Nonlinear Continua

Eduardo N. Dvorkin

Marcela B. Goldschmit

**Nonlinear** **Continua**

SPIN 10996775

— Monograph —

May 6, 2005

Springer

Berlin Heidelberg NewYork

Hong Kong London

Milan Paris Tokyo

To the Argentine system of public education

Preface

This book develops a modern presentation of Continuum Mechanics, oriented

towards numerical applications in the fields of nonlinear analysis of solids,

structures and fluids.

Kinematics of the continuum deformation, including pull-back/push-forward

transformations between different configurations; stress and strain measures;

objective stress rate and strain rate measures; balance principles; constitutive

relations, with emphasis on elasto-plasticity of metals and variational principles

are developed using general curvilinear coordinates.

Being tensor analysis the indispensable tool for the development of the

continuum theory in general coordinates, in the appendix an overview of tensoranalysisisalsopresented.

Embedded in the theoretical presentation, application examples are developed

to deepen the understanding of the discussed concepts.

Even though the mathematical presentation of the different topics is quite

rigorous; an effort is made to link formal developments with engineering physical

intuition.

This book is based on two graduate courses that the authors teach at the

Engineering School of the University of Buenos Aires and it is intended for

graduate engineering students majoring in mechanics and for researchers in

the fields of applied mechanics and numerical methods.

VIII Preface

Preface IX

I am grateful to Klaus-Jürgen Bathe for introducing me to Computational

Mechanics, for his enthusiasm, for his encouragement to undertake challenges

and for his friendship.

I am also grateful to my colleagues, to my past and present students at the

University of Buenos Aires and to my past and present research assistants at

the Center for Industrial Research of FUDETEC because I have always learnt

from them.

I want to thank Dr. Manuel Sadosky for inspiring many generations of

Argentine scientists.

I am very grateful to my late father Israel and to my mother Raquel for

their efforts and support.

Last but not least I want to thank my dear daughters Cora and Julia, my

wife Elena and my friends (the best) for their continuous support.

Eduardo N. Dvorkin

I would like to thank Professors Eduardo Dvorkin and Sergio Idelsohn for

introducing me to Computational Mechanics. I am also grateful to my students

at the University of Buenos Aires and to my research assistants at the Center

for Industrial Research of FUDETEC for their willingness and effort.

I want to recognize the permanent support of my mother Esther, of my

sister Mónica and of my friends and colleagues.

