Figure 8.8 the evolution of the vertical displacement under the load is drawn with
respect to the fibers orientation in the case of the homogeneous membrane (Figure 8.4, n=1).
The global minimum displacement is obtained for a value of 170°. When the starting point of
the optimization process of the problem (8.2) is close to 45°, 0° fibers orientation is found as
a local optimum. As -45° is chosen here for the initial design (i.e. 135°), the global optimum
can be reached. This illustrates the fact that a gradient based method is not able to reach the
global optimum, unless the starting point is in its vicinity. In Figure 8.8, the influence of the
mesh refinement on the solution is presented, as well.
8.3. Multi-objective Optimization
A symmetric laminate made of 4 plies and subjected to 2 in-plane load cases is considered.
N 2
3
N 1
1
θ
x
N 6
N 1
Figure 8.9. Laminate subjected to in-plane loads.
The applied loads and the initial configuration are reported in Table 8.2. The load case
(2) is variable : the factor k takes the values 0,1,2,…,8. The extreme load cases are, on one
hand (1000,0,0) and on the other hand the combination of (1000,0,0) and (0,2000,0) N/mm.
Table 8.2. Definition of the problem: load case and starting point
Load case (1) Load case (2) Initial orientations Initial thickness
( N 1,
N 2 , N6
) ( N 1,
N 2 , N6
) θ = ( θ1,
θ 2 )
t = ( t1,
t2
)
in N/mm
in N/mm
in degrés
en mm
(1000,0,0) (0, k × 250 ,0) (30,120) (1,2)
The performance of three approximation schemes are compared : GCMMA, MMA and
SAM. The optimization problem writes :
1
min max ε( ) ( j)
2 j Aε
j 1,2
T
θ, t =
TW ( j)
( θ i , ti
) ≤1
i
, j = 1,
2
2
N 2

Optimization of Laminated Composite Structures… 89
4
∑ t i ≤ 4
i=
1
(8.3)
0. 001 ≤ θ i ≤ 180 i = 1,
2
0. 001 t ≤ 10
i = 1,
2
≤ i
where j is the number of the load case. This problem is solved by resorting the its bound
formulation (Olhoff, 1989) including here 5 design variables (2 orientations, 2 thicknesses
and the multi-objective factor β) and 7 constraints:
1
min β
2
1 T
ε(
j)
Aε(
j)
2
TW ( j)
( i , ti
)
4
∑ t i
i=
1
0. 001 ≤ i ≤
0. 001 ≤ i ≤ 10
≤ β
2
j = 1,
2
θ ≤1
i , j = 1,
2
(8.4)
≤ 4
θ 180 i = 1,
2
t i = 1,
2
The results are reported in Figure 8.10 for the different values of k. The solution is
obtained when the relative variation of the design variables at 2 successive iterations is lower
than 0.01. It is seen that a large number of iterations is needed to reach the optimum when
MMA is used. GCMMA converges in a lower number of iterations. As for mono-objective
problems, SAM is the most effective optimization method.
+ MMA
o GCMMA
Δ SAM
Maximum strain energy density (N/mm)
4
3
2
1
0
0 2 4 6 8
80
60
40
20
Number of iterations
0
0 2 4 6 8
Load parameter k Load parameter k
Figure 8.10. Variation of the strain energy density and number of iterations needed to reach the solution
as a function of the parameter k.

90
10 5
10 0
15
10
10 -5
5
10 -10
Objective functions (N/mm)
0
0 10 20 30 40 50
3
2.5
2
1.5
1
Variations of the objective functions
0 10 20 30 40 50
Michaël Bruyneel
2 Maximum constraints violations
10
10 0
10 -2
10 5
10 0
10 -5
0 10 20 30 40 50
Maximum variables variation
0 10 20 30 40 50
Figure 8.11. Convergence history for MMA. k is equal to 3.
Objective functions (N/mm)
0.5
0 2 4 6 8 10 12
10 0
10 -2
10 -4
10 -6
Variations of the objective functions
2 4 6 8 10 12
10 1
10 0
10 -1
10 5
10 0
10 -5
Maximum constraints violations
0 2 4 6 8 10 12
Maximum variables variation
Figure 8.12. Convergence history for SAM. k is equal to 3.
2 4 6 8 10 12
Figure 8.13 illustrates the optimum stacking sequence for the different values of the load
parameter k. The solution corresponds to a [0/90]S with a variable proportion of 90° plies
(depending on k).
Figure 8.14 describes the design space for k = 4. The iso-values of both objective
functions are drawn. The arrow indicates the direction for an increase of the stiffness. The
optimal solution is characterized here by identical values of both objective functions.

Optimization of Laminated Composite Structures… 91
4
3.5
3
2.5
2
1.5
1
Evolution of the strain
energy density
Laminate configuration
for the several load
cases
0.5
0 1 2 3 4 5 6 7 8
Load parameter k
Figure 8.13. Variation of the strain energy density and configuration of the corresponding optimal
laminate.
Figure 8.14. Evolution of the strain energy densities for the [0/90]S laminate Subjected to
( N 1 , N 2 , N6
) = ( 0,
1000,
0)
N / mm and ( N 1 , N 2 , N6
) = ( 1000,
0,
0)
N / mm . t0° and t90° are the plies
thickness.
The variation of the strain energy density for each single load case is illustrated in
Figures 8.15 and 8.16. In those particular cases, the optimal solutions are given by only 90° or
0° orientations. This illustrates the need for a multi-objective formulation when several
functions are considered as objective.

92
Michaël Bruyneel
Figure 8.15. Evolution of the strain energy density in the [0/90]S laminate subjected to
N 2 = 1000N
/ mm .
Figure 8.16. Evolution of the strain energy density in the [0/90]S laminate subjected to
N 1 = 1000N
/ mm .
8.4. Optimal Design with Respect to Stiffness and Strength Restrictions
In this application a stiffened laminated composite panel subjected to a uniform pressure is
considered. The geometry, the boundary conditions and the stacking sequence of the different
parts of the panel are illustrated in Figure 8.17. The plies thickness is equal to 0.125 mm and
the base material is carbon/epoxy.

3.5
3
2.5
2
1.5
1
Optimization of Laminated Composite Structures… 93
laminate 1 :[(0/90/45/-45)2]S
laminate 2 : [0/90/45/-45]4
Figure 8.17. Geometry and initial stacking sequence of the stiffened panel.
Relative compliance
0.5
0 10 20 30
3
2.5
2
1.5
1
0.5
0 5 10 15
2
1.8
1.6
1.4
1.2
1
GCMMA
Relative mass
0.8
0 10 20 30
1.8
1.6
1.4
1.2
1
SAM
0.8
0 5 10 15
60
50
40
30
20
10
Number of violations
0
0 10 20 30
Relative compliance Relative mass Number of violations
2
60
Figure 8.18. Convergence history for GCMMA and SAM.
50
40
30
20
10
0
0 5 10 15

94
Michaël Bruyneel
The optimization problem consists in maximizing the structural stiffness for a given
maximum weight, knowing that a safety margin of 0.15 on the Tsai-Hill criterion on the top
and the bottom of each ply must be obtained at the solution. 64 strength restrictions are
defined at the plies level. The design variables are the orientations of the plies initially
oriented at 0, -45, +45 and 90 degrees and the related thicknesses. The problem includes 16
design variables. The convergence histories of GCMMA and SAM are compared in Figure
8.18.
The SAM approximation succeeds in finding a solution in a very small number of
iterations, with comparison to GCMMA. The optimal stacking sequence is illustrated in
Figure 8.19. As already observed by Grenestedt (1990) and Foldager (1999), the optimal
laminates include very few different orientations.
laminate 1 :[90]
laminate 2 : [0/93/96/93]4
Figure. 8.19. Optimal design of the stiffened panel.
8.5. Optimal Design under Buckling Considerations
2.48 mm
Anyone who has carried out optimal sizing with a buckling criterion has experienced an
undesirable effect of very slow convergence speed and possibly large variations of the design
functions during the iteration history. The reasons for the bad convergence of the buckling
optimisation problem are multiple, and make it difficult to solve: discontinuous character of
the problem due to the localized nature of local buckling, non differentiability of the eigenvalues
and related problems in the sensitivity computation, modes crossing, selection of a
right optimisation method, etc.
A curved composite panel including 7 hat stiffeners is considered. The load case consists
of a compression along the long curved sides, and in shear on the whole outline. The structure
is simply supported on its edges. Bushing elements are used to fasten the stiffeners to the
panel. In each super-stiffener (made of one stringer and the corresponding part of the whole
panel), 3 design variables are used for defining the thickness of the 0°, 90° and ±45° plies in
the panel and in the stiffener. 42 design variables are then defined. The goal is to find the
structure of minimum weight with a minimum buckling load larger than 1.2. The results
obtained in Bruyneel et al. (2007) are reported in Figures 8.20 for Conlin (Fleury and
Braibant 1986) and SAM (Bruyneel 2006). The 12 first buckling loads are the design
restrictions of the optimisation problem. In Figure 8.20, the evolutions of the weight and the

Optimization of Laminated Composite Structures… 95
first buckling load λ1 over the iterations are plotted, as well as some characteristic buckling
modes.
Figure 8.20. Convergence history for the buckling optimisation with Conlin (left) and SAM (right)
Bruyneel et al. (2007)
It is seen that when Conlin is used (Figure 8.20, left) a solution can not be reached. With
SAM (Figure 8.20, right), the solution is obtained after an erratic convergence history. Those
oscillations come from the fact that local buckling modes appear during the optimisation
process, and some parts of the structures are no longer sensitive to this criterion. A small
thickness is therefore assigned to those parts to decrease the weight, what makes them very
sensitive to buckling at the next iteration, leading to oscillations of the design variables and
functions values. It was observed in Bruyneel et al. (2007) that when a large number of
buckling loads are used in the optimization problem (say 100 for the problem of Figure 8.20),

96
Michaël Bruyneel
a solution with SAM is reached in 6 iterations, while Conlin is still no longer able to
converge.
8.6. Topology Optimization of Laminated Composite Structures
The topology optimization problem of Figure 4.3 is here considered. In topology optimization
of isotropic material (Bendsoe 1995), the design variable is a pseudo-density μi that varies
between 0 and 1 in each finite element i (Figure 4.2). The so-called SIMP material law
(Simply isotropic Material with Penalization) takes the following form:
Ei
= μ
p 0
i E
0
ρi = μi
ρ
(8.5)
where E 0 and ρ 0 are the Young modulus and the density of the base material (e.g. steel), E
and ρ are the effective material properties, and p is the exponent of the SIMP law, chosen by
the user (1

Optimization of Laminated Composite Structures… 97
The problem in Figure 8.21 is solved with this parameterization. It includes 3750 design
variables. The optimal topology and orientations obtained for an half of the structure are
given in Figure 8.22. A comparison of the convergence speed for several approximations is
provided in Figure 8.23.
?
Figure 8.21. Definition of topology optimization problem. The initial structure is full of material.
Figure 8.22. Optimal topology with orthotropic material. Only one half of the structure is drawn. The
fibers orientation is plotted in the few elements that contain full material at the solution
Figure 8.23. Convergence history for several approximation schemes for the topology optimization
problem including orthotropic material.

98
Michaël Bruyneel
8.7. An Industrial Solution for the Pre-design of Composite Aircraft Boxes
As reported in Krog et al. (2007), the pre-design of an aircraft wing is a large scale
optimization problem including (up to now) about 1000 design variables and about 300000
constraints. Those variables are linked to the total thickness of the laminate made of 0, ±45
and 90° plies in the panel and to the dimensions of the cross section for the composite
stiffener of each super-stringer defining the box structure (Figure 8.24). The constraints
expressed as reserve factors (RF) are amongst others related to buckling and damage
tolerance.
Figure 8.24. The principle of a composite wing made of super-stringers (from Krog et al., 2007).
Taking into account a so large number of design functions in the optimization problem
will dramatically increase the CPU time spent in the optimizer. In order to decrease the size
of the optimization problem, a technique for scanning the constraints (Figure 8.25) has been
implemented in Boss Quattro (www.samcef.com). It consists in feeding the optimizer with
the most critical constraints, based on their value at a given iteration. This leads to the
definition of 2 sets of active and inactive constraints. The optimizer can only see the active
restrictions. Those sets are not updated at each iteration but only when some inactive
constraints tend to become violated after a given number of iterations (FREQ in Figure 8.25).
When the SAM method (Bruyneel 2006) is used, the information at the previous design point
is lost when the sets are updated, and the approximation is therefore only built based on the
information at the current design point, that is with GCMMA (Svanberg 1995), for that
specific iteration.
The SAM approximation was found to be reliable in solving pre-design optimization
problems of composite aircraft box structures in wings, center wing box, vertical and
horizontal tail planes. Typically 30 iterations were enough to reach a stationary value of the
weight and a nearly feasible design where very few constraints (less than 10) were still
violated but of an amount of no more than 3 percents (RF larger than 0.97). Details of the
results and of the implementation can be found in Krog et al. (2007).

Optimization of Laminated Composite Structures… 99
Figure 8.25. Strategy for scanning the constraints in large scale optimization problems (Krog et al.
2007).
8.8. Optimal Design with Respect to Damage Tolerance
A simple DCB beam is considered (Figure 8.26). The energy release rates of modes I, II and
III are computed at the straight crack front with a specific virtual crack extension method
described by Bruyneel et al. (2006). The stacking sequence composed of 32 plies is given by:
[θ/−θ/0/−θ/0/θ/θ/04/θ/0/−θ/0/−θ/θ/d/−θ/θ/0/θ/0/−θ/04/−θ/0/θ/0/θ/-θ]
where d is the location of the interface where delamination will take place and θ is a variable.
The goal is to find the optimal value of the orientation that will decrease the maximum value
of GI along the crack front.
Figure 8.26. DCB beam and variation of GI along the crack front for the initial design. On the left the
displacements of the lips are multiplied by 50.

100
Michaël Bruyneel
The solution is provided in Figure 8.27. The optimal value for the angle θ is zero. The
convergence is achieved in 5 iterations with the SAM approximation and in 15 for MMA
(Figure 8.28). Although the solution of this problem is trivial, the procedure could be used for
more realistic structures subjected to several complex load cases.
Figure 8.27. DCB beam and variation of GI along the crack front for the optimal design. On the left the
displacements of the lips are multiplied by 50
Figure 8.28. Convergence history for the optimization with respect to damage tolerance. SAM
converges in 5 iterations while MMA needs 15 iterations to reach the solution
9. Conclusion
In this chapter the optimal design of laminated composite structures was considered. After a
review of the literature an optimization method specially devoted to composite structures was
presented. This review helped us in selecting a formulation of the optimization problem that
satisfies the industrial needs. In this context the fibers orientations and the ply thicknesses
were selected as design variables. It was shown on the proposed applications that the
developed solution procedure is general and reliable. It can be used for solving laminated
composite problems including membrane, shells, solids, single and multiple load cases, in

Optimization of Laminated Composite Structures… 101
stiffness, buckling and strength based designs. It is routinely used in an (European) industrial
context for the design of composite aircraft box structures located in the wings, the center
wing box, and the vertical and horizontal tail plane. This approach is based on sequential
convex programming and consists in replacing the original optimization problem by a
sequence of approximated sub-problems. A very general and self adaptive approximation
scheme is used. It can consider the particular structure of the mechanical responses of
composites, which can be of a different nature when both fiber orientations and plies
thickness are design variables.
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In: Composite Materials Research Progress ISBN: 1-60021-994-2
Editor: Lucas P. Durand, pp. 109-128 © 2008 Nova Science Publishers, Inc.
Chapter 3
MAJOR TRENDS IN POLYMERIC COMPOSITES
TECHNOLOGY
W.H. Zhong
Department of Mechanical Engineering and Applied Mechanics
North Dakota State University, Fargo, ND
R.G. Maguire
Boeing 787 Program/Phantom Works, The Boeing Co. Seattle WA
S.S. Sangari
Boeing Materials & Processes Technology, The Boeing Company, Seattle, WA
P.H. Wu
Spirit Co., Wichita KS
Abstract
Composites have been growing exponentially in technology and applications for decades.
The world of aerospace has been one of the earliest and strongest proponents of advanced
composites and the culmination of the recent advances in composite technology are realized in
the Boeing Model 787 with over 50% by weight of composites, bringing the application of
composites in large structures into a new age. This mostly-composite Boeing 787 has been
credited with putting an end to the era of the all-metal airplane on new designs, and it is
perhaps the most visible manifestation of the fact that composites are having a profound and
growing effect on all sectors of society.
It is generally well-known that composite materials are made of reinforcement fibers and
matrix materials, and light weight and high mechanical properties are the primary benefits of a
composite structure. Accordingly, the development trends in composite technology lie in 1)
new material technology specifically for developing novel fibers and matrices, enhancing
interfacial adhesion between fiber and matrix, hybridization and multi-functionalization, and
2) more reliable, high quality, rapid and low cost manufacturing technology.
New reinforcement fiber technology including next generation carbon fibers and organic
fibers with improved mechanical and physical properties, such as Spectra®, Dyneema®, and
Zylon®, have been developing continuously. More significantly, various nanotechnology
based novel fiber reinforcements have conspicuously and rapidly appeared in recent years.
Matrix materials have become as complex as the fibers, satisfying increasing demands for

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W.H. Zhong, R.G. Maguire, S.S. Sangari et al.
impact resistant and damage tolerant structure. Various means of accomplishing this have
ranged from elastomeric/thermoplastic minor phases to discrete layers of toughened materials.
Nano-modified polymeric matrices are mostly involved in the development trends for matrix
polymer materials. Technology for enhancing the interfacial adhesion properties between the
reinforcement and matrix for a composite to provide high stress-transfer ability is more
critically demanded and the science of the interface is expanding. Fiber/matrix interfacial
adhesion is vital for the application of the newly developed advanced reinforcement materials.
Effective approaches to improving new and non-traditional treatment methods for better
adhesion have just started to receive sufficient attention. Multi-functionality is also an
important trend for advanced composites, in particular, utilizing nanotechnology
developments in recent years to provide greater opportunities for forcing materials to play a
more comprehensive role in the designs of the future.
More reliable and low cost manufacturing technology has been pursued by industry and
academic researchers and the traditional material forms are being replaced by those which
support the growing need for high quality, rapid production rates and lower recurring costs.
Major trends include the recognition of the value of resin infusion methods, automated
thermoplastic processing which takes advantage of the unique advantages of that material
class, and the value of moving away from dependence on the large and expensive autoclaves.
Introduction
In sectors such as aerospace, wind energy, power transmission, marine, automotive and
trucking, composites have been moving into the primary structure of wings, fuselages,
chassis, hulls, and towers. In products such as sports goods and equipment, medical
equipment, civil infrastructure, and dentistry composites are contributing to a market growth
that will soon relegate homogeneous and isotropic materials to a niche category. The word
“composite” is becoming synonymous with greater design flexibility and optimized materials
utilization, leading to more opportunities for monolithic structural designs, less fasteners and
holes, optimization of overall structural element architecture, improved fatigue and corrosion
behavior, and high efficiency and maintainability. Composites are also particularly suitable
for structural health monitoring systems with the associated advantages of reduced
conservatism in designs.
As they have evolved over the past several decades, composites now are spreading out
and leaving their early material forms and traditional processes, and incorporating new
constituents from nano particles to smart additives to hybridizing to capture the best of all
technologies. This has led to lean and efficient automated processes that will enable these
new developments to be cost-effective in production and performance-enhanced in products
that serve us all.
Over the past several decades polymeric composites have matured and evolved,
sometimes fitfully, but much of the time in a steady development driven by the increasing
awareness among industries of the values available from combinations of matrices and fibers.
The world of aerospace has been one of the strongest proponents of advanced composites,
eventually converging on a combination of carbon fibers and thermosetting materials as the
preferred choice for the harsh environment and complex loading of aerostructures. Structural
materials applied for airplane structures from metallic materials, to composites and then nanomodified
composite materials are being developed, see Fig. 1.

• Mechanical performance
enhancements through alloying
& heat treatments
Isotropic Metal
Materials
Major Trends in Polymeric Composites Technology 111
Continuing Trend Toward Increasing
Capability of Engineering Materials
Anisotropic Composite
Materials
• Fiber orientation optimizing effects on
strength & stiffness
• Laminate tailoring for coupled
deformations
• Independent toughness
improvements through polymer alloying
and controlling phase morphology
Figure 1. Materials for airplane structures.
Nano-Modified
Composite Materials
• Broad mechanical property
improvements
• Flammability and solvent resistance
enhancements
• Conductivity and CTE tailoring
• Inherent color, optical qualities, etc.
• Multi-functionality
Composites have been applied in various areas including aerospace, automotives,
renewable energy structures (e.g. windmill blades as shown in Fig. 2), marines, sports and
construction. . The steps for pursuing composite materials with ultra-light weight, super
mechanical properties and multi-functionalities have been developing fast, in particular, the
nanotechnology speed up and provide revolutionary opportunities for the new trends of
composite development, which can be summarized in the Fig. 3.
Figure 2. 38-meter European fiber glass windmill blade.

112
Reinforcement:
Advanced materials
technology
a. New fibers: PBO,
UHMWPE fibers, etc.
b. Nano-tech based:
• 1D reinforcement:
(i) Spun nanofibers
(nanocomposites)
(ii) Nano-scaled fibers (e.g.
nano-PAN based CF)
(iii) Ropes, yarns, bundles
(e.g. CNT ropes, yarns)
• 2D reinforcement:
Film, sheet, mat, etc.
(e.g. Bucky paper,
nano paper)
c. Hybridization: e.g.ARALL
W.H. Zhong, R.G. Maguire, S.S. Sangari et al.
Improve properties for FRP composites
Interface:
a. Nano-coating fibers
(e.g., CNTs coated GF)
Improve interfacial adhesion
b. Reactive nano-matrix
(nanocomposites)
(e.g. reactive nano-epoxy:
improve wetting& adhesion)
Reliable and low cost
manufacturing technology
Matrix:
Nano-filled Resin:
(nanocomposites)
a. Simple physical mixture of
polymer and nano-fillers
b. Integration of nano
polymer systems
(e.g. CNTs-epoxy, GNFsepoxy)
c. multi-functionalization
Figure 3. Routes of property enhancement for macro-composites.
With the success and proliferation of state-of-the-art composite materials in many areas
including aerospace and renewable energy structures, there are new opportunities that are
being considered now that the bar has been successfully raised with the 787. Some of these
will be explored in this chapter.
I. Nanocomposites and Multifunctional Materials: carbon nanotubes (CNTs),
nanofibers, and nanoplatelets have a natural affinity for polymeric materials and their
inclusion in composites offers the promise of multi-functionality: electrical/thermal
conductivity, acoustic damping and optical functionalities may be combined with
load bearing capabilities.
II. Hybridization: we see a new generation of metal/composite combinations deriving
from early attempts such as GLARE®, ARALL®, titanium/graphite (TiGr), but
utilizing new processes for the applications of monodisperse metallic coatings of
high mechanical properties and durability, as well as multi-functionality.
III. Alternatives to Carbon Fibers and Next Generation Carbon Fibers: as the
understanding of the shortcomings of organic fibers such as UHMWPE increases,
new approaches to interface enhancements are being developed which offer the

Major Trends in Polymeric Composites Technology 113
benefit of a new supply chain for highly capable structural fibers. New generations
of continuous carbon fibers are also being developed both on the micro and nano
scales with improved performance and functionality.
IV. Processing Technologies: the autoclave has been the mainstay for many years, but
growing trends toward lean manufacturing make this a roadblock to improved
efficiency. Continuous and inexpensive processing is making the curing step a part
of the overall lean philosophy in composite manufacturing. A growing trend is the
move away from prepregs to the family of materials and processing called resin
infusion (RI). This includes Resin Transfer Molding (RTM), Vacuum-Assisted
Resin Transfer Molding (VARTM), Resin Film Infusion (RFI), and the development
of new continuous processes. All this is based on low-cost material forms of neat
resin and fiber preforms that allow the manufacturer to put the two together in
proprietary and efficient ways.
V. Other trends include a new generation of thermoplastics being developed to serve the
growing automation of TP parts, smart materials and structures which can de-couple
requirements and reduce weight, low-cost carbon fibers from bio sources, and others.
1. Nanocomposites and Multifunctional Materials
The definition of nanocomposites covers a variety of systems such as one-dimensional, twodimensional
and, three-dimensional materials made of distinctly dissimilar components and
mixed at the nanometer scale for achieving drastically enhanced properties. To obtain multifunctionality
in nanocomposites, nanoparticles with high aspect ratio have been successfully
employed. This denotes being functional in one property while either achieving new
properties that are unknown in the individual components or improving and maintaining other
intrinsic properties.
Nanocomposites can be used as matrices for nano/macro-composites, or traditional
composites when micro-scale fibers are included. Nano constituent composites have also been
made into nanoscale fibers through spinning methods. To date, the creation of such new
materials has resulted in:
• Enhanced mechanical properties: strength, stiffness; toughness, impact resistance,
structural durability, etc.
• Improved electrical conductivity
• Improved thermal conductivity and thermal management
• Improved flame resistance, thermal stability and increased service temperatures
• Enhanced acoustic damping
• Improved dimensional stability (low or tailored coefficient of thermal expansion)
• Enhanced tribological properties (wear, abrasion resistance, hardness)
• Improved barrier properties and environmental controls
• Decreased permeability
• Reduced shrinkage

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W.H. Zhong, R.G. Maguire, S.S. Sangari et al.
In many cases the product is a multifunctional material that has several of the above
characteristics and properties in combination. Nanocomposites can improve many kinds of
functionalities and typically can result in multi-functionalities. These functionalities include:
This can be extremely attractive for engineered applications where costs of testing,
evaluations, qualifications and certifications of each new material are high, and sometimes
prohibitive. The prospect of a single material for multiple functions can therefore be very
compelling, both for engineered performances and the non-recurring costs associated with
bringing the material to readiness for the designer.
Bulk form:
Enhanced
functionalities:
- electrical
- thermal
- flame resistant
- damping
- mechanical …
As loose fillers:
Nano-fillers + polymers
--> Nanocomposites
…
Nano-fillers: CNTs, CNFs, etc.
Spinning fibers: 1D
? 2D sheet/paper
Enhanced
functionalities:
- electrical
- thermal
- flame resistant
- damping
- mechanical…
+ Fiber
Combined further:
1D: ropes, yarns, bundles
2D: sheet/paper forms
+Matrix
+Matrix
FRP composites:
(Hybrid composites)
Enhanced
functionalities:
- electrical
- thermal
- flame resistant
- damping
- mechanical…
…
Figure 4. Nano-scale materials in different forms for applications in composites.
Nano-sized materials as nano-fillers for nanocomposites can be classified into three
categories according to the shape: particles (metals, metal compounds, organic and inorganic
particles), fibrous materials (nanotubes, nanofibers, nanowires), and layered materials

Major Trends in Polymeric Composites Technology 115
(graphene platelets, clay). These nano-fillers have exceptionally high specific surface areas so
the overall amount of interfacial area is enormous if the nano-fillers are adequately dispersed
within the matrix. This can result in creating various functionalities including mechanical,
physical and other classes of properties. The large specific surface areas are highly desirable
for stress transfer between the nano-fillers and the matrix, as well as providing increased
chemical reactivity and energy levels compared to conventional bulk materials. Almost all
nano-scale materials can be used as loose fillers for making nanocomposites. Fibrous nanofillers
can be made into yarns/bundles, mats, braids, sheets/papers, which could be used in
fiber reinforced polymer (FRP) composites. Nanocomposites can be applied in various forms
such as coatings, films/sheets, spinning fibers, bulk materials as well as matrices for FRP
composites due to the nano-fillers’ dramatic capability in enhancing functionalities for the
polymer materials. When these nano-fillers or nanocomposites are used in fiber reinforced
polymer (FRP) composites they become hybrid composite materials, which may exhibit
multifunctional properties as illustrated in Fig. 4.
Nanocomposites and hybrid composites with various functionalities have vast
applications in structural applications in aircraft, space vehicles and renewable energy
assemblies; impact protection systems; thermal management components; fuel cells;
electronic devices, sensors, actuators, various functional coatings, electrostatic dissipation
(ESD) and electromagnetic interference (EMI) radiation protection, lightning strike
protection, etc.
It has been established that improvements in the properties of nanocomposites are
strongly affected by many factors including nano-filler size distribution, shape, aspect ratio,
concentration, degree of dispersion, characteristics of the matrix, interactions between the
filler and the matrix, and interfaces between the nano-particles themselves.
According to the potential functionalities, nano-scale fillers can also be divided into (1)
carbon types, such as CNT, carbon nanofibers (CNF, VGCF, or GNF), and graphite
nanoplatelets (GNP), and (2) non-carbon types, such as nano-clay, POSS, nano-silica, metal
nanoparticles and nano metal oxide particles. Nanocomposites with carbon type nano-fillers
are mainly utilized for improvements of damping, mechanical, electrical and thermal
properties. Nanocomposites with non-carbon type nano-fillers are predominantly used for
flame retardency, improved barrier property, creep resistant, tribological properties and to
some extent in early works, mechanical property enhancements.
Nanocomposites have been undergoing rapid developments and significant progress has
been made in the fields of nanocomposites and nanocomposite multi-functionalities over the
past few decades. However, there are an abundant amount of questions and challenges left to
be solved before taking full advantage of nano-scale fillers for development of stable, highquality
nanocomposites. These include types, purity levels and polymer types
(thermoset/thermoplastic), structure characteristics, viscosity at room and/or elevated
temperature, appropriate treatment methods to be applied to the nano-fillers which will affect
the interaction between the nano-fillers and polymer matrix, etc. In order to create a blend
with controlled ratios of components and a well-dispersed nano-filler into the polymer matrix
effective mixing methods and processing parameters should be understood and applied. Only
when a complete understanding of these issues is established will the performance of
nanocomposites with desired properties/functionalities be fully realized.
Although many researchers have conducted remarkably successful experiments for
achieving high performance nanocomposites, and obtained many encouraging empirical

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results, there is still a critical lack of comprehensive mathematical modeling that is needed to
be used to make effective predictions for processing-structure-property relationship or when
evaluating the multi-functionalities. An example is the wide array of electrical conductivity
percolation threshold values for certain nano-fillers (e.g. CNT or CNF), when combined with
the goal of improved strength and modulus. Modeling can increase the speed of selection and
reduce the scale of actual testing, a huge advantage for bringing new products quickly to
market. Mathematical modeling also provides practical benefits to industry in developing
modeling capabilities for designing new materials. With the added dimensions of the
nanocomposites options, conventional testing and down-selection for choices between the
various and numerous materials can quickly become unfeasible.
Nanostructures have unique physical and chemical properties different from bulk
materials of the same chemical composition. The mechanical, electrical, thermal and
magnetic properties of composites consisting of an insulating matrix and dispersed
nanoparticles have been extensively studied over the past few decades. The significant
progress in the understanding of nanocomposite systems within recent years has shown that
multifunctional nanocomposites offer both great potential and great challenges, marking it as
a highly active field of research. The research is continuing at an increasing pace, as the
requirements for stronger and lighter materials are needed by a variety of industries.
However, much research effort is continuing toward the development of new processing
techniques that control the purity and dispersions of nanoparticles in the polymers.
2. Hybridization
As composites develop and improve, and their applications grow, there will be inevitably, the
realization that there are limitations and compromises in changing from metals to “nonmetals”.
In many cases, the logical thought process is to consider how the best of both
materials can be in included in hybrids. One of the major subsets of this segment of materials
is based on the concept of going with one of the composites most valuable characteristics, that
is, lamination of individual plies. In some cases the concept that comes most readily to mind
is the replacement of one or more composite plies with metal foil or sheet and these are
generally referred to as Fiber Metal laminates or FMLs.
FMLs were first developed at Delft University in the Netherlands in the early 1980s, and
marketed by Aluminum Company of America (Alcoa), combining sheets of aluminum in an
alternating pattern with plies of traditional composite. As a result of this hybridization FMLs
can theoretically combine the best of both the metal and the composite materials. The first
FML was ARALL® (ARamid-ALuminum Laminate), a combination of aluminum and
aramid/epoxy. This fiber-aluminum adhesive-bonded laminate is a super-hybrid composite
material, which has many attractive properties such as good damage tolerance property, very
high fatigue crack growth resistance, and high static strength along the fiber direction. The
characteristics of ARALL® also include low density and resistance to the effects of
temperature, humidity and acidity/alkalinity, etc.
But greater applications became apparent if the aramid fiber composites were replaced by
the ubiquitous glass fibers composites. In the 1980s, Delft developed a glass/epoxy FML
called GLARE (GLAss-REinforced) composed of thin layers of aluminum sheet or foil
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Major Trends in Polymeric Composites Technology 117
in various directions to accommodate the loading and because of this characteristic, it is truly
a composite laminate with tailorable in-plane properties, and with the capability to add the
aluminum plies in various locations and arrangements through the stack up, but with
processing properties similar to bulk aluminum sheet metal. Its major advantages over
conventional aluminum are lower density, better fatigue resistance (cracks are inhibited in
growth due to the restraining effects of the adjacent composite plies) and better resistance to
impact. An important consideration is the matching of the cure temperature of the composite
with the thermal effects on the metal. For example, if the composite requires a 350°F degree
cure, the aluminum alloy must be able to sustain its performace after exposure to that
temperature because it is the total laminate that must go through the autoclave process. The
original GLARE® used 250°F curing fiberglass composite, with an appropriate aluminum
alloy, but in situations where a higher Tg is required for design thermal exposure, a 350F
version was created and has been dubbed “New GLARE”.
Another interesting version of FML is titanium-graphite (TiGr) laminates which consist
of layers of titanium interleafed through the thickness of a Carbon Fiber Reinforced Plastic
(CRFP) laminate. TiGr offers advantages over metallic structures in terms of weight, fatigue
characteristics, damage tolerance, and design flexibility, and also advantages over traditional
composite materials through higher bearing capabilities, greater toughness, and an expanded
design space. There has been extensive testing in the industry to support development of
mechanical properties for TiGr with various epoxy prepreg systems and success in
optimization of surface preparation has resulted in extremely robust and environmentally
durable TiGr. Production-related issues such as scale-up, compound contours, drilling and
trimming, NDI, repair methods and even automation have been addressed for feasibility and
optimization.
These extreme hybrid composites have great potential for use in many applications
including aircrafts. However, due to the large difference in thermal expansion coefficient of
both the fiber and metal, and the anisotropy of the composites layers combined with isotropic
metals, large residual stresses can be built up during the curing cycle which could cause an
unsuitable residual stress system that may seriously hinder its outstanding performance. To
regulate the residual stresses with controlled layups is necessary. In addition, there are many
factors which influence the performance of these FMLs due to three kinds of material
constituents involved, and the two interfaces: fiber/resin and metal/resin. To control the
quality of the FML materials is critically important for the applications in practice.
Other types of hybrids include co-mingling of different fibers for multifunctionality of
the composite ply. Co-mingling of carbon fibers and glass fibers can add a softness to a
laminate in places that need dimensional flexibility. Co-mingling of carbon and
thermoplastic fibers can add toughness to strength and stiffness in a part. In some
commercial products, such as the Cytec Priform technology, the thermoplastic actually melts
during the cure and disperses in a controlled way throughout the matrix to form a minor phase
of toughening material.
A new trend is the coating of polymeric composites with metals for multiple purposes
ranging from electrical conductivity to thermal management to surface hardness. MesoScribe
Technologies has a Direct Write Thermal Spray in which fine powders are injected into a
small thermal plasma and accelerated through an aperture to make patterns on composites
without a cure cycle or masking, and is adaptable to large and highly contoured surfaces.

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Integran has its Nanovar technology in which finely grained metal is applied to a composite
surface to increase wear in leading edges and for Invar tool repair. The reduced grain size
leads to increased hardness and strength by the inverse relationship to the square root of the
grain size. Other methods are in development to apply monodisperse metallic coating to
composites for weight reduction, improved surface finish, alternatives to environmentally
unacceptable coatings, and greater design freedom.
In the nano-scale composites, hybridization of nanoparticles offers the potential of a rich
soup in which some additives e.g. CNTs, are added for mechanical and electrical properties,
some, e.g. POSS (Polyhedral Oligomeric Silsesquioxane) or nanoclay can be added for flame
resistance, and others, e.g. nano gold particles, can be added for color.
As we become more fluent in the use of new materials and less prejudiced in the use of
older materials hybridization will become a natural trend for those who want it all.
3. Alternatives to Carbon Fibers and Next Generation Carbon
Fibers
Alternatives to Carbon Fibers With the growth of applications of composites using the high
specific strength and stiffness of carbon fibers, comes also the growing demand for those
fibers in many sectors of industry: aerospace, energy, transportation, infrastructure, medical,
sports equipment, etc. There is also the issue of galvanic corrosion in some applications
where carbon forms one of the three elements of a circuit with metals like aluminum, and
water. As a result there is a growing interest in non-carbon fibers, many of which have been
looked at in the past, but now being considered with a hungrier eye, if some of their
shortcomings can be overcome without sacrificing their benefits, especially in tensile
strength.
One example is liquid crystal polymers (LCP). Celanese developed a thermotropic
polyester-polyarylate LCP in the mid 1970’s and commercialized the Vectra family of resins
in 1985. Vectran® is an example of an LCP fiber. The molecules of the liquid crystal
polymer are rigid and position themselves into randomly oriented domains. The polymer
exhibits anisotropic behavior in the melt state, leading to the term thus the term “liquid crystal
polymer.” The molten polymer is extruded through spinneret holes and the molecules align
parallel to each other along the fiber axis. The highly oriented fiber structure results in high
tensile properties.
Vectran differs from other high-performance non-carbon fibers, aramid and ultra-high
molecular weight polyethylene (UHMWPE), in that is thermotropic, melt-spun, and melts at a
high temperature. Aramid fiber is lyotropic, solvent-spun and does not melt at high
temperature. UHMWPE fiber is gelspun and melts at a relatively low temperature. In all
these fibers the high modulus/high tensile strength is achieved through the oriented linear
molecules called microfibrils. And all these fibers have an order of magnitude lower
compression strength than tensile strength.
In general, organic fibers, such as aramid fiber (e.g. Kevlar®), ultra high molecular
weight polyethylene (UHMWPE) fiber (e.g. Spectra®), and Poly(p-phenylene-2,6benzobisoxazole)
(PBO) fiber (e.g. Zylon®), have excellent mechanical and physical
properties. Kevlar® provides excellent impact resistance and is one of the lightest structural

Major Trends in Polymeric Composites Technology 119
fibers available on the market today, which has been widely used both in soft body armor
applications, and as reinforcement for hard armor, helmets and electronic housing protection.
UHWMPE fibers, such as Spectra® and Dyneema®, are a type of ultra lightweight, highstrength
polyethylene fibers. High damage tolerance, non-conductivity and flexibility, a much
higher specific strength and modulus and energy-to-break, low moisture sensitivity, and good
UV resistance, all make this fiber a good aramid alternative. These fibers are typically used in
ballistic and high impact composite applications. Zylon® consists of a rigid chain of
molecules of poly(p-phenylene-2,6-benzobisoxazole, PBO. It has excellent tensile strength
and modulus. Fabrics made from Zylon ® are found in both ballistic and composite
applications.
In addition to the attractive mechanical properties of organic fibers, albeit limited in their
current form, it should be noted that they are highly valuable by the key industries of the U.S.
and offer the significant advantage in the avoidance of galvanic corrosion with aluminum and
certain other metals. With organic fiber composites, the corrosion threat is avoided and the
cost and weight benefits would be enormous to the aerospace and other industries if other
critical properties could be enhanced. In addition, there is the economic challenge now of the
growing applications for carbon fibers and the resultant shortages of carbon composite
materials, which may impact the economy of the US. Engineering conferences for the past
two years have focused on that increasing shortage and what technical and scientific
alternatives are available. How to make organic fibers a viable alternative to carbon fibers for
structural applications is often discussed in these forums within the genre of nanocomposites.
This category of research holds great promise to enable that technology sector and can
accelerate the fulfillment of the promise of multi-functional materials.
Typical characteristics of these organic fibers include non-polar chemical structures and
crystalline chains. It is these structural characteristics that impart the advanced mechanical
properties on to the fibers. For example, UHMWPE fiber obtains its high strength from the
straightening of long polymer chains by taking advantage of the strong covalent bonds in the
backbone of the monomer. The modulus of the fiber is proportional to the draw ratio which
controls the degree of crystallinity. The main benefits of a UHMWPE continuous fiber
include high specific strength and moduli, leading to a lower weight for a given design load.
The chemical neutrality of the fiber surface leads to a high degree of corrosion resistance;
there are no places to allow for a concentrated attack on the surface. In addition, the
anisotropic nature of the fiber allows for low coefficients of thermal expansion, meaning
dimensional stability of the finished composite product.
However, the non-polar chemical structure and resulting lack of reactive groups on the
organic fiber surface lead to low surface energy, and thus leads to difficulty in obtaining good
wetting and adhesion at the fiber/matrix interface. This low surface energy requires that the
matrix material be of an even lower energy to achieve sufficient wetting and adhesion,
ultimately realizing strong bond at the fiber/matrix interface. This results in the limited
applications of the organic fibers because many properties of the composites are determined
by the transfer ability of the fiber/matrix interface. To tackle the problem, various surface
treatments to improve the interfacial wetting and adhesion, are applied. There appears to be
an absence of a good means to alter the fiber without sacrificing its desirable properties. It is
concluded that novel cost-effective methods for improving the interfacial adhesion between
the organic fibers and the polymer matrix are vital to the full realization of their potential as
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For aramid and UHMWPE fibers, silane coupling treatments (effective for glass fibers)
and oxidation treatments (effective for carbon fibers) are not effective in improving the
interfacial strength [1-6]. For UHMWPE fibers, many treatment methods including nitrogen
ion implantation, nitrogen plasma, fast atom beams, laser ablation, chain disentanglement,
high power ion beam treatments, and cold plasma [7-13] have been used. Such approaches
show some improvement in interfacial properties, but also can degrade the mechanical
properties of the fibers by damaging the chain structure of the UHMWPE fiber and result in
formation of amorphous hydrogenated carbon. Recently, the effectiveness of an atmospheric
plasma was demonstrated for dramatic improvement in the adhesion of polyetheretherketone
(PEEK) composites to epoxy [14]. This atmospheric plasma was shown to be an important
potential strategy for improving the interfacial adhesion between organic fibers and polymer
resins.
A nanotechnology approach was recently developed by Dr. Zhong’s group in which
conventional epoxy resins are converted into reactive nano-epoxy resins. Unlike the
conventional epoxy resins that require carbon fibers to be surface oxidized/treated before
being impregnated, the nano-epoxy resins contain reactive graphite nanofibers which can
improve the wettability and adhesion properties between UHMWPE fibers and the resin
matrix [15-19].
Next Generation Carbon Fibers – Continuous Nanoscale Carbon Fibers Traditional Carbon
fibers have high strength, high modulus and attractively low density. The high strength-toweight
ratio combined with superior stiffness has made carbon fibers the material of choice
for high performance composite structures in the aerospace, defense and other industries.
Polymer fibers, which leave a carbon residue and do not melt upon pyrolysis in an inert
atmosphere, are generally considered candidates for carbon fiber production. It is known that
the structural perfection of precursor fibers is the most crucial factors on the strength of
carbon fibers. Imperfections (such as surface defects, bulk defects and others) in the precursor
fibers are likely to be translated to the resulting carbon fibers, and the amounts and sizes of
structural imperfections directly determine the final fiber strength. The fundamental
approach/solution for improving the strength of carbon fibers is to reduce the amounts and
sizes of numerous types of defects in the precursor and there has been a clear and continuing
trend among commercial carbon fiber suppliers in achieving higher strengths through this
approach.
There is also a growing interest in having thinner plies of composite material without loss
of strength or stiffness and thinner plies means thinner fibers. Historically the means of
producing very small diameter (down to submicron range) has been electrospinning. Since the
1930s electrospinning has been used on nylon and other polymers to achieve small fibers to
provide filtering media and other applications. In the process a strong electric field acts on a
polymer solution resulting in a polymer stream which solidifies through the evaporation of
the solvent.
This can also be applied to a polyacrylonitrile (PAN) copolymer (precursor) to produce
nanofibers with diameters in the nanoscale range with the potential of ultimately producing
continuous nano-scaled carbon fibers with strengths and stiffness much higher than
conventional micro scale carbon fibers. Additionally, since diameters of the electrospun PAN
nanofibers can be further reduced by stretching and carbonization processes, the resulting
nano-scaled carbon fibers can have diameters of less than 100 nm. When incorporated into a