Marcela B. Goldschmit

Contents

1 Introduction ............................................... 1

1.1 Quantificationofphysicalphenomena ...................... 1

1.1.1 Observationofphysicalphenomena.................. 1

1.1.2 Mathematicalmodel............................... 2

1.1.3 Numericalmodel .................................. 2

1.1.4 Assessmentofthenumericalresults.................. 2

1.2 Linearandnonlinearmathematicalmodels ................. 2

1.3 Theaimsofthisbook.................................... 4

1.4 Notation ............................................... 5

2 Kinematics of the continuous media ....................... 7

2.1 The continuous media and its configurations ............... 7

2.2 Massofthecontinuousmedia............................. 9

2.3 Motionofcontinuousbodies .............................. 9

2.3.1 Displacements .................................... 9

2.3.2 Velocities and accelerations ......................... 10

2.4 Material and spatial derivatives of a tensor field ............. 12

2.5 Convectedcoordinates ................................... 13

2.6 Thedeformationgradienttensor........................... 13

2.7 Thepolardecomposition ................................. 21

2.7.1 TheGreendeformationtensor ...................... 21

2.7.2 Therightpolardecomposition ...................... 22

2.7.3 TheFingerdeformationtensor ..................... 25

2.7.4 Theleftpolardecomposition........................ 25

2.7.5 Physical interpretation of the tensors t ◦R , t ◦U and t ◦V 26

2.7.6 Numericalalgorithmforthepolardecomposition...... 28

2.8 Strainmeasures ......................................... 33

2.8.1 TheGreendeformationtensor ...................... 33

2.8.2 TheFingerdeformationtensor...................... 33

2.8.3 TheGreen-Lagrangedeformationtensor.............. 34

2.8.4 TheAlmansideformationtensor .................... 35

XII Contents

2.8.5 TheHenckydeformationtensor ..................... 35

2.9 Representation of spatial tensors in the reference

configuration(“pull-back”) ............................... 36

2.9.1 Pull-backofvectorcomponents ..................... 36

2.9.2 Pull-backoftensorcomponents ..................... 40

2.10 Tensors in the spatial configuration from representations in

the reference configuration(“push-forward”) ................ 42

2.11 Pull-back/push-forwardrelationsbetweenstrainmeasures .... 43

2.12 Objectivity ............................................. 44

2.12.1 Referenceframeandisometrictransformations ....... 45

2.12.2 Objectivity or material-frame indifference ............ 47

2.12.3 Covariance ....................................... 49

2.13 Strainrates............................................. 50

2.13.1 Thevelocitygradienttensor ........................ 50

2.13.2 The Eulerian strain rate tensor and the spin (vorticity)

tensor............................................ 51

2.13.3 Relations between differentratetensors .............. 53

2.14 TheLiederivative ....................................... 56

2.14.1 ObjectiveratesandLiederivatives .................. 58

2.15 Compatibility ........................................... 61

3 Stress Tensor .............................................. 67

3.1 Externalforces.......................................... 67

3.2 TheCauchystresstensor................................. 69

3.2.1 Symmetry of the Cauchy stress tensor (Cauchy

Theorem) ........................................ 71

3.3 Conjugatestress/strainratemeasures...................... 72

3.3.1 The Kirchhoff stresstensor ......................... 74

3.3.2 The first Piola-Kirchhoff stresstensor ............... 74

3.3.3 The second Piola-Kirchhoff stresstensor ............. 76

3.3.4 A stress tensor energy conjugate to the time derivative

oftheHenckystraintensor ........................ 79

3.4 Objectivestressrates .................................... 81

4 Balance principles ......................................... 85

4.1 Reynolds’transporttheorem.............................. 85

4.1.1 GeneralizedReynolds’transporttheorem............. 88

4.1.2 Thetransporttheoremanddiscontinuitysurfaces ..... 90

4.2 Mass-conservationprinciple............................... 93

4.2.1 Eulerian (spatial) formulation of the mass-conservation

principle ......................................... 93

4.2.2 Lagrangian (material) formulation of the mass

conservationprinciple.............................. 95

4.3 Balanceofmomentumprinciple(Equilibrium) .............. 95

Contents XIII

4.3.1 Eulerian (spatial) formulation of the balance of

momentumprinciple............................... 96

4.3.2 Lagrangian (material) formulation of the balance of

momentumprinciple...............................103

4.4 Balanceofmomentofmomentumprinciple(Equilibrium) ....105

4.4.1 Eulerian (spatial) formulation of the balance of

momentofmomentumprinciple ....................105

4.4.2 Symmetry of Eulerian and Lagrangian stress measures 107

4.5 Energybalance(FirstLawofThermodynamics) ............109

4.5.1 Eulerian (spatial) formulation of the energy balance . . . 109

4.5.2 Lagrangian (material) formulation of the energy balance112

5 Constitutive relations ......................................115

5.1 Fundamentalsforformulatingconstitutiverelations..........116

5.1.1 Principleofequipresence ...........................116

5.1.2 Principle of material-frame indifference ..............116

5.1.3 Application to the case of a continuum theory

restrictedtomechanicalvariables....................116

5.2 Constitutive relations in solid mechanics: purely mechanical

formulations ............................................120

5.2.1 Hyperelasticmaterialmodels .......................121

5.2.2 Asimplehyperelasticmaterialmodel ................122

5.2.3 Othersimplehyperelasticmaterialmodels............128

5.2.4 Ogdenhyperelasticmaterialmodels..................129

5.2.5 Elastoplastic material model under infinitesimal strains 135

5.2.6 Elastoplastic material model under finitestrains ......155

5.3 Constitutive relations in solid mechanics: thermoelastoplastic

formulations ............................................167

5.3.1 Theisotropicthermoelasticconstitutivemodel .......167

5.3.2 Athermoelastoplasticconstitutivemodel ............170

5.4 Viscoplasticity ..........................................176

5.5 Newtonian fluids ........................................180

5.5.1 Theno-slipcondition ..............................181

6 Variational methods .......................................183

6.1 ThePrincipleofVirtualWork ............................183

6.2 The Principle of Virtual Work in geometrically nonlinear

problems ..............................................186

6.2.1 IncrementalFormulations ..........................189

6.3 ThePrincipleofVirtualPower............................194

6.4 ThePrincipleofStationaryPotentialEnergy ...............195

6.5 Kinematicconstraints....................................207

6.6 Veubeke-Hu-Washizuvariationalprinciples .................209

6.6.1 KinematicconstraintsviatheV-H-Wprinciples ......209

6.6.2 ConstitutiveconstraintsviatheV-H-Wprinciples ....211

XIV Contents

A Introduction to tensor analysis ............................213

A.1 Coordinatestransformation...............................213

A.1.1 Contravarianttransformationrule ...................214

A.1.2 Covarianttransformationrule.......................215

A.2 Vectors ................................................215

A.2.1 Basevectors ......................................216

A.2.2 Covariantbasevectors .............................216

A.2.3 Contravariantbasevectors..........................218

A.3 Metricofacoordinatessystem ............................219

A.3.1 Cartesiancoordinates..............................219

A.3.2 Curvilinear coordinates. Covariant metric components . 220

A.3.3 Curvilinear coordinates. Contravariant metric

components ......................................220

A.3.4 Curvilinear coordinates. Mixed metric components .....221

A.4 Tensors ................................................222

A.4.1 Second-ordertensors...............................223

A.4.2 n-ordertensors....................................227

A.4.3 Themetrictensor .................................228

A.4.4 TheLevi-Civitatensor ............................229

A.5 Thequotientrule .......................................232

A.6 Covariantderivatives.....................................233

A.6.1 Covariantderivativesofavector ....................233

A.6.2 Covariantderivativesofageneraltensor .............236

A.7 Gradientofatensor .....................................237

A.8 Divergenceofatensor....................................238

A.9 Laplacianofatensor.....................................239

A.10Rotorofatensor ........................................240

A.11 The Riemann-Christoffeltensor ...........................240

A.12TheBianchiidentity ....................................243

A.13Physicalcomponents.....................................244

References .....................................................247

Index ..........................................................255

1

Introduction

The quantitative description of the deformation of continuum bodies, either

solids or fluids subjected to mechanical and thermal loadings, is a challenging

scientific field with very relevant technological applications.

1.1 Quantification of physical phenomena

The quantification of a physical phenomenon is performed through four different

consecutive steps:

1. Observation of the physical phenomenon under study. Identification of its

most relevant variables.

2. Formulation of a mathematical model that describes, in the framework of

the assumptions derived from the previous step, the physical phenomenon.

3. Formulation of the numerical model that solves, within the required accuracy,

the above-formulated mathematical model.

4. Assessment of the adequacy of the numerical results to describe the phenomenon

under study.

1.1.1 Observation of physical phenomena

This is a crucial step that conditions the next three. Making an educated

observation of a physical phenomenon means establishing a set of concepts

and relations that will govern the further development of the mathematical

model.

At this stage we also need to decide on the quantitative output that we

shall require from the model.

2 **Nonlinear** continua

1.1.2 Mathematical model

Considering the assumptions derived from the previous step and our knowledge

on the physics of the phenomenon under study, we can establish the

mathematical model that simulates it. This mathematical model, at least for

the cases that fall within the field that this book intends to cover, is normally

a system of partial differential equations (PDE) with established boundary

and initial conditions.

1.1.3 Numerical model

Usually the PDE system that constitutes the mathematical model cannot be

solved in closed form and the analyst needs to resort to a numerical model in

order to arrive at the actual quantification of the phenomenon under study.