Major Trends in Polymeric Composites Technology 121
matrix composite this could yield a prepreg ply about two orders of magnitude less thickness
with the same strength and stiffness as conventional prepreg offering benefits in weight
critical structure.
4. Processing Technologies
Out-of-Autoclave Processing: Although the autoclave has served the composite industry well
providing structural integrity as well as thermal curing, there is a growing demand for a leaner
means of curing parts. Being able to cure parts in a continuous stream like a pizza oven is the
Lean Manufacturing teams dream. But even a common oven can provide leaner flow.
The economic advantage of an oven process for structural composites is large.
Autoclaves are an order of magnitude more expensive than ovens with the same temperature
uniformity. The process flow and batch constraints of autoclaves can potentially be
eliminated with ovens. And liquid nitrogen systems used to prevent fires would not be
needed. Also there is the possibility of part growth that may limit the autoclave usefulness.
Such an example is seen in the windmill blades in which designs are growing faster than an
autoclave can be depreciated. Blades of necessity have been hot bag cured for many years
and as they surpassed the 38-meter length the designs have been using carbon fiber in place of
glass fiber composites. There is also the issue of large-scale complex parts such as racing
sailboat hulls that would require an autoclave of the scale that not many groups can afford.
These also have been “cooked in a tent” for many years.
In the past there have been attempts to utilize UV curing, e-beam curing, X-ray curing,
oven curing, tent curing, hot press curing, continuous pultrusion curing, many forms of resin
infusion, etc., all with various degrees of success or the lack of significant applications. The
key to success is the material. If they can be developed to have no voids in the uncured
laminate and no volatile components in the resin system then the pressure element becomes
moot. The concerns include matching the fiber volume associated with autoclave cures and
processing variable that can impact design allowables. One of the noted successes in this
trend is oven cured epoxy carbon fiber systems approved by the FAA for structural
applications on civil aircraft (Agate Program allowables database is FAA approved) [20].
Some of these systems are improved with high (>30 in Hg). Generally if a void free
uncured laminate can be achieved either through hot debulking, ultrasonic compaction or
other means, vacuum bag pressure in an oven will be sufficient to achieve a structural part.
There are many new methods that involve either single or double diaphragms to hold the
part while being vacuum-cured. One of these is the double diaphragm resin infusion process
RIDFT of Florida State University High Performance Materials Institute which should offer
lower tooling costs and shorter cycle times. Another novel approach is that of Quickstep®
developed jointly with the Australian research organization CSIRO, it is based on a liquid
filled container in which a lightweight mold floats on one of the flexible faces of the pressure
chambers [21]. The container is filled with a heat transfer fluid which is circulated through
the chamber to rapidly heat and cool the mold. The process enables the cure cycle to be
stopped and restarted at any time and parts of the laminate to be left uncured. Parts can be
consolidated and formed and then final cured in-situ at a later time. It is being developed
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In-situ compaction and curing is another approach that will have great value for lean and
rapid production of composites using unidirectional composite prepreg tape or tow forms.
The uni-prepreg is wound onto a mandrel and heat applied from several IR sources to the
material as it is placed. There is also work being done at NASA on E-Beam in-situ curing
using low energy processing [22].
Resin Infusion Processes: Resin infusion although it is a generally an out-of-autoclave set of
processes, is itself a highly significant trend in composites processing. It covers many
processes such as RTM, VARTM, RFI and each of these major subsections has many
variations (including some which go back into the autoclave), but the common factor among
all of these is that the resins and the fibers are marketed separately and the customer fulfills
the impregnation process within their own manufacturing planning. This can offer
opportunity for customization and optimization for specific user needs and as well a
significant cost savings in materials since the costly impregnation process is now done inhouse.
Several of the major composite material suppliers forecast growing markets for the
infusion resins and dry fiber preforms as compared to the more historical prepreg material
forms.
Resin infusion is a closed low pressure process for the manufacture of complex-shape,
high-strength and lightweight composite parts for a wide range of aerospace, automotive,
marine and satellite applications. In this process, a resin system is drawn into a dry fiber
laminate in a mold where it can cure to form the finished part. RTM is an infusion process
that employs an injection system to transfer a mixture of liquid resin and catalyst into a closed
mold containing a preform, which is preset fiber mats. The resin is injected under a controlled
pressure by a carefully designed pattern of inlet ports and vent holes. This guarantees that the
fibers become fully wet and produce a low void content and high fiber-volume composite
part. Fiber volumes approaching the 65% values, which is typical for prepreg lay-up
techniques, have been achieved with RTM. Subsequent curing of the resin forms a net-shape
part with good dimensional tolerances. The size and complexity of the part significantly
influences the cycle times.
VARTM is an adaptation of the RTM process and is generally used to manufacture parts
for marine, ground transportation and infrastructure applications. The process uses an open
mold cavity which is laid up with a preform and covered with a vacuum bag made of air
impervious films such as nylon or silicone film. The air is expelled from the preform
assembly using a vacuum pump. A liquid resin is allowed to infuse into the mold from an
external reservoir after all air leaks are eliminated and the system is equilibrated. A high
permeability resin distribution medium is placed on top of the preform to facilitate the resin
flow over the lateral extent of the part. The system is kept under vacuum until the resin is
completely gelled. The part may then be cured at room temperature or in an oven. Bag leaks
and bridging are common problems in VARTM. Bag leaks take place at the sealant-bag-film
interface or as a result of film failure due to improper handling. Bridging is the failure of the
sealing bag to conform to the shape of the mold. This leads to the part failing to receive
uniform pressure during the cure cycle.
Matrix viscosity and process time are the two main differences between RTM and
VARTM. For RTM, the resin must travel through the "X" and "Y" directions while for
VARTM it travels on top and only needs to impregnate the "T" or "Z" direction. This requires

Major Trends in Polymeric Composites Technology 123
shorter time and provides the advantage of the lower temperature and faster cure time
combined with reduced thermal stress.
Resin film infusion, RFI, is a composite manufacturing process which has advanced from
earlier work on vacuum impregnation and RTM. In this process, a semi-cured resin film is
liquefied and absorbed throughout the fiber. The mold filling is further assisted by vacuum to
reduce the air voids remaining in the fabricated part. The resin and the fiber are generally
placed together into the mold but are not initially combined. In some applications the fiber
and the resin are placed in the mold in separate steps and are combined by applying pressure.
Computer simulations are commonly used to determine processing details such as resin
viscosity, preform permeability, resin/preform interactions, and the time to completely cure
the composite part. The major difference between RFI and RTM is that former uses a hot
melt resin film while the later utilizes a liquid resin. RFI does not require low minimum
viscosity as in the RTM process.
Orthophthalic, isophthalic polyesters and vinyl esters are primarily used in RTM
processes. A variety of polymers are being developed specifically for RTM application, e.g.
low-shrink and low-profile polyesters for improved surface appearance. New resins including
epoxies, acrylic/polyester hybrids, urethanes, bismaleimides (BMI), and phenolic resins are
also produced which require changes in the equipment and conditioning the resin prior to
injection. These systems offer a whole new range of cost and performance options to the
RTM process.
Reinforcements used in RTM are normally glass fibers, continuous fiber mats and
chopped strand preforms. Special mats that contain thermoplastic binders are heated and thermoformed
into perfect preforms. Both woven and non-woven glass fibers and biaxial and
triaxial mats have been produced for the RTM applications. Other high performance
reinforcements such as carbon fiber and aramid can be incorporated in RTM laminates either
alone or as part of a hybrid system.
A typical RTM mold features oil connections, injection runner system, and self
clamping/load devices. RTM surfaces offer high quality through a combination of
appropriate resin reinforcements, molds and process conditions. A combination of mechanical
clamping arrangements with special presses is necessary to secure the mold halves. This is
required for applying pressure uniformly to the mold when pressurized resin is employed.
RTM processing requires accurate and reliable injection of liquid resin. Resins must balance
low viscosity at processing temperatures and long pot life without sacrificing the mechanical
properties to alter the flow characteristics. The resin is injected until the mold is completely
filled. Motionless or non-mechanical mixers are normally utilized to blend resin and catalyst.
After curing for a required time the composite part is moved from the mold. A mixer flush
system is also incorporated when required to purge non-disposable mixers.
RTM is employed to reduce fill times and to fabricate large-scale composite structures
with substantial laminate thicknesses. It fills the gap between hand lay-up and compression
molding of sheet or bulk moldings in matched metal molds. In comparison with lay-up and
spray-up processes RTM provides two finished surfaces on parts which can be similar or
dissimilar, highly reproducible thickness, low monomer loss and higher output since it is less
labor and material intensive. Compared with matched-metal-die compression molding, RTM
enables the use of parts such as ribs and inserts, decreases lead times for molds, and has
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RTM provides a lot of processing flexibility. Shrink control systems can be employed to
produce improved surfaces. The fibers and resins are utilized at their lowest cost plus resin
content can be controlled to a significant degree and reinforcements can be easily
incorporated. Cores and other insets can also be positioned prior to resin injection to yield
complex parts in one fabrication step. RTM can also be used to prototype parts for market
evaluation since the initial investment costs such as tooling and operating expenses are low.
In this case, RTM's short lead time and lower cost tooling is a real advantage. It also offers
production of near-net-shape parts which in turn leads to low material wastage. Closed RTM
molds release fewer volatile materials than open molds. Added benefits of this closed-mold
process are greatly reduced volatile organic compound (VOC) emissions.
VARTM can be used at room temperature with no heated mold and compared to RTM
more consistent and uniform coverage can be obtained. VARTM provides significant savings
in the tooling cost as it requires only a one-piece mold. The use of the vacuum bag eliminates
the need for making a precise matched metal mold as in the conventional RTM process and
thus reduces the cost and design difficulties associated with large metal tools. VARTM also
beats the 5-10% accuracy of hand lay-up.
For a given resin, mold filling using RFI is relatively faster than most other liquidcomposite-molding
processes since the RFI products are usually thin and the infusion
distance is short. In addition, RFI can provide parts with high mechanical properties due to
solid state of initial polymer material and elevated cure temperature.
Along with the advantages, RTM also has inevitable disadvantages including inability to
manufacture very large parts and permeability issues which lead to increased processing time.
It is an intermediate volume production process and its tooling and equipment costs are higher
than the hand lay-up and spray-up. Tooling design and fabrication to handle injection
pressures, clamping and sealing are more complex and manufacture of complex parts require
trial and error experimentations combined with flow simulation modeling.
VARTM also is a relatively complex process to perform well. The flexible nature of the
vacuum bag brings about the difficulty of controlling the final thickness of the preform, and
hence the fiber volume fraction of the composite. Due to the complex nature of VARTM, the
trial and error experimentation is not only inefficient but also expensive for the process design
and optimization.
RTM is being used for a number of products including aircraft components, recreational
vehicle, truck and sports car bodies, automobile panels, medical equipment, dish antenna,
storage tanks, electrical covers, windmill blades, plumbing parts, transportation seating,
chemical pumps, marine components such as hatches and small boats, bicycle frames and
doors.
Resin Infusion processes enable us to manufacture a wide range of complex as well as simple
fiber reinforced composite products. Improvements are being made on resins and tooling
developments which further expand market opportunities. The combination of pre-positioning
a variety of reinforcements and incorporating secondary reinforcements and other design
details with good surface control on both sides makes RTM a primary process candidate for
structural applications in the aerospace industry as well as other markets. Due to its
versatility, growing use in industry, and being environmentally friendly, experts believe that
the future of this useful composite process should continue to grow. In many applications its
replacement of hand layup/ prepreg material forms/autoclave processing production has

Major Trends in Polymeric Composites Technology 125
resulted in significant cost and rate benefits, further enhanced with the possibility of ganginfusion
production scenarios and automated preform production and handling.
Advanced Thermoplastic Composite Processing: Thermoplastics are having a resurgence of
interest, largely due to a growing list of successful new processing methods. Reinforced
Thermoplastic Laminates (RTL) is an economical means of producing a solid laminate if
there is no constraint against constant thickness. The pre-consolidated laminate is heated
above the melt temperature, usually by infrared lamps, and then automatically transferred into
a pair of cool tools that rapidly close to form and cool the part. This achieved very rapid
cycle time. Another process that has been growing in applications is the automated
thermoplastic pultrusion process that can produce high volumes of long straight parts of
various shapes [23]. These and other thermoforming processes are bringing new life to the
applications of thermoplastics in significant structural applications that promise to realize the
attractions of short cycle times and the possibility of recycling.
5. Other Trends
Smart Composites: Smart materials have the ability to perform both sensing and actuation
functions. The use of imbedded sensors such as piezoelectric, shape-memory alloys,
magneto-strictive, or fiber optics with Bragg gratings (FOBG) to sense and mitigate the
threats to the health of a structure, i.e. Structural Health Monitoring (SHM), holds great
promise for the future of composite primary structure through the elimination of designed-in
excess material for undetected damage events; being aware of damage when it happens and
where it happens can eliminate much design conservatism. Other possibilities are the
incorporation of self-healing or restorative abilities, active control of key functions such as
vibration, etc. Smart composites face the challenges of effective dispersion and interfacial
adhesion of the “smart” constituents. Smart composite materials can be obtained by mixing
the polymer matrix with smart material used for health monitoring, active control and selfrestoration
of structural and functional materials. Recent advances in optical glass fibers have
produced a form which has the approximate same diameter as a carbon fiber so can be
incorporated into a tape or fabric reinforcement without disruption of the load carrying
capability.
Bio-based composites: Increasing interest is developing in bio-based composite constituents.
With shortages developing for the traditional petroleum based products, there are activities in
US, China, Singapore and elsewhere to develop carbon fibers from renewable agricultural
sources such as corn, soy, rice, wheat and other biomaterials that do not deplete the petroleum
reserves. To date the efforts are still in their early stages of success, the quality is not that of
the PAN or pitch based fibers but the costs are very attractive and the growing interest in
greener processing will add impetus to these activities, particularly for those applications
where lower performance is not critical and which are suffering from the current carbon fiber
shortage. University of Delaware Affordable Composites from Renewable Resources
(ACRES) program is one source of development in this area, having been awarded a USDA
National Research Initiative to investigate the possibility of making circuit boards from soy
resins and chicken feather based carbon fibers rather than the conventional epoxy, PAN-based

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composites. Michigan State University has a center for biocomposites with many projects
underway on bio composites, green nanocomposites, biodegradeable thermoplastic polymers
and soy based bioplastics among others [24].
Short fiber composites: Originally short fibers in composites were very short, basically just
additives with aspect ratios only slightly greater than one. Subsequently, the long
discontinuous fiber (LDF) composites appeared with fibers either chopped, stretch broken, or
otherwise made discontinuous. These LDFs could approach the performance of continuous
fiber composites with fiber lengths of 2” to 4” and some degree of alignment. For applications
with complex geometry, these materials can offer relief from the limitations of continuous
fiber prepregs difficulties in conforming to bends and reentrant shapes. In many cases these
materials and processes can replace traditional metals in parts with complex shapes. In
aerospace applications the advantages offered in replacing small metal parts and in some
cases conventional composites materials and processes are significant and growing [25].
Chopped fibers can also be hybridized with continuous fibers to create an engineered form.
Phoenix Composites selectively uses continuous fiber uni-directional and woven
reinforcement locally introduced into a parent structure consisting of chopped random fiber
reinforcement for a form that is comparable to continuous fiber composites in strength and
stiffness but still retaining the geometric flexibility of a chopped fiber process. Chopped fiber
composites when combined with thermoplastics can be a very cost-effective process.
University of Alabama-Birmingham has produced effective bus seats using Long Fiber
Reinforced Thermoplastics (LFT: PP + glass fiber) in a compression molding process.
Conclusions (Summary)
Polymeric composites technology has been the vehicle of change in key industrial sectors for
the past 30 years, growing from fiberglass reinforcement to more sophisticated polymeric
fibers and the current champion, the carbon fiber/multi-phase matrix polymer composite
materials. As applications have grown both in breadth and scale, new needs and visions have
created strong and focused trends, in both materials and processing sciences and technologies,
and emerging at an increasing rate. Market-driven pull and science-based push mechanisms
have brought us to a richer landscape of increased dimensions and applications unimagined a
few decades earlier. The composites community, unlike other industrial technolgies, is not
complacent. Although autoclaves and prepreg have served us well, we want to get rid of
them and process in a more optimized way with resin infusion and leaner manufacturing.
While the current materials have enabled entire airplanes to be made of composite materials,
we want those materials to now serve multiple functions, behave with intelligence, be
greener, and are exploring the huge benefits of very small matters in the world of
nanotechnology. Conventional materials such as thermoplastics take on a new life as vastly
more efficient and focused processing methods are developed, and combining conventional
materials in unconventional and novel ways is opening new possibilities. The compositeer
has much to feel satisfied about but there is much to challenge them in the future.

Acknowledgements
Major Trends in Polymeric Composites Technology 127
Dr. Zhong acknowledges the support from NASA through grant NNM04AA62G and from
NSF through NIRT Grant 0506531.
References
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[3] Broutman, L. J.; Agarwal, B. D. Polym. Eng. Sci. 1974, 14, 581-588.
[4] William Jr, J. H.; Kousiounelos, P. N. Fibre. Sci. Tech. 1978, 11, 83-88.
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molecular weight polyethylene fibers, 50 th Intl. SAMPE, Long Beach, CA, 2005.
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Anaheim, CA, 1988.
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Treatment of Polyetheretherketone Composites for Improved Adhesion, SAMPE Fall
Technical Conference Proceedings: Global Advances in Materials and Process
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[15] Neema, S.; Salehi-Khojin, A.; Zhamu, A.; Zhong, W. H.; Jana, S.; Gan,Y. X. J. Colloid
Interf. Sci. 2006, 299, 332-341,
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[17] Salehi-Khojin, A.; Stone, J. J.; Zhong, W. H. J. Compos. Mater. 2007, 41, 1163-1176,.
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[22] http://www.tc.faa.gov/its/cmd/visitors/data/AAR-430/advanced.pdf
[23] http://www.quickstep.com.au
[24] http://www.sti.nasa.gov/tto/spinoff2001/ip7.html
[25] http://www.acm-fn.de/e_start.htm
[26] http://www.egr.msu.edu/cmsc/biomaterials/star/star
[27] http://www.hexcel.com/Products/Matrix+Products/Other+FRM/HexMC

In: Composite Materials Research Progress ISBN: 1-60021-994-2
Editor: Lucas P. Durand, pp. 129-164 © 2008 Nova Science Publishers, Inc.
Chapter 4
AN EXPERIMENTAL AND ANALYTICAL STUDY
OF UNIDIRECTIONAL CARBON FIBER REINFORCED
EPOXY MODIFIED BY SIC NANOPARTICLE
Yuanxin Zhou a , Hassan Mahfuz b , Vijaya Rangari a
and Shaik Jeelani a
a Center for Advanced Materials at Tuskegee University, Tuskegee, AL, 36088
b Department of Ocean Engineering, Florida Atlantic University, Boca Raton, FL 33431
Abstract
In the present investigation, an innovative manufacturing process was developed to
fabricate nanophased carbon prepregs used in the manufacturing of unidirectional composite
laminates. In this technique, prepregs were manufactured using solution impregnation and
filament winding methods and subsequently consolidated into laminates. Spherical silicon
carbide nanoparticles (β-SiC) were first infused in a high temperature epoxy through an
ultrasonic cavitation process. The loading of nanoparticles was 1.5% by weight of the resin.
After infusion, the nano-phased resin was used to impregnate a continuous strand of dry
carbon fiber tows in a filament winding set-up. In the next step, these nanophased prepregs
were wrapped over a cylindrical foam mandrel especially built for this purpose using a
filament winder. Once the desired thickness was achieved, the stacked prepregs were cut
along the length of the cylindrical mandrel, removed from the mandrel, and laid out open to
form a rectangular panel. The panel was then consolidated in a regular compression molding
machine. In parallel, control panels were also fabricated following similar routes without any
nanoparticle infusion. Extensive thermal and mechanical characterizations were performed to
evaluate the performances of the neat and nano-phased systems. Thermo Gravimetric Analysis
(TGA) results indicate that there is an increase in the degradation temperature (about 7 0 C) of
the nano-phased composites. Similar results from Differential Scanning Calorimetry (DSC)
and Dynamic Mechanical Analysis (DMA) tests were obtained. An improvement of about
5 0 C in glass transition temperature (T g) of nano-phased systems were also seen. Mechanical
tests on the laminates indicated improvement in flexural strength and stiffness by about 32%
and 20% respectively whereas in tensile properties there was a nominal improvement between
7-10%. Finally, micro numerical constitutive model and damage constitutive equations were
derived and an analytical approach combining the modified shear-lag model and Monte Carlo

130
Yuanxin Zhou, Hassan Mahfuz, Vijaya Rangari et al.
simulation technique to simulate the tensile failure process of unidirectional layered
composites were also established to describe stress-strain relationships.
Introduction
Carbon fiber reinforced polymer matrix composites due to their high specific strength and
specific stiffness have become attractive structural materials not only in the weight sensitive
aerospace industry, but also in marine, armor, automobile, railways, civil engineering
structures, sport goods, etc. Generally, the in-plane tensile properties of the fiber/polymer
composite are defined by the fiber properties, while the compression properties and properties
along the thickness dimension are defined by the characteristics of the matrix resin.
Epoxy resin is the most commonly used polymer matrix for advanced composite
materials. Over the years, many attempts have been made to modify the properties of epoxy
by the addition of either rubber particles [1-2] or fillers [3-4] so that the matrix-dominated
composite properties are improved. The addition of rubber particles improves the fracture
toughness of epoxy, but decreases its modulus and strength. The addition of fillers, on the
other hand, improves the modulus and strength of epoxy, but decreases its fracture toughness.
Usually, the typical filler content needed for significant enhancement of these properties can
be as high as 10-20% by volume. At such high particle volume fractions, the processing of
the material often becomes difficult, and since the inorganic filler has a higher density than
the resin, the density of the filled resin is also increased. Nanoparticle filled resins are
attracting considerable attention since they can produce property enhancement that are
sometimes even higher than the conventional filled polymers at volume fractions in the range
of 1 to 5%. It has been established that adding small amounts of nano-particles (

An Experimental and Analytical Study of Unidirectional Carbon Fiber… 131
Monte Carlo simulation on the tensile failure process of unidirectional carbon prepreg
laminates.
Materials
T700SC 12000-50C carbon fiber used in this research is manufactured by Toray Carbon
Fibers America, Inc., USA. It is the highest strength, standard modulus fiber in the form of
continuous filament tows with outstanding processing characteristics for filament winding,
weaving and prepregging produced using the PAN (polyacrylonitrile) process.
The epoxy system used was CR46T. It’s a high temperature cure prepreg resin system.
To avoid degradation of its properties, the resin is kept under sub-zero temperatures in a
sealed atmosphere. This resin system is well suited for prepreg applications, and its properties
are shown in Table 1.
Table 1. Resin properties
Density 0.0457 lb/in 3
Gell (min) @ 350 0 F 6 – 10
GIC
0.733
lb/in
± 0.3
2
Tg DRY 2hr @ 375 0 F 437 0 F
Tg WET 2hr @ 375 0 F 295 0 F
Tensile strength @ RT 10 ksi
Tensile modulus @ RT 643 ksi
Poisson ratio 0.36
Elongation 1.7 %
The nanosized fillers for this present investigation were chosen as nano-sized silicon
carbide particles. These are highly complex material existing primarily in amorphous or
crystalline states. The amorphous SiC is mainly used in coating industries. In functional and
structural applications, crystalline SiC are extensively used due to their excellent thermomechanical
properties such as high hardness and stiffness, good corrosion and oxidation
resistance, high thermal conductivity and high chemical and thermal stability. [10-12]. Such
SiC is available in two different phases, namely alpha (α) and beta (β) phases. The formation
of these two structures depends on the molecular organization of the basic structural unit, a 2layer
planner unit of Si and C in tetrahedral coordination. β−SiC is formed when the planes of
Si and C are rearranged in a cubic symmetry with a lattice constant a = 0.4358 nm. On the
other hand, heating of β−SiC to high temperature causes the transformation of the cubic
symmetry to a mixture of hexagonal (6H) and rhombohedral (15R) polytypes known as α-
SiC. The corresponding lattice constant parameters of α-SiCs are: a = 0.3082 nm and c =

132
Yuanxin Zhou, Hassan Mahfuz, Vijaya Rangari et al.
1.5117 nm [13-14]. The α-SiC is chemically unstable and as a result their application is very
limited. A schematic showing the crystal structure of α and β−SiC is given in Figure 1.
The nano sized β-SiC particles were obtained from MER Corporation, USA. These
particles are spherical in shape with average diameter of about 30 nm as shown in Figure 2.
The bulk material contains more than 95% of SiC with small traces of Oxygen and Carbon.
(a)
(b)
Figure 1. Crystal structure of (a) α-SiC, (b) β-SiC Particles
[www.a-e/englisch/lexikon/ siliciumcarbid-bild2.htm]

An Experimental and Analytical Study of Unidirectional Carbon Fiber… 133
Manufacturing
Figure 2. TEM micrograph of nano SiC sized particles.
Manufacturing of the Nano-phased Carbon Prepregs
Online solution impregnation and filament winding were used as the method of
manufacturing nano-phased unidirectional carbon prepregs. This method involves four
principle steps: (1) uniform dispersion of nano particles in the resin system; (2) application of
resin reaction mixture onto the reinforcing tows; (3) removal of solvent from the prepregs;
and (4) filament winding.
There are various techniques to disperse nanoparticles in the resin system. Acoustic
cavitation is one of the efficient ways to disperse nano-particles into the virgin materials. In
this case, the ultrasonic power supply (generator) converts 50/60 Hz voltage to a high
frequency electrical energy. This voltage is applied to the piezoelectric crystals within the
converter, where it is changed to small mechnical vibrations. The converter’s longitudinal
vibrations are amplified by the probe (horn) and transmitted to the liquid as ultrasonic waves
consisting of alternate compressions and rarefactions. These pressure fluctuations give rise to
microscopic bubbles (cavities), which expand during the negative pressure excursions, and
implode violently during the positive excursions. Some of these cavities oscillate at a
frequency of the applied field (usually 20 kHz) while the gas content inside these cavities
remains constant. As the bubble collapse, millions of shock waves, eddies and extremes in
pressures and temperatures are generated at the implosion sites. Although this phenomenon
known as cavitation, lasts but a few microseconds, and the amount of energy released by each
individual bubble is minimal, the cumulative amount of energy generated is extremely high.
During the operation, an active cavitation region is created close to the source of the
ultrasound probe and that the ultrasonic processing produces high pitched noise in the form of
harmonics which are above the human audible range, emanating from the container walls and
the fluid surface. The development of cavitation processes in the ultrasonically processed
melt creates favorable conditions for the intensification of various physio-chemical processes.

134
Yuanxin Zhou, Hassan Mahfuz, Vijaya Rangari et al.
Acoustic cavitation accelerates heat and mass transfer processes such as diffusion, wetting,
dissolution, dispersion and emulsification [15-16].
Optimization of the solution prepregging process begins with the appropriate choice of
solvent. A high degree of wetting can only be expected from solvents that possess favorable
thermodynamics regarding wetting of the particular solid material (carbon filaments, in this
case) [17-18]. The process of wetting entails the contact and spreading of the solvent over the
surface of the solid, i.e., liquids that possess a low contact angle for a particular solid show
considerable wetting behavior (as opposed to liquids that display high contact angles). This
solvent should be chosen from a list of candidate solvents capable of dissolving the matrix
polymer. The differences in wetting action, coupled with other relevant parameters such as
boiling point and general practicality of the particular solvent choice usage, will lead to an
appropriate choice of solvent. In particular, solvent characteristics should include a much
lower boiling point than melt flow point of the resin and a lower density then that of the resin
for ease of residual solvent removal [19]. An example of the preceding contact angle analysis
can be found in a study by Patel and Lee [17]. In their study, fiberglass tows were subjected
to contact angle analysis using the Wilhelmy plate method. A series of liquids was used (not
polymer solutions), each having differing values of viscosity and surface tension. The
equilibrium contact angles for all of these liquids were not observed to be a function of
solvent viscosity (viscosity range = 0.33 mPa – 1499.0). Furthermore, the liquid surface
tension was found to be positively correlated with the contact angle, i.e., increases in surface
tension generally yielded larger contact angle measurements. It should be stressed that these
results only indicate trends in contact angles; they may not imply favorable conditions for
capillary flow (in addition to wetting), which is another important consideration in the
prepreg process [15]. Once the appropriate solvent is identified for solution prepregging,
prepregged tapes can be manufactured. The objective in solution prepregging is to prepare a
uniform tape in which every fiber surface is uniformly wetted with the polymeric matrix
material. Another objective in solution prepregging is maximizing the amount of matrix
material pick-up. This is easily quantifiable as the amount of matrix material adhering to the
fiber surface after a single immersion into the resin bath. The nature of the relationship
between fiber dispersion and matrix pick up is expected to be competitive. This can be
inferred from the extremes of the process. In a polymer solution with a concentration
approaching zero, every filament can be expected to be wetted (resulting in a good fiber
dispersion), assuming that the thermodynamics are favorable. But the matrix pick up in this
case is nearly zero since there is no polymer in solution. At the other extreme, the polymer
weight fraction in solution approaches one. In this case, the fiber wetting upon dipping will be
very poor given the extremely high viscosity of the resin (kinetic limitation). But upon
wetting, a large amount of polymer will remain on the fiber surface (high matrix pick up).
Therefore, intuition states that there will exist an intermediate polymer solution concentration
in which a balance is obtained between the fiber dispersion and matrix pick up. The concepts
in the preceding paragraph can be more easily visualized by using a model that approximates
the wetting process of a fiber tow by a polymer solution. By combining the Kelvin equation,
which describes wetting of a solution in micro-capillaries and Darcy’s Law, which describes
flow in porous media, the following equation is obtained:
f
void
{ 2S
( 2 / R)
γ θ}
2
t =
l μV /
cos
b
sizing

An Experimental and Analytical Study of Unidirectional Carbon Fiber… 135
where: tf = tow wetting time
l = tow thickness
µ�= solution viscosity
Vvoid = tow void volume
Sb = tow permeability (perpendicular to fiber direction)
R = fiber-fiber separation
γsizing= solution surface tension
θ= contact angle
A quick survey of the equation reveals the following three trends:
• As the solution viscosity increases, the time of tow wetting increases.
• As the surface tension of the solution increases, the time of tow wetting decreases.
• As the contact angle increases from 0 0 �(complete wetting) to 90 0 �(mostly nonwetting),
the cosine term decreases and thus increases the time of tow wetting.
Prepreg residence time is also known to influence both the fiber dispersion and
efficiency. In a study by Lacroix et al. [20], ultra-high modulus polyethylene fiber bundles
were prepregged with a xylene/ low-density polyethylene solution. For a prepregging time
range of 8 min. – 19.5 hours, it was noted that increasing prepreg time increased the layer
thickness of deposited polymer around the fiber surfaces. Similar results were obtained in a
study by Moon et al. [19] in which solvent prepregged fiber bundles were prepared from glass
fibers and a high-density polyethylene/ toluene solution.
After the fiber tapes are prepregged with the nano-phased resin, the solvent has to be
driven off. In this case, since the tapes are not to be wound around a storage spool following
prepregging, solvent elimination should be complete. This represents a crucial step in the
overall composite manufacturing process, as residual solvent can result in voids during the
melt consolidation process. How the solvent interaction with the fiber/matrix/nanoparticle
interface is an important consideration, given the influence of the quality of the interface in
determining the final mechanical properties of the composite. The presence of solvent is
generally known to reduce the quality of the matrix/fiber interface. The reasons for this
phenomenon are unclear, but can be explained by the following hypothesis [21]:
• Solvent extraction can cause separation of the fiber/matrix interface
• Solvent concentration at the interface will interfere with fiber/matrix contact; and
• Phase separation of low molecular weight species at the interface may form a weak
interface between the fiber and matrix.
Solvent removal, in part, is regarded to proceed by solvent concentration at the interface,
followed by solvent traversing the fiber surface and escaping from the ends of the composite.
Obviously this will result in poor interfacial quality if this is to occur during melt
consolidation or autoclave processing, as the case maybe. A study conducted by Wu et al.
[22] illustrates how residual solvent negatively affects composite mechanical property
quality. Solution prepregged carbon fiber reinforced polyethersulphone composites were
prepared and compared with strictly hotmelt processed composites of the same nominal fiber

136
Yuanxin Zhou, Hassan Mahfuz, Vijaya Rangari et al.
content. The transverse flexural strength of the solution prepregged material was only half
that of the melt-processed material. Upon analysis of the solution prepregged material using
differential scanning calorimetry (DSC), it was found that residual solvent remained in the
sample, despite hotmelt consolidation of the prepreg. Residual solvent can most likely be
attributable to difficulty in solvent diffusion during the consolidation process. The reasons for
poor interfacial quality are thought to be attributable in the reasons outlined in the preceding
paragraph.
Commercially available high temperature prepreg resin CR46T was first dissolved in
acetone (Dimethyl ketone, class 1B, Fisher Scientific Co. LLC, USA), at a ratio of 65:35 by
mechanical stirring at 1500 RPM for about 4 hours as shown in Figure 3. Spherical shaped
silicon carbide nanoparticles were carefully measured to have a 1.5% loading by weight of
the resin and mechanically mixed with the liquid resin. The mixture was then irradiated with
high intensity Sonic Vibra Cell ultrasonic liquid processor (Ti-horn, 20 kHz, 100W/cm 2 ) at
50% amplitude for 30 minutes. This ensured uniform mixing of nanoparticles over the entire
volume of the resin. To avoid temperature rise during sonication, cooling was employed by
submerging the mixing beaker in a water bath maintained at 50 0 F as shown in Figure 4. The
nano-phased resin reaction mixture was then transferred into a heating bath maintained at a
constant temperature of 80 0 F throughout the fabrication as shown in Figure 5. A continuous
strand of carbon fiber from a spool attached in the spindle bracket assembly was allowed to
pass through the resin bath at a rate of about 1 meter per minute. In this case, the resin
reaction mixture individually wet each filament within the fiber tow. Once the fiber was
coated with nano-phased resin the excess solvent was removed from the prepreg by passing
the wet strand through a high temperature heater maintained at 160 0 F. The nano-phased
prepreg tow was then routed and fed through a fiber delivery system and was precisely hoopwound
on a rotating foam mandrel on the filament winding machine. Figure 6 represents the
schematic of solution impregnation and filament winding setup. During the fiber placement,
the winding angle was kept at 89.875 0 to avoid excessive gaps or overlaps between adjacent
courses.
Figure 3. CR46T resin mixed with acetone.

An Experimental and Analytical Study of Unidirectional Carbon Fiber… 137
Figure 4. Ultrasonic cavitation.
Figure 5. Nano-phased resin.
Figure 6. Schematic of solution impregnation and filament winding.

138
Yuanxin Zhou, Hassan Mahfuz, Vijaya Rangari et al.
Manufacturing of Nano-phased Carbon Prepreg Unidirectional Laminates
The process of unidirectional laminate fabrication started with online prepregging. By this it
means that when eight layers of the manufactured prepregs were hoop wound on the rotating
foam mandrel during prepreg manufacturing, the remaining tow was cut and the mandrel was
removed from the filament winding machine. During prepreg stacking, care was taken to
place tows without allowing the previous layer to dry. The prepregs on the cylinder was then
longitudinally cut open into a rectangular sheet as shown in Figure 7. These rectangular
sheets were arranged in a compression molding setup by putting symmetric layers of plastic
film, bleeder cloth and teflon on the top and bottom. The whole setup was then placed in
Tetrahedron MTP press compression molder as shown in Figure 8. Mold temperature was
ramped to get 350 0 F while the mold pressure was kept as 40 Psi and consolidated for about 4
hours to obtain a 2mm thick SiC-carbon-epoxy nanophased unidirectional laminate (as shown
in Figure 9). A typical consolidation cycle is shown in Figure 10.
Figure 7. Schematic of unidirectional laminate preparation.
Figure 8. Compression molder.

An Experimental and Analytical Study of Unidirectional Carbon Fiber… 139
Figure 9. Nano-phased unidirectional laminate.
Figure 10. Consolidation cycle for laminates.
Experimental Results and Discussion
Differential Scanning Calorimetry
The Differential Scanning calorimetric (DSC) studies have been carried out to understand the
effect of nanoparticles on glass transition temperature of cured carbon-epoxy prepregs based
on consolidated samples. Figure 11 represents the DSC curve of as-fabricated neat system.
The curve exhibits an endothermic baseline shift at about 220 0 C and highly exothermic peak
at about 390 0 C. The baseline shift at 220 0 C is assigned to the glass transition temperature
and exothermic peak at 390 0 C is assigned to the decomposition temperature of the epoxy
resin. These results match well with the supplier materials data sheet. Figure 12 represents the
DSC curve of as-fabricated 1.5 wt.% SiC nano-phased system. The curve showed only one
exothermic peak at about 397 0 C which is assigned to the decomposition temperature of the
cured epoxy resin. The baseline shift corresponding to glass transition temperature was
almost disappeared. This clearly indicated that the epoxy was highly cross-linked due to
catalytic effect caused by SiC nanoparticles. The shift was not observed, we believe, because
of the equipment sensitivity to the higher cross-linked polymers. To further validate the

140
Heat Flow (W/g)
Heat Flow (W/g)
0.1000
0.0812
0.0625
0.0437
0.0250
0.0062
-0.0125
-0.0313
-0.0500
-0.0688
-0.0815
-0.1063
-0.1250
-0.1438
-0.1625
-0.1812
Yuanxin Zhou, Hassan Mahfuz, Vijaya Rangari et al.
-0.2000
50 100
150
200
250
300
350
400
0.0500
0.0188
-0.0816
-0.1563
-0.2250
-0.2938
-0.3625
-0.4312
Temperature (°C)
Figure 11. DSC graph of neat prepreg system.
-0.5000
50 100
150
200
250
300
350
400
Temperature (°C)
Figure 12. DSC graph of 1.5 wt.% SiC nano-phased prepreg system.

An Experimental and Analytical Study of Unidirectional Carbon Fiber… 141
Figure 13. DMA graphs of neat and 1.5 wt.% SiC nano-phased prepreg systems.
results, dynamic mechanical analysis (DMA) tests were also carried out under single
cantilever test environment. The results indicated the existence of glass transition (Tg) for the
nano-phased system which is about 5 0 C higher than the neat counterpart as shown in Figure
13. The increase in Tg may be attributed to a loss in the mobility of chain segments of epoxy
resin resulting from the high nanoparticle/matrix interaction. Impeded chain mobility is
possible if the nanoparticles are well dispersed in the matrix. The particle surface-to-surface
distances (‘matrix bridges’) should then be relatively small and chain segment movement may
be restricted. Good adhesion of nanoparticles with the surrounding polymer matrix
additionally may have benefited the dynamic modulus by hindering molecular motion to
some extend. The hard particles incorporated into the polymer may also have acted as
additional virtual ‘‘network nodes’’. In either situation it can be deduced that Tg increased as
a result of more number of cross-linked polymer chains and restricted mobility of the chain
segments in the presence of SiC nanoparticles.
Thermo Gravimetric Analysis
Thermo gravimetric analysis (TGA) has been carried out to find the degradation temperature
or to estimate the thermal stability of neat and 1.5 wt.% nano-phased prepreg systems. Figure
14 shows that in as-fabricated neat system, the resin decomposed at about 390 0 C which is
represented by the peak of the derivative curve. The TGA curve shown in Figure 15 indicated
that the decomposition temperature of the nano-phased system was about 397 0 C, which is
almost 7 0 C higher than the neat counterpart. From the results it is clear that in nano-phased
systems, epoxy was amply cross-linked and had minimum particle-to-particle interaction,

142
Yuanxin Zhou, Hassan Mahfuz, Vijaya Rangari et al.
which resulted in increase in thermal stability of the system. These results are consistent with
the DSC results as well.
o
Deriv. of Weight (%/ C)
o
Deriv. of Weight (%/ C)
0.5
0.4
0.3
0.2
0.1
0.0
0.8
0.6
0.4
0.2
0.0
390 C
o
200 300 400 500 600
Temperature( oC)
Figure 14. TGA graph of neat prepreg system.
397 C
o
200 300 400 500 600
Temperature( oC)
Figure 15. TGA graph of 1.5 wt.% SiC nano-phased prepreg system.
110
100
90
80
70
110
100
90
80
70
Weight (%)
Weight (%)

An Experimental and Analytical Study of Unidirectional Carbon Fiber… 143
Flexure Response of Layered Composites
A typical flexure stress-strain plot of neat and nano-phased laminates is shown in Figure 16.
The curves show considerable non-linear deformation and the irregularities in the curves were
attributed to random fiber breakage with pinging noise during the test. The specimens failed
rapidly after reaching the point of maximum stress. In general, the composites exhibited
brittle-type failure. The curves also revealed that by infusing 1.5 wt.% SiC nanoparticles,
strength and modulus significantly improved. Nano-phased system showed approximately
32% increase in flexural strength and 20% in modulus when compared to the neat ones as
shown in Table 2. This result was as expected because of the strong bonding between filler
particles and matrix in nano-phased specimen which resulted in the static adhesion strength as
well as the interfacial stiffness to transfer stresses and elastic deformation efficiently from the
matrix to the fillers via the interface. In other words, large contact areas which translated into
high interfacial stiffness and homogeneous dispersion of nanoparticles assisted in an efficient
stress transfer between polymer and nanoparticles which lead the particles to carry a part of
the external load and resulted in improved flexural strength and stiffness. In addition, the
nanoparticles may have acted as stoppers to crack growth by pinning the cracks. It is also
observed for the nanocomposites in the present study that the strain-to-break tends rather to
slightly higher values in comparison with the neat systems. This increase suggests that the
nanoparticles are able to introduce additional mechanisms of failure and energy consumption
without blocking matrix deformation. Standard deviation for the set of neat and nano-phased
system experimental data were shown to have lower values (Table 3). Lower standard
deviation indicated the stability and consistency in the results as well.
Figure 16. Engineering Stress-Strain curves of flexure test.