1.1.4 Assessment of the numerical results

The analyst has to judge if the numerical results are acceptable. This is a very

important step and it involves:

• Verification of the mathematical model, that is to say, checking that the

numerical results do not contradict any of the assumptions introduced

for the formulation of the mathematical model and verification that the

numerical results “make sense” by comparing them with the results of a

“back-of-an-envelope” calculation (here, of course, we only compare orders

of magnitude).

• Verification of the numerical model, the analyst has to assess if the numerical

model can assure convergence to the unknown exact solution of the

mathematical model when the numerical degrees of freedom are increased.

The analyst must also check the stability of the numerical results when

small perturbations are introduced in the data. If the results are not stable

the analyst has to assess if the unstable numerical results represent an

unstable physical phenomenon or if they are the result of an unacceptable

numerical deficiency.

• Validation of the mathematical/numerical model comparing its predictions

with experimental observations.

1.2 Linear and nonlinear mathematical models

When deriving the PDE system that constitutes the mathematical model of

a physical phenomenon there are normally a number of nonlinear terms that

appear in those equations. Considering always all the nonlinear terms, even if

1.2 Linear and nonlinear mathematical models 3

their influence is negligible on the final numerical results, is mathematically

correct; however, it may not be always practical.

The scientist or engineer facing the development of the mathematical

model of a physical phenomenon has to decide which nonlinearities have to be

kept in the model and which ones can be neglected. This is the main contribution

of an analyst: formulating a model that is as simple as possible while

keeping all the relevant aspects of the problem under analysis (Bathe 1996).

In many problems it is not possible to neglect all nonlinearities because

the main features of the phenomenon under study lie in their consideration

(Hodge, Bathe & Dvorkin 1986); in these cases the analyst must have enough

physical insight into the problem so as to incorporate all the fundamental

nonlinear aspects but only the fundamental ones. The more nonlinearities are

introduced in the mathematical model, the more computational resources will

be necessary to solve the numerical model and in many cases it may happen

that the necessary computational resources are much larger than the available

ones, making the analysis impossible.

Example 1.1. JJJJJ

In the analysis of a solid under mechanical and thermal loads some of the nonlinearities

that we may encounter when formulating the mathematical model

are:

• Geometrical nonlinearities: they are introduced by the fact that the equilibrium

equations have to be satisfied in the unknown deformed configuration

of the solid rather than in the known unloaded configuration. When

the analyst expects that for her/his practical purposes the difference between

the deformed and unloaded configurations is negligible she/he may

neglect this source of nonlinearity obtaining an important simplification in

the mathematical model. An intermediate step would be to consider the

equilibrium in the deformed configuration but to assume that the strains

are very small (infinitesimal strains assumption). This also produces an

important simplification in the mathematical model. Of course, all the

simplifications introduced in the mathematical model have to be checked

for their properness when examining the obtained numerical results.

• Contact-type boundary conditions: these are unilateral constraints in

which the contact loads are distributed over an area that is a priori unknown

to the analyst.

• Material nonlinearities: elastoplastic materials (e.g. metals); creep behavior

of metals in high-temperature environments; nonlinear elastic materials

(e.g. polymers); fracturing materials (e.g. concrete); etc.

JJJJJ

4 **Nonlinear** continua

Example 1.2. JJJJJ

In the analysis of a fluid flow under mechanical and thermal loads some of

the nonlinearities that we may encounter when formulating the mathematical

model are:

• Non-constant viscosity/compressibility (e.g. rheological materials and turbulent

flows modeled using turbulence models).

• Convective acceleration terms for flows with Re>0 when the mathematical

model is developed using an Eulerian formulation, which is the standard

case.

JJJJJ

Example 1.3. JJJJJ

In the analysis of a heat transfer problem some of the nonlinearities that we

may encounter when formulating the mathematical model are:

• Temperature dependent thermal properties (e.g. phase changes).

• Radiation boundary conditions

JJJJJ

There are mathematical models in which the effects (outputs) are proportional

to the causes (inputs); these are linear models. Examples of linear models are:

• linear elasticity problems,

• constant viscosity creeping flows,

• heat transfer problems in materials in which constant thermal properties

are assumed and radiative boundary conditions are not considered,

• etc.

Deciding that the model that simulates a physical phenomenon is going to

be linear is an analyst decision, after first considering and afterwards carefully

neglecting, in the formulation of the mathematical model, all the sources of

nonlinearity.

1.3 The aims of this book

This book intends to provide a modern and rigorous exposition of nonlinear

continuum mechanics and even though it does not deal with computational

implementations it is intended to provide the basis for them.

In the second chapter of the book we present a consistent description of the

kinematics of the continuous media. In that chapter we introduce the concepts

of pull-back, push-forward and Lie derivative requiring only from the reader

1.4 Notation 5

a previous knowledge of tensor analysis. Objective and covariant strain and

strain rate measures are derived.

In the third chapter we discuss different stress measures that are energy

conjugate to the strain rate measures presented in the previous chapter. Objective

stress rate measures are derived.

In the fourth chapter we present the Reynolds transport theorem and then

we use it to develop Eulerian and Lagrangian formulations for expressing

the balance (conservation) of mass, momentum, moment of momentum and

energy.

In the fifth chapter we develop an extensive presentation of constitutive

relations for solids and fluids, with special focus on the elastoplasticity of

metals.

Finally, in the sixth chapter we develop the variational approach to continuum

mechanics, centering our presentation on the principle of virtual work and

discussing also the principle of stationary potential energy and the Veubeke-

Hu-Washizu variational principles.

The basic mathematical tool in the book is tensor calculus; in order to

assure a common basis for all the readers, in the Appendix we present a

review of this topic.

1.4 Notation

Throughout the book we shall use the summation convention; that is to say,

in a Cartesian coordinate system

3X

aα bα =

aαβ bβ =

aα bα

α=1

3X

aαβ bβ for α =1, 2, 3 ,

β=1

and in a general curvilinear system

3X

ai b i =

a ij bj =

ai b

i=1

i

3X

a ij bj for i =1, 2, 3 .

j=1

Also, our notation is compatible with the notation introduced in continuum

mechanics by Bathe (Bathe 1996). We shall define all notation at the point

whereweincorporateit.

2

Kinematics of the continuous media

In this chapter we are going to present a kinematic description of the deformation

of continuous media. That is to say, we are going to describe the

deformation without considering the loads that cause it and without introducing

into the analysis the behavior of the material.

Some reference books for this chapter are: (Truesdell & Noll 1965, Truesdell

1966, Malvern 1969, Marsden & Hughes 1983).

2.1 The continuous media and its configurations

Continuum mechanics is the branch of mechanics that studies the motion

of solids, liquids and gases under the hypothesis of continuous media. This

hypothesis is an idealization of matter that disregards its atomic or molecular

structure.

A continuous body is an open subset of the three-dimensional Euclidean

space ¡ < 3¢ (Oden 1979) 1 .Eachelement“χ” of that subset is called a point or

a material particle. The region of the Euclidean space occupied by the particles

χ of the continuous body B at time t is called the configuration corresponding

to t.

Above, we use the notion of time in a very general sense: as a coordinate

that is used to enumerate a series of events. Aninstant t is a particular

value of the time coordinate.

We can establish a bijective mapping (Oden 1979) between each point

of space occupied by a material particle χ at t and an arbitrary curvilinear

coordinate system { t x a ,a=1, 2, 3}.

Thefactthatateachinstantt the set { t x a } defines one and only one

particle χ implies that in a continuum medium, different material particles

1 The requirement of an open subset is introduced in order to eliminate the possible

consideration of isolated points, sets in < 3 with zero volume, etc.