144
Material
Yuanxin Zhou, Hassan Mahfuz, Vijaya Rangari et al.
Table 2. Flexure test data for carbon prepreg laminates
Flexural
Strength
(MPa)
Average
Strength
(MPa)
Gain/Loss
Strength
(%)
Flexural
Modulus
(GPa)
Neat Sample 1 765.32 82.14
Neat Sample 2 806.46 80.43
Neat Sample 3 789.38 787.45 -- 82.05
Neat Sample 4 782.62 80.29
Neat Sample 5 793.45
76.91
1.5wt% Sample1 1043.35 93.93
1.5wt% Sample2 995.28 97.02
1.5wt% Sample3 1012.59 1042.34 +32.37 98.25
1.5wt% Sample4 1088.92 90.65
1.5wt% Sample5 1071.55
99.36
Average
Modulus
(GPa)
80.36 --
Gain/Loss
Modulus
(%)
95.84 +19.26
Table 3. Standard deviation and Coefficient of variation of flexure test data
Standard Deviation (±) Co-eff. of Variation (%)
Neat system Strength 13.52 1.72
Neat system Modulus 1.89 2.36
+1.5wt% Nano-phased system Strength 34.99 3.36
+1.5wt% Nano-phased system Modulus 3.17 3.31
Tensile Response of Layered Composites
Typical curves for the tensile behavior of both neat and 1.5 wt.% nano-phased specimen are
shown in Figure 17. The in-plane tensile behavior of both the composites shows linear
behavior up to approximately 1.2% strain where initial fiber failure occurred. The behavior
continued to be linear again till the final specimen failure. Both the elastic modulus and the
strength of nano-phased composites were between 7-10% higher than their neat counterparts.
The reason for such small improvement could be visualized in the sense that, in tension the
fiber took maximum load and the nanoparticle infusion in the matrix did not contribute much
in improving the tensile properties. The improvement of modulus in this study was mainly
because of the improvement of the matrix modulus by filler dispersion. Therefore it can be
deduced that higher tensile properties in the nanocomposite is due to higher nano-phased
matrix properties. Average mechanical properties and their deviation are shown in Table 4
and Table 5, respectively.

Material
An Experimental and Analytical Study of Unidirectional Carbon Fiber… 145
Figure 17. Engineering Stress-Strain curves of tensile test.
Table 4. Tensile test data for carbon prepreg laminates
Tensile
Strength
(GPa)
Average
Strength
(GPa)
Gain/Loss
Strength
(%)
Tensile
Modulus
(GPa)
Neat Sample 1 1.32 86.44
Neat Sample 2 1.41 84.35
Neat Sample 3 1.39 1.38 -- 85.03
Neat Sample 4 1.42 88.57
Neat Sample 5 1.34
86.61
1.5wt% Sample1 1.48 96.39
1.5wt% Sample2 1.39 90.68
1.5wt% Sample3 1.51 1.48 +7.25 93.13
1.5wt% Sample4 1.49 97.90
1.5wt% Sample5 1.55
94.37
Average
Modulus
(GPa)
86.20 --
Gain/Loss
Modulus
(%)
94.49 +9.62
Table 5. Standard deviation and Coefficient of variation of tensile test data
Standard Deviation
(±)
Neat system Strength 0.04 2.86
Neat system Modulus 1.49 1.69
+1.5wt% Nano-phased system Strength 0.05 3.56
+1.5wt% Nano-phased system Modulus 2.51 2.66
Co-eff. of Variation
(%)

146
SEM Analysis
Yuanxin Zhou, Hassan Mahfuz, Vijaya Rangari et al.
SEM analysis was carried out on JSV 5800 JOEL Scanning Electron Microscope. Specimen
from failed samples of flexure tests were selected, prepared and attached to the sample holder
with a silver paint and coated with gold to avoid charge build-up by the electron. Figures 18
and 19 show the SEM micrographs obtained. It is observed from Figures 18a-18b that most of
the damage was located in the loading zone, including large intra and inter-layer delamination
cracks as well as fiber/bundle failures. The damage and the fracture processes were mainly
due to local shear components. They also show interfiber micro-cracks and delamination
cracks. These micrographs also reveal that the carbon fibers were highly oriented with
uniform resin distribution. Figure 19b shows the SEM micrograph in which SiC nanoparticles
(white dots) are distributed uniformly without agglomeration. Also revealing the size of the
filament to be 8-10 microns in diameter and SiC nanoparticle to be of about 30-40 nm range.
(a) (b)
Figure 18. SEM of failed flexure samples in thickness direction.
(a) (b)
Figure 19. SEM of failed flexure samples.

An Experimental and Analytical Study of Unidirectional Carbon Fiber… 147
Numerical Simulation of Tensile Failure Process of Layered
Composites
The tensile failure of fiber reinforced composite material involves a complicated damage
accumulation process resulting from random fiber breakage, stress transfer form broken to
intact fiber, and interface debonding between the fiber and matrix. It is difficult to analyze
such a complicated probabilistic failure phenomenon precisely by means of analytical
methods. The Monte Carlo simulation technique coupled with a stress analysis method is one
of the most effective tools for understanding the tensile failure process [23-27]. In past
Monte Carlo simulations, a micro-composite unit with a coarse mesh and a few fibers of short
length was always used as the numerical model. In practice, structural composites usually
contain large quantities of fibers, when such micro-composite unit is applied to simulate the
failure process of practical composites, it may result in a lack of statistical effects and
magnification of boundary effects, causing errors in calculations of the stress concentration.
Yuan et al. [28] presented a two-dimensional large-fine numerical micro-composite model
with fine mesh, sufficient fibers and adequate length instead of the aforementioned model and
developed a new Monte Carlo simulation method to study the tensile failure process of
unidirectional composites. Based on the new model and method, the average statistical
evolution of the composites deformation and failure, caused by the accumulation of the
random breakages of large quantities of fibers, matrices and interfaces, is successfully
simulated. By taking account of the inertial effect, strain-rate effect of components and the
softening effect caused by the thermo-mechanical coupling in the simulation model, the
tensile stress–strain curves of unidirectional fiber reinforced resin matrix composites CFRP
and GFRP at different high strain-rates were successfully predicted, which agree well with the
experimental results [29-31].
All above Monte Carlo simulations were coupled with the classical shear-lag model. It is
assumed that the fibers bear the whole axial load and the matrix only carries the shear stress.
Ochiai et al. [32, 33] proposed a modified shear-lag model, which takes the axial load born by
the matrix into account, to study the stress concentration in the elastic and elastic-plastic
matrix caused by single fiber breakage. In the present study, Monte Carlo numerical
constitutive model according to Ochiai's modified shear-lag model with fine mesh, sufficient
fibers and adequate length was established to study the failure process of unidirectional
layered composites, to predict the mechanical behavior of these composites with the prepreg
epoxy matrix and to study the relationship between the interface and composite strengths.
Model of Composites
Figure 20 shows the large-fine numerical model of unidirectional composite that consists of n
fibers and n+1 matrices. Each fiber or matrix, at the length of L, is composed of m elements
of length Δx=L/m in the longitudinal direction. The cross-section of the fiber and the matrix
are simplified as rectangle, considering that the simplified fiber has the same sheared area as
that of the actual one, we have

148
Yuanxin Zhou, Hassan Mahfuz, Vijaya Rangari et al.
H = πD
/ 2 df = D / 2
dm =
V
1
( −V
f )
df
where H, df and dm are the thickness of composite, the width of fiber and the width of matrix,
respectively (as shown in Figure 20). D and Vf are the diameter and the volume fraction of
fiber. The displacement component at node (i,j) is expressed as u i,
j .
Figure 20. Model of Unidirectional carbon fiber reinforced matrix resin.
The composite is pulled at one end and fixed at the other end. In figure 20, the left
boundary is fixed, and the right moves with a constant speed V, namely
The initial condition is
⎪⎧
u
⎨
⎪⎩ u
k
i,
0
k
i,
m
= 0
= VkΔt
0
u 0 ( 1 ≤ ≤ 2n
+ 1)
, = i j
Constitutive Assumptions
( 1 ≤ i ≤ 2n
+ 1)
( 1 ≤ i
≤ 2n
+
i and ( ≤ j ≤ m)
f
(1)
(2)
0 (3)
It is assumed that the fibers are homogeneous and linear elastic, the fiber strength is described
statically by single Weibull distribution [34]:

An Experimental and Analytical Study of Unidirectional Carbon Fiber… 149
β
⎡ L ⎛ σ ⎞ ⎤
P ( σ ) = exp⎢−
⎜
⎟ ⎥
(4)
⎢ L0
⎣ ⎝σ
0 ⎠ ⎥⎦
where, P is the survive probability of fiber at stress σ , and the σ 0 and β are Weibull scale
L length and reference length of fiber.
In addition, it is assumed that epoxy matrix is homogeneous and linear elastic.
parameter and Weibull shape parameter, L and 0
⎧σ
m = Emε
⎨
⎩τ
m = Gmγ
where, E m and Gm are tensile modulus and shear modulus.
Shear Stress on the Interface
The shear stress at i−1/i,j interface (shown in Figure 21) τ i−1/i,j can be expressed as a function
of the fiber displacement ui,j and the interface displacement ui−1/i,j:
( u − u ) /( df / 2)
τ =
(6a)
i−1 / i,
j G f i,
j i−1
/ i,
j
where G f is the shear modulus of the fiber. τ i−1/i,j can be also expressed as :
( u − u ) / ( dm / 2)
τ i−1 / i,
j = Gm i−1
/ i,
j i−1,
j
(6b)
If the interface does not break, combine (6A) and (6B) to eliminate ui−1/i,j, then we get
when the interface breaks, we have
2GmG
f
τ i−1
/ i,
j =
( ui,
j − ui−1,
j )
(7)
G df + G dm
m
τ = τ
i−
1 / i,
j
where, τ c is the friction between the fiber and the matrix when the matrix cracks or the
interface debonds.
f
c
(5)

150
Governing Equation
For the fiber element
where,
For the matrix
Yuanxin Zhou, Hassan Mahfuz, Vijaya Rangari et al.
Figure 21. Interface shear stress between fiber element and matrix element.
d u
dfE τ τ
(8a)
f
2
i,
j
+ 2
dx
i / i+
1,
j i−1
/ i,
j
( − ) = 0
d u
dmE τ τ
(8b)
m
2
i,
j
+ 2
dx
i / i+
1,
j i / i−1,
j
( − ) = 0
d u
dmE τ (8c)
m
2
1,
j
+ 2
dx
1/
2,
j
( − 0)
= 0
2
d u2n
+ 1,
j
dmEm + ( 0 −τ
2 / 2 1,
) = 0
2
n n+
j
(8d)
dx
Equation 8A-D can be expressed by the governing equation as follow
A i
A dfE
i
f
d
u
2G
G
2
i,
j
2 +
dfGm
m f
+ dmG f
i+
1,
j i,
j i−1,
j
dx
⎛1
+ μ ⎞ ⎛1
+ μ ⎞
= ⎜ ⎟ + ⎜ ⎟( −1)
⎝ 2 ⎠ ⎝ 2 ⎠
i
( u − 2u
+ u ) = 0
( 1
− )
E dm E V
m
μ = =
E df E V
f
m f
f f
(9)

An Experimental and Analytical Study of Unidirectional Carbon Fiber… 151
Finite Difference Method
We define the non-dimensional displacement coordinate as
where
U = u
i,
j i,
j
/ ξ
, X x
ξ =
The Equation 9 can be rewriten as
A
i
d
2
U
dX
= /ξ ,
The second-order derivative at point (i,j) is:
2 k
dU i,j
d X
no element break
element (i,j)~(i,j+1) break
element (i,j-1)~(i,j) break
2
τ i / i+
1,
j = τ i / i+
1,
j
( + )
dfE f G fdm G mdf
2G
G
f m
( U − U + U ) = 0
i,
j
+ 1,
2 2 i+
j i,
j i−1,
j
Substituting Equation (13) into (12), one can obtain:
UL =
⎧
⎨
⎩
⎪⎧
UR
= ⎨
⎪⎩
U
0
ξ /
E f
df
(10)
(11)
(12)
⎧ 1 k
k k
⎪ 2 ( U i,j − 1 − 2U i,j + U i,j + 1)
( )
⎪
Δx
⎪ 4 k k
= ⎨ 2 ( U i,j − U i,j −1)
⎪ 3(
Δx)
⎪ 4 k k
2 ( U i,j + 1 − U i,j )
⎩⎪
3(
Δx
)
(13)
( UL + UR)
+ ( ΔX
)
C
C
2
k Ai
(
2
4
U = i,
j
C1Ai
+ C2C
3 Δ
k
i,
j−1
U , + 1
k
i j
0
element (i,j-1)~(i,j) unbroken
element (i,j-1)~(i,j) broken
( ) 2
X
UD +
C
5
UU )
(14)

152
⎧
UU = ⎨
⎩
⎧
UD = ⎨
⎩
C1
C 2
C3
⎧
= ⎨
⎩
⎧
= ⎨
⎩
*
τ / + 1,
k
i i j
0
*
τ −1/
,
k
i i j
1
2
1
0
0.
75
⎧0
⎪
= ⎨1
⎪
⎩2
Yuanxin Zhou, Hassan Mahfuz, Vijaya Rangari et al.
element (i,j)~(i,j+1) unbroken
element (i,j)~(i,j+1) broken
interface (i,j)~(i+1,j) unbroken
interface (i,j)~(i+1,j) broken
interface (i-1,j)~(i,j) unbroken
interface (i-1,j)~(i,j) broken
element broken
no element broken
element broken
no element broken
both side matrix broken
single side matrix brokes
no matrix broken
C 4
C
5
1 i ≠ 1
= ⎨⎧
⎩0 i = 1
1 i ≠ n
= ⎨⎧
⎩0 i = n
Using the successive over-relaxation, we have
[ ] [ ] ( )[ ] 1
k q k q
k q−
U = U + 1−
λ U
i,
j
λ (15)
i,
j
where, λ is the relaxation factor, which controls the convergence speed of solution, and q is
the times of iteration.
k
k
i,
j
U i,
j is the right side of Equation (14). After U i,
j is obtained, the
stress of the segment of fiber and matrix can be calculated from following expression:

An Experimental and Analytical Study of Unidirectional Carbon Fiber… 153
For fiber element:
For matrix element
U i+
1,
j −U
i,
j
σ i,
j = E f
(16a)
X
U i+
1,
j −U
i,
j
σ i,
j = Em
(16b)
X
The strain and stress of composites were calculated from average stress of all elements.
c
n m n m
1 ⎡
⎤
= ×
( )
( ) ⎢∑∑σ
2i+
1,
j 1−
V f + ∑∑σ
2i,
jV
f
2N
+ 1 M
⎥
⎣ i−0
1 1 1 ⎦
σ (17)
Strength of Fiber Element and the Failure Criterion
VKΔt
ε c =
(18)
L
Strength assignment to the fiber elements
In simulating, the strength of the fiber elements should be predetermined. According to
the Weibull statistical constitutive model the strength of the fiber follow Equation (4). If we
assume L in Equation (4) equal to mesh length Δx, here σΔx can be obtained from the scale
parameter σ0 at experimental length Lo. n×m random array ηi,j, equally distributed in the
range of (0,1), are produced by the computer, and we let
η
i,
j
β
⎡ Δx
⎛ S ⎤
i,
j ⎞
= P ( Δx,
S ) = ⎢−
⎜
⎟
i,
j exp
⎥
(19)
⎢ L0
⎣ ⎝ σ 0 ⎠ ⎥
⎦
From the Equation (19), we can get the strength of fiber element
The failure criterion
The failure criterion of fiber is
i,
j ≥ S i,
j
1
i,
j
⎡ L0
= − ln j σ
⎤ β
( ηi,
) 0
S ⎢ ⎥
(20)
⎣ Δx
⎦
σ fiber element broken (21a)

154
Yuanxin Zhou, Hassan Mahfuz, Vijaya Rangari et al.
σ fiber element unbroken (21b)
i,
j ≤ S i,
j
The failure criterion of interface is
τ ≥ τ
i,
j m
interface broken (22a)
τ ≤ τ
i,
j m
interface unbroken (22b)
where, τm is the ultimate shear stress of matrix.
The failure criterion of matrix is
ε ≥ ε
i,
j m
matrix element broken (23a)
ε ≤ ε
i,
j m
matrix element unbroken (23b)
where, εm is the failure strain of the matrix
Simulation Procedure
Based on the above numerical constitutive model, a computational program is compiled to
simulate the microscopic dynamic failure process of unidirectional composites. The
simulation procedure is illustrated as following:
(A) Randomly assign a statistical strength Si,j (i=2,4,...2n, J=1,2,…m) to the fiber
element, definitely assign the tensile strength to the matrix and the shear strength to
the interface.
(B) Solve Equation (14) by iteration formula (15) using Ui,j k−1 , Ui,j k−2 and the boundary
conditon at time t=kΔt, obtain the displacement field Ui,j k (i=1,2,...2n+1,j=1,2,...m).
(C) Determine whether the element or the interface breakage (or the matrix element
unloading) has happened, if new breakage occurs, take the breakage into account and
repeat steps (B) and (C) until no new breakage occurs, else calculate the apparent
stress σc and strain εc using Equation (17) and (18).
(D) Increase a time step and repeat step (B) and (C) till the composites failure happens.
Here the "composites failure" is defined as the state when the stress σc drops from
σmax to about 50% of σmax.
Experimental Results of Fiber, Matrix and Composite
The statistical parameters of fiber were obtained from tension tests of T700 fiber bundles. The
tensile stress-strain curve of fiber bundle in Figure 22 shows considerable amount of nonlinearity.
The specimen failed gradually after reaching the maximum stress due to the tensile
strength distribution of fibers. Three parameters were determined from each stress-strain

An Experimental and Analytical Study of Unidirectional Carbon Fiber… 155
curve: elastic modulus (E), tensile strength ( σ b ), and failure strain ( ε b ). Elastic modulus or
Young's modulus is the initial slope of the stress-strain curve. Tensile strength is the
maximum stress at the peak load and the strain corresponding to the tensile strength is the
failure strain. The average values of these three properties are:
E = 210GPa
σ = 1.
93GPa
ε = 1.
07%
b
Based on fiber bundles model and statistical theory of fiber strength [34], the Weibull
parameters for tensile strength of carbon fibers also can be obatined:
Stress (GPa)
2.00
1.60
1.20
0.80
0.40
0.00
σ 2.
70GPa
β = 9.
03 L 100mm
0 =
T700 Carbon Fiber
Experimental Results
Simulated Results
0.000 0.005 0.010 0.015 0.020
Strain
Figure 22. Stress strain curve of carbon fiber.
b
0 =

156
Stress (MPa)
120
80
40
0
Yuanxin Zhou, Hassan Mahfuz, Vijaya Rangari et al.
Neat Epoxy
+1.% SiC Nano Particle Filled Epoxy
0.00 0.02 0.04 0.06
Strain (mm/mm)
Figure 23. Stress-strain curves of the epoxy.
Figure 23 shows the stress-strain curves of epoxy and nanophased epoxy. It can be
observed that both the modulus and strength have been improved by filling nano particle into
matrix system. All parameters of fiber and matrix were listed in Table 6.
Table 6. Parameters of T700 fiber and matrix
Material T700 carbon fiber Neat epoxy Nano-phased epoxy
E (GPa) 210 2.45 3.32
G (GPa) 87.5 1.02 1.38
D ( μ m ) 5 -------- --------
Vf (or Vm) 49% 51% 51%
β 9.03 -------- -------σ
0 ( GPa)
2.70 -------- -------σ
b (MPa)
-------- 89 110
τ ( ) -------- 45 55
max GPa

An Experimental and Analytical Study of Unidirectional Carbon Fiber… 157
Simulation Result and Discussion
Figure 24 shows the simulation results of neat and 1.5wt.% SiC nano-phased layered
composites along with the experimental tensile results. It can be observed that the numerical
simulation results agree well with the experimental results. This indicates that the numerical
model, the simulation method and the program are reliable. Simulated results and
experimental results also show that the strength of nanocomposite was about 7-10% higher
than that of the neat one. As already discussed in the previous chapter, it may have been
contributed from the improvement of matrix and interface properties.
Figure 24. Simulated Stress-strain curves of tensile test.
The failure strain of matrix is higher than that of fiber in unidirectional composites, the
fiber element with the lowest strength is first broken. Then, with increasing applied load, the
breakage of fiber elements occurs randomly. On the other hand, high shear stress are
generated in the matrix elements due to the fiber breakages, so that the matrix cracking and
interfacial debonding occur, leading to the final failure of composites. Figure 25 shows the
simulated failure process of composite from the above results. Figure 25a and 25b indicate
the initial fracture occurring at fiber at low stress level. Stress concentration in the matrix

158
Yuanxin Zhou, Hassan Mahfuz, Vijaya Rangari et al.
elements results in matrix cracking and interfacial debonding. One can observe the damage
zone of nano-phased composite is smaller than that of neat system. Figure 25c and 25d show
the failure appearances at about 60% Peak load of the composites. Figure 25e and 25f show
the finial failure appearances of composites. In these figures, the fiber element breakage of
two composite are almost same, but matrix element breakage of nano-phased composite is
smaller than that of neat system.
Figure 25. Continued on next page.
(a) (b)
(c) (d)

An Experimental and Analytical Study of Unidirectional Carbon Fiber… 159
(e) (f)
Figure 25. Simulated tensile failure process of neat and nano-phased composites.
Figure 26. Stress strain curves of CFRP with different interface strengths.

160
Yuanxin Zhou, Hassan Mahfuz, Vijaya Rangari et al.
Figure 27. Relationship between composite’s tensile strength and interface strength.
Interface strength is a very important parameter affecting the strength of composite.
Based on the current numerical model, the effect of interface strength on the tensile failure
process of unidirectional composites has been studied. Figure 26 shows the tensile stress–
strain curves of composite with different interface strength (100, 50, 20 and 10MPa). The
relationship between the strength of composites and interface strength are shown in Figure 27.
From Figure 26 and 27, it can be observed that the tensile strength of the composites
increases with interface strength, and when the value of interface strength is over 50 MPa, the
tensile strength of composites tends to be a constant and not affected by interface strength.
Figure 28 shows the simulated micro damage patterns of the composites with weak interface
and strong interface. Comparing these patterns with the macro stress–strain curves, it is
evident that: when the interface strength gets weaker, the adjacent interfaces get easier to
break after the fibers break, the load transfer along the traverse direction gets more
ineffective, the reinforced function of fiber gets more insufficient, the damage area gets
larger, the damage pattern gets more disordered, the negative effect gets greater, thus the
composite’s strength gets lower; when the interface strength gets stronger, the fiber breakage
propagates throughout the composite along the straight line and the composites strength gets
higher.

Conclusion
An Experimental and Analytical Study of Unidirectional Carbon Fiber… 161
(a) (b)
Figure 28. Simulated micro damage patterns. (a) Weak interface, (b) Strong interface.
Using the solution impregnation and filament winding techniques, an innovative
manufacturing method was introduced to manufacture nano-phased carbon/epoxy prepreg
tapes/tows from a nanoparticle modified epoxy resin system.
It was seen from TGA that the as-prepared nanocomposites were thermally more stable
than their neat counterparts. An improvement of about 7 0 C was noticed. The improvement in
thermal stability was believed to have been caused by increased cross-linking and better
interactions between the epoxy and SiC nanoparticles.
As seen with DSC and DMA, an improvement in glass transition temperature (Tg) of the
nano-phased system was about 5 0 C. The improvement is due to the restricted mobility of the
chain segments in the presence of SiC nanoparticles, as a result of a greater number of crosslinked
polymer chains.
Response of nano SiC infused composites under flexure loading showed significant
improvements in strength as well as stiffness over the neat ones. On an average the increment
in strength and stiffness was 32% and 20% respectively over the neat systems. It is believed
that the higher strength of the SiC systems is attributed to better interfacial bonding and the
resistance offered by the nanoparticles to crack propagation.
In respect of tensile strength and stiffness, the nanocomposite systems offered
improvements between 7 and 10%. It was seen that all the test coupons (neat and nanophased)
failed in the gage length and failure modes were as described in the ASTM standard.
This nominal improvement in the tensile properties was believed to have occurred because in
tension, stresses in fibers were more dominant than in the nano-phased matrix and the
property improvement is merely due to the improved properties of the resin.
Monte Carlo simulation technique has been established to simulate unidirectional layered
neat and nano-phased composites. The simulation results are in agreement with the
experimental results. Tensile failure process was simulated and the damage zone and micro-

162
Yuanxin Zhou, Hassan Mahfuz, Vijaya Rangari et al.
damage patterns were studied. The damage zone for the nano-phased system was found to be
less than that of the neat system. It was also shown that the interface strength played an
important role in the composite’s tensile failure.
Acknowledgements
The authors would like to gratefully acknowledge the support of USACERL through grant
no.:W9132T-07-p-0011 and National Science Foundation.
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In: Composite Materials Research Progress ISBN: 1-60021-994-2
Editor: Lucas P. Durand, pp. 165-207 © 2008 Nova Science Publishers, Inc.
Chapter 5
DAMAGE EVALUATION AND RESIDUAL STRENGTH
PREDICTION OF CFRP LAMINATES BY MEANS
OF ACOUSTIC EMISSION TECHNIQUES
Giangiacomo Minak 1 and Andrea Zucchelli 2
Department of Mechanical Engineering, Alma Mater Studiorum - Università di Bologna,
Viale Risorgimento 2, 40135 Bologna, Italia
Abstract
A new approach that integrates acoustic emission (AE) and the mechanical behaviour of
composite materials is presented. Usually AE information is used to evaluate qualitatively the
damage progression in order to assess the structural integrity of a wide variety of mechanical
elements such as pressure vessels. From the other side, the mechanical information, e.g. the
stress-strain curve, is used to obtain a quantitative description of the material behaviour. In
order to perform a deeper analysis, a function that combines AE and mechanical information
is introduced. In particular, this function depends on the strain energy and on the AE events
energy, and it was used to study the behaviour of CFRP composite laminates in different
applications: (i) to describe the damage progression in tensile and transversal load testing; (ii)
to predict residual tensile strength of transversally loaded laminates (condition that simulates a
low velocity impact).
Introduction
Long fibre reinforced composite laminates are a complex structure at the meso-scale. The
fibres embedded in the matrix constitute the lamina and the overlapping of different laminas
makes the composite laminate. A consequence of this architecture is the complex behaviour
during loading and servicing of components realized by such material, and the multiplicity of
different failure mechanisms that determine the damage progression.
1 E-mail address: giangiacomo.minak@unibo.it
2 E-mail address: a.zucchelli@unibo.it

166
Giangiacomo Minak and Andrea Zucchelli
In particular the prevision of the performances (e.g. stiffness, damping) and the strength
limits (e.g. tensile, compressive and fatigue) of this kind of material is an important task for
the material scientists and engineers.
Numerical models for the composite laminate progressive failure are currently developed
by the researchers for different applications [1-3] .
These models require an experimental validation by means of tests in which damage
progression is monitored in a suitable way.
Nowadays, different techniques are proposed such as electrical resistance [4], fibre Bragg
grating sensors [5], photo-elasticity [6] and acoustic emission [7-9].
On the other hand, residual strength evaluation after fatigue or impact loading is
important for the determination of composite components reliability. In fact, laminate
composite materials have a wide application in light-weight structural members. In particular
fibre-reinforced plastics are increasingly used in airborne structures and the long range
passenger airplanes of the future may include many important parts of the fuselage and
components made with Carbon Fibre Reinforced Plastic (CFRP). This class of materials is
characterized by outstanding strength-to-weight and stiffness-to-weight ratios. Nevertheless
their resistance to accidental damage is an important issue for the designer. In particular,
CFRPs are very susceptible to internal damage caused by transverse loads such as indentation
and impact, while the probability of such loadings occurring during the manufacture, service
or maintenance of composite structures is very high [10]. This lack of resistance to low
velocity and low energy impact damage [11-13] is one of the main obstacles to a more
widespread application of these composite materials, especially in the case of a thermo-set
matrix like epoxy.
A threshold, conventionally located at 20 m/s, divides the impact problems into two
fields, high and low velocity, due to the different types of induced damage [14].
In the low-velocity impact field, a quasi-static loading can simulate the actual behaviour,
since the vibrational effects are negligible [15, 16]. In fact, many researchers [17-23] use
load-displacement histories to compare structural responses from impact and quasi-static tests
and they find that both the dynamic and static responses have corresponding load drops due to
failures in the laminates.
Low velocity and low energy impact damage usually consists of matrix cracking [24, 25]
and delamination [23, 26, 27], while debonding and fibre breaking occur for higher impact
energy values [28, 29].
As said, besides the behaviour of the material during an impact, an issue of great interest
is the evaluation of the post-impact resistance characteristics of CFRP. In fact, damage due to
impact often can be present in the component before it is put into service and loaded.
To detect the damage level present in the laminate and the damaged zone area, several
techniques are used, such as simple visual inspection , C-scan and X-ray. AE event counts are
also utilized to predict the residual tensile strength (RTS) after impact [30].
Due to the importance of delamination, which decreases locally the buckling load, much
effort has been spent in researching the compression after impact (CAI) performances of
composites [31, 32]. Nevertheless tensile [30, 33, 35] and fatigue properties [36, 37] are also
important to predict the component failure.

Damage Evaluation and Residual Strength Prediction of CFRP Laminates … 167
In this chapter after a brief introduction about AE, we define a function of the acoustic
energy and of the strain energy that allowed us to characterize damage in two different load
configurations:
tensile testing, on different types of CFRP laminate specimens;
transversal loading of quasi-isotropic CFRP laminate plates.
Moreover in the second case, the residual tensile strength was related to the AE recorded
during the transversal loading phase.
Acoustic Emission Technique
Nearly every scientist who makes mechanical experimental testing is familiar with the
acoustic emission produced by the material during the loading phase, which sometimes can be
heard simply by naked ears.
In fact, during a material test or in general when a component is subject to external loads,
a rapid stress redistribution can occur due to permanent and irreversible phenomena, caused
by damage mechanisms, such as a matrix crack onset or growth, delamination and fibre
fracture. During this redistribution, part of the strain energy stored in the material is released
in the form of heat and of elastic waves that propagate in the material until they reach the free
surface. These transient elastic waves are commonly detected as acoustic waves.
Some acoustic emission can be also produced by mechanisms different from damage
(such as sliding and friction of two surfaces in contact) and this must be taken into account.
The elastic waves propagating at the component surfaces are detected by means of
piezoelectric devices that convert the mechanical signal into an electrical one.
Even if the AE physical principle is very simple and immediate, the use of this technique
is not so straightforward because the acoustic wave propagation in solids, especially in the
anisotropic ones as CFRP, is quite complicated. Multiple waves that propagate with different
velocities, reflection, refraction, dispersion, and attenuation, may affect the measured signal.
Nevertheless some advantages with respect to other non destructive testing techniques can be
found in the possibility to monitor a large volume of material by means of few sensors able to
locate the damage by triangulation and to make it continuous during real life service. In
reality, the acoustic emission is produced within the material itself once loaded at a level that
produces some form of damage. In this sense, it is not strictly a non destructive testing
method since it is based on passive monitoring of acoustic energy released by the material or
structure itself while under load.
AE is also used to monitor damage onset and progression in laboratory tests [7-9, 38, 39].
The most difficult AE analysis task is the identification of the damage mechanism,
particularly when multiple damage mechanisms are present as in the case of CFRP.
In fact, changes in AE due to propagation in the material and to the measurement system
may mask characteristics that are related to the damage mechanism. If the AE source is
known, as in the case of uniaxial specimens, the quality of AE data can be improved by noise
discrimination and rejection.

168
Giangiacomo Minak and Andrea Zucchelli
Numerous methods have been attempted to identify damage mechanisms from AE data
[40,41] and in general many carefully controlled laboratory experiments are necessary to
develop relationships between measured AE signals and a specific damage mechanism.
The results from AE monitoring have been used in attempts to estimate the residual
strength or life of a structure [34]. Most strength assessments from AE are based on empirical
correlations developed from failure tests on a large number of nominally identical structures
[42].
Even if recently the research on AE, especially as regards composite laminates focused
on modal analysis [43-45] in this chapter we consider classical feature-based (also known as
parametric) AE analysis [38], in which for each acoustic emission a set of meaningful
parameters (shown in figure 1) are detected such as:
- progressive event number
- counts per event
- maximum amplitude within the event
- event duration
- event energy
Figure 1. Acoustic emission parameters.
This approach has been used in composite laminates with different AE interpretations.
Many authors (e.g. Siron and tsuda [40] ) report that fibre breakage produces large amplitude
signals while matrix cracking results in much smaller amplitudes and delamination is thought
to produce medium amplitude signals. Other studies conclude that matrix cracking causes
large amplitude signals while fibre breakage produced low amplitudes [47].
In reality the amplitude depends on a number of factors including the local stress
conditions and the energy released. In fact, for example, in [43] a very small increment of
matrix crack growth produces a much smaller amplitude signal than a large matrix crack.
Moreover, long duration events are attributed to delamination [44].

Damage Evaluation and Residual Strength Prediction of CFRP Laminates … 169
More details on AE methods can be found for example in [46].
Our contribution in this field is the introduction of a novel function able to combine the
acoustic energy released during an event and the strain energy stored in the material in that
moment. This function will be called “sentry” function because it signals important material
damage events during the tests, and its integral is related to the total damage and to the
residual strength as it will be shown in the next paragraph and in the examples.
The Sentry Function
In order to perform a deeper analysis of the laminate behaviour, a function that combines both
the mechanical and acoustic energy information [35, 47] is introduced. This function is
expressed in terms of the logarithm of the ratio between the strain energy (Es) and the
acoustic energy (Ea),
( )
( ) ⎟⎟
x ⎞
x
⎛ Es
f ( x)
= Ln ⎜
(1)
⎝ Ea
⎠
where x is the test driving variable (usually displacement or strain).
The function f(x) takes into account the continuous balancing between the stored strain
energy and the released acoustic energy due to damage. The function f(x) is generally
discontinuous and can be described by the combinations of four types of function, shown in
figure 2: (I) an increasing function PI(x), (II) a sudden drop function PII(x), (III) a constant
function PIII(x) and (IV) a decreasing function PIV(x).
These functions are defined over an “acoustic emission domain” (ΩAE) that correspond to
the displacement range over which the AE events were recorded. For all AE quantities ΩAE
represents the definition domain and outside the function of AE cumulative events,
cumulative counts, events energy and all other quantities related to the AE information are
null.
Dividing the AE domain ΩAE in sub-domain as reported in figure 2 it possible to write the
following relation:
Ω = Ω U Ω U Ω U Ω
(2)
AE AE,
I AE,
II AE,
III AE,
IV
In that condition the function f can be written as follow:
( x)
( x)
( x)
( x)
⎧ PI
⇔ x ∈ ΩAE,
I
⎪
⎪
PII
⇔ x ∈ ΩAE,
II
f = ⎨PIII
⇔ x ∈ ΩAE,
III
(3)
⎪PIV
⇔ x ∈ ΩAE,
IV
⎪
⎪⎩
0 x ∉Ω
AE

170
⎟ ⎞
⎠
s
⎜ ⎛
⎝
=
a
E
E
Ln
f
Giangiacomo Minak and Andrea Zucchelli
PI
Ω AE,I
….
PII
….
Ω AE,II
Ω AE
PIII
Ω AE,III
….
PIV
Ω AE,IV
Displacement
Figure 2. The basic functions P I , P II , P III and P IV, used to describe the function f.
If there is more than one sub-domain ΩAE,k, k∈{I,II,III,IV}, it is possible to write the
following relation:
Ω
n I ⎛
= ⎜
UΩ
⎝ i=
1
n II
n III
n IV
⎞ ⎛ i ⎞ ⎛ i ⎞ ⎛ i ⎞
⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
, I ⎟
U
⎜U
ΩAE,
II ⎟
U
⎜U
ΩAE,
III ⎟
U
⎜U
ΩAE
VI ⎟
(4)
⎠ ⎝ i=
1 ⎠ ⎝ i=
1 ⎠ ⎝ i=
1 ⎠
AE
i
AE
,
where nI, nII, nIII, and nIV are the number sub-domain corresponding to the trend type I,II, III
and IV respectively. Then functions PI,PII, PIII and PIV can be written as follow:
P
P
I
( x)
III
( x)
⎧ PI
⎪
= ⎨
⎪
⎩PI
,
, 1
n I
⎧ P
⎪
= ⎨
⎪
⎩
PIII
( x)
( x)
III,
1
, n III
/ x ∈ Ω
...
/ x ∈Ω
( x)
( x)
1
AE,
I
n I
AE,
I
/ x ∈Ω
...
/ x ∈ Ω
1
AE,
III
n III
AE,
III
P
P
II
( x)
IV
( x)
⎧ P
⎪
= ⎨
⎪
⎩PII
II,
1
, n II
⎧ P
⎪
= ⎨
⎪
⎩
P
( x)
IV,
1
( x)
IV,
n IV
/ x ∈ Ω
( x)
...
/ x ∈ Ω
( x)
1
AE,
II
/ x ∈ Ω
...
n II
AE,
II
/ x ∈ Ω
1
AE,
IV
n IV
AE,
IV
From the physical point of view the part of f(x) characterized by an increasing trend, type
I, represents the strain energy storing phases. The slope of PI,i (x) functions decreases during
(5)

Damage Evaluation and Residual Strength Prediction of CFRP Laminates … 171
the test because the energy stored in the material tends to its limit and the AE cumulate
energy (Ea) increases due to the progression of material damage.
When a significant internal material failure happens there is an instantaneous release of
the stored energy that produces an AE event with a high energy content. This fact is
highlighted by the sudden drops of the function f(x) that can be described by functions of type
II: PII(x).
In figure 3 are reported some examples of combinations of parts of the function f(x), and
in particular in figure 3a it possible to note three trends of type I (PI,1, PI,3 and PI,2)
respectively connected by two functions of type II (PII,1 and PII,2).
f
f
f
PI,1
Displacement
PI,4
Displacement
Displacement
S1+
PI,5
S1-
PI,2
PIII,
PII,1
S2+
S2-
1 PIV,1
PIV,2
PI,3
PII,2
PII,3 PII,4
Figure 3. Examples of compositions of the basic function describing common parts of the function f.
PII,5
A
B
C

172
Giangiacomo Minak and Andrea Zucchelli
During the first phase, PI,1, the material is storing the strain energy, but when an internal
limit is reached a failure happens and the function f presents a first sudden drop (PII,1). After
this first drop the material starts again to store strain energy and it is interesting to notice that
the slope S1- of PI,1, before the event, is equal to the initial lope of PI,2 (S1+). The failure that
caused the first fall in this case does not affect significantly the material integrity (i.e. PII,1 is
not related to an important material modification). So the value of f(x) is reduced due to the
internal energy release, but the material strain energy capability is not compromised.
Different is the case represented by the second fall PII,2 in which the slope before (S2-)
and after (S2+) are different. This means that after PII,2 the material strain energy storing
capability is changed and because of an important material modification. In particular it is
interesting to notice that S2->S2+. After this event it is also reasonable to hypothesize that the
material damage increases. It was also previously observed [47], that typically after one or
two drops of f(x) characterized by the slope variation of PI(x) (S->S+) the next drop is
followed by a function of type III or IV. In figure 3B and figure 3C are reported two possible
situations that happen when the material damage has reached an important level. In the case
reported in figure 3B at the end of the storing energy phase PI,4 there is an instantaneous strain
energy release that causes the falling phase PII,3. The following constant behaviour of f(x),
described by PIII,1, is due to a progressive strain energy storing phase that is superimposed to
an equivalent material damage progression. The next fall PII,4 is followed by a decreasing
function PIV,1. The decreasing function type PIV, is related to the fact that the AE activity is
greater then the material strain energy storing capability: the damage has reached a maximum
and the material has no resources to bear the load. Then the phase PIV,1 in figure 3B indicates
that the material is totally damaged. In figure 3B between PIII,1 and PIV,1 there is a drop
function PII,4 that is due to an important failure inside the material, but this situation is not the
general one. In fact, sometimes it happens that after a constant trend of f(x) there are no more
drops and f(x) gradually decreases. This behaviour is due to a gradual damage progression
inside the material and no important failure events happen. At the opposite, the case
represented in figure 3C is related to a critical event inside the material: the strain energy
storing phase PI,5 is followed by a sudden drop PII,5 and then it follows a decreasing phase
PIV,2. This situation generally happens at the end of a test or, if it happens at the beginning it
reveals the presence of some defects inside the material.
The described analysis shows that the function f(x) can be usefully implemented to
describe the material damage progression because it takes into account both mechanical and
acoustic information. Summarizing we have that the increasing part of f(x) reveals the
material strain energy storing capability, the falls reveal the instantaneous release of the
stored energy due to failures, the constant and decreasing trends of f(x) prelude and describe
an important failure of the material structure. In the following sections two examples
regarding the application of different strategies about the use of the function f(x) to describe
the composite material damage are developed. In the first example the function f(x) is used to
determine the damage progression of five different types of laminates under in plane tensile
loading. In particular considering the function f(x) it was possible to identify the most
important material failure highlighted by the sub function PII,i. Then considering the stressstrain
information limited in the sub-domain ΩAE,I, ΩAE,III and ΩAE,IV, it was determined the
progressive stiffness reduction of the material.
In the first example the local structure of the function f(x) is used to interpret the stressstrain
data in order to determine the material damage model. While in the second example

Damage Evaluation and Residual Strength Prediction of CFRP Laminates … 173
the function f(x) is used to estimate the overall material damage when composite laminate are
subjected to an out-of-plane load. Because of its synthesis capability the function f(x) can be
used to summarize the whole material damage history and in the case of the transversal load
tests the integral of the function f(x), Int(f), over the domain ΩAE was calculated:
Int
⎛ Es
⎞
f = ∫ Ln⎜ ⎟dΩ
(6)
⎝ Ea
⎠
( )
Ω
AE
The values of Int(f) are influenced by the complete trend of f(x) and by the extension of
the AE domain.
Applications
Case study 1: materials and method
The composite laminates used for the tensile tests were different in terms of lay-up
(unidirectional laminates (UD), angle-ply laminates (AP) and quasi-isotropic laminates (QI),
fibre volume percentage and laminates thickness.
The pre-pregs were made by T-300 graphite fibres and epoxy matrix. Specimens were
cured in autoclave then cut by a diamond saw.
The characteristics of these three types of laminates are reported in Table 1. For each type
ten specimens were tested.
Table 1. Types of laminates used for the tensile tests
Laminate type ID Lay-up
Fibre
Volume
(%)
Thickness
(mm)
Unidirectional UD [0°]8 60 1.4
Angle ply AP [±45°]4S 30 2.8
Quasi isotropic
QI1 [0°,90°, ±45°]4S 60 1.4
QI2 [0°, ±45°, 90°]4S 60 1.4
QI3 [0°,90°, ±45°]4S 30 2.8
Specimen dimensions were 250 mm in length and 25 mm in width for all types of
laminates as recommended by ASTM 3039M for AP and QI lay-ups and the gage length was
140 mm, as shown in figure 4A.
Uniaxial tensile tests were done under displacement control using an INSTRON 8032
with a 100 kN load-cell, and speed of 0.05 mm/sec.
In order to reduce the acquisition of spurious acoustic external signals [47] two noise
gates were assembled in a series configuration with the specimens as shown in figure 4B.

174
Giangiacomo Minak and Andrea Zucchelli
During the test, the AE were monitored by a Physical Acoustic Corporation (PAC) PCI-
DSP4 device with two transducers PAC R15 setting up the amplitude threshold at 40 dB.
A
B
140 mm
110 mm
AE transducers
Grips
AE transducers
Specimen
Grips
250 mm
Figure 4. (A) specimen for tensile test and test set-up with AE transducers position, (B) the specimen
fixing grips and external AE noise insulation devices.
25 mm

Damage Evaluation and Residual Strength Prediction of CFRP Laminates … 175
The AE transducers were placed in a linear configuration located at a distance of 110
mm, as shown in figure 4A. The Elastic modulus was measured by means of HBM LY41-
6/350 strain-gauges.
Case study 1: results and discussion
The experimental results analysis and discussion is here organized in two phases: the first
one in which the only classical mechanical information (stress and strain) are considered, and
the second one in which the AE information is analyzed and related to the material
mechanical response.
The stress-strain curves in Figure 5 show the effect of the fibre orientation and volume
percentage and also the small, but not negligible, influence of the plies sequence in the
laminates.
In particular the AP diagram was qualitatively different from the other laminate ones
because of the different failure mechanism that was fibre-dominated for UD and QI and
matrix-dominated for AP.
In fact, the mechanical response of AP laminates with respect to in plane tensile load was
strongly nonlinear and the ultimate stress value was about 60 MPa. The two aspects that
determined the behaviour of AP laminate were the fibre direction (±45°) and the low volume
percentage of fibres. Visual inspection revealed also a marked necking due to the high
percentage of matrix forming the composite and the sliding of fibres in it.
Stress (MPa)
2500
2000
1500
1000
500
UD
QI2
QI1
QI3
AP
0
0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090
Strain
Figure 5. Examples of stress-strain diagram for the different types of laminates.
From the stress-strain diagram it was possible to determine some of the most significant
information regarding the mechanical response of laminates. In particular the elastic modulus
(EX) and the ultimate strength were estimated considering all tested specimens. In table 2 are
summarized the theoretical Elastic modulus, estimated by means of the lamination theory, and

176
Giangiacomo Minak and Andrea Zucchelli
the corresponding experimental values, while in table 3 the ultimate strain and stress values
are reported.
Table 2. Theoretical calculation and experimental measure of Elastic modulus of each
laminate type
Ex (MPa)
Theoretical Experimental
ID M.V. M.V. S.D.
UD 197000 190000 16000
AP 23000 22000 3000
QI1 75000 74000 6000
QI2 75000 76000 3000
QI3 77000 78000 3000
Table 3. Ultimate strain and stress values of tested laminates
Ultimate Strain Ultimate Stress (MPa)
ID M.V. S.D. M.V. S.D.
UD 0.019 0.001 2157 104
AP 0.084 0.008 62 7
QI1 0.012 0.002 571 35
QI2 0.010 0.001 600 44
QI3 0.012 0.001 322 19
The main acoustic parameters that have been considered are: counts per AE event and AE
event energy (Ea). Both parameters have been related by means of double entry diagram with
the stress-strain behaviour. In the following figures 6-10 examples of these diagrams, one for
each type of laminate, are reported.

Damage Evaluation and Residual Strength Prediction of CFRP Laminates … 177
Figure 6. UD laminate type, cumulative counts, AE energy, function f and stress versus displacement.

178
Giangiacomo Minak and Andrea Zucchelli
Figure 7. AP laminate type cumulative counts, AE energy, function f and stress versus displacement.

Damage Evaluation and Residual Strength Prediction of CFRP Laminates … 179
Figure 8. QI1 laminate type, cumulative counts, AE energy, function f and stress versus displacement.

180
Giangiacomo Minak and Andrea Zucchelli
Figure 9. QI2 laminate type, cumulative counts, AE energy, function f and stress versus displacement.

Damage Evaluation and Residual Strength Prediction of CFRP Laminates … 181
Figure 10. QI3 laminate type, cumulative counts, AE energy, function f and stress versus displacement.

182
Giangiacomo Minak and Andrea Zucchelli
From figures 6 to 10 it is possible to observe that the five types of laminates have
different behaviours from the AE point of view. A preliminary observation can be done
considering the AE domain that can be used to identify the Free Failure Domain (FFD), i.e.
the strain domain over which no failures are detected. Considering the ratio between ΩAE and
the strain at rupture it can be seen that the percentages of the FFD over the all strain domain
are the following: 11% in the case of UD laminates, 0.5% in the case of AP laminates, 1% in
the case of QI1 laminates, 30% in the case of QI2 laminates and 1.4% in the case of QI3
laminates. The estimated percentage values of FFD indicate the different attitude to the
damage onset of each type of laminate, and in particular it is interesting to note that the QI2
laminate type is the one that has the greater capability to be strained without significant
damage. On the contrary the AP laminate types are the most sensitive to the applied strain and
reveal an early damage onset, probably due to the high matrix percentage content and to fibre
orientation (±45°).
Considering the diagram of AE event cumulative counts, figures 6A to 10A, it can be
observed that only in the case of AP laminates the slope of the diagram is quite constant
during the all test. For the other laminate types, UD and QI, the cumulative counts reveal an
initial trend with low slope values that progressively increase during the test. Such behaviour
can be interpreted considering the different damage attitude of the laminates and their
structure. In the case of AP laminates the mechanical behaviour was dominated by the matrix
deformation and cracking. The effect of fibres in AP laminates did not influence the material
behaviour and, on the contrary, as observed during experiments at the early stage of tests,
fibres promoted matrix breakage and spalling. Such interpretation of AP laminates behaviour
is also supported by the AE energy diagram, figure 7B, where it is noticeable the presence of
AE events with an energy content (the maximum AE event energy is about 4.0⋅10 -4 J) that is
typical of composite laminate matrix failures [47]. Different behaviour was observed in the
case of UD, QI1 and QI2 laminates. Considering, for example, diagrams of figures 6A, 8A
and 9A, for the UD, QI1 and QI2 laminates respectively, it is possible to note the presence of
a strain domain where the cumulative count rate is quite low. For these laminates, during the
initial test stage no significant failures can be detected and considering also the energy
diagrams, figures 6B, 8B and 9B it is possible to assume that the sources of AE event are
mainly due to matrix cracks onset. Comparing in particular the cumulative counts and energy
diagrams of QI1 and QI2 laminates it is interesting to note that in the case of QI2 the
maximum number of cumulative counts (∼ 3⋅10 5 counts) is lower than the one of QI1
laminate (∼ 2⋅10 6 counts), but, at the same time, the maximum AE event energy of QI2 (∼
1.2⋅10 -3 J) is comparable to the one of QI1 laminate (∼ 2.2⋅10 -3 J). This behaviour can be
understood considering the different delamination strength of the two laminates. In fact, as
reported in [47, 48], delamination is a possible failure mechanism for laminates of type QI1,
and, on the contrary, it is not a typical failure for laminates of type QI2. So in the case of QI1
the maximum number of counts is greater than in the case of QI2 thanks to the contribution of
events caused by inter-laminar fractures and delamination. Nevertheless the maximum AE
energies for the QI1 and QI2 laminates are comparable because the final crisis of both
materials is characterized by a fibre breaking process that determines the release of an AE
event with an high energy content. The behaviour of QI3 laminate is quite different if
compared to QI1 and QI2. In particular the cumulative counts trend, figure 10A, shows a
consistent release of AE events at the early test stage, but the total number of cumulative

Damage Evaluation and Residual Strength Prediction of CFRP Laminates … 183
counts over all test (∼ 1.4⋅10 4 counts) is lower than in the case of QI1 and QI2. Considering
the AE energy diagram of QI3, figure 10B, it is possible to observe that events with a high
energy content happens also in the early-middle stage of the test, differently than in the case
of QI1 and QI2 where events with a high energy content appear only at the final test stage.
All these facts are related to the QI3 ply composition: the matrix volumes of each ply
influences the dominant material failure mode, the matrix cracking, as in the case of AP.
All the previous considerations can be effectively summarized and completed considering
the diagrams of the sentry functions f.
In the case of UD laminate the construction of the function f reveals the initial material
damage, figure 6C, that is highlighted by the sub-function PI,1 and PII,1. This initial damage
can be related to some material internal adjustment and to the onset of some internal cracks
but, as revealed by the following PI,2, such damages do not compromise the material
capability of storing strain energy. The following PIV,1 sub-function reveals that the internal
cracks propagate and that the material storing energy capability is progressively reduced until
a sequence of drops due to splitting (PII,2, PII,3 and PII4) mixed with some strain energy storing
phases (PI,3, PI,4, PI,5) that prelude to the first and most critical drop PII,5. This drop is mainly
related to the free edge delamination and after this a new but small strain energy storing phase
starts: PI,6. During this phase all the laminate plies work as springs in parallel and go on
storing strain energy. After this phase next drops, PII,6 and PII,7, mixed with a slowly
increasing part of f, PI,7, and two constant trends for f, PIII,1 and PIII,2, prelude the final crisis of
the laminate.
Considering the diagram in figure 7C of the AP laminate it can be observed that the
structure of the sentry function is simpler if compared to the UD case. In fact only three subdomains
of strain energy storing phases, PI,1, PI,2 and PII,6, and three drops, PI,1, PI,2 and PI,3,
characterize the structure of the sentry function for the AP laminate reported in figure 10C.
The smooth trend of the sub-function of type I reveal the modest capability of the material to
store the strain energy and this fact is due to the high matrix volume percentage in each ply
and to the fibre orientation.
The initial trend of the sentry function of QI1 is characterized by a first material crisis,
PIV,1, followed by a drop, PII,1. Such behaviour is due to an initial material adjustment and to
some inter-laminar cracks onset that will contribute to delamination process. The next phase
is characterized by two strain energy storing phases, PI,1 and PI,2, respectively followed by a
sudden drop, PI,2, and a decreasing sub-function, PIV,2. In particular the sudden drop and the
decreasing sub-function indicate the onset of the material crisis due to delamination and
transversal cracks. After the sub-function PIV,2 the sentry function is characterized by a
complex combination of sudden drops and constant sub-functions. This behaviour indicates
that the material integrity is compromised and that each ply is progressively damaged until
the final crisis. In this way it is interesting to notice that after the PIV,2 there are four sudden
drops indicating the important crisis of each basic ply type (0°, +45°, -45°, 90°) that
constitutes the original laminate QI1.
In the case of QI2 the sentry function has a different trend with respect to the case of QI1.
In fact at the test beginning damage is not appreciable and a sub-function of type I, PI,1,
indicates a strain energy storing phase. After the first drop, PII,1, a consistent material damage
indicates the reduced strain energy storing capability. Such material damage is mainly due to
a global laminate loss of strength: the absence of delamination contributes to the cracks
distribution and growth inside all the deformed material volume and this creates the condition

184
Giangiacomo Minak and Andrea Zucchelli
for a general crisis of the laminate. In fact the PII,1 is followed by a small series of strain
energy storing phases, PI,2 and PI,3, and a combination of sub-function of type II, III and IV
highlighting a great material damage.
The QI3 is characterized by a sentry function that mixes the behaviour of the studied UD
and AP laminates. At the test beginning is visible a combination of sub-functions of type I
and of type II with a predominant strain energy storing phase. This phase is characterized by
the sequence of the following sub-functions: PI,1, PII,1, PI,2, PII,2 and PI,3. A delamination
failure is revealed by the slope change of f corresponding to PII,3. After this failure a reduced
energy storing capability is revealed by the PI,4 and the following drop PII,4 is due to the
failure of the weakest ply inside the laminate. This first ply crisis is followed by a subfunction
of type I, PI,5, that has a smooth trend due to the previous material damage. The
damage corresponding to the sub-function PII,6 is due to the crisis of one of the stronger plies.
After this laminate crisis the energy storing phase represented by the PI,6 is due to the stresses
redistribution between the undamaged plies that are now working as springs in parallel.
The analysis here described has been used to perform a quantitative estimation of the
laminate damage and in particular the sentry function of each laminate has been used to
perform a discretization of the stress-strain curve. As an example of this analysis we report
the case of UD laminate type.
f
40
35
30
25
20
15
10
5
f Stress-Strain
Drops
0
0
0.000 0.003 0.005 0.008 0.010
Strain
0.013 0.015 0.018 0.020
Figure 11. Stress and f diagram versus strain, the most key drops of f are highlighted by means of gaps
diagram.
The analysis of the sentry function in figure 11 allows the identification of 10 drops on
the basis of which the stress-strain curve was divided in order to calculate the Elastic
modulus.
2500
2000
1500
1000
500
Stress (MPa)

Stress (MPa)
Stress (MPa)
Damage Evaluation and Residual Strength Prediction of CFRP Laminates … 185
2500
2000
1500
1000
500
0
2500
2000
1500
1000
Experimental
Model
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02
Strain
Figure 12. Stress-strain diagram and its discretization according to the key-drops.
500
Stress - Strain Model
Young Modulus
0
0.0E+00
0 0.0025 0.005 0.0075 0.01 0.0125 0.015 0.0175 0.02
Strain
2.0E+05
1.8E+05
1.6E+05
1.4E+05
1.2E+05
1.0E+05
8.0E+04
6.0E+04
4.0E+04
2.0E+04
Figure 13. Discretized stress-strain curve and Elastic modulus according to the key-drops analysis.
By means of a linear regression the Elastic modulus of the stress-strain curve
corresponding to each strain interval was estimated. In figure 12 the discretized stress-strain
curve is reported and overlapped to the original diagram. In figure 13 the discretized stressstrain
curve and Elastic modulus values are reported.
Young modulus (MPa)

186
Giangiacomo Minak and Andrea Zucchelli
Table 4. Lower, upper and intermediate strain values used for the stress-strain curve
discretization, Elastic modulus (Young) and Damage (D)
UD
AP
QI1
QI2
QI3
Low
strain
Up
strain
Mean Values
Ref.
Strain
Young
[Mpa]
0.0000 0.0030 0.0015 191200 0.029
0.0031 0.0097 0.0064 137500 0.302
0.0104 0.0109 0.0106 105653 0.464
0.0120 0.0154 0.0137 88522 0.551
0.0154 0.0190 0.0172 31211 0.842
0.0000 0.0006 0.0003 22865 0.006
0.0012 0.0037 0.0025 6340 0.724
0.0085 0.0137 0.0111 1666 0.928
0.0252 0.0307 0.0280 329 0.986
0.0308 0.0839 0.0573 182 0.992
0.0000 0.0015 0.0008 74300 0.047
0.0015 0.0031 0.0023 71600 0.082
0.0031 0.0040 0.0035 62000 0.205
0.0060 0.0072 0.0066 45175 0.421
0.0081 0.0086 0.0083 32976 0.577
0.0089 0.0115 0.0102 22151 0.716
0.0000 0.0029 0.0015 72100 0.051
0.0034 0.0041 0.0038 67050 0.118
0.0044 0.0075 0.0060 59050 0.223
0.0080 0.0087 0.0084 41022 0.460
0.0087 0.0099 0.0093 28195 0.629
0.0000 0.0003 0.0001 77608 0.005
0.0003 0.0007 0.0005 53487 0.314
0.0008 0.0016 0.0012 45048 0.422
0.0016 0.0035 0.0025 27500 0.647
0.0041 0.0048 0.0045 24973 0.680
0.0050 0.0114 0.0082 20900 0.732
0.0114 0.0120 0.0117 12997 0.833
D

Damage Evaluation and Residual Strength Prediction of CFRP Laminates … 187
It has to be noticed that building up the diagram in figure 13 the Elastic modulus values
have been associated to the mean values of the strain range that have been used to discretize
the stress-strain curve.
Using the calculated values of the Elastic modulus it was also estimated the values of the
damage by means of its conventional definition: D = 1- E(s)/E0, where E0 is the Young
modulus of the undamaged material.
In Table 4 are summarized the strain intervals, low-strain and up-strain, that have been
used to discretize the stress-strain curves of all specimens for each type of laminates, the
corresponding strain mean value to which the estimated Elastic modulus and Damage values
are associated.
Stress (MPa)
2500
2000
1500
1000
500
0
0
0 0.0025 0.005 0.0075 0.01 0.0125 0.015 0.0175 0.02
Strain
Stress - Strain Model
Damage
Figure 14. Discretized stress-strain curve and Damage according to the key-drops analysis.
The information summarized in Table 4 can be usefully implemented in FEA software to
model the considered composite laminate progressive failure behaviour.
In figure 15 are graphically represented the Elastic modulus and the Damage values
plotted considering as strain values the mean values reported in Table 4
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Damage

188
A
B
E (MPa)
D
250000
200000
150000
100000
50000
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Giangiacomo Minak and Andrea Zucchelli
0
0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090
Strain
0.0
0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090
Strain
E
D() s = 1−
E
Figure 15. Trends of (A) Elastic modulus, E, and (B) Damage, D, versus strain for each type of
laminate; for the damage calculation, of each laminate, E 0 is equal to the mean value of the Young
modulus as reported in Table 2 that correspond to the Young modulus of the undamaged laminate.
Details about the damage of each type of laminate are reported in figures 16 to 20.
( s)
0
UD
AP
QI1
QI2
QI3
UD
AP
QI1
QI2
QI3

D
D
Damage Evaluation and Residual Strength Prediction of CFRP Laminates … 189
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Strain
Figure 16. Damage for UD laminate types.
0.0
0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070
Strain
Figure 17. Damage for AP laminate types.
UD
AP

190
D
D
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Giangiacomo Minak and Andrea Zucchelli
0.0
0.000 0.002 0.004 0.006 0.008 0.010 0.012
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Strain
Figure 18. Damage for QI1 laminate types.
0.0
0.000 0.002 0.004 0.006 0.008 0.010 0.012
Strain
Figure 19. Damage for QI2 laminate types.
QI1
QI2

D
Damage Evaluation and Residual Strength Prediction of CFRP Laminates … 191
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014
Strain
Figure 20. Damage for QI3 laminate types.
Case study 2: materials and method
Eighteen graphite/epoxy composite square laminate plates 250x250 mm 2 were studied.
Their thickness was 1.6 mm. They have been made in autoclave from pre-pregs by stacking
eight unidirectional plies with quasi-isotropic orientations [0,90,45,-45]s.
The specimens were placed in a circular clamping fixture with an internal diameter of
200 mm and they were loaded at the centre: an hemispherical hardened steel ball with a radius
of 7 mm was indented in the top centre point of the laminate by means of a servo-hydraulic
Instron 8033 testing machine controlled by an MTS Teststar II system and equipped by a
25kN load cell.
The specimens were loaded monotonically out-of-plane in control of displacement and
the head speed was 0.05 mm/s.
The tests were stopped at three different damage levels, one (Low) corresponding to the
load value of 2 kN, the second (Medium) corresponding to the first load drop in the loaddisplacement
curve and the third (High) to the complete perforation of the plate.
During the test, the AE has been monitored by a Physical Acoustic Corporation (PAC)
PCI-DSP4 device with four transducers PAC R15 setting up the amplitude threshold at 40 dB.
In figure 21a it is possible to see the fixture system equipped with AE piezoelectric
sensors and in figure 21b the complete experimental setup.
After each quasi-static test, the damaged plate was sliced by a diamond saw to obtain
tensile specimens with the same geometry suggested by ASTM D 5766 for open hole testing
of CFRP, a width of 40 mm and a length of 250 mm. The indented zone was in the centre of
these tensile specimens and the whole damaged zone was included in the specimen width.
The damaged zones size was previously identified by means of the localization tool of the
AE system as it is shown in figure 22 for the High damage level.
QI3

192
Giangiacomo Minak and Andrea Zucchelli
Figure 21. (A) fixture system, (B)experimental device.
AE sensors
Cutting directions
40 mm
250 mm
Detected AE
sources
External
fibers direction
Figure 22. Damaged zone area detected by AE emission and tensile specimens cutting directions for the
High damage level.
Some plates were also analyzed by C-Scan and MicroCT and in these cases the value of
damaged area evaluated by AE was confirmed.
Nine plates had the external ply fibres oriented in the direction of the specimen axis and
other nine had the external ply fibres orthogonal to the specimen axis, so that two different

Damage Evaluation and Residual Strength Prediction of CFRP Laminates … 193
stacking sequences were produced, respectively [0,90,45,-45]s and [90,0,45,-45]s , as shown
in figure 22.
Analogous tensile tests were run, for each stacking sequence, on nine undamaged
specimens cut from undamaged zones of the same plates in order to get reference values.
Tensile tests were performed by means of the same servo-hydraulic machine in
displacement control with a head speed of 0.02 mm/s and a gauge length of 150 mm. Also in
the case of tensile tests the AE have been monitored by a Physical Acoustic Corporation
(PAC) PCI-DSP4 device with two transducers PAC R15, setting the amplitude threshold at
40 dB.
Case study 2: Results and Discussion
Transversal load test
In figure 23 are shown macro photos of the loaded side (a) and of the back side (b) of
damaged plates for the three different damage levels.
Figure 23. Damaged zones (inside the dotted circles) for the three damage levels (Low, Medium, High)
on the loaded surface (a) and on the back surface (b).

194
Giangiacomo Minak and Andrea Zucchelli
In the picture of the low damage level lamina, both in the loaded side (L-a) and in the
back side (L-b) the indentation is barely visible by the naked eye. The medium level damage
laminas present a slightly larger mark on the loaded side (M-a) and some fibre breakage with
matrix leakage on the back side (M-b). Finally the highly damaged lamina pictures (H-a) and
(H-b) show fibre failure on both sides.
Figure 24. Fibres failure on the loaded surface for the Low load level: fractures on fibres are evidenced
by the arrows.
Figure 25. Tensile fibre failure on the back surface for the Medium load level: fracture on one fibre is
evidenced by the arrows.

Damage Evaluation and Residual Strength Prediction of CFRP Laminates … 195
Investigating more deeply the loaded side of the low damage level laminate it was
possible to find a number of broken fibres due to local shear [29], as it is shown in the SEM
image of figure 24.
A different failure mode for the fibre is shown in the SEM image of figure 25.
3
2.5
2
1.5
1
0.5
Indentation: Maimum Load (kN)
Load (kN)
B
3
2.5
2
1.5
1
0.5
0
0 2 4 6 8 10 12 14
0
0 2 4 6 8 10 12 14
M
3
2.5
2
1.5
1
0.5
P
0
0 2 4 6 8 10 12 14
Displacement (mm)
Figure 26. Transversal load-displacement curves for the three damage levels.
3.0
2.5
2.0
1.5
1.0
0.5
Load
Energy
0.0
0.0
0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0
Indentation: Maximum displacement (mm)
30.0
25.0
20.0
15.0
10.0
5.0
Indentation: Total Strain Energy
(J)
Figure 27. Maximum displacement, maximum load and total strain energy of the transversal loading
tests.

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Giangiacomo Minak and Andrea Zucchelli
In this case, referred to the back side of a medium damage level lamina, matrix-fibre
debonding is evident and the tensile fibre fracture surfaces are orthogonal to their axis and
widely separated.
Load-displacement curves for the three damage levels are reported in figure 26 while the
maximum load, maximum displacements and total strain energy recorded in each test can be
found in figure 27.
Acoustic emission analysis of the transversal load test
A parametric AE analysis was performed considering the acoustic energy, the cumulative
events and the cumulative counts per event. As an example, in figure 28 these three AE
parameters are plotted together with the respective load-displacement diagrams.
For the AE diagrams reported in figure 28, the ΩAE is defined in the displacement range
[0.7; 11.1] mm. It is interesting to notice the presence of the FFD that represents a phase of
the test during which no damages are induced in the laminate (i.e. [0; 0.7]mm). The diagrams
of cumulative events and cumulative counts versus the displacement are characterized by a
general monotonic increasing trend, a quite similar shape and, for each diagram, at the same
displacement value there are significant slope variations. In particular from figure 28A and
figure 28B, inside the ΩAE, it is possible to notice a first part of the cumulative events and
counts diagram characterized by low values and a smooth trend and dominated by AE events
with a low number of counts and a low AE event rate. In figure 28 this first test phase has
been identified in an AE sub-domain marked as Z1 limited, for the considered example, in the
displacement range of [0.7; 6.3] mm. Considering also the energy diagram, figure 28C, it is
possible to observe that only one event in Z1, at the displacement value of 3.6 mm, has an
appreciable acoustic energy (over 3.0 10 -6 J) and from the statistical analysis it was observed
that only 5 events have an energy over 1.0 10 -6 J, that can be considered typical for fibres
breakage in bending. The sub-domain Z1 is physically dominated by material adjustment
(especially fibre alignment) and by matrix deformation and matrix crack onset that are
typically related to AE events with a small number of counts and low energy content [47].
The presence of few AE events with a high value of energy (over 1.0 10 -6 J) can be physically
related to the breakage of some fibres as was previously noticed by the SEM image (figure
23) even if probably the energy content of these events in most cases should be low since
these fibres are not loaded in tension. So during this first test phase (Z1) no significant
damage is induced to the material, as it was noticed during the visual inspection of laminate
surfaces, a small hemispherical mark in the matrix is appreciable (figure 23 L-a & L-b), and
as pointed out by means of the SEM image, figure 24, a certain amount of fibres are broken.
After this first phase there is a considerable increase of the AE activity represented by
increased values of the slope in both cumulative event and counts diagrams. The test phase
characterized by a high AE activity presents two other slope variations that can be used to
define three other sub-domains: Z2 over the displacement range [6.3; 7.2] mm, Z3 over the
displacement range [7.2; 8.1] mm and the Z4 over the displacement range [8.1; 11.1] mm. It is
important to point out, as initially noticed, that all the slope variations in the cumulative event
and counts diagram correspond to the same indenter displacement values. This is mainly
related to the fact that, generally, the damage growth inside the material causes an increase of
the total AE activity (events with an increased rating and an increased number of counts per

Damage Evaluation and Residual Strength Prediction of CFRP Laminates … 197
Event CUM
Count CUM
Eac (J)
1.0E+04
9.0E+03
8.0E+03
7.0E+03
6.0E+03
5.0E+03
4.0E+03
3.0E+03
2.0E+03
1.0E+03
0.0E+00
3.0E+05
2.5E+05
2.0E+05
1.5E+05
1.0E+05
5.0E+04
Event CUM
Load
0 2 4 6 8 10 12 14
4.5E+05
Displacement (mm)
3.0
4.0E+05
3.5E+05
Count CUM
Load
B
2.5
6.0E-04
5.0E-04
4.0E-04
3.0E-04
2.0E-04
1.0E-04
0.0E+00
Z 1 Z 2 Z 3 Z 4
0.0E+00
0.0
9.0E-04
8.0E-04
7.0E-04
0 2 4 6 8 10
Eac Displacement (mm)
Load
12
C
14
3.0
2.5
Ω AE
0 2 4 6 8 10 12 14
Displacement (mm)
Figure 28. Main AE parameter and load diagram versus displacement; (A) cumulate of AE events, (B)
cumulate of AE counts per event, (C) AE energy of each event.
A
3.0
2.5
2.0
1.5
1.0
0.5
0.0
2.0
1.5
1.0
0.5
2.0
1.5
1.0
0.5
0.0
Load (kN)
Load (kN)
Load (kN)

198
Giangiacomo Minak and Andrea Zucchelli
event [47]). The slope variations in all diagrams are associated to events with a high energy
content. In fact, as shown in figure 28C, the transitions between Z1 and Z2 and then between
Z2 and Z3 are determined by two events with an energy content higher then 1.0 10 -6 J
probably caused by fibres breakage. It is also interesting to notice that in the sub-domain Z2
there is a considerable number of events with a high energy content (22 events with an energy
higher then 1.0 10 -6 J) probably related to the breakage of quite large number of fibres. This
fact was also confirmed by the visual and SEM analysis (figure 23 M-a & M-b, figure 25).
Similar considerations can be developed considering the sub-domains Z3 and Z4 with a
number of 27 and 100 events with energy content higher then 1.0 10 -6 J respectively, that
probably reveal the progressive fibres breaking process.
Besides the analysis of the AE information it is interesting to relate the AE diagrams to
the mechanical response of the laminate.
In particular it is possible to notice that both the first zone Z1 and the second zone Z2 end
at the two important load drops. In the sub-domain Z1 the load-displacement diagram has a
monotonic increasing trend with an increasing slope and this is a direct consequence of the
fact that the system stiffness is increased by the transition from the bending to the membrane
behaviour [35]. So this sub-domain characterizes the test phase during which no important
damage is induced and the main part of the mechanical energy is stored in the material as
strain energy, in fact only a small part of the mechanical energy is dissipated by fibres
adjustment or alignments and matrix crack onset.
After the first load drop, the sub-domain Z2 begins, where the load displacement diagram
is again increasing monotonically, but contrary to what is observed in Z1 the slope is
decreasing. This is related to the material damage corresponding to the first load drop. In fact,
as noticed by visual inspection and SEM analysis, after the first load drop the fibres breakage
and the brittle matrix leakage reduce the local resistance of the laminate. So in the subdomain
Z2 the strain energy storing capability of the laminate is reduced if compared with the
laminate behaviour in Z1.
In the sub-domain Z3 the load-displacement diagram is characterized by a monotonic
increasing trend with a consistent decreasing of the slope. After the second load drop
delamination and fibre breakages compromise the local out-of-plane resistance of the
laminate and the energy storing capability is significantly reduced. The third zone ends when
the load reaches a relative maximum value and then it decreases. The sub-domain Z4 is
characterized by a slowly decreasing trend of the load with the total penetration of the
indenter in the laminate and the AE event are mainly caused by the delamination, matrix
cracking and leakage, fibre breaking and bending-pull-out. At the end of the loaddisplacement
diagram there is a new increasing trend due to the contact of the support of the
spherical indenter with the laminate surface.
The physical evidence of the failure modes that happens during the loading history in Z4
can be reconstructed by visual and SEM inspection as shown in figure 23 (H-a& H-b) and
figure 24.
An example of implementation of the function f(x) for this case study is shown in figure
29 where the strain energy (Es), the cumulative AE event energy (Ea), the load and the f(x)
diagrams relative to the same test reported in figure 28 are shown.

Damage Evaluation and Residual Strength Prediction of CFRP Laminates … 199
Es (J)
Ea Cumulate (J)
f
1.8E+01
1.6E+01
1.4E+01
1.2E+01
1.0E+01
8.0E+00
6.0E+00
4.0E+00
2.0E+00
0.0E+00
Es
Load
2.5E-03
0 2 4 6 8 10 12 14
3.0
Displacement (mm)
2.0E-03
Eac CUM
Load
B
2.5
1.5E-03
1.0E-03
5.0E-04
0.0E+00
2.5E+01
2.0E+01
1.5E+01
1.0E+01
5.0E+00
0.0E+00
Z1
0 2 4 6 8 10 12 14
PII,1 PII,2
PI,1 PI,2 PI,3
ΩAE
Displacement (mm)
PIII,1
0 2 4 6 8 10 12 14
Displacement (mm)
f
Load
Figure 29. (A) f general behaviour, (B) example of f diagram for an experimental indentation test.
In figure 29A the sub-domain Z1,Z2, Z3 and Z4, as previously analysed and in figure 29B
the AE domain, ΩAE, are reported. The strain energy diagram, figure 29A, is continuous,
Z2 Z3
PIV,1
PII,3
PII,4
PIII,2
Z4
A
C
3.0
2.5
2.0
1.5
1.0
0.5
0.0
2.0
1.5
1.0
0.5
0.0
3.0
2.5
2.0
1.5
1.0
0.5
0.0
Load (kN)
Load (kN)
Load (kN)

200
Giangiacomo Minak and Andrea Zucchelli
monotonic with an increasing trend and it is characterized by four main parts: the first part
has an exponential trend, the other parts have nearly linear trends with different slopes. The
slope variations coincide to the sub-domain Z2, Z3 and Z4 limits. The cumulative AE event
energy is a discrete function that monotonically increases. The sub-domain previously cited
delimits some of the Ea diagram variations: the first part of Ea is characterized by low values
and a smooth trend (inside Z1), the second domain, Z2, is characterized by an Ea increasing
trend and slope and at the end of Z2 there is an evident gap in the Ea value. This gap, as it is
clear in figure 28C, is due to an AE event with a high energy content and from the mechanical
point of view is related to the second load drop: the material damage has reached a high level.
In the sub-domain Z3 and Z4 the Ea diagram is characterized by an increasing trend but no
particular behaviour can be noticed. On the contrary, it is interesting to notice that in the Z2
domain the Ea trend is increasing with a general slope greater than the Ea slope in Z3 and Z4.
In particular comparing the diagrams in figure 28 and the Ea diagram in figure 29B it is
evident that the AE activity in Z2 is characterized by an increased AE event rate than in Z1
and in Z2 the events have a mean number of counts and a mean energy content higher than the
ones in Z1. This analysis of the AE information indicates that the test phase coinciding to the
sub-domain Z2 is the prelude to the main material crisis.
Considering the f(x) diagram it is possible to notice the presence of three increasing
diagram parts (type I: PI,1; PI,2; PI,3), four falls (diagram of type II: PII,1; PII,2; PII,3; PII,4), two
constant parts (PIII,1; PIII,2) and one decreasing part (PIV,1). The three increasing diagram parts
of f(x) are limited in the sub-domain Z1 indicating that at this first test stage the material has a
moderate attitude in storing the mechanical energy. Analysing the f(x) diagram in the subdomain
Z1 it is possible to note that the first fall PII,1 that connect PI,1 and PI,2 is not related to
an important material failure: there is no slope variation of f(x) before and after the fall PII,1
(S1-=S1+). On the contrary, considering the second fall (PII,2) it is possible to note that the final
slope of PI,2 and the starting slope of PI,3 are different (S2->s2+). This fact is due to the first
important material damage and, as noticed during 28C analysis, it is related to the fibres
breakage. It is worth noting that that the simple analysis of the load-displacement diagram
does not single out this first damage even though it is important because it definitely
influences the material strain energy storing capability and indicates the displacement value at
which the damage significantly begins. The other three sub-domains are characterized by a
general decreasing and constant trends of f(x). In particular in Z2, after the third fall (PII,3), the
f(x) is characterized by an initial constant trend directly followed by a decreasing trend (no
falls connect PIII,1 and PIV,1). This behaviour can be explained by the fact that the cumulated
damage during the test phase in sub-domain Z1 and the first fall is great enough to
compromise the material strain energy storing capability. The fourth fall, PII,4, that follows the
decreasing trend of f(x) (PIV,1) is due to the final material local degradation. It is interesting to
notice that in sub-domains Z3 and Z4 the f(x) is characterized by a constant trend while the
load-displacement diagram reveals the presence of a local load maximum value. So even if
the load diagram indicates a residual stiffness of the laminate the f(x) clearly indicates that the
material damage is definitive. The constant behaviour of f(x) indicates that despite the local
load maximum the material energy release is continuous and great enough to compensate the
material strain energy storing attitude: the mechanical energy propagates the material damage.

Damage Evaluation and Residual Strength Prediction of CFRP Laminates … 201
3.3. Residual Tensile Strength
From the experimental results it was verified that the RTS of the laminates with [0,90,+45,-
45]s lay-up was greater than the [90,0,+45,-45]s one for each damage level. In [30] this result
is put into relation with the greater tensile fibre damage of the ply adjacent to the most
external one on the back side.
In figure 30 the tensile tests results are summarized in terms of RTS and the
correspondent values of the displacement.
RTS tests showed a reduction respect to undamaged specimens [35] even for the barely
visible indentations corresponding with Low damage levels. This is explained by the shear
fibre breakage shown in figure 24. This fibre failure mode, different from the tensile one, is
characterized by low strain energy level and, consequently, low acoustic energy emission.
Nevertheless the study of the function f can be useful to identify this kind of failure and an
example of this analysis was previously cited and it is reported in figure 29 B. In particular
from the diagram in figure 29B the important drop of f at a displacement of 3.5 takes into
account a low value of strain energy stored in the laminate (Es = 1.3 J) and an AE event with
low energy content (Ea = 3.810-6 J). In order to take into account the material damage it is
necessary to evaluate all events that cause loss of structural integrity. Since the function f
amplifies the most important material damage events and it is able consider at the same time
the strain energy storing capability and the released internal energy, its integral was utilized
as a damage indicator. In figure 31 the RTS data are plotted versus the respective values of
Int(f) for each laminate type. In particular it is evident the negative relation between the RTS
and the values of the f integrate, confirming, as presented by other authors using different
damage indicators [49-52], that the variable Int(f) is a reliable instrument to evaluate the
material damage during the indentation process.
To represent mathematically the relations between RTS and the damage indicator many
different approaches are utilized: discontinuous relations are composed by linear[49] or non
linear [50] equations and they present a threshold at the damage indicator, so values of
damage indicator lower to a specified threshold value do not change the RTS that is so equal
to the virgin material tensile strength; on the contrary continuous relations[51, 52] have a
plateau which value is equal to the virgin material tensile strength when the damage indicator
in zero and they have a curvature inversion.
In the present work in order to relate the Int(f) and the RTS a continuous relation was
considered having the following form:
C
BInt ( ( f)
)
RTS Ae −
= (2)
Where the constant A is related to the ultimate load of the virgin material, and the
constant B and C can be obtained by means of a linear regression based on the experimental
data. Implementing the model in (2) to the experimental data it was estimated the following
values for the coefficient of the continuous model:
- A = 39 kN
- laminate configuration [0,90,+45,-45]s: B = 8.8 10 -5 ; C = 2.0 (mm -1 );
- laminate configuration [90,0,+45,-45]s: B = 8.3 10 -7 ; C = 2.8 (mm -1 );

202
Giangiacomo Minak and Andrea Zucchelli
In figure 32 the mathematical continuous models implementing the previous coefficient
are represented by means of the continuous line showing a good fit.
Maximum Displacement (mm)
Maximum Event Cum
14.0
12.0
10.0
8.0
6.0
4.0
2.0
0.0
1.8E+04
1.6E+04
1.4E+04
1.2E+04
1.0E+04
8.0E+03
6.0E+03
4.0E+03
2.0E+03
A
0.0 25.0 50.0 75.0 100.0 125.0 150.0
Int(f) (mm)
A
MAX Event
Max Count
0.0E+00
0.0E+00
0.0 25.0 50.0 75.0 100.0 125.0 150.0
Int(f) (mm)
MAX Displacement
Max Load
3.0
2.5
2.0
1.5
1.0
0.5
0.0
1.2E+06
1.0E+06
8.0E+05
6.0E+05
4.0E+05
2.0E+05
Figure 30. Scatter diagrams of Int(f) and the main mechanical variables (A) and AE parameters (B).
Maximum Load (kN)
Maximum Count Cum

Tensile Test: Load at rupture (kN)
Tensile Residual Strength (kN)
40
35
30
25
20
15
10
5
Damage Evaluation and Residual Strength Prediction of CFRP Laminates … 203
0
0 1 2 3 4 5 6
Tensile Test: Displacement at rupture (mm)
[0,90,+-45]
[90,0,+-45]
Figure 31. Ultimate load and ultimate displacement from the residual strength tensile tests.
45.0
40.0
35.0
30.0
25.0
20.0
15.0
10.0
5.0
[0,90,+-45]
[90,0,+-45]
0.0
0.0 20.0 40.0 60.0 80.0 100.0 120.0 140.0 160.0
Int(f ) (mm)
Figure 32. Tensile residual strength versus the integrate of the function f and the representation of the
respective correlation model for the two damaged laminate configurations [0°,90°.+45°,-45°]s and
[90°,0°.+45°,-45°] s

204
Conclusion
Giangiacomo Minak and Andrea Zucchelli
In this chapter, a new approach to the evaluation of damage progression and of the residual
strength of CFRP was presented.
This approach is based on standard parametric AE and in particular on the acoustic
energy.
A function of the acoustic energy and of the strain energy, called Sentry function, was
introduced and its application was illustrated in the case of:
1) damage progression in tensile testing of different types of CFRP laminates;
2) damage progression and residual strength evaluation in the case of CFRP plates loaded
at the centre.
In the first case, the Sentry function allowed us to single out important material failures
and to calculate the corresponding damage values, while in the second case, after the damage
identification phase, the residual tensile strength was related to the integral of the Sentry
function over the acoustic domain defined in the transversal load test.
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In: Composite Materials Research Progress ISBN: 1-60021-994-2
Editor: Lucas P. Durand, pp. 209-236 © 2008 Nova Science Publishers, Inc.
Chapter 6
RESEARCH DIRECTIONS IN THE FATIGUE TESTING
OF POLYMER COMPOSITES
W. Van Paepegem * , I. De Baere, E. Lamkanfi,
G. Luyckx and J. Degrieck
Dept. of Mechanical Construction and Production, Sint-Pietersnieuwstraat 41,
9000 Gent, Belgium
Abstract
For a long time, fatigue testing of composites was only focused on providing the S-N
fatigue life data. No efforts were made to gather additional data from the same test by using
more advanced instrumentation methods. The development of methods such as digital image
correlation (strain mapping) and optical fibre sensing allows for much better instrumentation,
combined with traditional equipment such as extensometers, thermocouples and resistance
measurement. In addition, validation with finite element simulations of the realistic boundary
conditions and loading conditions in the experimental set-up must maximize the generated
data from one single fatigue test.
This research paper presents a survey of the authors’ recent research activities on fatigue
in polymer composites. For almost ten years now, combined fatigue testing and modelling has
been done on glass and carbon polymer composites with different lay-ups and textile
architectures. This paper wants to prove that a synergetic approach between instrumented
testing, detailed damage inspection and advanced numerical modelling can provide an answer
to the major challenges that are still present in the research on fatigue of composites.
1. Introduction
The research on fatigue in composites in general has been largely inspired by the research on
fatigue in metals. Despite the advantages that this knowledge transfer has provided, it has also
brought about that there is still a widespread belief that the fatigue behaviour of metals and
composites is indeed very similar. As a consequence the aim of most fatigue tests on
* E-mail address: Wim.VanPaepegem@UGent.be

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composites is still to establish the S-N curve for that particular composite. The efforts to
combine such fatigue tests with a variety of online and offline monitoring techniques and
detailed numerical simulations of the experimental boundary conditions and observed
material degradation, are much more limited.
This paper wants to give a general overview of the different types of fatigue tests, the
available online and offline monitoring techniques and the indispensable need of finite
element calculations to understand the outcomes of these tests. As such, it should become
clear that one single experimental fatigue test, if properly instrumented and simulated, can
provide a lot more information about the fatigue behaviour of the tested composite material.
The next paragraphs will discuss:
• the different fatigue test set-ups and related online monitoring techniques,
• the inspection of fatigue damage,
• the finite element simulation of experimental boundary conditions.
2. Fatigue Test Set-ups and Online Monitoring Techniques
In this paragraph, a general overview of the most relevant fatigue test set-ups is given:
(i) tension-tension fatigue, (ii) bending fatigue, and (iii) shear dominated fatigue. The related
online monitoring techniques are discussed and some examples of measurements are briefly
presented.
An elaborate discussion of all types of fatigue testing, including tension-compression
fatigue, biaxial fatigue and torsional fatigue, can be found elsewhere [1].
2.1. Tension-Tension Fatigue
The uni-axial tension-tension fatigue test is the most widely used fatigue test. The coupon
geometry is a parallel-sided specimen, instrumented with tabs. The choice of the tabbing
material differs among the testing laboratories. Some prefer steel or aluminium tabs, but most
of them use glass/epoxy tabs, where the glass reinforcement has a [+45°/-45°]ns stacking
sequence. In most cases, the tabs are straight-sided non-tapered tabs.
A fatigue test is usually conducted with a servo-hydraulic testing machine, equipped with
grips that clamp the specimen. The alignment of the specimen is very important. No bending
loads must be induced in the specimen due to misalignment.
In tension-tension fatigue tests, the stress ratio R (= σmin/σmax) is often chosen to be 0.1.
The test frequency is always chosen as high as possible to limit the duration of the test and
minimize the cost, but the fatigue response of some composites strongly depends on the
frequency (especially in case of fibre-reinforced thermoplastics).
In the international standards, the number of cycles to failure is considered as the main
outcome of the tension-tension fatigue test. Yet it is worth the effort to use online
instrumentation methods.
The most simple and effective online measurement is the axial stiffness evolution. The
axial stiffness can be directly calculated from the axial stress (loadcell) and the axial strain
(extensometer). The axial strain must never be calculated from the axial displacement and the

Research Directions in the Fatigue Testing of Polymer Composites 211
gauge length, because the inevitable slip in the clamps can lead to serious errors in the strain
calculation. Depending on the fibre and matrix type and the stacking sequence, the stiffness
degradation can range from a few percent to several tens of percent [2-7].
If the transverse strain is measured additionally, the Poisson’s ratio νxy can be followed as
well. It has been recently showed by Van Paepegem et al. [8] that the evolution of the
Poisson’s ratio is a very sensitive parameter for fatigue damage. Figure 1 shows the evolution
of the Poisson’s ratio for a [0°/90°]2s unidirectional glass fabric/epoxy composite in tensiontension
fatigue. The νxy – εxx curves in strain-controlled fatigue between 0.0006 and 0.006
show a highly nonlinear behaviour and are upper-bounded by the static degradation of the
Poisson’s ratio.
ν xy [-]
0.20
0.15
0.10
0.05
-0.00
0.000 0.005 0.010 0.015 0.020
-0.05
-0.10
-0.15
-0.20
ν xy versus ε xx for [0°/90°] 2s fatigue test W_090_8
ε xx [-]
static [0°/90°] 2s test IF4
static [0°/90°] 2s test IF6
[0°/90°] 2s fatigue test W_090_8: cycle 600 + 5
[0°/90°] 2s fatigue test W_090_8: cycle 3600 + 5
[0°/90°] 2s fatigue test W_090_8: cycle 37200 + 5
Figure 1. Evolution of the Poisson’s ratio ν xy in function of the longitudinal strain ε xx for a glass/epoxy
[0°/90°] 2s specimen at three chosen intervals in the fatigue test [8].
Another online technique is the use of embedded optical fibre sensors with a Bragg
grating. The Bragg grating is a periodical variation of the optical refractive index that is
written in the core of the glass fibre and is typically a few millimetres in length (Figure 2).
When broadband light is transmitted into the optical fibre, the Bragg grating acts as a
wavelength selective mirror. For each grating only one wavelength, the Bragg wavelength, λB
is reflected with a Full Width at Half Maximum of typically 100 pm, while all other
wavelengths are transmitted. The Bragg wavelength is directly proportional with the period of
the Bragg grating. If the optical fibre sensor is embedded in a composite laminate, the strain
in the loaded laminate will cause the period of the Bragg grating to change, and hence the
value of the reflected Bragg wavelength.