8 **Nonlinear** continua

cannot occupy the same space location and that a material particle cannot be

subdivided:

t a t a t a

x = x (χ, t) ; χ = χ( x ) . (2.1)

We assume that two arbitrary coordinate systems defined for the configuration

at time t are related by continuous and differentiable functions:

t b t b t a t a t a t b

˜x = ˜x ( x ) ; x = x ( ˜x ) . (2.2)

In a formal way, we say that the configuration of the body B corresponding

to time t is an homeomorphism of B onto a region of the three dimensional

Euclidean space (< 3 ) (Truesdell & Noll 1965). An homeomorphism (Oden

1979) is a bijective and continuous mapping with its inverse mapping also

continuous.

The coordinates { txa } are called the spatial coordinates of the material

particle χ in the configuration at time t.

We call the motion of the body B the evolution from aconfiguration at

an instant t1 to aconfiguration at an instant t2.

We select any configuration of the body B as the reference configuration

(e.g. the undeformed configuration, but not necessarily this one); also we can

set the time origin so that in the reference configuration t = 0 .Inthe

reference configuration we define an arbitrary curvilinear coordinate system

{ ◦x A ,A =1, 2, 3}: thematerial coordinates. For the reference configuration

Eqs.(2.1) are

◦ A ◦ A ◦ A

x = x (χ) ; χ = χ( x ) . (2.3)

From Eqs.(2.1) and (2.3) we obtain the bijective mapping tφ between the

configuration at time t and the reference configuration,

t a t a ◦ A ◦ A £

x = φ ( x ,t) ; x =

tφ−1 ¤A t a

( x ) . (2.4)

In a regular motion the inverse mapping tφ −1 exists and if tφ ∈ Cr also

t −1 r r φ ∈ C (Marsden & Hughes 1983), where C is the set of all functions

with continuous derivatives up to the order “r”. The formal concept of regular

motion agrees with the intuitive concept of a motion without material

interpenetration.

From Eqs. (2.2) and (2.4) we get

t a t a t b t a

˜x = ˜x ( x ) = ˜x h i

t b ◦ A

φ ( x ) = t˜

a ◦ A

φ ( x ) . (2.5)

The mapping tφ is a function of:

• the reference configuration,

• the configuration at t,

• the material coordinate system,

• the spatial coordinate system.

Since we restrict our presentation to the Euclidean space < 3 ,wecanconsider

t φ as a vector. Hence, in Sect. 2.3.1 we are going to define the position

and displacement vectors.

2.2 Mass of the continuous media

2.3 Motion of continuous bodies 9

The continuous media have a non-negative scalar property named mass.

Our knowledge of Newton laws makes us relate the mass of a body with a

measure of its inertia.

After (Truesdell 1966) we are going to assume a continuous mass distribution

in the body B. Concentrated masses do not belong to the field of Continuum

Mechanics (therefore, Rational Mechanics is not part of Continuum

Mechanics).

We define the density “ tρ ” corresponding to the configuration at time t

as

Z

t t

m = ρ dV , (2.6)

t V

where, m: massofbodyB, t V :volumeofB in the configuration at time t.

Equation (2.6) incorporates an important postulate of Newtonian mechanics:

the mass of a body is constant in time.

2.3 Motion of continuous bodies

2.3.1 Displacements

In the mapping tφ (< 3 −→ < 3 ) schematized in Fig. 2.1, at a given point

(particle) χ, the vectors ◦g are the covariant base vectors

A 1 (Green & Zerna

1968) of the material coordinates { ◦x A } (reference configuration; t =0)and

the vectors tg are the covariant base vectors of the spatial coordinates {

a txa }

(spatial configuration corresponding to time t).

In the 3D Euclidean space we also define a fixed Cartesian system

{ ◦z α ≡ tzα α =1, 2, 3} with a set of orthonormal base vectors eα. For the Cartesian coordinates of a particle χ in the reference configuration

we use the triad { ◦z α } and for the Cartesian coordinates of the same particle

in the spatial configuration we use the triad { tzα }.