212
The advantages are numerous:
W. Van Paepegem, I. De Baere, E. Lamkanfi et al.
Figure 2. Measurement principle of the optical fibre sensor.
• the measurement is absolute and does not drift in time,
• fibre optic sensors are rugged passive components resulting in a high lifetime
(>20 years) and are insensitive to electromagnetic interference,
• the fibre Bragg grating forms an intrinsic part of the optical fibre and has very
small dimensions which makes it very suitable for embedding in composite plates,
• many fibre Bragg gratings can be multiplexed employing only one optical line so
more sensing points can be read out at the same time.
Doyle et al. [9] experimented on the use of fibre optic sensors for tracking the cure
reaction of a fibre reinforced epoxy, with success. They also successfully demonstrated the
feasibility of these sensors for monitoring the stiffness reduction due to fatigue damage, for
thermosetting matrix.
De Baere et al. [10,11] have shown that the optical fibre sensors also survive the
production process for carbon fabric thermoplastics (both autoclave and compression
moulding) and that the correspondence between the axial strain measurements from the
extensometer and the optical fibre sensor were identical in tension-tension fatigue tests (see
Figure 3). That means that the adhesion of the embedded optical fibre sensor to the
surrounding thermoplastic material is very good.
The accumulation of permanent strain is another important phenomenon to monitor.
Especially in composites with large residual stresses built up during manufacturing, the relief
of thermal stresses due to fatigue cracking can result in accumulation of permanent strain.
There again, optical fibre sensors are very sensitive sensors to measure these permanent
strains. Figure 4 shows the stress-strain curves of intermediate static tensile tests during a
tension-tension fatigue test on a carbon thermoplastic. The accumulation of permanent strain
can be clearly seen.

Research Directions in the Fatigue Testing of Polymer Composites 213
Figure 3. Comparison of the longitudinal strain ε xx measurement from the optical fibre and the
extensometer in a tension-tension fatigue test of a carbon fibre-reinforced thermoplastic [11].
Figure 4. Intermediate static tests in a tension-tension fatigue experiment of a carbon fibre-reinforced
thermoplastic [11].
Resistance measurement is a well-established damage detection technique for
unidirectional carbon composites [12]. For a long time, there has been disagreement between
researchers whether the resistance should increase or decrease when local fibre fractures
occur [13-15]. In a recent series of articles, it has been clearly demonstrated that the
resistance must increase with increasing damage to the fibre yarns, but a lot of researchers
observe a decrease of resistance, due to bad contact of the electrodes.

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Recently, De Baere et al. [16,17] showed that resistance measurement also works very
well for monitoring damage in carbon fabric reinforced thermoplastics under tension-tension
fatigue loading. Current injection has been done with an innovative technique. Behind the
tabs, in the strain-free area of the specimens, the current is injected by means of two rivets at
both ends of the specimen, as shown in Figure 5.
Figure 5. Use of rivets for electrical resistance measurement in carbon fibre composites [17].
Figure 6 shows the evolution of relative resistance change ρ and axial fatigue stress σxx
during fatigue cycles 4025 till 4030.
Figure 6. Correspondence between applied sine wave of stress σ xx and measured resistance in a tensiontension
fatigue test of a carbon fabric/PPS composite [17].
2.2. Bending Fatigue
Uni-axial fatigue experiments in tension/compression are most often used in fatigue research
[18-20] and accepted as a standard fatigue test, while bending fatigue experiments are
scarcely used to study the fatigue behaviour of composites [21-23]. Bending fatigue tests
differ in several aspects:

Research Directions in the Fatigue Testing of Polymer Composites 215
• the bending moment is (piecewise) linear along the length of the specimen (3-point
bending, 4-point bending, cantilever beam bending). Hence stresses, strains and
damage distribution vary along the gauge length of the specimen. On the contrary,
with tension/compression fatigue experiments, the stresses, strains and damage are
assumed to be equal in each cross-section of the specimen,
• due to the continuous stress redistribution, the neutral fibre (as defined in the classic
beam theory) is moving in the cross-section because of changing damage
distributions. Once a small area inside the composite material has moved for example
from the compressive side to the tensile side, the damage behaviour of that area is
altered considerably,
• the finite element implementation of related damage models gives rise to several
complications, because each material point is loaded with a different stress, strain
and possibly stress ratio, so that damage growth can be different for each material
point. In tension/compression fatigue tests, the stress- or strain-amplitude is constant
during fatigue life and differential equations describing decrease of stiffness or
strength, can often be simply integrated over the considered number of loading
cycles,
• smaller forces and larger displacements in bending allow a more slender design of
the fatigue testing facility.
Basically, three types of bending fatigue tests can be distinguished: (i) three-point
bending [24,25], (ii) four-point bending [26], and (iii) cantilever bending [22,27-30]. The
success of these tests for fatigue of polymer composites is quite limited, because the
interpretation of the results is more difficult and in case of stiffness degradation, stress
redistribution across the specimen height comes into play.
Moreover, as long as the bending stiffness of the laminate is high enough (e.g. sandwich
composites), the deflections are small and linear beam theory still applies, but once that the
bending stiffness of the composite decreases (e.g. thin laminates), the deflections are large
and geometric nonlinearities and friction at the roller supports affect the fatigue results.
The authors designed a test set-up for cantilever bending fatigue tests as depicted in
Figure 7.
The power of the motor is transmitted by a V-belt to a second shaft. The second shaft
bears a mechanism with crank and connecting rod, which imposes an alternating
displacement on the hinge (point C in Figure 7) that connects the connecting rod with the
lower clamp of the composite specimen. At the upper end the specimen is clamped (point A
in Figure 7). Hence the sample is loaded as a composite cantilever beam.
A full Wheatstone bridge on the connecting rod is used to measure the force acting on the
composite specimen. Due to the (bending) stiffness degradation of the specimen during
fatigue life, the measured force will gradually decrease as the amplitude umax of the prescribed
displacement remains constant. In order to record the out-of-plane displacement profile, it
was necessary to develop a mechanism to hold the specimen fixed in this state, because
recording the profile while the test keeps running at a frequency of 2.2 Hz, gives rise to some
practical problems. A rotary digital encoder was attached to the second shaft. Its angular
position (relative to a certain reference angle) is directly related with the loading path of the

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composite specimen. The frequency inverter reads the signal from the rotary encoder and can
stop and hold the motor at a predetermined angular position of the encoder.
Figure 7. Test set-up for cantilever bending fatigue [28].
A digital photograph of the out-of-plane displacement profile is taken from the side view.
To enhance the contrast, the edge of the composite specimen has been painted white. An
example of such a digital photograph is given in Figure 8. When the number of pixels for a
known distance is counted, the out-of-plane displacement profile can be calculated. Thereto
an edge-detection algorithm is used which detects the edges of the composite specimen on the
digital photograph. Figure 8(right) shows an example of the edge detection algorithm. Of
course, the calculated out-of-plane displacement profile applies to the deformation of the
specimen surface, not to the out-of-plane displacement of the midplane of the laminate.
Figure 8. Use of image processing algorithms to track the out-of-plane displacement profile in
cantilever bending fatigue.

Research Directions in the Fatigue Testing of Polymer Composites 217
2.3. Shear Dominated Fatigue
Fatigue testing in pure shear is very difficult. Lessard et al. [31] modified the static three-rail
shear test (ASTM D 4255/D 4255M – 01) to do fatigue testing on carbon/epoxy plates.
A much more common method are the tension-tension fatigue tests on a [+45°/-45°]ns
laminate, This test is based on the ASTM D3518/D3518M-94(2001) Standard Test Method
for ‘In-Plane Shear Response of Polymer Matrix Composite Materials by Tensile Test of a
±45° Laminate’. This standard explains how the shear stress-strain curve can be derived from
a static tensile test on a ±45° laminate, by measuring the longitudinal and transverse strain.
The test is also called a bias tension test, because the bias (or cross-grain) direction is the 45°
direction between warp and weft direction in case of fabric reinforced composites.
In both pure shear and shear dominated fatigue, the test frequency is a very important
parameter. The shear stresses can lead to significant autogeneous heating and once the
temperature exceeds the glass transition temperature, the deformations can be very large.
Figure 9 shows the localized yielding of a [(+45°,-45°)]2s carbon fabric/PPS composite in
tension-tension fatigue at 2 Hz. Temperature rises up to 90 °C were measured with a
thermocouple at the top surface.
Figure 9. Localized yielding of a [(+45°,-45°)] 4s carbon fabric/PPS composite in tension-tension fatigue
at 2 Hz.
Shear stress τ 12 [MPa]
60
50
40
30
20
10
Shear stress-strain curve for cyclic [+45°/-45°] 2s test IH2
IH2 cyclic test
IH4 static test
0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07
Shear strain γ12 [-]
Figure 10. Shear stress-strain curve for the cyclic tensile test on a [+45°/-45°] 2s glass/epoxy specimen
and the envelope of the corresponding static test [32].

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A hardly studied phenomenon is the accumulation of permanent strain during shear
dominated fatigue loading. For composite materials with a thermoplastic matrix, creep effects
seem to be dominant, while in case of thermosetting materials, permanent strain is simply
neglected in most reported literature. Moreover, for both types of material, the phenomenon is
not understood quite well.
Van Paepegem et al. [32,33] studied the accumulation of permanent shear strain in
[+45°/-45°]2s glass/epoxy laminates under cyclic loading. They showed that the shear
modulus significantly degrades, but that the accumulation of permanent shear strain is even
more important. Figure 10 shows the accumulation of permanent shear strain in cyclic loading
of unidirectional glass fabric/epoxy composites.
3. Visualization of Fatigue Damage
3.1. Micrographs
The most easy inspection technique is visual inspection. Depending on the difference in
optical refractive index of the matrix and fibre materials, the transparency of the composite
laminate can be very high. Gagel et al. [34] reported an extraordinary high transparency of Eglass
multi-axial non-crimp fabric epoxy laminates. Matrix cracks, voids and inclusions could
be detected easily by transmitted light.
Optical or light microscopy provides a direct path from observations made with the naked
eye, to what is visible at magnifications up to about 1000 × [35]. Fracture surfaces are
embedded in resin and polished before observation. Figure 11 shows a microscopic image of
the damage in a plain weave glass/epoxy composite loaded in bending fatigue [36].
1 mm
Figure 11. Micrograph of the fatigue damage at the clamped end of a composite specimen loaded in
cantilever bending fatigue [36].
3.2. Ultrasonic Inspection
A very common inspection technique for fatigue damage in (textile) composites is
ultrasonics. Ultrasonics can be performed in various modes of operation, but the most
common for fatigue damage detection is the through-transmission (C-scan) technique.

Research Directions in the Fatigue Testing of Polymer Composites 219
Through-transmission ultrasonics basically consists of a transducer for emitting ultrasonic
pulses that is placed at or near one surface and a receiver sensor that is located at the opposite
surface. The technique applies to relatively low frequency sound beams, typically 0.5 MHz to
15 MHz, having a small aperture. The transducer and receiver are coupled to the surfaces or
they are immersed in water together with the composite. The ultrasound waves are attenuated
by defects in the composite and the acoustic attenuation is monitored using the receiver [37].
Figure 12 shows the C-scan of a thermoplastic composite specimen tested in three-point
bending fatigue. The central area is clearly damaged.
Figure 12. C-scan of the central damaged zone in a composite specimen loaded in three-point bending
fatigue.
With classical ultrasonic C-scans, the surface of the object under investigation is scanned
point by point in order to detect and to localise possible defects or possible anomalies. In a Cscan
the transducer is normally kept perpendicular and at a constant distance to the surface of
the object.
A less known but promising technique is the ultrasonic polar scan. With the use of polar
scans we do not aim at the detection and localisation of defects or anomalies, but rather at the
characterisation of the material. Therefore in a polar scan a single representative point of the
object is scanned, under all possible angles θ and ϕ of incidence of the ultrasonic beam, as is
shown in Figure 13. Due to the dimensions of a real ultrasonic beam, a small zone, rather than
a single point of the object is scanned. The distance between transducer and scanned point is
again kept constant, and an acoustic coupling medium, such as water, is used. As is also the
case with classical C-scans, scanning is performed using pulsed signals. Obliquely incident
ultrasonic waves have already been used more or less frequently for purposes of material
characterisation. In each case wave velocities or arrival times of ultrasonic pulses were
measured [38-40]. In a polar scan however, the amplitude of the transmitted beam is
measured. Amplitude measurements are much easier to perform, and can be done with the
most simple ultrasonic apparatus, an advantage for the possible application of the technique in
industrial circumstances.
In the early eighties Van Dreumel and Speijer [41] have shown that ultrasonic polar scans
in principle can visualise in a non-destructive way fibre orientations of the layers in laminates
stacked from unidirectional layers. Unfortunately, after these experiments, polar scans have
been hardly studied or used any more, the reasons for this being mainly the complexity of the
"formation" of a polar scan, and the lack of means at that time for the numerical simulation of
a polar scan.

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Figure 13. Schematic drawing of the polar scan set-up (left) and example of an experimentally
measured polar scan of a unidirectional carbon/epoxy composite [42].
Yet, Maes [43] showed that the recorded polar scans of a glass fabric/epoxy composite
before and after fatigue damage clearly differ, as shown in Figure 14. Due to the degradation
of the elastic properties, the propagation speed of ultrasound in the respective directions has
changed.
Figure 14. Polar scan of a glass fabric/epoxy composite before fatigue loading (left) and after fatigue
loading (right) [43].
3.3. X-ray Micro-tomography
High-resolution 3D X-ray micro-tomography or micro-CT is a relatively new technique
which allows scientists to investigate the internal structure of their samples without actually
opening or cutting them [44]. Without any form of sample preparation, 3D computer models
of the sample and its internal features can be produced with this technique. In order to
perform tomography, digital radiographs of the sample are made from different orientations
by rotating the sample along the scan axis from 0 to 360 degrees. After collecting all the

Research Directions in the Fatigue Testing of Polymer Composites 221
projection data, the reconstruction process is producing 2D horizontal cross-sections of the
scanned sample. These 2D images can then be rendered into 3D models, which enable to
virtually look into the object.
Figure 15 shows the micro-tomography images of a fatigue damaged 5-harness satin
weave carbon/PPS (left) and the embedded optical fibre sensors in a carbon thermoplastic
composite (right).
Figure 15. Micro-tomography images of a fatigue damaged carbon/PPS composite (left) and three
composite samples with embedded optical fibre sensor (right) [11].
4. Finite Element Simulation of Experimental Boundary
Conditions
In this paragraph it is shown that finite element simulations should support the experimental
work in order to be able to discriminate between intrinsic material behaviour and induced
effects by (insufficiently understood) experimental boundary conditions.
Four examples are given where the strong correlation between experimental
measurements and finite element simulations is proven.
4.1. Clamping Conditions in Tension-Tension Fatigue
As stated before, the composite coupons for tension-tension fatigue testing are parallel-sided
specimens, instrumented with aluminium or composite tabs. One of the main concerns in
tension-tension fatigue testing of composites is tab failure, i.e. the specimen fails just next to
or inside the tabbing area. Such failures are due to the inevitable stress concentration near the
clamped edges.
In Figure 16, two types of standard tensile machine fixtures are shown, with the
dimensions of the grips.
In order to optimize the shape and length of the tabs, it is important to simulate the stress
state near the clamped region. Therefore a simulation of part of the clamping mechanism has
been done in ABAQUS/Standard v6.6-2.

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Figure 16. Instron TM mechanical grips (left) and Instron TM servohydraulic grips (right).
Figure 17 illustrates the simulated parts in the finite element model. Because of
symmetry, only half of the clamps is modelled, which reduces calculation time. The
corresponding symmetry boundary conditions have been imposed on the specimen. To further
reduce computation time, a rigid body constraint is placed on part of the housing cylinder of
the wedge grips, only the area where the cylinder makes contact with the grip is left
deformable. Furthermore, a part that models the wedge is added, also with a rigid body
constraint to reduce calculation time. The reference point of this part is given a certain
downward displacement. This part represents the hydraulic plunger in the hydraulic clamps.
Figure 17. The simulated clamps in the finite element model [45].

Research Directions in the Fatigue Testing of Polymer Composites 223
Two time steps were implemented: in the first, the wedge was given a downward motion
of 0.75 mm, simulating the pre-stressing of the grips; in the second, the bottom of the
specimen was pulled down over 1 mm, simulating a tensile test.
Contact conditions were imposed between the surfaces of the specimen and the grip, the
grip and the cylinder and the grip and the wedge. Since the grip first follows the movement of
the wedge and then the movement of the specimen, the slave surfaces of all contact conditions
mentioned, were placed on the grips. Between specimen and grip, the tangential behaviour
‘rough’ was implemented, which means that no slip occurs once nodes make contact. For the
other contact conditions, the ‘lagrange’ condition was used, which means that the tangential
force is μ times the normal force, μ being the friction coefficient. The same friction
coefficient was used for both conditions.
The grip was meshed with a C3D8R element, a linear brick element with reduced
integration, whereas all other parts were meshed with C3D20R, a quadratic brick element
with reduced integration. The C3D8R of the grip is required instead of the C3D20R, since the
slave surfaces require midface nodes and the C3D20R do not have one.
For the grip, the wedge and the cylinder, steel was implemented with a Young’s modulus
of 210000 MPa and a Poisson’s ratio of 0.3. The specimen was modelled in a composite
material with the following elastic properties (Table 1).
Table 1. The implemented engineering constants in the finite-element model for the
specimen.
E11 [MPa] 56000 ν12 [-] 0.033 G12 [MPa] 4175
E22 [MPa] 57000 ν13 [-] 0.3 G13 [MPa] 4175
E33 [MPa] 9000 ν23 [-] 0.3 G23 [MPa] 4175
In [46], the authors have derived an analytical formula for the clamping setup illustrated
in Figure 18 that describes the interaction between the load F on the specimen, the force RA of
the plunger (see Figure 18) represented by part A and the contact force P on the specimen.
Figure 18. Symbolic representation of the gripping principle of a clamp [45].

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The following equation was derived, with μij the coefficient of friction between parts i
and j (i,j=A,B,C):
F cosα −μBCsin α (1 −μACμBC)cos α −( μBC −μAC
)sinα
P= + RA
2 sinα + μ cosα sinα + μ cosα
BC BC
For both grips displayed in Figure 16, the angle α is set equal to 10 degrees.
In [45] it was shown that the grips in Figure 17 can be replaced by their equivalent
contact pressure, calculated from Equation (1), because the simulated contact pressure with
the finite element model of Figure 17 perfectly corresponds with the analytically calculated
contact pressure.
Next a detailed finite element model of the end tab region has been developed. Figure 19
shows the model of this setup, both mesh and boundary conditions are illustrated.
The specimen is meshed using C3D20R elements using a global element size of 2 mm.
Where stress concentrations were expected, the element size was reduced to 0.5 mm. The
thickness of the specimen was 2.4 mm, which is also the thickness of the tabs, as has already
been mentioned. The material properties for the composite specimen are given in Table 1.
For the boundary conditions, the displacement along the 1 and 2 axis was inhibited for
planes B1 (on top) and B2 (at the bottom), simulating the ‘rough’ boundary condition from the
previous paragraph. Since contraction of the specimen is possible in the 3-direction due to the
Poisson effect, the movement along the 3-axis was allowed for both planes. In order to
prevent movement of the entire sample along the 3 axis, the central line of plane C (at the
back) was fixed.
Figure 19. Illustration of the model for the end tab region [45].
Two time steps were modelled. In the first, the contact pressure p, calculated from
Equation (1), was imposed. In the second, a tensile stress of 600 MPa was applied on surface
A. The exact value of the stress does not matter, since the stress concentration factors are
compared.
(1)

Research Directions in the Fatigue Testing of Polymer Composites 225
In that way, a detailed analysis of the stress concentration factors in the end tab region
was studied for several clamping conditions and end tab geometries. Detailed results can be
found in [45].
4.2. Friction in Single-Sided 3-Point Bending Fatigue
In uni-axial fatigue tests on fibre-reinforced composites, the stress-strain hysteresis loop can
be used as a measure for stiffness degradation and energy dissipation. In case of three-point
bending fatigue tests, the hysteresis loop of the bending force versus midspan displacement
can yield similar information. In this numerical study, it is shown that the shape of the
hysteresis loop can be affected significantly by friction at the supports, especially for large
deflections. As such, the area of the closed hysteresis loop is no longer a measure for energy
dissipation and damage growth.
Figure 20 shows typical hysteresis loops of the bending force versus midspan deflection
at several times during fatigue life for the [90°/0°]2s carbon/PPS laminate. The amplitude of
the midspan deflection was 14.5 mm and the testing frequency was 2.0 Hz. The hysteresis
loops are gone through in clockwise direction (loading – unloading).
The problem treated here, is the typical shape of the hysteresis curve. One would expect
that the dissipated energy during such a hysteresis cycle is used for initiation and propagation
of microscopic fatigue damage, but in the case of this material, ultrasonic C-scans could not
detect any significant fatigue damage in the specimens (apart from the last stage in fatigue
life). Therefore it was assumed that the effect could be induced by friction at the supports,
given the very large midspan displacements for a short span length.
Bending force [N]
Typical hysteresis curves in bending for [90°/0°] 2s carbon/PPS laminate
700
600
500
400
300
200
100
Cycle 1
Cycle 80 000
Cycle 160 000
Cycle 191 000
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Deflection [mm]
Figure 20. Typical hysteresis curves in bending for [90°/0°] 2s carbon/PPS laminate.

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Finite element simulations have been done to prove the hypothesis of friction affecting
the shape of the hysteresis loop.
The simulations have been done with the commercial implicit finite element code
SAMCEF TM . The finite element mesh is shown in Figure 21. Eight layers of composite have
been modelled with isoparametric volumic elements, one element per layer through the
thickness. The end supports and the load striking edge have been modelled as rigid body
cilinders with radius 5 mm. The contact conditions between supports and composite elements
can have a different friction coefficient.
The material is assumed to behave in a linear elastic manner, but the geometric
nonlinearity is taken into account.
Figure 21. SAMCEF TM finite element model of the three-point bending test.
Figure 22. Simulated displacement contours for a three-point bending test on a [90°/0°] 2s carbon/PPS
laminate.
Figure 22 shows the simulated deflection of the [90°/0°]2s specimen for a prescribed midspan
displacement of 14.5 mm (in agreement with the imposed displacement in the three-point bending
fatigue tests).

Bending force [N]
Research Directions in the Fatigue Testing of Polymer Composites 227
700
600
500
400
300
200
100
Simulated hysteresis curves for different friction conditions
μ = 0.0
μ = 0.1
μ = 0.2
μ = 0.3
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Deflection [mm]
Figure 23. Simulated hysteresis curves for a [90°/0°] 2s carbon/PPS laminate with different friction
conditions at the supports: (i) μ = 0.0, (ii) μ = 0.1, (iii) μ = 0.2 and (iv) μ = 0.3.
700
600
500
400
300
200
Bending force [N] Simulated force-displacement history for three-point bending test (μ = 0.3)
100
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Deflection [mm]
Figure 24. Detailed simulation of the force-displacement curve of the [90°/0°] 2s carbon/PPS laminate
for μ = 0.3.
In Figure 23, the simulated hysteresis curves are plotted for different friction conditions.
The complete loading-unloading path has been simulated, where the imposed midspan

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W. Van Paepegem, I. De Baere, E. Lamkanfi et al.
displacement increases from 0.0 to 14.5 mm and decreases back to 0.0 mm. The curve of
bending force versus midspan deflection is shown for four different friction conditions at the
two end supports: (i) μ = 0.0, (ii) μ = 0.1, (iii) μ = 0.2 and (iv) μ = 0.3.
It can be clearly seen that for μ = 0.0, there is no hysteresis. However, the curve is
slightly nonlinear due to the geometric nonlinearity (large deflection). For μ = 0.3, the typical
shape of the hysteresis curve is found back, although no material damage was taken into
account. As a consequence, the shape variation is only due to the friction coefficient.
The simulation for μ = 0.3 has been done again with a very small time step at the
transition from loading to unloading. The effect is even more pronounced, as can be seen in
Figure 24. It is worth to mention that the value of the maximum bending force is in very good
agreement with the experimentally measured one during the three-point bending fatigue tests
(see Figure 20).
Finally, the simulated stress-strain history in Figure 25 for one of the integration points of
the finite element at the tensile side in midspan proves that there is no material hysteresis.
It has been shown that the friction between the composite specimen and the supports was
the predominant cause of this phenomenon. Static bending tests with different support
conditions were performed and three-dimensional finite element analyses were done with
different friction coefficients. These tests confirmed the hypothesis.
Stress σ 11 [MPa]
Simulated stress-strain history for three-point bending test (μ = 0.3)
700
600
500
400
300
200
100
0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
Strain ε11 [%]
Figure 25. Simulated stress-strain history of [90°/0°] 2s carbon/PPS laminate for μ = 0.3.
Therefore, it can be concluded that the information from hysteresis loops in bending
fatigue must be considered very carefully.

Research Directions in the Fatigue Testing of Polymer Composites 229
4.3. Fully-Reversed 3-Point Bending Fatigue
This study investigates whether a three-point bending setup with fully reversed loading can be
used for the validation of (a combination of) damage models for thin composite laminates in
static or fatigue loading conditions.
When fully reversed bending is used, each side of the specimen is successively loaded in
tension as well as in compression. As a result, the material in the beam sees alternating
tension and compression, which makes this setup ideal for the validation of tensioncompression
fatigue models.
If fully reversed bending is considered, some changes must be made to the original threepoint
bending setup in Figure 26.
Figure 26. Single-sided three-point bending test [47].
Figure 27. The central roll and the rotating outer support (left) and the mounted fully reversed threepoint
bending setup [47].
(i) for the central indenter as well as the outer supports, two contact cylinders are
required, one for the upward and one for the downward motion. Since the centre of the
specimen remains horizontal (see Figure 26, right), no additional modifications are needed for
the central indenter; ii) because the specimen rotates at its ends (see Figure 26, right), the
outer supports need to allow for this rotation. Otherwise, this would induce unwanted reaction
forces in the specimen, corrupting the fatigue data. The indenter and used supports for the

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W. Van Paepegem, I. De Baere, E. Lamkanfi et al.
developed fully-reversed bending fatigue set-up are shown in Figure 27 and the total
assembly in Figure 28.
Figure 28. Total assembly of the fully reversed three-point bending setup [47].
The correct modelling of the boundary conditions applied in this fully-reversed threepoint
bending set-up is not straightforward. Below the detailed finite element model is
discussed.
Since the central indenting rolls do not require any rotation, they are modelled by two
separate rolls (Figure 29, left). To reduce calculation time, rigid body conditions are applied
on all areas that do not make contact with the specimen. The element type is the same linear
brick element with reduced integration, C3D8R, as before, the element size is 0.5 mm.
The easiest way to model the rotating support is by modelling it as a single part, which is
depicted in Figure 29, right. The rolls are slightly longer than the width of the specimen, so
that the specimen does not make contact with the connecting part between the two rolls. Extra
partitions are created resulting in a better mesh. The distance between the two rolls is equal to
the thickness of the specimen, the rolls have a diameter of 10, as was the case in the
experimental setup.
Figure 29. The model of the central indenter as two separate rolls (left) and the model of the rotating
support as one part (right) [47].

Research Directions in the Fatigue Testing of Polymer Composites 231
Again there is a rigid-body constraint on all the partitions that do not make contact with
the specimen, in order to save calculation time (Figure 29, left and right). The part is meshed
with C3D8R elements; the element size is also 0.5 mm. The latter is done to assure that the
calculation does not diverge as a result of contact problems.
Figure 30 shows the final model of the fully-reversed three-point bending set-up.
For the boundary conditions of the rotating support, only the rotation of the support
around its ‘natural axis’ is allowed, all other movement is constrained.
For the contact conditions, the slave surface is put on the support and the master surface
is on the specimen. The latter helps the rotating of the support, since normally, the slave
surface follows the movement of the master surface.
The specimen is a beam with dimensions 2.4 mm x 15 mm x 80 mm and it is meshed
with quadratic brick elements with reduced integration, C3D20R. The global size of the
elements is 3 mm. However, in the zones of contact, the size is 1 mm to ensure that no
convergence problems occur due to the contact conditions. The material model is the same
linear elastic model as in paragraph 4.1 (see Table 1).
Figure 30. Illustration of the mesh and the boundary conditions for the three-point bending setup with
rotating supports [47].
With this model, the bending stresses in the experimental fatigue tests could be simulated
very well. More details can be found in [47].
4.4. Cantilever Bending Fatigue
This paragraph deals with the correct modelling of the set-up for cantilever bending fatigue
that was already shown in Figure 7. It is tempting to model the clamped side of the specimen
by a number of fully constraint nodes in the finite element mesh. However, this model
appears to add unwanted stiffness to the specimen and the predicted force is considerably
higher than the measured one.
The calculations were done for a plain weave glass/epoxy specimen. The prescribed
displacement umax (= 34.4 mm) was chosen rather large in order to assess the effect of
geometrical non-linearities. The corresponding maximum bending force for the specimen,

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W. Van Paepegem, I. De Baere, E. Lamkanfi et al.
measured by a strain gauge bridge, was 117.5 Newton at the first loading cycle. Table 2
illustrates the influence of several modelling assumptions. 2-D and 3-D meshes have been
used with “complete fixation” (fixing all nodes in the clamped cross-section) and “clamping
surfaces” (modelling of the clamping plates with prestressing force). Due to symmetry
conditions, the 3-D simulations were performed for the half width of the specimen and they
were indicated in the table as “3-D symmetry models”. All simulations are quasi-static
analyses, except the fourth simulation, which takes into account the inertia forces during
fatigue loading.
From the third and fourth simulation it is confirmed that a quasi-static analysis is
sufficient. Indeed, since the fatigue experiments are performed at a frequency of 2.23 Hz and
the mass of the reciprocating parts is very small due to the limited forces in bending, the
inertia forces are negligible.
Table 2. Comparison of the different finite element models for the bending fatigue setup.
FE model type
No. of
elements
Bending
force [N]
CPU time
2D plane strain, complete fixation 445 155.1 0’17’’
2D plane strain, clamping surfaces 517 141.2 0’46’’
3D symmetry model, complete fixation 1461 139.2 32’45’’
3D symmetry model, complete fixation, inertia forces 1461 138.9 4 h 35’57’’
3D symmetry model, clamping surfaces, no 1765 146.7 21’53’’
geometrical non-linearities
3D symmetry model, clamping surfaces 1765 120.8 40’44’’
Z
Y
X
Figure 31. Finite element model of the bending fatigue setup [36].

Research Directions in the Fatigue Testing of Polymer Composites 233
Figure 31 shows the finite element mesh for a 3-D analysis with full modelling of the
clamped surfaces and the prescribed displacement. The diagonal lines in the left part of the
mesh are used by the SAMCEF preprocessor to indicate the presence of clamping conditions.
Due to the symmetry conditions with respect to the (x,y)-plane, only one half of the specimen
width has to be modelled. The lines in the bottom right part make up the rigid body part
where the prescribed bending displacement is applied.
5. Conclusion
This paper has presented a collection of research efforts in the field of (i) fatigue test set-ups
and related online monitoring techniques, (ii) inspection of fatigue damage and (iii) the finite
element simulation of experimental boundary conditions.
It has been shown that an integrated approach of these three research fields can benefit
the knowledge and insight into the fatigue testing of fibre-reinforced composites.
Acknowledgements
The author W. Van Paepegem gratefully acknowledges his finance through a grant of the
Fund for Scientific Research – Flanders (F.W.O.), and the advice and technical support of the
Ten Cate company. The author I. De Baere is highly indebted to the university research fund
BOF (Bijzonder Onderzoeksfonds UGent) for his research grant.
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In: Composite Materials Research Progress ISBN: 1-60021-994-2
Editor: Lucas P. Durand, pp. 237-256 © 2008 Nova Science Publishers, Inc.
Chapter 7
DAMAGE VARIABLES IN IMPACT TESTING
OF COMPOSITE LAMINATES
Maria Pia Cavatorta and Davide Salvatore Paolino
Mechanical Engineering Department – Politecnico di Torino,
Corso Duca degli Abruzzi, 24 – 10129 Torino (Italy)
Abstract
The Chapter presents an overview of the damage variables proposed in the literature over
the years, including a new variable recently introduced by the Authors to specifically address
the problem of thick laminates subject to repeated impacts. Numerous impact data are used as
a basis to address and comment potentials and limitations of the different variables. Impact
data refer to single impact events as well as repeated impact tests performed on laminates with
different fiber and matrix combinations and various lay-ups. Laminates of different thickness
are considered, ranging from tenths to tens of millimeters.
The analysis shows that some of the variables can indeed be used for assessing the
damage tolerance of the laminate. In single impact tests, results point out the existence of an
energy threshold at about 40-50% of the penetration energy, below which the damage threat is
quite negligible. Other variables are not directly related to the amount of damage induced in
the laminate but rather give an indication of the laminate efficiency of energy absorption.
Introduction
Composite laminates are expected to absorb low velocity impacts either during assembling or
use. Even when the impact damage is barely visible, the incurred micro-damage may have a
significant effect on the laminate strength and durability. The impact energy can be absorbed
at any point of the laminate, well away from the point of impact, and by means of various
laminate level failure mechanisms including front face indentation (indicative of local matrix
crushing and local fiber breakage), interlaminar delamination, back face splitting and fiber
peeling. In the literature [1-11], it is acknowledged that matrix cracking is the first type of
damage introduced during impact; however, the presence of matrix cracks per se does not
significantly change the overall laminate stiffness. Rather, the matrix crack tips may act as

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Maria Pia Cavatorta and Davide Salvatore Paolino
initiation point for delaminations and fiber breaks which may dramatically reduce the local
and or global laminate stiffness thus affecting the load-time response. The literature also
acknowledges that more damaging energy absorption mechanisms (such as delamination,
fiber pull-out, fiber/matrix debonding and fiber fracture) follows matrix cracking and that
they significantly reduce the strength and stiffness of the laminate. Considering the
importance of damage assessment, there have been several attempts in the literature to look
for measurable test quantities that could be correlated to the damage process [6, 12-19].
Under low-velocity impact loading conditions, the time of contact between the impactor
and the target is relative long. Even though vibratory load responses from the composite
sample, the impactor and the specimen supporting fixture are common features of impact
loading history, the load history can still yield important information concerning damage
initiation and growth [20]. Several authors have used the force-time history to compare the
structural response from impact tests: in particular, values of the First Damage Force (FDF)
[14-21], the Delamination Threshold Load (DTL) [6] or Hertzian Failure Load (Ph) [15] as
well as of the Peak Force (Fpeak) [2-3, 17, 22-24] have often been used to rank laminate
performance. Identification of the FDF poses no troubles as it corresponds to the first load
drop which can be detected in the load-time history. However, comparison of laminate
performance on such basis can be risky since the level of laminate damage associated with the
first load drop may be quite different for a given laminate tested under different impact
energies, or for different laminates tested at a given impact energy. In this respect, definition
of the DTL and of the Ph appear more suitable for ranking laminate performance as they are
intended to identify a more specific damage condition, that is the initiation of significant
damage. In the case of the DTL, significant damage is defined as predominately delamination,
while for the Ph energy absorption mechanisms other than matrix cracking are considered.
The DTL and Ph do not necessarily correspond to the first load drop; rather, they are
associated to the load drop at which a significant change in the slope of the forcedisplacement
curve may be detected and which signals a change in laminate stiffness.
Experimental determination of these load thresholds, which are shown to vary with the
laminate thickness to the 3/2 power, may prove helpful for damage tolerant design: no
significant damage threat is associated to impact events for which Fpeak is below the laminate
DTL or Ph. On the contrary, for impact events for which Fpeak is above the laminate DTL or
Ph, a damage threat exists, even if no information can be obtained on the final amount of
cumulative damage that will occur.
Difficulties and possible ambiguities in determining the DTL or the Ph have often led
researchers to use Fpeak instead, considering it as the turning point between rather limited and
more significant forms of damage. In [17,25], Liu suggested that for any composite laminate
there exists a maximum value of Fpeak. When the impact energy is such that Fpeak is below this
maximum value, the laminate suffers indentation and local matrix cracking, whereas when
loaded by the maximum Fpeak significant delamination starts to take place.
The idea of Fpeak as the signaling point of significant damage initiation is the basis of the
dimensionless parameter introduced in 1975 by Beaumont et al. [16] called the Ductility
Index (DuI). The DuI, which is proposed as a useful tool for ranking the impact performance
of different materials under similar testing conditions, is defined as the ratio between the
propagation energy Epropagation and the initiation energy Einitiation and it is given by the
expression:

Damage Variables in Impact Testing of Composite Laminates 239
E
propagation
DuI = (1)
E
initiation
where Einitiation and Epropagation correspond to the energies absorbed before and after Fpeak,
respectively.
The ductility index is small for brittle materials, where most of the energy is absorbed
before Fpeak, and high for ductile materials, where most of the energy is absorbed after Fpeak is
exceeded. The energy absorption mechanisms before Fpeak are crazing and microcracking of
the matrix; whereas, after Fpeak, crack growth is observed via fiber pull-out, fiber/matrix
debonding and fiber fracture [26-29].
Other energy variables have been introduced since the DuI to rank impact performance.
In [12-13] Belingardi and Vadori introduced the Damage Degree (DD) defined as the ratio
between the absorbed energy (Ea) and the impact energy (Ei). Ei is the kinetic energy of the
impactor right before contact takes place and it is indeed the energy introduced into the
specimen. Ea can be calculated from the force-displacement curve as the area surrounded by
the curve in case of closed force-displacement curves (impact event with rebound) or the area
bounded by the force-displacement curve up to a constant level of force and the horizontal
axis in case of open force-displacement curves (impact event with no rebound). Based on the
energy viewpoint, penetration should take place the first time Ea reaches Ei. Therefore the DD
is below one for impact events with rebound while it reaches the value of one in case the
impactor is stopped with no rebound or specimen penetration is achieved. In [13-14], it was
shown that the relationship between the DD and the impact energy increases monotonically
until saturation and a fairly good data interpolation was achieved by a linear regression curve
[14]. A saturation energy level (Esa) was defined as the impact energy at which the DD
regression curve reaches the value of one. This energy threshold is of practical and theoretical
interest since it defines the maximum energy level the laminate can dissipate with no
penetration and by means of internal damage mechanisms only [12]. In synthesis, the DD is
defined as:
In [17,18], Liu proposed a second-order polynomial regression curve to describe the
absorbed energy vs. impact energy curve up to penetration (named by Liu the energy profile):
2
Ea = aEi
+ bEi
+ c
Depending on the laminate under study, the linear term and the constant c can be smaller
than the quadratic term so that equation (2) can be simplified as:
a
2
i
(2)
E ≅ aE
(3)
From the energy profile, Liu was able to define a Penetration Threshold (Pn) (in a series
of “continuous” impacts at increasing impact energies, it represents the first condition of no
impactor rebound and therefore of equality between impact and absorbed energy), and a
Perforation Threshold (Pr) (first condition of laminate complete perforation). Between the
penetration and perforation thresholds, there exists a range, named by Liu “the range of the
penetration process”, in which the impact energy and the absorbed energy are equal to each

240
Maria Pia Cavatorta and Davide Salvatore Paolino
other but which represent different stages of the penetration process with the impactor
moving deeper and deeper into the specimen as the impact energy increases.
Penetration and perforation thresholds increase with thickness, so does the range of the
penetration process. In other words, while for thin laminates the difference between the
penetration and the perforation thresholds can be negligible, for thick laminates the same can
become quite significant. For cross-ply glass-epoxy composite laminates, Liu [17] found:
Pn
= 0.
8t
P
r
0.
0247
where t is the laminate thickness. Equation (4) indicates that for the investigated glass/epoxy
laminates, the penetration threshold is about 80% of the perforation threshold. In case of 3mm
thin laminates, the range of penetration process (Pr–Pn) is less than 2 J. For 6-mm
laminates, a difference of 15J is found, while for 12-mm thick laminates, (Pr –Pn) exceeds
100J and by far can not be neglected.
In addition to identification of the laminate Pn, Pr and range of penetration process, the
energy profile was used by Liu to define a coefficient η, named the Efficiency of Energy
Absorption. The coefficient is defined as the ratio between the area bounded by the
polynomial regression line of equation (2) up to Pn and the horizontal axis and the area of the
rectangular triangle having for hypotenuse the bisector from zero to Pn. The bisector of the
energy profile represents the equal energy between impact and absorption; therefore, the
triangular zone corresponds to the highest energy-area the material can possibly have. As all
materials have an energy absorption capability less than 100%, the regression curve is always
below the bisector. However, the closer the regression curve to the bisector, the higher the
energy absorption capability of the laminate.
An interesting analysis of the energy profile was provided in [19]. By normalizing the
impact energy and the absorbed energy by the laminate Pn, Mian and Quaresimin were able to
obtain a single master curve which proved to work very well when thin laminates were
investigated. A direct consequence of the existence of a master curve is that, when normalized
by the laminate Pn, the efficiency of energy absorption is basically constant for all laminates,
i.e. η varies linearly with the laminate Pn.
The range of penetration process yet remained to be investigated. To this aim, a new
variable, named the Damage Index (DI), was recently introduced by the Authors [30-32]. The
DI definition aroused considering that in the range of the penetration process, the impactor
moves deeper and deeper into the specimen as the impact energy increases. On the contrary,
pure energy variables as the DD by definition saturates to one over the entire penetration
process.
s
≈
0.
8
(4)
MAX
DI = DD
(5)
sQS
The value sMAX in equation (5) refers to the displacement value recorded at the instant
when the force approximately reaches a constant value, in case of impact tests that cause

Damage Variables in Impact Testing of Composite Laminates 241
specimen perforation; while it corresponds to the maximum displacement recorded during the
test below the perforation threshold Pr.
In order to make the DI a non-dimensional quantity, the displacement sMAX was
normalised by the corresponding displacement sQS measured in quasi-static perforation tests.
The choice of normalising by sQS was taken so to define an absolute reference test and leave
apart possible strain-rate effects on the sMAX values. For all the laminates investigated by the
Authors, sMAX of perforation tests was constant regardless of the impact velocity and equal to
the sQS value.
Experimental
Experimental impact tests were performed according to ASTM 3029 standard [33] using an
instrumented free-fall drop dart testing machine. The impactor has a total mass of 20 kg; its
head is hemispherical with a radius of 10 mm. Stainless steel was chosen for its high hardness
and resistance to corrosion. The maximum falling height of the testing machine is 2 m, which
corresponds to a maximum impact energy of 392 J. The drop-weight apparatus was equipped
with a motorized lifting track. By means of a piezoelectric load cell, force-time curves were
acquired. The acceleration history was calculated dividing the force term by the impactor
mass. The displacement was obtained by double integration of the acceleration and thus
force-displacement curves were plotted. By integration of the force-displacement curves,
deformation energy-displacement curves were then obtained. Initial conditions were given
with the time axis having its origin at the time of impact. At time t=0, the dart coordinate is
zero and its initial velocity can be obtained by the well known relationship:
v = 2gΔh
0 (6)
where Δh is defined as the height loss of the centre of mass of the dart with respect to the
reference surface [12]. The impact velocity was also measured by an optoelectronic device.
Agreement between measured and theoretical values was very good.
The collected data were stored after each impact and the impactor was returned to its original
starting height. Using this technique, the chosen impact velocity was consistently obtained in
successive impacts. Because, the target holder was rigidly attached to the frame of the testing
device, the tup struck the specimen each time at the same location.
Square specimen panels, with 100 mm edge, were clamped through rigid plates having a
central hole 76.2 mm in diameter, and fixed to a rigid base to prevent slippage of the
specimen. The clamping system was designed to provide an uniform pressure all over the
clamping area.
Prior to impact tests, a series of quasi-static perforation tests were performed to get
information on the laminate strength characteristics. Specimens were tested using a servohydraulic
machine with maximum loading capacity of 100 kN. The hydraulic actuator was
electronically controlled in order to perform constant velocity tests. Signals of the force
applied by the actuator and of the actuator displacement were acquired in time with an
appropriate sampling rate.
Table 1 reports main characteristics of the laminates used in the study.

242
Maria Pia Cavatorta and Davide Salvatore Paolino
Table 1. Main characteristics of the laminates analyzed in the chapter
Ref. Fiber /Matrix Lay-up
Thickness
[mm]
Acronym
used
[30] Glass/Vinylester [random/02/90/random 12.31 GVP90_12.31
+Polyester
/90/02/random]
[0/90] 15
4.00 GE90s_4.00
[32] Glass/Epoxy
[0/90] 30
8.00 GE90s_8.00
[03 /903] 5
4.00 GE90m_4.00
[03 /903] 10
8.00 GE90m_8.00
[21] Glass/Epoxy [random/-45/+45/02] 2 4.50 GE45_4.50
[0/90] 4
0.35 CE90_0.35
[0/90] 8
0.75 CE90_0.75
[14] Carbon/Epoxy
[0/90] 16
1.55 CE90_1.55
[0/60/-60] 4
0.40 CE60_0.40
[0/60/-60] 8
0.85 CE60_0.85
[0/60/-60] 16
1.75 CE60_1.75
Results for Single Impact Tests
Figures 1-3, 5-6 reports the results obtained for the analyzed laminates. For sake of
comparison among the different laminates, in all graphs the impact energy Ei is divided by the
penetration threshold Pn [34].
F peak/F pen
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
E i/P n
GVP90_12.31
GE90s_8.00
GE90m_8.00
GE45_4.50
GE90s_4.00
GE90m_4.00
CE60_1.75
CE90_1.55
CE60_0.85
CE90_0.75
CE60_0.40
CE90_0.35
1st linear trend
2nd linear trend
polynomial trend
Figure 1. Normalized peak force plotted against non-dimensional impact energy E i/P n. Linear and
polynomial trends.
Figure 1 reports data for Fpeak vs. the non-dimensional impact energy Ei/Pn. Values of
Fpeak are normalized by the peak force Fpen registered for an impact energy equal to Pn.

Damage Variables in Impact Testing of Composite Laminates 243
Considering the two-order difference in laminate thickness, data scattering is fairly limited.
Data for the thinner laminates are the most dispersed and show the lowest values. In [35], the
impact force was shown to depend on the flexibility of the laminate: values of Fpeak decrease
with increasing laminate flexibility.
A general trend for Fpeak can be envisaged. In [36], Found et al. proposed a relationship
between Fpeak and the square root of the impact velocity, i.e. between Fpeak and the impact
energy to the ¼ power. The proposed relationship (dotted curve in Figure 1) well interpolates
the experimental data.
Interpolation by two straight lines also appears rather good, allowing to point out that, for
impact energies above 40%-50% Pn, the rate of increase of Fpeak with increasing impact
energies slows down, with the value of Fpeak approaching an asymptote. The asymptotic trend
of Figure 1 well agrees with the idea of a maximum value for Fpeak [17,25]. In this respect,
data of Figure 1 seems to suggest that no real damage threat is associated to impact events for
which the impact energy is below 40%-50% of the laminate Pn. A concept of impact threshold
energy has been put forward by many researchers [35, 37-39]. This threshold has been
defined as a measure of the ability of a composite laminate to resist initial strength
degradation [35].
DD
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
E i/P n
GVP90_12.31 GE90s_8.00
GE90m_8.00 GE45_4.50
GE90s_4.00 GE90m_4.00
CE60_1.75 CE90_1.55
CE60_0.85 CE90_0.75
CE60_0.40 CE90_0.35
Figure 2. DD values plotted against non-dimensional impact energy E i/P n.
Figure 2 reports data for the DD, which appear fairly more dispersed, apart from the data
points of the three thicker laminates (GVP90_12.31, GE90s_8.00; GE90m_8.00) that are
basically overlapping. As a general rule, it can be said that the DD increases for increasing
impact energies and shows notably higher values for thicker laminates. In this respect, it is
important to note that high values of the DD do not imply severe damage within the
laminates. Indeed, the DD is a measure of the percentage of impact energy absorbed by the
laminate whereas no distinction is made on the absorption mechanisms as it is the case for the
DuI. High values of the absorption energy Ea can indeed be desirable, for example in crash

244
Maria Pia Cavatorta and Davide Salvatore Paolino
events [40]. Strictly speaking, the DD is not a damage variable but rather a point by point
measurement of the laminate efficiency of energy absorption, whereas the coefficient η
proposed by Liu averages the efficiency of energy absorption over a wide range of impact
energies (from very low energies to Pn).
E a/P n
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
GVP90_12.31
GE90s_8.00
GE90m_8.00
GE45_4.50
GE90s_4.00
GE90m_4.00
CE60_1.75
CE90_1.55
CE60_0.85
CE90_0.75
CE60_0.40
CE90_0.35
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
E i/P n
Figure 3. Energy absorption Master Curve.
Figure 3 plots the Master Curve proposed in [19]. Data for the three thicker laminates are
again superimposed and very close to the bisector, meaning that the absorbed energy is about
equal the impact energy (DD almost one). For thin laminates, the difference is more enhanced
pointing out a lower efficiency of energy absorption [3].
As said, no indications can be evinced from Figures 2 and 3 in terms of laminate damage
tolerance. In this respect, definition of the DuI, which differentiates between the nature of
energy absorption mechanisms, appears rather appealing. In its original definition, the DuI is
meant to rank different laminates tested under similar impact conditions, on the basis of a
more fragile or more ductile behavior under impact loading. It is worthwhile noticing that in
the literature the DuI is basically used to rank different laminates at perforation or fracture
(Charpy tests), for which determination of Einitiation and Epropagation poses no trouble. The forcedisplacement
curves are open curves and Einitiation is calculated without ambiguity as the area
bounded by the force-displacement curve up to Fpeak and the horizontal axis, while Epropagation
is calculated as the area bounded by the force-displacement curve from Fpeak to a constant
level of force and the horizontal axis (Figure 4a). In the attempt of proving the DuI against
other damage variables, computation of the DuI was also applied to impact tests with
rebound. When the dart rebounds, the force-displacement curves are closed curves and
computation of Einitiation and Epropagation becomes troublesome as different areas could be
considered. No references of DuI computation in impact tests with rebound were found in the
literature. Considering that in impact events with rebound the energy absorbed by the
laminate is equal to the area bounded by the force-displacement curve, in computing the DuI

Damage Variables in Impact Testing of Composite Laminates 245
it was decided to calculate Einitiation as the area bounded by the force-displacement curve up to
Fpeak and Epropagation as the area bounded by the force-displacement curve after Fpeak so that the
sum of the two energies is equal to the overall energy absorbed by the laminate (Figure 4b).
In this way, only the portion of impact energy in fact dissipated by the laminate was taken
into account and differentiated by the nature of energy absorption mechanisms.
Force [kN]
12
10
8
6
4
2
0
Initiation
Energy
F peak
0 5 10 15 20 25
Displacement [mm]
Propagation
Energy
Figure 4a. An example of force-displacement curve with perforation: filled areas correspond to E initiation
(Initiation Energy) and Epropagation (Propagation Energy).
Force [kN]
30
25
20
15
10
5
0
Initiation
Energy
0 2 4 6 8 10
Displacement [mm]
F peak
Propagation
Energy
Figure 4b. An example of force-displacement curve with rebound: filled areas correspond to E initiation
(Initiation Energy) and Epropagation (Propagation Energy).

246
DuI
2.5
2.0
1.5
1.0
0.5
0.0
Maria Pia Cavatorta and Davide Salvatore Paolino
GVP90_12.31
GE90s_8.00
GE90m_8.00
GE45_4.50
GE90s_4.00
GE90m_4.00
CE60_1.75
CE90_1.55
CE60_0.85
CE90_0.75
CE60_0.40
CE90_0.35
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
E i/P n
Figure 5. DuI values plotted against non-dimensional impact energy E i/P n.
Figure 5 reports data for the DuI plotted against the non-dimensional impact energy Ei/Pn.
By looking at the DuI values at penetration (Ei=Pn), thicker laminates appear to exhibit a
more ductile behavior. However, considering the elevated heterogeneity of the laminates
under study (in terms of type of fiber and matrix, orientation and percentage of fibers as well
as laminate thickness), it is the Authors’ opinion that caution must be taken in ranking
laminate performance. It should also be reminded that for the thicker laminates, penetration
and perforation thresholds do not coincide but are quite distant from each other. Significance
of Figure 5 is to show that, by extending computation of the DuI to impact energies below Pn,
the DuI can be used as a damage variable. In particular, data on Figure 5 show that for impact
energies up to 40% Pn, the amount of Epropagation is almost null meaning that the main energy
absorbing mechanism is matrix cracking. Above 40% Pn, and especially in the case of thick
laminates, the contribution of Epropagation becomes more and more important. This implies that
Fpeak occurs at a value of displacement significantly lower than the maximum displacement
reached by the laminate before dart rebound. As the impact energy increases, contribution of
delamination and of fiber breakage to the energy absorption mechanisms becomes more and
more important.
Figure 6 reports data in terms of the DI. By taking into account the value of the maximum
displacement, the DI is more of a damage variable than the DD was. Indeed, Figure 6 shows
that at very low impact energies the DI is almost null to then increase monotonically for
increasing impact energies. Up to impact energies of about 40-50% Pn, signaled by graphs of
Fpeak and DuI as energy thresholds, the difference in the value of the DI for different
laminates is quite limited. Above this threshold, the difference in DI data for different
laminates increases significantly. Data on Figure 6 also show that DI values do not saturate to
one at Pn, thus allowing to monitor the range of the penetration process. The effectiveness of
the variable in distinguishing between the penetration and the perforation thresholds can be
evinced from Figures 7 and 8 which report DD and DI data for two different laminates.

Damage Variables in Impact Testing of Composite Laminates 247
Interestingly, up to Pn the DI increases linearly with the impact energy to then grow quite
abruptly over the range of the penetration process. A linear relationship also exists between
the DD data and the impact energy; however, the DD data can not be used beyond penetration
as (by definition) the DD stays at the value of one over the entire range of the penetration
process.
DI
DD, DI
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
GVP90_12.31 GE90s_8.00
GE90m_8.00 GE45_4.50
GE90s_4.00 GE90m_4.00
CE60_1.75 CE90_1.55
CE60_0.85 CE90_0.75
CE60_0.40 CE90_0.35
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
E i/P n
Figure 6. DI values plotted against non-dimensional impact energy E i/P n
y = 0.68x + 0.28
R 2 = 0.96
y = 0.52x - 0.14
R 2 = 0.99
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
E i/P n
DD DI
Figure 7. Comparison between DD and DI values for impact tests on glass/epoxy 6.25 mm thick
laminates. Impact data taken from reference [17].

248
DD, DI
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Maria Pia Cavatorta and Davide Salvatore Paolino
y = 0.80x + 0.21
R 2 = 0.90
y = 0.58x - 0.01
R 2 = 0.98
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
E i/P n
DD DI
Figure 8. Comparison between DD and DI values for impact tests on GE45_4.50 laminates.
Results for Repeated Impact Tests
Apart from monitoring the range of the penetration process, the DI has proven to provide
important pieces of information in case of repeated impact tests, a loading conditions of
particular relevance in marine applications [4,30-32,38-39,41-42]. Figures 9-14 report data
obtained on two different laminates (GVP90_12_31, CE90m_4.00) tested under repeated
impacts. The two depicted impact energies were selected to represent tests of no laminate
perforation within 40 impacts and tests of laminate perforation. Figures 9-10 reports data for
Fpeak. As it can be observed, for impact energies that cause no perforation within test duration,
values of Fpeak slightly increase in the first few impacts to then reach an asymptote. On the
contrary, for energies that cause perforation, values of Fpeak decrease impact after impact as a
consequence of damage accumulation. For a given laminate, initial values of Fpeak reported in
Figure 10 are obviously higher than those of Figure 9 due to the higher impact energy used in
the test (Figure 1), while values just before perforation are lower than the asymptotic values
of Figure 9 due to the significant damage induced in the laminate. With respect to the initial
values of Fpeak, it is worthwhile noticing that for the 25 J tests and the 98 J test performed on
the GVP90_12.31 laminate, the maximum in Fpeak is not reached at the first impact. This
effect has already been observed in the literature. In a series of repeated impact tests run on
carbon/epoxy composite laminate, Wyrick and Adams [22] commented the initial increase in
Fpeak as the result of the compaction process of the thin layer of unreinforced resin at the
impacted surface. At low impact energy levels, damage to the fibers near the surface is
minimal and the compaction process provides a harder surface for the next impact. In this
respect, it is worthwhile noticing that this initial increase in Fpeak was observed when the
impact energy was below 40% Pn, once more confirming the existence of an energy threshold
level.

F peak [kN]
16
14
12
10
8
6
4
2
0
Damage Variables in Impact Testing of Composite Laminates 249
1 4 7 10 13 16 19 22 25 28 31 34 37 40
Impact Number
GVP90_12.31
CE90m_4.00
Figure 9. Values of peak force in repeated impact tests performed on GVP90_12.31 and CE90m_4.00
laminates. Impact energy: 25 J.
F peak [kN]
30
25
20
15
10
5
0
1 3 5 7 9 11 13 15 17 19
Impact Number
GVP90_12.31
CE90m_4.00
Figure 10. Values of peak force in repeated impact tests performed on GVP90_12.31 and CE90m_4.00
laminates. Impact energy: 98 J.
Figures 11-12 report data in terms of the DuI. As in Figure 5, for impact tests with
rebound, contribution of Einitiation and Epropagation is calculated according to the definition
illustrated in Figure 4b. Figure 11 shows that for impact energies that cause no perforation,
the DuI is very low and constant throughout the test, signaling no significant damage

250
Maria Pia Cavatorta and Davide Salvatore Paolino
accumulation. For impact energies that cause perforation, the DuI maintains a very low value
up to a few impacts before perforation when it rapidly increases (Figure 12).
DuI
0.10
0.08
0.06
0.04
0.02
0.00
1 4 7 10 13 16 19 22 25 28 31 34 37 40
Impact Number
GVP90_12.31
CE90m_4.00
Figure 11. DuI data for repeated impact tests performed on GVP90_12.31 and CE90m_4.00 laminates.
Impact energy: 24.5 J.
DuI
2.0
1.5
1.0
0.5
0.0
1 3 5 7 9 11 13 15 17 19
Impact Number
GVP90_12.31
CE90m_4.00
Figure 12. DuI data for repeated impact tests performed on GVP90_12.31 and CE90m_4.00 laminates.
Impact energy: 98 J.
Data in terms of DD and DI are reported in Figures 13 and 14. To avoid confusion, data
are organized for single impact energies and single laminates. DD and DI data are plotted
together to favor a comparison between the two variables. For energies that cause no

Damage Variables in Impact Testing of Composite Laminates 251
perforation (Figure 13), the DD data show an initial decrease thus suggesting a reduction in
the percentage of impact energy that the laminate is able to absorb. For the thicker laminate
(GVP90_12.31), this initial reduction is quite significant, going from a percentage of energy
absorption of about 85% in the first impact to a quite stable value of about 70% in subsequent
tests. Visual observation of the laminate after each impact pointed out that in the first few
impacts the impactor indents the laminate and that the size of this indentation does not
significantly change in subsequent tests. Existence of an initial localized damage that does not
appear to significantly grow in subsequent tests is also what is conveyed by the DuI as well as
the DI data, which keep to a constant low level throughout the test.
DD, DI
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
DD DI
1 4 7 10 13 16 19 22 25 28 31 34 37 40
Impact Number
Figure 13a. Comparison between DD and DI data in repeated impact tests performed on CE90m_4.00
laminates. Impact energy: 24.5 J.
DD, DI
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
DD DI
1 4 7 10 13 16 19 22 25 28 31 34 37 40
Impact Number
Figure 13b. Comparison between DD and DI data in repeated impact tests performed on GVP90_12.31
laminates. Impact energy: 24.5 J.

252
DD, DI
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Maria Pia Cavatorta and Davide Salvatore Paolino
y = 1.1E-01x + 3.2E-01
R 2 = 1.00
1 2 3 4 5
Impact Number
DD DI
Figure 14a. Comparison between DD and DI data in repeated impact tests performed on CE90m_4.00
laminates. Impact energy: 98 J.
DD, DI
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
y = 9.3E-03x + 1.9E-01
R 2 = 0.96
1 3 5 7 9 11 13 15 17 19
Impact Number
DD DI
Figure 14b. Comparison between DD and DI data in repeated impact tests performed on GVP90_12.31
laminates. Impact energy: 98 J.
Figure 14a reports data for the CE90m_4.00 laminate tested at an impact energy of 98J.
Perforation is achieved at the 5 th impact. As it can be observed, the DD constantly increases
impact after impact to reach a value of about one at the 4 th impact, where laminate penetration
is achieved. DI values increase at a constant rate up to the 3 rd impact to then grow very
rapidly and reach the value of one at laminate perforation. This trend is more evident in
Figure 14b, for which perforation is achieved at the 19 th impact. Up to the 17 th impact, the DI
slowly increases at a constant rate impact after impact. In the last two impacts before

Damage Variables in Impact Testing of Composite Laminates 253
perforation, the increase is on the contrary very rapid. Differently from the DuI which keeps
to a constant low level up to a few impacts before perforation (Figure 12), the DI allows to
monitor the initial phase of steady damage accumulation helping foreseen perforation. The
initial slow decrease of DD values from the first to the second impact is followed by a
constant phase up to the 17 th impact, after which the DD increases rapidly and reaches a value
of one at perforation. Likewise the DuI, DD data are not very sensitive for predicting laminate
perforation as, apart from the last 2-3 impacts, the constant phase of Figure 14b does not
differ from the asymptotic trends of Figures 13a and 13b, where no perforation is achieved.
Also, DD values at the first impact are about the same, regardless of the level of impact
energy.
Conclusion
Impact test data obtained on different laminates are used to compare damage variables which
have been proposed in the literature over the years. To this aim, definition of the two energy
contributions used to compute the DuI has been extended to analyze impact tests with
rebound.
In single impact tests performed at different impact energies, data for Fpeak and DuI point
out the existence of an impact energy threshold at about 40-50% Pn, below which the energy
absorption mechanism is mainly matrix cracking. Graphs of Fpeak vs. impact energy show a
bi-linear trend with a change in slope around the energy threshold; while values of the DuI are
almost null below the energy threshold to then increase quite abruptly up to penetration.
Interestingly, the energy threshold is about the same for all the laminates analyzed in the
study, whose thickness varies from tenths to tens of millimeters. DI data increase
monotonically for increasing impact energies and show very limited scattering up to the
energy threshold. DD values and data in the Master Curve give no indications on the laminate
damage tolerance; rather, they provide a measure of the absorption capability of the laminate.
Results show that thicker laminates are characterized by a higher efficiency of energy
absorption.
Main advantage of the DI variable is the possibility to distinguish between the
penetration and perforation energy thresholds. The distinction is essential when dealing with
thick laminates, for which the impact energy that causes laminate perforation can by far
exceed the penetration energy. In the range of the penetration process, the DI effectively
monitors the impactor moving deeper and deeper into the laminate.
Also in case of repeated impact tests, the DI provides important pieces of information.
For impact energies that cause no laminate perforation within test duration, the DI stays at a
constant low value throughout the test, owing to a negligible damage accumulation besides
initial laminate indentation. For impact energies that cause perforation, the DI shows an initial
phase of linear growth with the number of impacts, owing to a steady accumulation of
damage. A few impacts before perforation, the DI starts raising quite abruptly, helping
foreseeing laminate failure.
Results for Fpeak show that it maintains a constant value when perforation is not achieved
while it decreases rapidly otherwise. However, graphs of Fpeak versus impact number do not
signal any change in the rate of damage accumulation. DuI values are almost null throughout
the test when no perforation occurs. Low and constant values also characterize tests at higher

254
Maria Pia Cavatorta and Davide Salvatore Paolino
energies up to a few impacts before perforation, when data show a rapid increase. Therefore,
by looking at the DuI values in the first impacts, no prediction can be made as to whether or
not laminate perforation will occur within test duration. In case of repeated impacts, DD data
can monitor the efficiency of energy absorption impact after impact. Results show that DD
values at the first impact are about the same, regardless of the impact energy. Moreover,
regardless of the final output of the test (no perforation/perforation), DD values show an
initial slight decrease followed by a rather constant phase. When perforation is to be achieved,
DD values start to increase a few impacts before final failure.
Acknowledgments
The Authors wish to acknowledge valuable discussions with professor Giovanni Belingardi.
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In: Composite Materials Research Progress
Editor: Lucas P. Durand, pp. 257-273
Chapter 8
ISBN 1-60021-994-2
c○ 2008 Nova Science Publishers, Inc.
ELECTROMECHANICAL FIELD CONCENTRATIONS
AND POLARIZATION SWITCHING BY ELECTRODES
IN PIEZOELECTRIC COMPOSITES
Yasuhide Shindo and Fumio Narita
Department of Materials Processing, Graduate School of Engineering,
Tohoku University
Abstract
The electromechanical field concentrations due to electrodes in piezoelectric composites
are investigated through numerical and experimental characterization. This
work consists of two parts. In the first part, a nonlinear finite element analysis is carried
out to discuss the electromechanical fields in rectangular piezoelectric composite
actuators with partial electrodes, by introducing models for polarization switching in
local areas of the field concentrations. Two criteria based on the work done by electromechanical
loads and the internal energy density are used. Strain measurements are
also presented for a four layered piezoelectric actuator, and a comparison of the predictions
with experimental data is conducted. In the second part, the electromechanical
fields in the neighborhood of circular electrodes in piezoelectric disk composites are
reported. Nonlinear disk device behavior induced by localized polarization switching
is discussed.
1. Introduction
Sensor and actuator applications take advantage of the piezoelectric coupling converting
electrical energy into mechanical energy and vice versa. Piezoelectric ceramics and composites
play a significant role as active electronic components in many areas of science and
technology, such as smart and MEMS devices. In some actuator applications, high values
of stress and electric field arise in the neighborhood of an electrode tip in piezoelectric
ceramics [1] and composites [2], and the field concentrations can result in electromechanical
degradation [3, 4]. One of the limitations for practical use of piezoelectric ceramics
and composites is also their nonlinear behavior, which occurs due to polarization switching
and/or domain wall motion at high electromechanical field levels near the electrode
tip. In order to optimize the performance of the piezoelectric devices, it is important to

258 Yasuhide Shindo and Fumio Narita
understand the electromechanical field concentrations due to electrodes in piezoelectric ceramics
and composites. Recently, Yoshida et al. [5] discussed the electromechanical field
concentrations due to circular electrodes in piezoelectric ceramics through theoretical and
experimental characterizations. Their model quantitatively predicted the nonlinear electromechanical
fields induced by polarization switching near the circular electrode tip. Also,
numerical predictions of strain concentration were in relatively good agreement with measured
values.
The main aim of this work is to evaluate the electromechanical fields in the neighborhood
of surface and internal electrodes in piezoelectric composites. First, we study the
effect of applied voltage on the electromechanical field concentrations near the electrodes
in rectangular piezoelectric composite actuators. A nonlinear finite element analysis is performed
to calculate the strain, stress, electric field and electric displacement by introducing
models for polarization switching in local areas of the field concentrations. Two criteria
based on the work done by electromechanical loads and the internal energy density are
used and compared. Strain measurements are also presented to validate the predictions
using a four layered piezoelectric actuator. A comparison of strain concentration is made
between measurements and calculations, and a nonlinear behavior induced by localized polarization
switching is discussed. The device performance and polarization switching zone
near the electrodes are further predicted for some electrode configurations in the rectangular
piezoelectric composites. Next, we discuss the electromechanical field concentrations due
to circular electrodes in piezoelectric disk composites. The effects of applied voltage and
localized polarization switching on the disk device performance are examined.
2. Basic Equations
Consider a piezoelectric material with no body force and free charge. The governing equations
in the Cartesian coordinates xi(i =1, 2, 3) are given by
σji,j =0 (1)
Di,i =0 (2)
where σij is the stress tensor, Di is the electric displacement vector, a comma denotes
partial differentiation with respect to the coordinate xi, and the Einstein summation convention
over repeated indices is used. The relation between the strain tensor εij and the
displacement vector ui is given by
εij = 1
2 (uj,i
and the electric field intensity vector is
+ ui,j) (3)
Ei = −φ,i (4)
where φ is the electric potential. In a ferroelectric, polarization switching leads to a change
in the remanent strain εr r
ij and remanent polarization Pi . The total strain and electric displacement
are
εij = ε l ij + ε r ij (5)
Di = D l i + P r
i
(6)

Eletromechanical Field Concentrations and Polarization Switching... 259
where the superscript l denotes the linear contribution to the strain and electric displacement,
and the linear piezoelectric relationships are given by
ε l ij = sijklσkl + dkijEk (7)
D l i = diklσkl + ɛikEk (8)
In Eqs. (7) and (8), sijkl, dkij and ɛik are the elastic compliance tensor, direct piezoelectric
tensor and dielectric permittivity tensor, which satisfy the following symmetry relations:
sijkl = sjikl = sijlk = sklij, dkij = dkji, ɛik = ɛki
σij and D l i are related to εl ij and Ei by
σij = cijklε l kl − ekijEk (10)
D l i = eiklε l kl + ɛikEk (11)
where cijkl and eikl are the elastic and piezoelectric tensors, and
cijkl = cjikl = cijlk = cklij, ekij = ekji
For piezoceramics which exhibit symmetry of a hexagonal crystal of class 6 mm with respect
to principal x1,x2, and x3 axes, the constitutive relations can be written in the following
form:
⎧
⎪⎨
⎪⎩
⎧
⎪⎨
⎪⎩
σ1
σ2
σ3
σ4
σ5
σ6
where
D l 1
D l 2
D l 3
⎫
⎡
⎢
⎪⎬
⎢
= ⎢
⎣
⎪⎭
⎫
⎪⎬
⎪⎭ =
⎡
⎢
⎣
c11 c12 c13 0 0 0
c12 c11 c13 0 0 0
c13 c13 c33 0 0 0
0 0 0 c44 0 0
0 0 0 0 c44 0
0 0 0 0 0 c66
0 0 0 0 e15 0
0 0 0 e15 0 0
e31 e31 e33 0 0 0
⎤ ⎧
⎥ ⎪⎨
⎥
⎦
⎪⎩
σ1 = σ11, σ2 = σ22, σ3 = σ33
ε l 1
ε l 2
ε l 3
ε l 4
ε l 5
ε l 6
(9)
(12)
⎫ ⎡
0
⎢
⎪⎬
⎢ 0
⎢ 0
− ⎢ 0
⎢
⎣
⎪⎭
e15
0
0
0
0
e15
0
0
e31
e31
e33
0
0
0
⎤
⎥ ⎧
⎥ ⎪⎨
⎥ ⎪⎩ ⎥
⎦
E1
E2
E3
⎫
⎪⎬
⎪⎭
(13)
⎧
ε
⎤
⎪⎨
⎥
⎦
⎪⎩
l 1
εl 2
εl 3
εl 4
εl 5
εl ⎫
⎡
⎤ ⎧ ⎫
⎪⎬ ɛ11 0 0 ⎪⎨ E1 ⎪⎬
⎢
⎥
+ ⎣ 0 ɛ11 0 ⎦ E2
⎪⎩ ⎪⎭
0 0 ɛ33 E3
⎪⎭
6
(14)
σ4 = σ23 = σ32, σ5 = σ31 = σ13, σ6 = σ12 = σ21
ε l 1 = ε l 11, ε l 2 = ε l 22, ε l 3 = ε l 33
ε l 4 =2εl 23 =2εl 32 ,εl 5 =2εl 31 =2εl 13 ,εl 6 =2εl 12 =2εl 21
�
�
(15)
(16)

260 Yasuhide Shindo and Fumio Narita
c11 = c1111 = c2222, c12 = c1122, c13 = c1133 = c2233, c33 = c3333
c44 = c2323 = c3131, c66 = c1212 = 1
2 (c11 − c12)
e15 = e131 = e223, e31 = e311 = e322, e33 = e333
The direction of the spontaneous polarization P s of each grain can change by 90 ◦ or
180 ◦ for ferroelectric switching induced by a sufficiently large electric field. In order to
develop a non-linear model incorporating the polarization switching mechanisms with the
electromechanical fields calculations, two criteria are used. The first criterion for polarization
switching is based on work down, and the second is internal energy density switching
criterion.
The first criterion [6] states that a polarization switches when the electrical and mechanical
work exceeds a critical value
σij∆εij + Ei∆Pi ≥ 2P s Ec
where ∆εij and ∆Pi are the changes in the spontaneous strain and polarization during
switching, respectively, and Ec is a coercive electric field. The changes in ∆εij = ε r ij and
∆Pi = P r
i for 180◦ switching can be expressed as
⎫
⎬
⎭
(17)
(18)
(19)
∆ε11 =0, ∆ε22 =0, ∆ε33 =0, ∆ε12 =0, ∆ε23 =0, ∆ε31 =0 (20)
∆P1 =0, ∆P2 =0, ∆P3 = −2P s (21)
For 90 ◦ switching in the x3x1 plane, there results
∆ε11 = γ s , ∆ε22 =0, ∆ε33 = −γ s , ∆ε12 =0, ∆ε23 =0, ∆ε31 =0 (22)
∆P1 = ±P s , ∆P2 =0, ∆P3 = −P s
For 90 ◦ switching in the x2x3 plane,
(23)
∆ε11 =0, ∆ε22 = γ s , ∆ε33 = −γ s , ∆ε12 =0, ∆ε23 =0, ∆ε31 =0 (24)
∆P1 =0, ∆P2 = ±P s , ∆P3 = −P s
The polarization switching criterion based on internal energy density (second criterion)
[7] is defined as
U = Uc
where U is the internal energy density and Uc is a critical value of internal energy density
corresponding to the switching mode. The internal energy density associated with 180 ◦
switching can be written as
U = 1
2 D3E3
In the case of 90 ◦ switching in the x3x1 plane, the internal energy density is
(25)
(26)
(27)
U = 1
2 (σ11ε11 + σ33ε33 +2σ31ε31 + D1E1) (28)

Eletromechanical Field Concentrations and Polarization Switching... 261
For 90 ◦ switching in the x2x3 plane,
U = 1
2 (σ22ε22 + σ33ε33 +2σ32ε32 + D2E2) (29)
The critical value of internal energy density is assumed in the following form:
where ɛ T 33
Uc = 1
2 ɛT 33(Ec) 2
is the dielectric permittivity at constant stress.
The constitutive equations (10) and (11) during polarization switching are
The new piezoelectric constant e ′ ikl
by
(30)
σij = cijklε l kl − e ′ kijEk (31)
D l i = e ′ iklε l kl + ɛikEk (32)
is related to the elastic and direct piezoelectric constants
e ′ 111 = d′ 111 c11 + d ′ 122 c12 + d ′ 133 c13
e ′ 122 = d ′ 111c12 + d ′ 122c11 + d ′ 133c13
e ′ 133 = d′ 111 c13 + d ′ 122 c13 + d ′ 133 c33
e ′ 123 =2d ′ 123c44
e ′ 131 =2d′ 131 c44
e ′ 112 =2d ′ 112c66
e ′ 211 = d′ 211 c11 + d ′ 222 c12 + d ′ 233 c13
e ′ 222 = d ′ 211c12 + d ′ 222c11 + d ′ 233c13
e ′ 233 = d′ 211 c13 + d ′ 222 c13 + d ′ 233 c33
e ′ 223 =2d ′ 223c44
e ′ 231 =2d′ 231 c44
e ′ 212 =2d′ 212 c66
e ′ 311 = d ′ 311c11 + d ′ 322c12 + d ′ 333c13
e ′ 322 = d′ 311 c12 + d ′ 322 c11 + d ′ 333 c13
e ′ 333 = d ′ 311c13 + d ′ 322c13 + d ′ 333c33
e ′ 323 =2d′ 323 c44
e ′ 331 =2d ′ 331c44
e ′ 312 =2d′ 312 c66
The components of the piezoelectricity tensor d ′ ikl are
d ′
�
ikl = d33ninknl + d31(niδil − ninknl)+ 1
2 d15(δiknl
�
− 2ninknl + δilnk)
where ni is the unit vector in the poling direction, δij is the Kroneker delta, and d33 = d333,
d31 = d311, d15 =2d131 are the direct piezoelectric constants.
(33)
(34)

262 Yasuhide Shindo and Fumio Narita
3. Rectangular Piezoelectric Composite Actuators
3.1. Computational Model
We performed 3D finite element calculations to present the electromechanical fields distributions
around the electrode tip. The geometry used was a four layered piezoelectric
composite actuator, as shown in Fig. 1. A rectangular Cartesian coordinate system ( x,y,z)
is used with the z-axis coinciding with the poling direction. Electrodes with length a and
width W are embedded in the piezoelectric actuator of length L and width W . An external
electrode is attached on both sides of the actuator to address each electrode. The
thickness of the layer h is chosen, and a L − a tab region exists on both sides of the layer.
The total thickness is 4h. Because of the geometric and loading symmetry, only a half
of the specimen needs to be analyzed. The electric potential on two electrode surfaces
(−L/2 ≤ x ≤−L/2 +a, |y| ≤W/2, z =0, 2h) equals the applied voltage, φ = V0.
The electrode surface (L/2 − a ≤ x ≤ L/2, |y| ≤W/2, z = h) is connected to the
ground, so that φ =0. The normal displacement and shear stress on the surface (|x| ≤L/2,
y = −W/2, |z| ≤2h) are zero. The surface (|x| ≤L/2, y = W/2, |z| ≤2h) is stress free.
W
h
h
Electrode
Poling
O
y
z
O
a
a
L
a
Figure 1. A rectangular piezoelectric composite actuator.
Each element consists of many grains, and each grain is modeled as a uniformly polarized
cell that contains a single domain. The model neglects the domain wall effects and
interaction among different domains. In reality, this is not true, but the assumption does
not affect the general conclusions drawn. The polarization of each grain initially aligns as
closely as possible with the z- direction. The polarization switching is defined for each
x
x

Eletromechanical Field Concentrations and Polarization Switching... 263
element in a material. The electric potential φ is applied, and the electromechanical fields
of each element are computed from the finite element analysis (FEA). The switching criterion
of Eq. (19) or (26) is checked for every element to see if switching will occur. After
all possible polarization switches have occurred, the piezoelectric tensor of each element
is rotated to the new polarization direction. The electroelastic fields are re-calculated, and
the process is repeated until the solution converges. The macroscopic response of the material
is determined by the finite element model, which is an aggregate of elements. The
spontaneous polarization P s and strain γ s are assigned representative values of 0.3 C/m 2
and 0.004, respectively. Our previous experiments [8] verified the accuracy of the above
scheme, and showed that the results obtained are of general applicability.
3.2. Experiments
The actuator discussed in this section was fabricated using a soft lead zirconate titanate
(PZT) C-91 [9]. The material properties are listed in Table 1, and the corcive electric field
is approximately Ec = 0.35 MV/m. The dimensions of the specimen are L = 30 mm, W =
10 mm, and 4h = 20 mm. The electrode length is a = 20 mm. The specimen was placed
on the rigid floor.
The high-voltage amplifier was limited to 1.25 kV so that a 0.25 MV/m field corresponded
to a layer thickness of 5.0 mm. Strain gauges were placed around the electrode tip
region. The sensors have an active length of 0.2 mm.
Table 1. Material properties of C-91.
Elastic stiffnesses Piezoelectric coefficients Dielectric permittivities
(×1010N/m2 ) (C/m2 ) (×10−10C/Vm) c11 c12 c13 c33 c44 e31 e33 e15 ɛ11 ɛ33
C-91 12.0 7.7 7.7 11.4 2.4 −17.3 21.2 20.2 226 235
3.3. Results and Discussion
We first present analytical and experimental results for L = 30 mm, W = 10 mm, and
4h =20mm. The electrode length is a = 20 mm. Fig. 2 shows the finite element analysis
results for the strain εzz versus electric field E0 = V0/h at the face of the actuator (at
y =5mm plane) for x = 5 mm and z = 0.8 mm. For the polarization switching effect, the
predictions based on work (Eq. (19)) and energy density (Eq. (26)) are shown. Also plotted
are the experimental data in the range approximately ± 0.18 MV/m. Calculation results
show that a monotonically increasing negative electric field causes polarization reversal.
Polarization switching in a local region leads to a significant increase of compressive strain
within the actuator when compared to the linear case. After the electric field reaches about
−0.20 (0.24) MV/m, local polarization switching, based on work (energy density), can
cause an unexpected decrease in compressive strain near the electrode tip during switching.

264 Yasuhide Shindo and Fumio Narita
Strain,� zz (�10 -6 )
200
100
0
-100
-200
L = 30 m m
W = 10 m m
4h = 20 m m
a = 20 m m
Test
FEA
x = 5.0 m m
y = 5.0 m m
z = 0.8 m m
W ork
Energy density
-0.3 -0.2 -0.1 0 0.1 0.2
Electric field, E0 (M V/m )
Figure 2. Strain versus electric field for laminated actuator.
E = -0.34 M V/m
0
W ork
Energy density
Poling
o
90 switching
o
180 switching
Figure 3. Polarization switching zone induced by electric field for laminated actuator.
Fig. 3 shows the 180 ◦ and 90 ◦ switching zones near the electrode tip of the actuator under
E0 = −0.34 MV/m. Predictions by different criteria are presented. 90 ◦ switching zone
based on energy density are larger than that based on work. Fig. 4 displays the distribution
of the normal component of stress σzz as a function of x at y =0mm and z =0, 1 and
9 mm for laminated actuator with L = 30 mm, W = 10 mm, 4h =20mm and a =20
mm under E0 =0.2 MV/m from the finite element analysis. The solid line represents the

Eletromechanical Field Concentrations and Polarization Switching... 265
normal stress at the interface, the dashed line represents the normal stress near the internal
electrode tip, and the alternate long and short dashed line denotes the value near the surface
electrode tip. The normal stress at the interface is singular at the electrode tip. The stress
ahead of the electrode tip is tensile, while the stress behind the electrode tip is compressive.
The values of the normal stress near the internal electrode tip are higher than those near the
Displacem ent,u z (�m )
2
1
0
-1
-2
FEA
W ork
x = 0 m m
y = 0 m m
z = 10 m m
L = 30 m m
W = 10 m m
4h = 20 m m
a = 20 m m
24
28
-1000 0 1000
Voltage,V 0 (V)
Figure 4. Normal stress versus x for laminated actuator under E0 =0.2 MV/m.
surface electrode tip. Fig. 5 shows the comparison of the distribution of the shear stress σzx
against x near the internal electrode tip with that near the surface electrode tip for the same
laminated actuator. The shear stress peaks at about x =5.5 mm, and the peak value near
the internal electrode tip is higher than that near the surface electrode tip.
Surface electrode
h
h
x
z
b
O
b �
b
c
r
Internal electrode
Figure 5. Shear stress versus x for laminated actuator under E0 =0.2 MV/m.
y

266 Yasuhide Shindo and Fumio Narita
!
( )
"# $
%& ' &
Figure 6. Displacement versus voltage for laminated actuator.
Next, the effect of electrode length on the performance of the actuator with L = 30 mm,
W = 10 mm and 4h = 20 mm is discussed. Fig. 6 shows the predictions of displacement
uz at x =0mm, y =0mm and z =10mm as a function of applied voltage V0, based on
work, for a =20, 24 and 28 mm. Non-linearity in the displacement versus voltage curves
depends on the electrode length. The displacement increases with an increase of a from 20
mm to 24 mm. Little difference is observed between the results for a =24mm and 28 mm.
4. Piezoelectric Disk Composites
4.1. Problem Statement and Solution Procedure
Consider a two-layered piezoelectric disk composite with radius c and thickness h as shown
in Fig. 7. The origin of the coordinates (r,θ,z) is located at the center of the interface
considered as z =0, 0 ≤ r ≤ c. The z axis is assumed to coincide with the six fold
axis of symmetry in the class of a 6mm crystal class, or with the poling axis in the case
of poled piezoelectric ceramics. Three parallel circular electrodes of radius b lie in the
planes z =0, ±h. The bonding at z =0is assumed to be perfect so that the stresses and
displacements are continuous along the interface of the composite. Let the voltage applied
to the internal electrode surface be denoted by V0. The surface electrodes are grounded.
The axisymmetric models were generated using the commercial FE method software
package. The electrode layers were not incorporated into the model.

Eletromechanical Field Concentrations and Polarization Switching... 267
" # $ %" #
Figure 7. A piezoelectric disk composite actuator.
4.2. Numerical Results and Discussion
We consider C-91 + /C-91 − and C-91 + /C-91 + with c =10mm and 2h =2mm, corresponding
to the tension and bending actuator models, respectively. The superscripts − and
+ denote, respectively, the situations for negative and positive poling directions.
Plotted in Fig. 8 are the numerical values of radial strain εrr near the circular internal
electrode tip (r =9mm and z =0.2 mm) as a function of electric field E0 = V0/h for
Strain,� rr (�10 -6 )
60
40
20
0
C-91 + /C-91 +
c= 10 m m
2h = 2 m m
b = 8 m m FEA
W ork
Energy density
-0.2 0 0.2
Electric field, E0 (M V/m )
!
r= 9 m m
z= 0.2 m m
Figure 8. Strain versus electric field for disk composite tension actuator.

268 Yasuhide Shindo and Fumio Narita
E 0=
-0.22 M V/m
-0.30 M V/m
Poling
0.5 m m
+0.22 M V/m
+0.30 M V/m
W ork
o
90 switching
o
180 switching
Figure 9. Normal stress versus r for disk composite tension actuator under E0 =0.2
MV/m.
C-91 + /C-91 − disk tension actuator with b =8mm. The predictions based on work (Eq.
(19)) and energy density (Eq. (26)) are shown. As the electric field is reduced from zero,
the compressive strain increases. Local polarization switching can cause a decrease in
compressive strain near the circular electrode tip. Little difference is observed between
two criteria. As the positive electric field increases, polarization switching did not occur.
Fig. 9 shows the distribution of the normal stress σzz as a function of r at z =0and 0.2
mm for C-91 + /C-91 − disk tension actuator with b =8mm under E0 =0.2 MV/m. Near
! " # $! " #
Figure 10. Shear stress versus r for disk composite tension actuator under E0 = 0.2

MV/m.
Eletromechanical Field Concentrations and Polarization Switching... 269
" # $ %" #
Figure 11. Displacement versus voltage for disk composite tension actuator.
the circular electrode tip, the normal stress at the interface is singular, and the stress ahead of
the circular electrode tip is tensile, while the stress behind the electrode tip is compressive.
The normal stress, apart from the interface, near the electrode tip has smaller value than the
interface stress. Fig. 10 gives the distribution of the shear stress σzr as a function of r at
z =0.01 and 0.2 mm for the same disk tension actuator. The magnitudes of the shear stress
increase toward the circular electrode tip as is expected. Fig. 11 shows the predictions of
displacement uz at r =0mm and z =1mm as a function of applied voltage V0, based
on work, for b =8and 10 mm. There is a small influence of the electrode radius on the
displacement versus voltage curves.
Fig. 12 shows the computed strain εrr of C-91 + /C-91 + disk bending actuator corresponding
to Fig. 8. The negative electric field increases the compressive strain, similar to
C-91 + /C-91 − disk tension actuator. After the electric field reaches about −0.25 MV/m,
polarization switching leads to a decrease in the compressive strain. As the electric field
E0 continues to be reduced, the strain becomes tensile. On the other hand, as the positive
electric field is increased, the strain near the electrode tip increases gradually due to the
piezoelectric effect and then sharply increases as switching occurs due to electromechanical
field concentrations. Little difference is observed between two criteria. Fig. 13 shows
the predicted switching zones, based on work, near the circular electrode tip. As the electric
fields increase, the area of the switched region grows. Fig. 14 shows the normal stress
distribution σzz as a function of r at z =0and 0.2 mm for C-91 + /C-91 + disk bending
actuator with b =8mm. The interface normal stress of the disk bending actuator is singular
at the circular electrode tip, similar to the disk tension actuator. The stress ahead of
the circular electrode tip is tensile, while the stress behind the circular electrode tip changes
from tensile to compressive in the neighborhood of the electrode tip.
!

270 Yasuhide Shindo and Fumio Narita
" # $ " # $
%
%
% & ' (
Figure 12. Strain versus electric field for disk composite bending actuator.
Figure 13. Polarization switching zone induced by electric field for disk composite bending
actuator.
!
% #
%

Eletromechanical Field Concentrations and Polarization Switching... 271
Figure 14. Normal stress versus r for disk composite bending actuator under E0 =0.2
MV/m.
Figure 15. Shear stress versus r for disk composite bending actuator under E0 =0.2
MV/m.
!

272 Yasuhide Shindo and Fumio Narita
! " # $! " #
Figure 16. Tip deflection versus voltage for disk composite bending actuator.
Fig. 15 shows the similar results for the shear stress distribution σzr. A singularity in the
interface shear stress also develops at the circular electrode tip. Note that for the C-91 + /C-
91 + disk bending actuator, since the problem is unsymmetrical to the r-axis, the shear stress
does not become zero along the whole r-axis. Fig.16 gives a plot of the tip deflection uz at
r =10mm and z =0mm with applied voltage V0, based on work, for C-91 + /C-91 + disk
bending actuator with b =8and 10 mm. The curve rises steeply at first when the voltage is
increased from zero. The tip deflection then gradually levels off when the voltage reaches
about 220 V, because of switching in the lower layer (see Fig. 13). A similar phenomenon
can be observed for negative voltage. The bending actuator for b =10mm exhibits higher
deflection.
5. Conclusions
The electromechanical field distributionsin the neighborhood of the electrodes in piezoelectric
composites were investigated. Two criteria for polarization switching in piezoelectric
materials were incorporated into a finite element procedure. The results indicated that high
values of electromechanical fields cause the localized polarization switching near the electrode
tip, and the strain vs electric field curves show the non-linear behavior. Also, the
size of the switching zone in the piezoelectric composites increased with increasing electric
fields. As a remark, we note that this study may be useful in designing advanced piezoelectric
composite actuators.

Eletromechanical Field Concentrations and Polarization Switching... 273
Acknowledgements
This work was partially supported by the Grant-in-Aid for Scientific Research (B) and
Young Scientists (B) from the Ministry of Education, Culture, Sports, Science and Technology,
Japan.
References
[1] Shindo, Y., Narita, F. & Sosa, H. (1998). Electroelastic analysis of piezoelectric ceramics
with surface electrodes. Int. J. Eng. Sci., 36, 1001-1009.
[2] Narita, F., Yoshida, M. & Shindo, Y. (2004). Electroelastic effect induced by electrode
embedded at the interface of two piezoelectric half-planes. Mech. Mater., 36, 999-
1006.
[3] Dos Santos e Lucato, S. L., Lupascu, D. C., Kamlah, M., Rödel, J. & Lynch, C. S.
(2001). Constraint-induced crack initiation at electrode edges in piezoelectric ceramics.
Acta Mater., 49, 2751-2759.
[4] Qiu, W., Kang, Y.-L., Qin, Q.-H., Sun, Q.-C. & Xu, F.-Y. (2007). Study for multilayer
piezoelectric composite structure as displacement actuator by Moiré interferometry
and infrared thermography experiments. Mater. Sci. Eng. A, 15, 452-453.
[5] Yoshida, M., Narita, F., Shindo, Y., Karaiwa, M. & Horiguchi, K. (2003). Electroelastic
field concentration by circular electrodes in piezoelectric ceramics. Smart Mater.
Struct., 12, 972-978.
[6] Hwang, S. C., Lynch, C. S. & McMeeking, R. M. (1995). Ferroelectric/ferroelastic
interactions and a polarization switching model. Acta Metall. Mater., 43, 2073-2084.
[7] Kalyanam, S. & Sun, C. T. (2005). Modeling of electrical boundary condition and
domain switching in piezoelectric materials. Mech. Mater., 37, 769-784.
[8] Narita, F., Shindo, Y. & Hayashi, K. (2005). Bending and polarization switching of
piezoelectric laminated actuators under electromechanical loading. Comput. Struct.,
83, 1164-1170.
[9] Shindo, Y., Yoshida, M., Narita, F. & Horiguchi, K. (2004). Electroelastic field concentrations
ahead of electrodes in multilayer piezoelectric actuators: experiment and
finite element simulation. J. Mech. Phys. Solids, 52, 1109-1124.

In: Composite Materials Research Progress ISBN: 1-60021-994-2
Editor: Lucas P. Durand, pp. 275-296 © 2008 Nova Science Publishers, Inc.
Chapter 9
RECENT ADVANCES IN DISCONTINUOUSLY
REINFORCED ALUMINUM BASED METAL MATRIX
NANOCOMPOSITES
S.C. Tjong *
Department of Physics and Materials Science, City University of Hong Kong,
Tat Chee Avenue, Kowloon, Hong Kong
Abstract
Aluminum-based alloys reinforced with ceramic microparticles are attractive materials
for many structural applications. However, large ceramic microparticles often act as stress
concentrators in the composites during mechanical loading, giving rise to failure of materials
via particle cracking. In recent years, increasing demand for high performance materials has
led to the development of aluminum-based nanocomposites having functions and properties
that are not achievable with monolithic materials and microcomposites. The incorporation of
very low volume contents of ceramic reinforcements on a nanometer scale into aluminumbased
alloys yields remarkable mechanical properties such as high tensile stiffness and
strength as well as excellent creep resistance. However, agglomeration of nanoparticles occurs
readily during the composite fabrication, leading to inferior mechanical performance of
nanocomposites with higher filler content. Cryomilling and severe plastic deformation
processes have emerged as the two important processes to form ultrafine grained composites
with homogeneous dispersion of reinforcing particles. In the present review article, recent
development in the processing, structure and mechanical properties of the aluminum-based
nanocomposites are addressed and discussed.
Introduction
Discontinuously reinforced aluminum (DRA) based metal matrix composites are of
increasing interest because of their high specific stiffness and strength, high isotropic and
excellent wear resistance as well as cost effective manufacturing. DRA composites have been
* E-mail address: aptjong@cityu.edu.hk

276
S.C. Tjong
developed in the past two decades for various automobile, aerospace, electronic packaging
and other structural applications. Many factors affect the mechanical properties of DRA
composites including matrix alloy composition, reinforcement material, reinforcement size,
shape, volume fraction and distribution, nature of the matrix-reinforcement interface, etc. The
reinforcement materials generally should possess significantly higher specific and specific
strength, as well as high melting temperature compared to the matrix alloy. Ceramic
reinforcement has the advantage of a relatively low density and high elastic modulus. Typical
ceramic particles commonly used to reinforce aluminum and its alloys including SiC, B4C,
Si3N4, AlN, Al2O3, TiC, TiB2, etc Particle reinforced composites are conventionally prepared
either via powder metallurgy (PM) or liquid metallurgy, in which the reinforcing particles
with sizes of several microns are directly incorporated into solid or liquid aluminum,
respectively. The composites thus prepared can be viewed as ex-situ MMCs. However,
ceramic microparticles fracture readily during mechanical loading, leading to low toughness
of the composites [1-3]. Figs. 1(a)-1(b) show typical fracture morphology of ceramic
microparticles in Al-based composites during tensile loading. Furthermore, reinforcement
material such as SiC is not thermodynamically stable and thus can react with aluminum
matrix during the composite fabrication and service at elevated temperatures. Efforts have
been made to overcome the occurrence of such difficulties by developing novel in-situ
processing. In the process, the reinforcing particles are directly formed in a metallic matrix by
chemical reactions between constituent elements during the composite fabrication [4, 5].
Accordingly, very fine in-situ particles with diameters down to submicrometer scale ( > 100
nm) can be synthesized and dispersed more uniformly within aluminum matrix [6]. The
formation of clean, ultrafine and thermally stable ceramic reinforcements rendering the in-situ
composites exhibit excellent mechanical properties.
Figure 1. SEM fractographs showing fracture and decohesion of alumina particles of (a) 6061
Al/20vol.%Al2O 3 and (b) 7005 Al /10vol.% Al 2O 3 composites tensile tested at room temperature [3].
The successful synthesis of large-scale ceramic, metallic and intermetallic nanoparticles
in recent years has motivated materials scientists to develop novel metal-matrix
nanocomposites with excellent mechanical properties for advanced structural engineering

Recent Advances in Discontinuously Reinforced Aluminum… 277
applications. Nanoparticles can be synthesized by several processes such as gas phase
condensation, laser ablation, aerosol route, mechanochemical processing is well established
[5, 7-9]. They reveal unique physical and mechanical properties that are different from those
of bulk solids and microparticles. Due to their high specific surface area, nanoparticles exhibit
a high reactivity and strong tendency towards agglomeration. It is necessary to disperse exsitu
nanoparticles more uniformly in aluminum matrix in order to obtain desired mechanical
properties. In the case of liquid metallurgy processing, high-intensity ultrasonic waves can be
employed to disperse the SiC nanoparticles more uniformly in molten aluminum alloy [10,
11]. In powder metallurgy route, mechanical alloying, particularly cryomilling has been used
to refine and disperse the ceramic phase in the Al matrix [12 -16]. In most cases, ceramic
particles with original sizes of several micrometers can be reduced to nanometer level after
cryomilling [17].
Recently, there has been a growing interest in the application of severe plastic
deformation (SPD) such as high pressure torsion (HPT) and equal channel angular pressing
(ECAP) for producing materials with ultrafine grain structure in submicrometer levels [18 -
29]. ECAP is more attractive for industrial applications because it can be employed to
produce large fully-dense samples or products. It consists of pressing the sample through a
die into an L-shaped channel without changing its cross-section. The sample deforms by
simple shear, thereby inducing a high density of dislocations that are subsequently arranged to
the meta-stable sub-grains of high-angle boundaries. By repeating the pressing process, the
strain is accumulated during each increment cycle. The ultra-fine grained composites
processed by ECAP exhibit high yield strength and good ductility [27].
Agglomeration of Particles
Generally, ceramic particles of micrometer sizes are prone to cluster during the composite
fabrication. Particle clustering is more prevalent in cast than in PM microcomposites [30, 31].
This leads to the mechanical properties of microcomposites are far below the theoretical
values. For the PM microcomposites, the particle size ratio of the matrix and reinforcement is
the main factor controlling the degree of microstructural homogeneity [32-35]. Furthermore,
secondary processing technique such as ECAP and HPT are reported to be very effective to
improve the dispersion of reinforcing ceramic particles in the PM DRA composites [20, 24,
27]. Figs. 2(a) -2(c) show the effect of ECAP extrusion cycles on the particle distribution in
PM 6061 Al/20% Al2O3 composite. The composite in the as-fabricated condition shows
extensive particle clustering as expected. The clusters are aligned along the extrusion
direction (Fig. 2(a)). These clusters begin to dissolve and disperse into individual particles
after four ECAP passes at 370 ºC. The particle distribution appears homogeneous after
pressing for seven passes. In addition to declustering, ECAP treatment also yields grain
refinement of the aluminum alloy matrix.
It is well recognized that nanoparticles tend to agglomerate into large clusters during
composite processing even under low loading levels of reinforcement. In this respect,
appropriate processing procedures are needed to improve the dispersion of nanoparticles in
aluminum matrix. Recently, Yang et al. used high-intensity ultrasonic waves to assist the
dispersion of SiC nanoparticles (average size ≤ 30 nm) in molten aluminum alloy A356 [10,
11]. Fig. 3 shows a typical experimental setup for the ultrasonic assisted melting. The

278
S.C. Tjong
ultrasonic waves generate nonlinear effects in molten metal such as transient cavitation and
acoustic streaming. Acoustic cavitation involves the formation, growth, pulsating and
collapsing of tiny bubbles, thereby yielding transient local hot spots and implosive impacts to
break up the clustered particles. The strong impact and local high temperatures enhance the
wettability between molten metal and nanoparticles. Consequently, cast Al-based
nanocomposite with better dispersion of ceramic nanoparticles can be prepared (Fig. 4).
(a)
(b)
Figure 2. Continued on next page.

Recent Advances in Discontinuously Reinforced Aluminum… 279
(c)
Figure 2. Microstructure of the PM 6061 Al /20% Al 2O 3 composite: (a) as-extruded condition. The
reinforcing alumina size is ~ 1-5 μm, (b) after four ECAP passes and (c) after seven ECAP passes [24].
Figure 3. Experimental setup of ultrasonic assisted melting [10].
In PM nanocomposites, clustering of nanoparticles often occurs during processing and
the degree of agglomeration increases with increasing filler content [36] (Fig. 5). Through a
solid-state cryomilling route, better dispersion of nanoparticles in aluminum matrix can be
achieved. In the process, collisions between the grinding media lead to repeated fracture and
welding of the raw powders in a high-energy ball mill. Low temperature (liquid nitrogen)
environment suppresses the recovery and recrystallization of matrix grains during milling,
thereby yielding finer grain structures. The nature of the process allows the incorporation of
large volume fractions of reinforcement into aluminum matrix with a homogeneous
distribution [12,13, 15, 37, 38]. Consolidation of cryomilled powders via hot pressing, cold
isostatic pressing (CIP), hot isostatic pressing (HIP), extrusion, spark plasma sintering, etc. is
necessary to produce bulk composites with full density, useful shapes and sizes for practical

280
S.C. Tjong
applications. Goujon et al. prepared the Al 5000/AlN (4-30 vol.%) nanocomposites though
cryomilling of 5000 Al powder (380 nm) and AlN particle (150 nm) followed by hot pressing
[12, 13]. Cryomilling for 6 h is required to obtain a good homogeneity of powder mixtures.
The crystallite sizes of Al and AlN in the powders are reduced to about 49 and 30 nm,
respectively. Hot pressing leads to homogeneous dispersion of the AlN phase in the Al alloy
matrix and to an increase of the crystallite size of Al to submicrometer regime but not of AlN
(Fig. 6). The microstructure of this composite consists of UFG aluminum grains free of AlN
particles and regions dispersed with AlN nanoparticles.
Figure 4. SEM micrograph showing the microstructure of as-cast A356/2%SiC nanocomposite [10].
Figure 5. TEM micrograph showing agglomeration of particulates at the grain boundary of Al/5 vol.%
Al2O 3 nanocomposite. The mean size of alumina nanoparticles is ~ 50 nm [36].

Recent Advances in Discontinuously Reinforced Aluminum… 281
Figure 6. Microstructure of Al 5000/20.6vol.% AlN nanocomposite prepared by cryomilling and hot
pressing [13].
Structure-Property Relationship
Aluminum-based nanocomposites can be classified into two categories according to the size
dimensions of reinforcing particle and aluminum matrix employed, i.e. micrograined matrix
composites reinforced with nanoparticles and UFG matrix composites reinforced with
submicron- or nanoparticles. In the former case, ceramic nanoparticles are introduced directly
into aluminum matrix having grain sizes in micrometer level via PM or ingot casting. The
latter relates the use of cryomilling to refine the reinforcing particles and aluminum matrix
down to submicrometer of nanoscale regime. Alternatively, the matrix grains of the
composites can also be refined to submicrometer level using the SPD process.
It is well recognized that the deformation behavior of nanocrystalline metals is quite
different from their micro-grained counterparts. According to the Hall-Petch relation, a
substantial increase in yield strength can be achieved by reducing the grain size of metals
to the submicrometer or nanometer regime. Nanocrystalline metals generally have very low
tensile ductility, and exhibit creep and superplasticity at lower temperatures compared to their
micro-grained counterparts [9]. This is attributed to large volume (more than 50%) of atoms
are located at the grain boundaries or interfacial boundaries of nanometals. Consequently,
grain boundary activity is a dominant factor for controlling the mechanical properties. It is of
practical interest to understand the effect of particle additions on the mechanical properties of
aluminum and its alloys having submicrometer or nanometer grain sizes.
Micro-grained Matrices
Tjong et al. investigated the microstructure and mechanical properties of pure aluminum
reinforced with low loading levels of Si3N4 (15 nm) or Si-N-C (25 nm) nanoparticles. Such

282
S.C. Tjong
nanoparticles were prepared by means of the laser induced gas-phase reactions [39-41]. They
reported that the mechanical strength of nanoparticle strengthened composites is far superior
to that of microparticle reinforce composite with a similar volume content of particulate. In
other words, the tensile strength of Al/1vol% Si3N4 (15 nm) and Al/1vol.% Si-N-C (25
nm)nanocomposites is comparable to that of Al/15vol% SiC (3.5μm) composite, but the yield
stress of such nanocomposites is significantly higher than that of the microcomposite. The
tensile ductility of nanocomposites is also higher than that of microcomposite (Table 1).
However, increasing the Si-N-C nanoparticle content to 5 vol.% leads to deterioration of
mechanical properties as a result of the particle agglomeration. The strengthening mechanism
of nanocomposites is derived from the Orowan stress. It is well known that the Orowan
strengthening results from interaction between dislocation and the dispersed particles during
mechanical loading. Recently, Kang and Chan [36] also reported that the tensile strength of
the Al/1vol.% Al2O3 nanocomposite is similar to that of the Al/10 vol.%SiCp (13 μm)
composite, and the yield strength of the former is higher than that of the latter (Fig. 7). This
figure reveals that the yield and tensile strengths of Al reinforced with Al2O3 nanoparticles
increase with increasing filler content up to 4 vol.% Al2O3 at the expense of tensile ductility.
Above 4 vol.%, the strengthening effects level off owing to the agglomeration of alumina
nanoparticles as shown in Fig. 5. The main strengthening effect in such nanocomposites also
arises from the Orowan stress. It is worth-noting that both the tensile strength and tensile
ductility of cast Al-based composites are improved considerably as a result of better
dispersion of nanoparticles in the alloy matrix via laser assisted melting [10,11].
Figure 7. Tensile properties of Al/Al 2O 3 nanocomposites prepared by conventional powder metallurgy
method. The tensile properties of Al/10 vol.% SiC (10 μm) microcomposites are also show for the
purposes of comparison [36].

Recent Advances in Discontinuously Reinforced Aluminum… 283
Table 1. Tensile properties of Al-based micro- and nanocomposites [41].
Specimen
Tensile Strength,
MPa
Yield Strength,
MPa
Elongation at
Break, %
Pure Al 70 30 ---
Al/15 vol.% SiC (3.5 μm) 176 94 14.5
Al/1 vol.% Si3N4 (15 nm) 180 144 17.4
Al/1 vol.% Si-N-C (25 nm) 178 134 19.7
Al/5 vol.% Si-N-C (25 nm) 153 114 6.2
It is widely known that DRA microcomposites exhibit higher creep resistance than their
unreinforced matrix materials because the particulates acting as barriers to dislocation
movement. There is no plastic flow occurs within ceramic reinforcing particles. Accordingly,
plastic deformation of DRA composites is controlled exclusively by flow in the metallic
matrices. The high temperature creep behavior of coarse-grained Al and its alloys reinforced
with microparticles is characterized by high values of n and Q. The creep activation energy of
microcomposites is often much larger than that for aluminum lattice self-diffusion (142
kJ/mol) [42-45]. Such anomalous behavior can be rationalized by introducing a threshold
stress (σo) opposing creep flow. In this respect, the observed creep deformation is not driven
by the applied stress σ but rather by an effective stress σc (σc = σ -σo). The threshold stress
may originate from several sources such as Orowan bowing between particles, attractive
attraction between dislocations and particles as well as back-stress associated with local
dislocation climb [5, 42]. The rate controlling equation can be written as follows:
σ −σ
o n Q
ε& = A(
) exp( − )
[1]
G RT
where ε& is the creep rate, A is a constant, G is shear modulus, R is Universal gas constant
and T is absolute temperature. The creep behavior of Al-based microcomposites is related to
modified creep behavior of aluminum solid solution alloys, and the equations developed for
solid solution alloys can be used to described the creep behavior of composites provided that
the applied stress is replaced by an effective stress. Thus, the threshold stress for creep in Albased
microcomposites is associated with interactions between dislocations and fine
dispersion of particles. These particles may be fine oxides in PM MMCs or precipitates in the
matrix alloy of cast composites [42-44]. Introducing a threshold stress and its temperature
dependence into the creep rate analysis yields a true stress exponent, n, of 3, 5 or 8, and true
creep activation energy. For composites with a true exponent close to 3, dislocation viscous
glide is rate controlling with an activation energy for creep is associated with interdiffusion of
the solute atoms. On the other hand, dislocation climb process predominates for n = 5 in
which the activation energy for creep is associated with aluminum lattice self-diffusion. For
the composites with a true exponent close to 8, creep deformation is controlled by the lattice
diffusion and its rate is proportional to the third power of substructure grain size λ [45, 46].
Mathematically, the phenomenological creep rate equation for n = 8 can be written as:
ε& = S (DL/b 2 ) (λ/b) 3 [(σ- σo)/E] 8 [2]

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S.C. Tjong
where DL is lattice self-diffusion coefficient, λ sub-grain size, E Young’s modulus and S a
numerical constant. A substructure is formed due to an increase in the dislocation density as a
consequence of the thermal mismatch between the matrix and the reinforcement. The size of
substructure is controlled by the interparticle spacing. Generally, subgrains can be generated
more easily in pure Al than in the Al solid solution alloys during creep deformation [45, 46].
Figure 8. Creep rate vs applied stress for the Al/1vol.% Si-N-C nanocomposite at 573 -673 K [41].
Figure 9. Arrhenius plot of steady creep rate against 10 3 /T for Al/1vol.%Si-N-C (25 nm)
nanocomposite [41].

Recent Advances in Discontinuously Reinforced Aluminum… 285
Figure 10. Comparison of creep behavior between Al/1vol.%Si-N-C (25 nm) nanocomposite (open
symbol) and Al/15vol% SiC (3.5μm) composite (solid symbol) at 573 and 623 K [41].
The creep behavior of the nanoparticle reinforced composites is mainly depended on the
matrix materials selected, i.e pure aluminum and aluminum solid solution alloy. The high
temperature creep strength of micrograined aluminum is also greatly improved by the
addition of low volume content of ceramic nanoparticles. Tjong et al. demonstrated that the
creep resistance of the Al/1vol.%Si3N4 (15nm) and Al/1vol.%Si-N-C (25 nm)
nanocomposites is about two orders of magnitude higher than that of the [40, 41]. Fig. 8
shows the variation of steady creep rate vs applied stress for the Al/1vol.%Si-N-C (25 nm)
nanocomposite. The Arrhenius plot of creep rate against 10 3 /T for this nanocomposite is
shown in Fig. 9. The nanocomposite exhibits an apparent stress exponent (n) varying from
15.7 to 23.0 and an apparent creep activation energy (Q) of 248 kJ/mol. The apparent
activation energy of the Al/1vol.%Si-N-C nanocomposite is much higher than that for lattice
diffusion of aluminum (142 kJ/mol). Similar high apparent values of n and Q values are also
observed for the Al/1vol.% Si3N4 nanocomposite. For the purposes of comparison, the creep
rates of the Al/15vol.% SiCp (3.5 µm) microcomposite and the Al/1vol.%Si-N-C (25 nm)
nanocomposite at 573 and 623 K are presented in Fig. 10. It is evident that the creep rate of
the Al/1vol.%Si-N-C nanocomposite is about two orders of magnitude lower comparing to
the Al/15vol.% SiCp (3.5 µm) microcomposite. To rationalize the high apparent values of n
and Q of the Al/1vol.%Si-N-C nanocomposite, a slip creep mechanism of constant
substructure as given in Eq (2) is applied to the Al/1vol.%Si-N-C nanocomposite (Fig. 11). It
appears that the datum points at three temperatures can be fitted linearly. It is considered that
the nanoparticles of very small volume content (1 vol.%) pin the subgrain boundaries
effectively. Consequently, the microstructure of nanocomposite remains unchanged during
creep deformation. The threshold stress can be determined from Fig. 12 by extrapolating the
linear regression line to zero strain rates. The values of threshold stress are determined to be
36.3, 25.7 and 17.3 MPa at 573, 623 and 673 K, respectively. It is obvious that the threshold

286
S.C. Tjong
stress is temperature dependent. By plotting the lattice diffusion compensated creep rate
(ε& /DL) against the modulus-compensated effective stress (σ -σo/E), the datum points for all
temperatures merge into a straight line with a slope of 8 (Fig. 12). This implies that the creep
rate of the Al/1vol.%Si-N-C nanocomposite is subgrain forming dislocation creep controlled
by lattice-diffusion.
1 / 8
Figure 11. Variation of ε& with applied stress on double linear scales for Al/1vol.%Si-N-C (25 nm)
nanocomposite [41].
Figure 12. Variation of diffusivity normalized steady creep rate, ε& /D L, with modulus-compensated
effective stress, σ -σ o/E, on double logarithmic coordinates for Al/1vol.%Si-N-C (25 nm)
nanocomposite [41].

Recent Advances in Discontinuously Reinforced Aluminum… 287
Figure 13. TEM micrograph showing interaction between dislocations and alumina nanoparticles
during creep of PM 2024 Al/Al2O 3 nanocomposite at 10 MPa, 678 K [47].
Figure 14. Creep rate vs effective stress on a logarithmic scale for PM 2014Al/Al 2O 3 nanocomposites
prepared at (a) 0.3 and (b) 1.0 % oxygen levels [47].
Recently, Mohamed and coworkers studied the creep mechanism of the PM aluminum
solid solution alloy (2014 Al) reinforced with alumina nanoparticles [47]. The alumina
nanoparticles of 30 and 35 nm are intentionally formed in 2014 Al during sintering via the
introduction of water moisture with oxygen levels controlled at 0.3 and 1.0 wt%, respectively.
Analysis of the creep data of this alloy reveals the presence of temperature- dependent

288
S.C. Tjong
threshold stress resulting from the interaction between moving dislocations and alumina oxide
nanoparticles (Fig. 13). Such dislocation-particle interaction would impede lattice dislocation
movement, thereby reducing creep rate of the composite. By incorporating the threshold
stress into analysis, plots of creep rate versus effective stress yield straight lines with different
slopes, i.e. n = 3 for low stress region and 5 for high stress regime (Fig. 14). Hence, the creep
behavior of nanocomposite is consistent with the behavior of Al-Cu solid solution alloy (2014
Al) that exhibits a transition from viscous glide (n = 3) to the high-stress region (n = 5) where
dislocations break away from the solute atom atmosphere. They indicated that the true creep
characteristics of PM 2024 Al/Al2O3 nanocomposite are consistent with those reported for
aluminum solid-solution alloys [43]. Therefore, the creep deformation of nanocomposite with
a matrix containing solutes is controlled by a viscous glide slip mechanism.
Ultrafine Grained Matrices
As mentioned above, conventional PM blending method yields Al nanocomposites with
inhomogeneous distribution of reinforcing particles within the metal matrix. Cryomilling can
provide a homogeneous dispersion of reinforcing particles in submicrometer or
nanocrystalline matrix. The subsequent hot consolidation of cryomilled nanopowders into
final bulk products causes the composites to have an UFG structure as a result of grain growth
of the matrix. Schoenung and coworkers investigated the microstructure and tensile behavior
of bulk nanostructures 5083 Al/5 vol.% SiC (25 nm) composite prepared by cryomilling
followed by hot isostatic pressing and hot rolling [48]. They reported that the hot rolled
composite consists of regions dispersed with SiC nanoparticles (100 -200 nm) and regions
free of SiC nanoparticles (~ 700 nm). Fig. 15 shows the TEM micrograph of the SiCdispersed
region in which SiC nanoparticles are distributed homogeneously within the
ultrafine grains of the 5053 al matrix. The tensile properties of such composite from room
temperature to 573 K are shown in Fig. 16. The composite exhibits very high tensile strength
at room temperature but extremely low ductility. The strength decreases but the ductility
increases with increasing test temperatures. In a nanocomposite with an UFG matrix, the
dislocation movement in the matrix is restricted by the high density of grain boundaries.
Consequently, the composite exhibits high tensile strength but very low tensile ductility.
It is well known that nanocrystalline (NC) materials exhibit very low tensile ductility and
toughness due to the lack of strain hardening [9]. The presence of coarser grains within the
nanocrystalline matrix can enhance the ductility of nanostructured materials at the expense of
mechanical strength [49-51]. Different toughening approaches have been proposed to enhance
the ductility of NC materials either via thermomechanical treatment or cryomilling. Recently,
Lavernia and coworkers reported that the UFG Al-Mg alloys with a bimodal microstructure
exhibit a combination of high strength and good ductility [52- 55]. Such alloys were
synthesized by consolidation of a mixture of cryomilled Al-Mg and unmilled powders.
Consequently, strain hardening is regained in CG regions while maintaining high strength in
NC regions. The CG grains can provide more dislocation activity than the NC grains. Ductilephase
toughening in bi-modal structured Al-Mg alloys is attributed to the occurrence of crack
bridging as well as delamination between UFG and CG regions during plastic deformation
[51].

Recent Advances in Discontinuously Reinforced Aluminum… 289
Figure 15. Bright field TEM micrograph of the hot rolled 5053 Al/5vol.% SiC composite showing the
dispersion of SiC nanoparticles in ultrafine grains [48].
Figure 16. Tensile stress-strain curves for the hot rolled 5053 Al/5 vol.% SiC composite at various
temperatures [48].
Based on this approach, Schoenung and coworkers prepared bimodal 5083Al/10 wt%B4C
composite by blending cryomilled composite powders with an equal amount of CG 5083 Al
followed by CIP and extrusion [15]. Figs. 17(a) -17(b) show the microstructure of bimodal
composite consisting of UFG (NC) Al and CG Al. The B4C are uniformly distributed in the

290
S.C. Tjong
NC Al, and the NC Al and the CG Al are alternately distributed. (Fig.17(a)). This bimodal
composite exhibits very high compressive yield strength of 1065 MPa comparing to 504 MPa
of bimodal 5083 alloy. However, the composite still exhibits low compressive ductility
(0.8%). Annealing the composite at 723 K improves its ductility to 2.5%. Fig. 18 shows the
temperature dependence of the yield strength for the tri-modal composite. The yield strength
decreases rapidly with test temperatures up to 473 K, followed by a relatively slow decrease
at higher temperatures. At 473 K, the compressive yield stress of the tri-modal composite is
282 MPa, being higher than that of the heat-treated 5083 alloy at room temperature [15]. In
another study, Scheonung and coworkers fabricated bimodal 5083Al/6.5vol.% SiC (25 nm)
composite by blending cryomilled composite powders with an equal amount of CG 5083 Al
followed by HIP and hot rolling [56]. Such nanocomposite exhibits improved tensile ductility
of 2.6% when compared with the nanocomposite consolidated from 100% of the cryomilled
composite powders having a tensile ductility of 0.5%. This is because the ductile coarsegrains
can undergo larger extent of plastic deformation, while ultrafine grains exhibit limited
deformation.
Figure 17. Bright field TEM images for the bimodal 5083Al/10 wt% B 4C composite in the (a) extrusion
direction and (b) transverse direction, with the inset being the selected area diffraction patterns taken at
the interface between the NC Al and B 4C [15].
The creep behavior of the UFG composites is now considered. Presently, little is known
regarding the high temperature creep behavior and deformation mechanism of such
composites. What is the effect of reinforcing particles on the UFG composites having large
grain boundary areas? Would these particles act as effective obstacles to the dislocation
movement and hinder grain boundary sliding and diffusional flow during high temperature
creep? If they do, the creep rates of UFG composites would reduce dramatically. More
research work is needed in this area in near future to elucidate these problems. Nevertheless,
proper understanding of the creep behavior of near-nanostructured Al-based alloy shed light
on the creep deformation of UFG composites. Very recently, Chauhan et al. investigated the
creep behavior of an UFG Al 5083 alloy at 573 – 648 K [57]. The alloy was prepared by
consolidating cryomilled powders via HIP and extrusion. Analysis of the creep date reveals
the presence of a temperature dependence threshold stress. Incorporation of this threshold

Recent Advances in Discontinuously Reinforced Aluminum… 291
stress into a modified creep equation yields a true stress of ~ 2 and a true activation energy
close to that for boundary diffusion for Al, indicating that the rate controlling process is
related to grain boundary sliding. In other words, the grain boundary activity becomes
dominant in an UFG 5083 Al alloy during creep deformation at high temperatures.
Processing of DRA composites through SPD is an effective route to refine the grain size
of composites to submicrometer level and to disperse the reinforcing particles homogeneously
within the UFG matrix. Langdon and coworkers studied microstructural development in an
Al-6061 composite reinforced with 10 vol.% Al2O3 particulates by means of the HPT and
ECAP techniques [23]. The average size of particulates is ~ 10 μm. For HPT, the samples
were strained at room temperature to a total strain of ~ 7 under a pressure of 3.5 GPa. For
ECAP, samples were pressed for eight passes at 673 K, and two additional passes at 473 K,
giving a total strain of ~ 10. Substantial grain refinement of aluminum alloy matrix can be
achieved using both techniques, i.e., a mean grain size of ~0.2 μm is attained after HPT and
~0.6 μm after ECAP. The microstructures of strained composites consist of an array of very
small grains with poorly defined boundaries. There is no refinement for the alumina
microparticles after HPT or ECAP treatment. The strength of the ECAP 6061 Al/Al2O3
composite is increased by almost two fold by ECAP and close to a factor of ~3 by torsion
straining due to the grain refining of aluminum alloy matrix [23]. In general, ECAP treatment
does not cause fracture of reinforcing particles during plastic straining, especially for finer
particulates [28]. However, limited particle cracking is found for the 6061 Al composite
reinforced with large alumina particulate (7.4 μm) subjected to ECAP at room temperature
[58].
Figure 18. Temperature dependence of the compressive yield strength for the tri-modal 5083Al/10 wt%
B4C composite. The inset shows true stress-strain curves tested at elevated temperatures [15].

292
S.C. Tjong
(a)
(b)
Figure 19. TEM micrograhs of (a) unreinforced pure Al after eight passes and (b) Al/5 vol.% Gr
composite after four passes at room temperature [29].
ECAP treatment of Al-based composites at room temperature is particularly attractive
from the economic viewpoint. Proper selection of reinforcing particulates that experience no
cracking is of technological interest. Very recently, Saravanan et al. used ECAP to refine the
matrix grains of the Al/5 vol.% Gr (65 μm) composite at room temperature [29]. The soft and
self lubricating nature of graphite can prevent the fracture of particulates during ECAP
treatment. Figs. 19(a) shows a typical TEM micrograph of pure aluminum after eight ECAP
passes at room temperature. The microstructure is characterized by well defined subgrains
with a size of ~ 620 nm. In contrast, a significant grain refinement, down to the
submicrometer level of ~ 300 nm can be achieved by pressing the Al/5 vol.% Gr composite at

Recent Advances in Discontinuously Reinforced Aluminum… 293
room temperature for only four passes (Fig. 19(b)). Moreover, the grain boundaries of
submicron grains of the composite are diffused comparing to a well defined structure of Al
grains. The selected area electron diffraction (SAD) patterns of pure Al and the composite
reveal numerous spot features indicating the presence of an array of many ultarfine grains
having random distribution of orientations (insets of Figs. 1(9a)-(b)). The tensile strength of
the composite increases from 97 to 249 MPa after four ECAP passes.
Figure 20. TEM micrograph of Al/5 vol.% Al 2O 3 nanocomposite fabricated by HPT consolidation of
raw material powders under 1.5 GPa [21].
Figure 21. Tensile stress-strain curves of Al samples (1,2) and Al/5 vol.% Al 2O 3 samples (3, 4, 5)
fabricated by HPT consolidation under the pressure of 1.5 GPa and tested at 300 ºC at strain rates of 10 -
4 s -1 (1,3) and 10 -3 s -1 (2, 4) and at 400 ºC at a strain rate of 10 -4 s -1 (5) [21].

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S.C. Tjong
Apart from forming ultrafine grains in composites, ECAP treatment can also consolidate
ultrafine raw powders to produce fully dense (> 98%) bulk composite materials. Alexandrov
et al. used the HPT technique to consolidate the Al powder (50 μm) and Al2O3 nanoparticle
(50 nm) to form the Al/5 vol.% Al2O3 nanocomposite under a pressure of 1.5 GPa at room
temperature. The powder mixture of nanocomposite was ball-milled for 30 min to ensure a
uniform distribution of ceramic particles [21]. Fig. 20 shows the TEM micrograph of the HPT
consolidated Al/5 vol.% Al2O3 nanocomposite. The nanocomposite exhibits an UFG structure
having an average gain size of 120 nm. Room temperature tensile tests showed that the Al/5
vol.% Al2O3 nanocomposite have limited ductility of 1 to 2%. At 300 ºC, the nanocomposite
tested at a strain rate of 10 -3 s -1 had a plastic flow stress of ~ 66 MPa and a tensile ductility of
~ 20 % (Fig. 21). In contrast, pure Al had a flow stress of ~ 60 MPa and a tensile ductility of
~ 40 % tested at the same strain rate. However, the Al/5 vol.% Al2O3 nanocomposite showed
a high strain-rate sensitivity of flow stress at 400 K; the strain-rate sensitivity (m) was 0.35.
Strain rate sensitivity defined as the slope of logarithmic plot of the flow stress vs. strain rate.
It is an inverse of stress exponent (n) and an important parameter in superplasticity. The Al/5
vol.% Al2O3 nanocomposite exhibited a low flow stress of 20 MPa but a high tensile ductility
of ~ 200 %. The enhanced tensile ductility observed in the HPT consolidated nanocomposite
with a total elongation of ~ 200 % indicating the occurrence of superplastic-like flow
behavior. According to the literature, high strain rate super-plasticity can be achieved in
ECAP processed aluminum alloys with UFG structures [59]. High strain rate superplasticity
in the sub-micron metals is often characterized by very high flow stresses or pronounced
strengthening. Grain boundary sliding is considered to be the dominant deformation mode for
superplasticity in the sub-micron and nanocrystalline metals [60]. Future challenges for
materials scientists are to elucidate the underlying creep and superplastic deformation
mechanisms of aluminum based nanocomposites having UFG and nanocrystalline matrices.
Conclusions
The development of aluminum nanocomposites is still in embryonic stage and there are many
challenges in this field in the years ahead. Considerable progress has been made in the
fabrication, microstructural and mechanical characterization of novel aluminum-based metal
matrix nanocomposites in recent years. The nanocomposites can be simply prepared by
incorporating very low volume contents of ceramic nanoparticles into aluminum matrix via
PM or ingot casting. The nanocomposites thus prepared exhibit excellent mechanical
properties including high yield strength and superior creep resistance. However,
agglomeration of nanoparticles occurs readily during the composite fabrication, leading to
poorer mechanical performance of composites with higher filler content. This problem can be
eliminated in cast nanocomposites by using high-intensity ultrasonic waves to disperse the
nanoparticles in molten aluminum. In the case of PM nanocomposites, cryomilling and severe
plastic deformation processes have emerged as the two major processes to produce aluminum
based composites having ultrafine grained matrix structures and homogeneous dispersion of
reinforcing particles within the matrices.

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A
Aβ, 283
accounting, 44
accuracy, 69, 124, 263
acetone, 136
acidity, 116
acoustic waves, 167
activation energy, 283, 285, 291
actuation, 125
actuators, xi, 106, 115, 257, 258, 272, 273
adaptation, 122
additives, 110, 118, 126
adhesion, ix, 11, 110, 112, 119, 120, 125, 141, 143,
212
adhesion properties, 120
adhesion strength, 143
adhesives, 162
adjustment, 183, 196, 198
aerospace, vii, viii, 2, 109, 110, 111, 112, 118, 119,
120, 122, 124, 126, 130, 276
age, ix, 109
Alabama, 126
algorithm, 52, 54, 65, 73, 75, 81, 103, 105, 216
alkalinity, 116
alloys, xi, 14, 47, 125, 163, 275, 276, 281, 283, 284,
288, 294
alternative(s), 6, 13, 36, 58, 118, 119
aluminium, 4, 162, 210, 221
aluminium alloys, 4
aluminum, xi, 14, 116, 117, 118, 119, 164, 275, 276,
277, 279, 280, 281, 283, 285, 287, 288, 289, 291,
292, 293, 294, 295
ambiguity, 244
amplitude, 136, 168, 174, 191, 193, 206, 215, 219,
225
anisotropy, 22, 42, 46, 117
annealing, 290
INDEX
AP, 173, 175, 176, 178, 182, 183, 184, 186, 188, 189
Arborite, vii
arithmetic, 5, 6, 41
ash, 255
aspect ratio, 44, 113, 115, 126, 130
asphalt, vii
assessment, 206, 238
assignment, 154
assumptions, 20, 23, 232
asymptotic, 243, 248, 253
atmosphere, 120, 131, 288
atoms, 281, 283
attention, ix, 110, 130
automation, 113, 117
averaging, 6
avoidance, 119
awareness, 110
B
barriers, 283
beams, 120, 219
behavior, xi, 4, 13, 49, 73, 75, 78, 105, 110, 118,
134, 144, 164, 244, 246, 257, 258, 272, 281, 283,
285, 288, 290, 294
Belgium, 51, 101, 102, 104, 105, 106, 209, 234, 235,
236
bending, 49, 59, 64, 72, 106, 196, 198, 210, 214,
215, 216, 218, 219, 225, 226, 227, 228, 229, 230,
231, 232, 233, 235, 236, 267, 269, 270, 271, 272
benefits, ix, 109, 116, 118, 119, 121, 124, 125, 126
bias, 217
biomaterials, 125, 128
blends, 255
BMI, 123
bonding, 143, 266
bounds, 12, 69
Bragg grating, 125, 204, 211, 212

298
braids, 115
broadband, 211
bubbles, 133, 278
bulk materials, 115
C
candidates, 38, 120
capillary, 134
carbon, vii, viii, ix, x, 2, 4, 6, 10, 11, 12, 18, 19, 22,
23, 27, 31, 36, 38, 42, 44, 46, 47, 48, 49, 50, 58,
81, 92, 109, 110, 112, 113, 115, 117, 118, 119,
120, 121, 123, 125, 126, 127, 129, 130, 131, 132,
133, 134, 135, 136, 137, 138, 139, 141, 143, 144,
145, 147, 149, 151, 153, 155, 156, 157, 159, 161,
162, 163, 164, 166, 205, 206, 207, 209, 212, 213,
214, 217, 220, 221, 225, 226, 227, 228, 234, 242,
248, 255, 256
carbon nanotubes, 44, 46, 49, 112
carbonization, 120
case study, 104, 127, 198
cast(ing), 277, 278, 280, 281, 282, 283, 294
catalyst, 122, 123
catalytic effect, 139
cell, 173, 191, 241, 262
cellulose, vii
ceramic(s), vii, xi, 2, 6, 163, 257, 258, 266, 273, 275,
276, 277, 278, 281, 283, 285, 294
chain mobility, 141
chemical composition, 116
chemical properties, vii, 116
chemical reactions, 276
chemical reactivity, 115
chemical structures, 119
chicken, 125
China, 125
civil engineering, 130
classes, 115
clustering, 277, 279
clusters, 277
coatings, 112, 115, 118
collagen, vii
collisions, 279
commercial, 72, 106, 117, 120, 163, 226, 266
communication, 234
community, 126
compatibility, 26
complexity, 105, 122, 219
compliance, 72, 84, 85, 93, 259
complications, 215
components, vii, 7, 18, 19, 22, 23, 28, 32, 33, 36, 38,
42, 113, 115, 121, 124, 147, 148, 165, 166, 212,
257, 261
Index
composites, vii, viii, ix, x, xi, 1, 2, 3, 4, 6, 10, 13, 14,
15, 21, 22, 28, 31, 38, 41, 42, 43, 44, 45, 46, 47,
48, 50, 51, 52, 61, 69, 70, 101, 102, 103, 106, 109,
110, 112, 113, 114, 115, 116, 117, 118, 119, 120,
121, 122, 125, 126, 129, 130, 135, 143, 144, 148,
149, 154, 155, 156, 157, 158, 159, 160, 161, 162,
163, 164, 166, 204, 205, 206, 207, 209, 210, 212,
213, 214, 215, 217, 218, 221, 225, 233, 234, 235,
236, 254, 255, 256, 257, 258, 272, 275, 276, 277,
279, 281, 282, 283, 285, 288, 290, 291, 292, 294
composition(s), 43, 171, 183, 276
compounds, 114
computation, 28, 67, 94, 222, 244, 246
computer simulation, 123
computing, 104, 244
concentration, viii, 1, 2, 25, 26, 28, 115, 134, 135,
148, 149, 158, 164, 221, 224, 225, 258, 273
conception, 101
concrete, vii
concurrent engineering, 104
condensation, 277
conditioning, 123
conductivity, 112, 113, 119, 132
confidence, 34
configuration, 3, 34, 65, 69, 73, 78, 81, 88, 91, 173,
175, 201
confusion, 250
Congress, 101, 106, 107
consolidation, 135, 136, 138, 288, 293
constant rate, 252
constraints, 52, 53, 54, 55, 56, 67, 69, 72, 73, 80, 81,
89, 90, 98, 99, 102, 107, 121
construction, 51, 111, 183
continuity, 71
control, x, 62, 116, 117, 124, 125, 129, 173, 191, 193
conventional composite, 126
convergence, 74, 77, 78, 83, 85, 94, 95, 97, 100, 102,
154, 231
convex, viii, 51, 52, 54, 55, 61, 64, 65, 66, 72, 74,
75, 79, 101, 103
cooling, 136
corn, 125
correlation(s), x, 168, 203, 209, 221
corrosion, 2, 110, 118, 119, 132, 241
cosine, 135
cost saving, 122
costs, ix, 110, 114, 121, 124, 125
Coulomb, 42
coupling, viii, 1, 44, 45, 59, 63, 64, 120, 148, 164,
219, 257
covalent bond, 119
coverage, 124
CPU, 56, 73, 98, 232

crack, 52, 99, 100, 116, 143, 162, 167, 168, 196,
198, 237, 239, 273, 288
creep, xi, 115, 218, 275, 281, 283, 284, 285, 286,
287, 288, 290, 291, 294
critical value, 260, 261
cross-linked polymers, 139
crystal polymers, 118
crystal structure, 132
crystalline, 47, 119, 132, 163
crystals, 133
curing, 113, 117, 121, 122, 123
cybernetics, 104
cycles, 54, 73, 84, 210, 214, 215, 277
cycling, 207
D
damping, 112, 113, 114, 115, 166
database, 121
DD, 190, 205, 206, 239, 240, 243, 244, 246, 247,
248, 250, 251, 252, 253, 254
decomposition, 139, 141
decomposition temperature, 139, 141
defects, 120, 172, 219
defense, 120
definition, 3, 32, 59, 61, 63, 66, 77, 80, 82, 98, 113,
169, 187, 238, 240, 244, 247, 249, 253
deformability, 162
deformation, 105, 143, 148, 164, 182, 196, 216, 241,
277, 281, 283, 284, 285, 288, 290, 291, 294
degradation, x, 129, 131, 141, 200, 211, 215, 220,
225, 233, 236, 243, 257
degree of crystallinity, 119
Delaware, 125
delivery, 136
demand, xi, 118, 121, 275
Denmark, 102, 103, 104, 234
density, 11, 26, 44, 62, 76, 77, 79, 82, 96, 116, 117,
120, 130, 134, 135, 260, 264, 267, 276, 277, 279,
284, 288
derivatives, 75, 76, 77, 78, 103
detection, 204, 213, 216, 218, 219, 234, 254
deterministic, 52, 54, 56, 72
deviation, 10, 18, 41, 143, 144, 147
diamond, 173, 191
diaphragm, 121
dielectric, 259, 261, 263
dielectric permittivity, 259, 261
differential equations, 215
differential scanning calorimetry (DSC), x, 129, 136,
139, 140, 142, 161
differentiation, 258
diffraction, 45, 46, 48, 290
Index 299
diffusion, 2, 25, 28, 44, 46, 134, 136, 283, 284, 285,
286, 291
diffusion process, 28, 44
diffusivity, 286
discontinuity, 235
discrete variable, 101
discretization, 184, 185, 186
discrimination, 167
discs, 102, 106
dislocation, 282, 283, 284, 286, 288, 290
dispersion, xi, 69, 115, 125, 133, 134, 135, 143, 144,
167, 275, 277, 278, 279, 280, 282, 283, 288, 289,
294
displacement, 26, 27, 59, 64, 87, 88, 149, 150, 152,
155, 166, 169, 173, 177, 178, 179, 180, 181, 191,
193, 195, 196, 197, 198, 200, 201, 203, 210, 215,
216, 222, 224, 225, 226, 227, 228, 231, 233, 238,
239, 240, 241, 244, 245, 246, 258, 259, 262, 266,
269, 273
distribution, 67, 68, 115, 122, 147, 156, 164, 183,
215, 264, 265, 268, 269, 272, 276, 277, 279, 288,
293, 294
doors, 124
double logarithmic coordinates, 286
dream, 121
ductility, 239, 277, 281, 282, 288, 290, 294
durability, 112, 113, 237
duration, 168, 210, 248, 253, 254
dynamic mechanical analysis, 141
E
ears, 167
EI, 5, 7, 15, 22
Einstein, Albert, 258
elastic deformation, 143
elasticity, 12, 23, 48, 96, 166
elastomers, 44, 47
electric field, 120, 257, 258, 260, 263, 264, 267, 268,
269, 270, 272
electric potential, 258, 262, 263
electrical conductivity, 113, 116, 117
electrical properties, 118, 163
electrical resistance, 166, 204, 214, 234
electrodes, xi, 213, 234, 257, 258, 266, 272, 273
electromagnetic, 115, 212
electron, 147
electrospinning, 120
elongation, 294
embryonic, 294
emission, x, 165, 166, 167, 168, 169, 192, 196, 204,
205, 206
emulsification, 134

300
endothermic, 139
endurance, 32
energy, x, xi, 61, 62, 72, 76, 77, 79, 82, 83, 84, 89,
91, 92, 99, 105, 110, 115, 118, 119, 122, 133, 143,
165, 166, 167, 168, 169, 170, 171, 172, 176, 177,
178, 179, 180, 181, 182, 183, 184, 195, 196, 197,
198, 199, 200, 201, 204, 205, 225, 237, 238, 239,
240, 241, 243, 244, 245, 246, 248, 249, 250, 251,
252, 253, 254, 255, 256, 257, 258, 260, 261, 263,
264, 268, 279
energy consumption, 143
energy density, xi, 61, 62, 76, 82, 83, 89, 91, 92, 257,
258, 260, 261, 263, 264, 268
energy emission, 201
environment, 110, 141, 279
environmental conditions, 2, 44
environmental control, 113
epoxy, vii, ix, 2, 6, 10, 11, 12, 13, 18, 19, 20, 22, 23,
27, 31, 34, 35, 36, 39, 41, 42, 44, 46, 48, 58, 81,
84, 92, 112, 116, 117, 120, 121, 125, 129, 130,
131, 138, 139, 141, 149, 150, 157, 161, 162, 163,
166, 173, 191, 204, 205, 206, 207, 210, 211, 212,
217, 218, 220, 231, 233, 235, 236, 240, 247, 248,
254, 255, 256
epoxy resins, 31, 34, 48, 120
equal channel angular pressing, 277
equality, 107, 239
equilibrium, 26, 27, 134
equipment, x, 110, 118, 123, 124, 139, 209
esters, 123
estimating, 5, 6, 8, 10, 12, 13, 42, 43, 47
European, viii, 49, 51, 52, 101, 102, 111, 162, 163,
206, 234, 235
evaporation, 120
evidence, 198
evolution, 44, 82, 88, 148, 204, 210, 211, 214
exclusion, 5
exothermic, 139
exposure, 117
extraction, 135
extrusion, 277, 279, 289, 290
F
FAA, 121
fabric, 125, 205, 207, 211, 212, 214, 217, 218, 220,
234, 236, 255, 256
fabrication, xi, 124, 136, 138, 275, 276, 277, 294
failure, viii, x, xi, 2, 31, 32, 33, 34, 35, 36, 38, 39,
40, 41, 43, 46, 47, 48, 53, 122, 130, 131, 143, 144,
148, 149, 155, 156, 158, 159, 160, 162, 164, 165,
166, 168, 171, 172, 175, 182, 183, 184, 187, 194,
Index
195, 198, 200, 201, 204, 205, 207, 210, 221, 237,
253, 254, 275
family, 72, 101, 113, 118
fatigue, x, 2, 29, 110, 117, 166, 204, 209, 210, 211,
212, 213, 214, 215, 216, 217, 218, 219, 220, 221,
225, 226, 228, 229, 230, 231, 232, 233, 234, 235,
236, 254, 256
fiber bundles, 135, 156
fiber optics, 125
fibers, vii, viii, ix, 2, 3, 10, 11, 12, 13, 14, 18, 19, 20,
23, 27, 28, 29, 31, 35, 36, 42, 44, 49, 51, 52, 53,
54, 56, 57, 58, 61, 62, 65, 66, 67, 69, 70, 72, 73,
75, 81, 84, 85, 86, 88, 96, 97, 100, 101, 109, 110,
112, 113, 114, 115, 116, 117, 118, 119, 120, 122,
123, 124, 125, 126, 127, 135, 147, 148, 149, 150,
156, 161, 162, 192, 246, 248
filament, ix, 48, 129, 131, 133, 134, 136, 137, 138,
147, 161, 205
filled polymers, 130
fillers, 112, 114, 115, 116, 130, 132, 143
film(s), 50, 115, 122, 123, 138, 164
finance, 233
finite element method, 67
fires, 121
First World, 106, 107
fishing, vii
fixation, 232
flame, 113, 114, 115, 118
flexibility, 110, 117, 119, 124, 126, 243
flexural strength, x, 129, 130, 136, 143
fluctuations, 133
fluid, 25, 27, 121, 133
focusing, 8
Formica, vii
fracture processes, 147
fractures, 182, 194, 213
France, 1, 235
freedom, 118
friction, 151, 167, 215, 223, 224, 225, 226, 227, 228
fuel cell, 115
fulfillment, 119
functionalization, ix, 109, 112
G
GAO, 206, 254, 255
gas phase, 277
gauge, 193, 211, 215, 232
generation, 44, 112, 113
genetic algorithms, 52, 71, 104, 106
genre, 119
Germany, 50, 102, 106, 107, 234

glass, x, 47, 48, 84, 111, 116, 117, 120, 121, 123,
125, 126, 129, 135, 139, 141, 161, 204, 205, 206,
209, 210, 211, 217, 218, 220, 231, 234, 235, 236,
240, 247, 254, 255, 256
glass transition, x, 129, 139, 141, 161, 217
glass transition temperature, x, 129, 139, 161, 217
GNP, 115
gold, 118, 147
grain boundaries, 281, 288, 293
grain refinement, 291, 292
grains, 262, 277, 279, 280, 281, 288, 289, 290, 291,
292, 293, 294
graph, 140, 142
graphite, 112, 115, 117, 120, 173, 191, 205, 206,
233, 256, 292
gravimetric analysis, 141
Greece, 236
groups, 121
growth, 110, 116, 117, 118, 121, 143, 167, 168, 183,
196, 204, 215, 225, 235, 238, 239, 253, 256, 278,
288
hardness, 113, 117, 118, 132, 241
head, 191, 193, 241
healing, 125
health, 110, 125
heat(ing), 111, 121, 122, 132, 134, 136, 167, 217,
290
heat transfer, 121
height, 215, 241
hemicellulose, vii
heterogeneity, 246
high fat, 116
homogeneity, 277, 280
Hong Kong, 275
hot pressing, 279, 280
hot spots, 278
house(ing), 105, 119, 222
humidity, 116
hybrid, vii, 115, 116, 117, 123, 163, 234
hybridization, ix, 109, 116, 118
hydroxyapatite, vii
hypothesis, 35, 135, 226, 228
hysteresis, 225, 226, 227, 228
hysteresis loop, 225, 226, 228
H
identification, viii, 1, 2, 3, 14, 15, 17, 30, 31, 39, 40,
42, 43, 44, 167, 184, 204, 240
I
Index 301
identity, 7
images, 221, 290
imaging, 205
immersion, 134
impact energy, 237, 238, 239, 240, 241, 242, 243,
244, 245, 246, 247, 248, 251, 252, 253, 254
implementation, 98, 198, 215, 236
impregnation, ix, 122, 123, 129, 133, 136, 137, 161,
163
in situ, 117, 164
incidence, 219
inclusion, 8, 9, 11, 43, 45, 49, 112
independent variable, 63
indication, xi, 237
indicators, 201
indices, 258
industrial application, 52, 67, 73, 277
industrial sectors, 126
industry, ix, 110, 116, 117, 118, 121, 124, 130
inelastic, 44
inertia, 232
infinite, 9, 11, 28, 33, 40
infrastructure, 110, 118, 122
initiation, 225, 238, 256, 273
insight, 233
inspection, 218
Instron, 191
insulation, 174
integration, 223, 228, 230, 231, 241
integrity, x, 121, 165, 172, 183, 201
intelligence, 126
intensity, 136, 258, 277, 294
interaction(s), 4, 9, 32, 33, 34, 48, 50, 115, 123, 135,
141, 161, 223, 262, 273, 282, 283, 287, 288
interface, ix, 99, 110, 112, 119, 122, 127, 130, 135,
143, 148, 149, 150, 151, 153, 155, 156, 157, 160,
161, 162, 164, 265, 266, 269, 272, 273, 276, 290
interfacial adhesion, ix, 109, 110, 112, 119, 120
interfacial bonding, 162
interfacial properties, 120
interference, 115, 212
intermetallic nanoparticles, 276
international standards, 210
interpretation, 14, 182, 215
interval, 185
intuition, 134
invariants, 58, 63
inversion, 14, 15, 16, 31, 201
investment, 124
ion implantation, 120
IR, 122, 204
iron, 14, 48
irradiation, 163

302
isostatic pressing, 279, 288
Italy, 237, 255
iteration, 52, 55, 77, 83, 84, 94, 95, 98, 154, 155
Japan, 273
kinetic energy, 239
knowledge transfer, 209
J
K
L
labor, 123
laminar, 182, 183
laminated composites, 42, 103, 106, 205
lamination, 56, 63, 64, 65, 66, 67, 73, 101, 103, 104,
116, 175
laser, 120, 277, 282
laser ablation, 120, 277
laws, 24, 45, 47
lead, 15, 22, 74, 77, 119, 123, 124, 134, 143, 211,
217, 263, 279
leakage, 194, 198
leaks, 122
lifetime, 212
lignin, vii
limitation, 134
liquid nitrogen, 121, 279
liquids, 134
literature, viii, x, 1, 2, 3, 4, 10, 14, 18, 21, 22, 31, 32,
33, 35, 38, 42, 43, 44, 45, 51, 52, 71, 72, 73, 100,
218, 237, 238, 244, 248, 253, 294
localization, 10, 11, 13, 17, 20, 30, 42, 45, 191
location, 60, 99, 241
London, 46, 163, 234, 235
M
machine learning, 103
magnetic properties, 116
management, 113, 115, 117
manipulation, 80
manufacturer, 113
manufacturing, ix, 2, 3, 43, 61, 69, 102, 104, 105,
109, 110, 112, 113, 122, 123, 126, 129, 130, 133,
135, 138, 161, 212, 275
mapping, x, 209
market(s), 110, 116, 119, 122, 124
Index
masking, 117
mass loss, 44
mass transfer process, 134
material degradation, 210
materials science, 6, 14
mathematical programming, 54, 65, 72, 102
mathematics, 5, 46
matrix, vii, viii, ix, x, 1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 28, 29,
30, 31, 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 48,
56, 57, 58, 63, 96, 109, 110, 112, 115, 116, 117,
119, 120, 121, 126, 130, 134, 135, 141, 143, 144,
148, 149, 150, 151, 153, 154, 155, 156, 157, 158,
162, 163, 164, 165, 166, 167, 168, 173, 175, 182,
183, 194, 196, 198, 211, 212, 218, 233, 237, 238,
239, 246, 253, 254, 256, 275, 276, 277, 279, 280,
281, 282, 283, 284, 285, 288, 291, 292, 294
measurement, x, 44, 46, 167, 209, 210, 212, 213,
214, 234, 244
mechanical behavior, 46, 149, 162, 164
mechanical degradation, 236
mechanical energy, 198, 200, 257
mechanical properties, ix, xi, 44, 61, 69, 109, 111,
112, 113, 117, 119, 124, 130, 135, 144, 207, 255,
275, 276, 277, 281, 282
mechanical stress, 27
media, 120, 279
melt(s), 117, 118, 120, 123, 125, 133, 134, 135, 136,
163
melting, 276, 277, 279, 282
melting temperature, 276
membranes, 59, 66, 73, 86, 105
MEMS, 257
metal oxide, 115
metallurgy, 276, 277, 282
metals, vii, 2, 47, 69, 114, 116, 117, 118, 119, 126,
209, 281, 294
Micarta, vii
micrometer, 277, 281
microscopy, 218
microstructure(s), 2, 4, 22, 28, 41, 42, 43, 280, 281,
285, 288, 289, 291, 292
military, vii
Ministry of Education, 272
mixing, 78, 115, 125, 136
MMA, 76, 77, 78, 79, 83, 88, 89, 90, 100, 101, 106
MMCs, 276, 283
mobility, 141, 161
modeling, 4, 43, 52, 105, 116, 124, 204
models, vii, viii, xi, 1, 2, 3, 4, 5, 13, 14, 18, 20, 30,
41, 42, 43, 44, 45, 47, 50, 103, 105, 166, 202, 215,
220, 221, 222, 229, 232, 236, 257, 258, 266, 267

modulus, 5, 12, 26, 42, 44, 46, 49, 96, 116, 118, 119,
120, 130, 131, 135, 141, 143, 144, 150, 156, 157,
175, 176, 184, 185, 186, 187, 188, 218, 223, 276,
283, 284, 286
moisture, vii, 1, 2, 4, 5, 8, 9, 10, 11, 13, 15, 17, 19,
20, 21, 23, 25, 26, 27, 42, 43, 44, 45, 48, 119, 287
moisture content, 5, 9, 11, 20, 21, 27, 43
moisture sorption, 44
mold, 121, 122, 123, 124, 138
moldings, 123
molecular weight, 118, 127, 135
molecules, 118, 119
monomer, 119, 123
Monte Carlo, x, 129, 131, 148, 149, 162, 163, 164
Moon, 135, 163
morphology, 2, 4, 7, 10, 11, 22, 42, 43, 46, 111, 276
Moscow, 163
motion, 141, 223, 229, 257
moulding, 212
movement, 141, 223, 224, 231, 283, 288, 290
MTS, 191
multiple factors, 43
multiplicity, 31, 165
multiwalled carbon nanotubes, 46
N
nanocomposites, xi, 112, 113, 114, 115, 116, 119,
126, 130, 143, 161, 162, 163, 275, 276, 279, 280,
281, 282, 283, 285, 287, 288, 294
nanocrystalline metals, 281, 294
nanofibers, 112, 114, 115, 120
nanofillers, 115
nanometer, xi, 113, 275, 277, 281
nanometer scale, xi, 113, 275
nanoparticles, ix, xi, 113, 115, 116, 118, 129, 130,
133, 136, 139, 141, 143, 147, 161, 162, 275, 277,
278, 279, 280, 281, 282, 285, 287, 288, 289, 294
nanostructured materials, 288
nanostructures, 116, 288
nanotechnology, ix, 109, 110, 111, 120, 126
nanotube(s), 48, 50, 114, 130
nanowires, 114
NASA, 122, 127, 204
National Science Foundation, 162
negative relation, 201
negativity, 80
Netherlands, 49, 103, 105, 106, 116, 127
network, 46, 141
neutrons, 45
New Orleans, 103
New York, 47, 101, 127, 163, 295
Newton, 72, 232
Index 303
next generation, ix, 109
nickel, 163
nitrogen, 120
nodes, 223, 231, 232
noise, 133, 143, 167, 173, 174
non-linear, 143, 231, 232, 260, 272
O
observations, 34, 218
oil, 123
one dimension, 80
optimization, viii, 51, 52, 53, 54, 55, 61, 64, 65, 66,
67, 68, 69, 70, 71, 72, 73, 74, 75, 78, 79, 80, 81,
82, 83, 84, 85, 88, 89, 94, 95, 96, 97, 98, 99, 100,
101, 102, 103, 104, 105, 106, 110, 117, 122, 124
optimization method, 52, 61, 69, 89, 100
optoelectronic, 241
organic fibers, 112, 118, 119, 120
organization, 121, 132
orientation, 42, 57, 58, 61, 62, 65, 66, 67, 70, 72, 73,
75, 78, 87, 88, 96, 97, 99, 103, 105, 111, 175, 182,
183, 246
oxidation, 44, 120, 132
oxide(s), 14, 283, 288
oxygen, 132, 287
P
packaging, 276
PAN, 112, 120, 125, 131
parameter, 64, 66, 80, 89, 90, 91, 103, 150, 154, 160,
197, 204, 211, 217, 238, 294
Paris, 46, 101, 103, 235
particles, xi, 9, 110, 114, 115, 118, 130, 132, 133,
141, 143, 162, 275, 276, 277, 278, 280, 281, 282,
283, 288, 290, 291, 294
passive, 167, 212
PEEK, 120
pendulum, 49
percolation, 116
perforation, 191, 239, 240, 241, 244, 245, 246, 248,
249, 250, 251, 252, 253, 254, 255
performance, vii, xi, 2, 88, 110, 111, 113, 115, 117,
118, 120, 123, 125, 126, 130, 238, 239, 246, 257,
258, 266, 275, 294
permeability, 113, 122, 123, 124, 135
phenolic resins, 123
physical and mechanical properties, 277
physical properties, ix, 109
piezoelectric, xi, 125, 133, 167, 191, 241, 257, 258,
259, 261, 262, 263, 266, 267, 269, 272, 273

304
piezoelectricity, 261
pitch, 125
planning, 122
plasma, 117, 120, 279
plastic deformation, xi, 275, 283, 288, 290, 294
plastic strain, 291
plasticity, 294
plastics, 166, 233
platelets, 115, 130
plywood, vii
PM, 276, 277, 279, 281, 283, 287, 288, 294
PMMA, 255
Poisson ratio, 131
polarization, xi, 257, 258, 260, 261, 262, 263, 268,
269, 272, 273
polyarylate, 118
polycarbonate, 163
polycrystalline, 4, 14, 15, 48, 163
polyester(s), 47, 118, 123, 204, 254, 255
polyetheretherketone, 120
polyethylene, vii, 50, 118, 119, 127, 135, 163
polymer(s), vii, ix, x, 1, 2, 4, 6, 15, 41, 44, 48, 110,
111, 112, 114, 115, 116, 118, 119, 120, 123, 124,
125, 126, 130, 134, 135, 141, 143, 161, 209, 215,
236, 255
polymer chains, 119, 141, 161
polymer composites, x, 209, 215, 255
polymer materials, ix, 110, 115
polymer matrix, 6, 41, 44, 115, 119, 125, 130, 141
polymer solutions, 134
polymer systems, 112
polymer-based composites, 44
polymeric composites, 110, 117
polymeric materials, 112
polymeric matrices, ix, 110
polystyrene, 163
poor, 78, 134, 135, 136
porous media, 134
ports, 122
Portugal, 102
power, 110, 120, 133, 215, 238, 243, 283
PPS, 214, 217, 221, 225, 226, 227, 228
prediction, 13, 30, 31, 32, 35, 44, 47, 206, 254
pressure, x, 64, 92, 104, 121, 122, 123, 133, 138,
165, 205, 224, 241, 277, 291, 293, 294
probability, 150, 166
probe, 133
production, ix, 110, 120, 122, 124, 125, 212
program(ming), viii, 51, 52, 56, 101, 105, 106, 125,
155, 157
proliferation, 112
propagation, 69, 162, 167, 220, 225, 238, 256
proposition, 205
Index
prototype, 124
PTFE, 162
pulses, 219
pultrusion, 121, 125
pumps, 124
PVC, 46
pyrolysis, 120
Q
qualifications, 114
quantitative estimation, 184
R
radial distance, 28
radiation, 115
radius, 28, 53, 191, 226, 241, 266, 269
range, 6, 13, 31, 34, 42, 44, 49, 120, 122, 123, 124,
130, 133, 134, 135, 147, 154, 166, 169, 187, 196,
211, 239, 240, 244, 246, 247, 248, 253, 255, 263
reactive groups, 119
reactivity, 277
realism, 4, 44
reality, 167, 168, 262
reasoning, 17, 22
recalling, 52
recognition, ix, 110
reconstruction, 221
recovery, 279
recrystallization, 279
recycling, 125
redistribution, 167, 184, 215
reduction, 172, 201, 212, 233, 251
reference frame, 8, 32
refining, 291
reflection, 167
refractive index, 211, 218
regression, 185, 201, 239, 240, 285
regression line, 240, 285
reinforcement, ix, 3, 15, 17, 104, 109, 110, 112, 119,
125, 126, 210, 254, 276, 277, 279, 284
reinforcing fibers, 18, 41, 42
rejection, 167
relationship(s), x, 116, 118, 130, 134, 149, 160, 168,
239, 241, 243, 247, 259
relaxation, 81, 154
relevance, 248
reliability, viii, 2, 32, 43, 44, 166
renewable energy, 111, 112, 115
repair, 117, 118, 163
reserves, 125

esin reaction, 133, 136
resins, 39, 41, 43, 46, 118, 120, 122, 123, 124, 125,
130
resistance, x, xi, 2, 104, 111, 113, 116, 117, 118,
119, 132, 162, 166, 198, 205, 209, 213, 214, 234,
241, 255, 275, 283, 285, 294
resolution, 220
resources, 54, 172
revolutionary, 111
rheological properties, 50
Rhode Island, 104
rice, 125
robust design, 102
rods, vii
rolling, 288, 290
room temperature, 122, 124, 276, 288, 290, 291, 292,
293
Royal Society, 46
RTS, 166, 201
rubber, 130, 162
S
SA, 81, 102, 105, 163, 205, 233
SAD, 293
safety, 94
sample(ing), 8, 31, 34, 136, 147, 215, 220, 221, 224,
238, 241, 277
satellite, 122
satisfaction, 54
saturation, 239
savings, 70, 124
sawdust, vii
scattering, 243, 253
science, ix, 110, 126, 257
scull, vii
search, 66, 101, 103
selected area electron diffraction, 293
selecting, 100
SEM micrographs, 147
sensing, x, 125, 209, 212
sensitivity, 17, 56, 61, 66, 81, 94, 103, 105, 119, 139,
294
sensors, 115, 125, 166, 167, 191, 192, 211, 212, 221,
234, 263
separation, 135
series, 134, 173, 184, 213, 239, 241, 248
shape, 22, 42, 49, 53, 54, 69, 74, 101, 103, 104, 114,
115, 122, 124, 125, 132, 150, 196, 221, 225, 226,
228, 276
shape-memory, 125
shear, x, 29, 31, 32, 33, 35, 46, 48, 49, 59, 94, 103,
104, 129, 147, 148, 149, 150, 151, 155, 158, 164,
Index 305
195, 201, 210, 217, 218, 235, 255, 262, 265, 269,
272, 277, 283
shear strength, 155
shock waves, 133
shortage, 119, 125
Si3N4, 276, 281, 282, 283, 285
SIC, 129
sign, 32
signaling, 238, 249
signals, 168, 169, 173, 219, 238
silane, 120
silica, 115
silicon, ix, 14, 129, 132, 136, 163
Silicon carbide, 163
silver, 147
simulation, x, 10, 39, 45, 124, 130, 131, 148, 155,
157, 162, 163, 164, 210, 219, 221, 227, 228, 232,
233, 273
sine wave, 214
Singapore, 125
sintering, 279, 287
sites, 133
smart materials, 113
society, ix, 109
software, 45, 72, 106, 187, 266
solid state, 124
solvent(s), 111, 118, 120, 133, 134, 135, 136
Spain, 235
species, 135
specific surface, 115, 277
speed, 85, 94, 97, 111, 116, 150, 154, 173, 191, 193,
205, 220, 255
spindle, 136
sports, 110, 111, 118, 124
stability, 71, 113, 119, 143
stages, 125, 240
statistical analysis, 196
statistics, 5, 6
steel, 96, 191, 210, 223, 241
storage, vii, 124, 135
strain, x, 9, 10, 11, 13, 14, 21, 23, 31, 32, 34, 35, 36,
39, 43, 44, 45, 47, 56, 61, 62, 67, 72, 82, 89, 91,
92, 105, 130, 143, 144, 148, 154, 155, 156, 157,
158, 160, 161, 164, 165, 167, 169, 170, 172, 175,
176, 182, 183, 184, 185, 186, 187, 188, 195, 196,
198, 199, 200, 201, 204, 209, 210, 211, 212, 213,
214, 215, 217, 218, 225, 228, 232, 234, 235, 241,
254, 258, 259, 260, 263, 267, 268, 269, 272, 277,
285, 288, 291, 293, 294
strategies, 172
stratification, 68
strength, viii, xi, 2, 3, 10, 18, 19, 30, 31, 32, 33, 34,
35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 51, 53, 66,

306
67, 69, 70, 71, 72, 73, 84, 94, 101, 102, 103, 105,
106, 111, 113, 116, 117, 118, 119, 120, 121, 122,
126, 130, 131, 143, 144, 150, 154, 155, 156, 157,
158, 160, 161, 162, 163, 164, 166, 168, 169, 175,
182, 183, 203, 204, 206, 207, 215, 233, 237, 238,
241, 243, 275, 276, 277, 281, 282, 285, 288, 290,
291, 294
stress, viii, ix, x, xi, 2, 21, 24, 25, 27, 29, 31, 32, 33,
34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48,
49, 56, 59, 71, 96, 102, 110, 115, 117, 123, 130,
143, 148, 149, 150, 151, 154, 155, 156, 157, 158,
161, 165, 167, 168, 172, 175, 176, 177, 178, 179,
180, 181, 184, 185, 186, 187, 210, 212, 214, 215,
217, 221, 224, 225, 228, 235, 257, 258, 261, 262,
264, 265, 268, 269, 271, 272, 275, 282, 283, 284,
285, 286, 287, 288, 289, 290, 291, 293, 294
stress-strain curves, 157, 175, 187, 212, 289, 291,
293
stretching, 120
structural characteristics, 119
suffering, 125
Sun, 127, 273
superplasticity, 281, 294
suppliers, 120, 122
supply, 113, 133
supply chain, 113
surface area, 130
surface energy, 119
surface tension, 134, 135
surface treatment, 205
Sweden, 103
switching, xi, 257, 258, 260, 261, 262, 263, 264,
268, 269, 270, 272, 273
symbols, 76, 78
symmetry, 7, 132, 222, 232, 233, 259, 262, 266
synthesis, 106, 173, 239, 276
systems, vii, x, 110, 113, 115, 116, 117, 121, 123,
124, 129, 130, 141, 143, 162, 254
T
tanks, vii, 124
Taylor series, 55, 74, 75
technology, viii, ix, 109, 110, 112, 117, 118, 119,
126, 204, 233, 257
teflon, 138
TEM, 133, 280, 287, 288, 289, 290, 292, 293, 294
temperature, ix, x, 2, 14, 21, 43, 44, 115, 116, 117,
118, 121, 123, 124, 125, 129, 131, 132, 136, 138,
141, 217, 279, 283, 285, 286, 287, 288, 290, 294
temperature dependence, 290
Index
tensile strength, x, 28, 44, 46, 118, 119, 155, 156,
160, 161, 162, 164, 165, 166, 167, 201, 204, 282,
288, 293
tensile stress, 28, 33, 148, 156, 160, 224
tension, 32, 33, 35, 36, 41, 48, 134, 144, 156, 162,
196, 210, 212, 213, 214, 215, 217, 221, 229, 234,
236, 267, 268, 269
test data, 31, 144, 145, 147, 253
Texas, 163
textiles, vii
TGA, x, 129, 141, 142, 161
theory, 9, 25, 34, 43, 46, 56, 59, 79, 156, 175, 215,
233
thermal expansion, vii, 1, 4, 10, 13, 14, 16, 17, 42,
43, 45, 113, 117, 119
thermal properties, viii, 1, 13, 42
thermal stability, 113, 132, 141, 142, 161
thermodynamics, 44, 134
thermo-mechanical, 11, 12, 14, 43, 49, 148, 164
thermomechanical treatment, 288
thermoplastic(s), vii, ix, 110, 113, 115, 117, 123,
125, 126, 163, 210, 212, 213, 214, 218, 219, 221,
234, 235, 256
Third World, 101
threat(s), xi, 119, 125, 237, 238, 243
threshold(s), xi, 35, 116, 166, 174, 191, 193, 201,
205, 237, 238, 239, 240, 241, 242, 243, 246, 248,
253, 254, 256, 283, 285, 288, 290
titanium, 14, 46, 112, 117
Tokyo, 105
toluene, 135
topology, 52, 54, 67, 68, 71, 72, 74, 78, 81, 96, 97,
101, 102, 106
tracking, 212
transducer, 219
transformation, 11, 21, 45, 132, 163
transition(s), vii, viii, 1, 2, 3, 4, 6, 8, 9, 10, 12, 13,
14, 17, 18, 19, 20, 21, 35, 36, 41, 42, 43, 44, 45,
47, 198, 228, 288
transmission, 110, 218, 219
transparency, 218
transport, 44
transportation, 118, 122, 124
treatment methods, ix, 110, 115, 120
trend, ix, 110, 113, 117, 118, 120, 121, 122, 170,
172, 173, 182, 183, 184, 196, 198, 200, 242, 243,
252, 253
trial and error, 124
triangulation, 167
UK, 234, 235, 254
U

ultrasonic waves, 133, 219, 277, 278, 294
ultrasound, 133, 163, 219, 220
uniaxial tension, 31
uniform, 21, 45, 64, 92, 122, 124, 133, 134, 136,
147, 241, 294
universities, 121
updating, 77
USDA, 125
UV, 119, 121
V
vacuum, 121, 122, 123, 124
validation, x, 166, 209, 229, 236
values, viii, 2, 4, 8, 13, 18, 19, 20, 41, 52, 54, 58, 61,
64, 65, 66, 70, 76, 77, 79, 81, 84, 88, 89, 90, 95,
110, 116, 122, 134, 143, 156, 166, 173, 176, 182,
185, 186, 187, 193, 196, 200, 201, 204, 238, 241,
243, 246, 247, 248, 252, 253, 254, 257, 258, 263,
265, 267, 272, 277, 283, 285
vapor, 46
variable(s), viii, x, xi, 26, 51, 52, 53, 54, 55, 61, 62,
65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79,
80, 81, 82, 83, 84, 88, 89, 90, 94, 95, 96, 97, 98,
99, 100, 101, 103, 121, 169, 201, 202, 237, 239,
240, 244, 246, 250, 253
variation, 44, 57, 61, 82, 83, 84, 89, 90, 91, 99, 100,
144, 147, 172, 200, 211, 228, 285
vector, 25, 52, 59, 258, 261
vehicles, 115
velocity, x, 165, 166, 205, 207, 236, 237, 238, 241,
243, 254, 255
versatility, 124
vessels, x, 165, 205
vibration, 53, 64, 66, 71, 125
Index 307
Vietnam, 105
vinylester, 254
Viscoelastic, 49
viscosity, 115, 122, 123, 134, 135
visualization, 234
W
Washington, 105
wave propagation, 167
wavelengths, 211
wear, 113, 118, 162, 275
Weibull distribution, 150
weight ratio, 2, 51, 166
weight reduction, 118
welding, 279
wettability, 120, 278
wetting, 112, 119, 134, 135
wheat, 125
wind, 110
wood, vii
workers, 130
writing, 6, 22, 45
X
X-ray, 47, 121, 166, 220, 236
X-ray diffraction (XRD), 47
xylene, 135
Y
yield, 32, 42, 45, 121, 124, 225, 238, 277, 281, 282,
288, 290, 291, 294