◦ In the Cartesian system, the position vector x of a particle χ in the

reference configuration is

◦ x(χ) = ◦ z α (χ) eα , (2.7)

and the position vector t x of the particle χ in the spatial configuration is

t x(χ, t) = t z α (χ, t) eα . (2.8)

The displacement vector of the particle χ from the reference configuration

to the spatial configuration is,

1 See Appendix.

10 **Nonlinear** continua

Fig. 2.1. Motion of continuous body

t u(χ, t) = t x(χ, t) − ◦ x(χ) (2.9a)

and the Cartesian components of this vector are,

2.3.2 Velocities and accelerations

t u α (χ, t) = t z α (χ, t) − ◦ z α (χ) . (2.9b)

During the motion t φ,thematerial velocity of a particle χ in the t-configuration

is

t v(χ, t) = ∂ t x(χ, t)

∂t

= ∂t u(χ, t)

∂t

(2.10)

assuming that the time derivatives in Eq. (2.10) exist.

The material velocity vector is definedinthespatialconfiguration (see

Fig. 2.2).

We can have, alternatively, the following functional dependencies:

t v = t v( ◦ x A ,t) , (2.11a)

t v = t v( t x a ,t) . (2.11b)

Equation (2.11a) corresponds to a Lagrangian (material) description of

motion, while Eq. (2.11b) corresponds to a Eulerian (spatial) description of

Fig. 2.2. Material velocity of a particle χ

2.3 Motion of continuous bodies 11

motion. In general, the motion of solids is studied using Lagrangian descriptions

while the motion of fluids is studied using Eulerian descriptions; however,

this classification is by no means mandatory and combined descriptions have

also been used in the literature (Belytschko, Lui & Moran 2000).

In any case, we can refer the velocity vector either to the spatial coordinates

or to the fixed Cartesian coordinates,

t v = t v a t ga , (2.12a)

t v = t v α eα . (2.12b)

Assuming an arbitrary tensor field t η = t η(χ, t) = t η( ◦ x,t) we define its

temporal material derivative (D t η/Dt) asthetimerateofthetensor t η when

we keep constant the particle χ or, equivalently, when we keep constant the

position vector ◦ x in the reference configuration.

We call the material acceleration of a particle χ,

t D

a = tv .

Dt

(2.13)

In what follows, we determine the material acceleration vector considering

the different combinations of:

⎧

⎫

⎨ Lagrangian⎬

⎧

⎨ Spatial

⎫

⎬

description

⎩

Eulerian

⎭

+ coordinates

⎩

⎭

FixedCartesian • Lagrangian description + spatial coordinates

12 **Nonlinear** continua

where the t Γ a bc

∙ t a

t ∂ v

a =

∂t + t Γ a bc t v b t v c

¸

tga

, (2.14a)

are the Christoffel symbols of the second kind of the spatial

coordinates { txa } 2 .

• Lagrangian description + fixed Cartesian coordinates

t ∂

a = tvα ∂t

eα . (2.14b)

• Eulerian description + spatial coordinates

The material particle that at time t is at the spatial location { txa },at

time t + dt will be at { txa + tva dt}. Hence,

t+dt t ∂

v = v + tv ∂t dt + ∂tv ∂txb t b

v dt

Considering that the base vectors, in a general coordinate system, are

functions of the position, we have

t

a =

∙ t a ∂ v

∂t + ∂tva ∂txb t b

v

t a

+ Γbc t b

v

¸

t c

v

tga

. (2.15a)

• Eulerian description + fixed Cartesian coordinates

t a =

∙ t α ∂ v

∂t + ∂tv α

∂tz β

¸

t β

v

e α . (2.15b)

2.4 Material and spatial derivatives of a tensor field

Let t η be an arbitrary tensor field, a function of time, and using a Lagrangian

description of motion we get,

t η = t η( ◦ x A ,t) . (2.16a)

Now, using an Eulerian description of motion, we get

t η = t η( t x a ,t) . (2.16b)

We indicate the first material time derivative (following the particle) as

2 See Appendix.

t D

˙η = tη Dt = ∂tη( ◦x A ,t)

|◦xA ∂t

. (2.17)

2.6 The deformation gradient tensor 13

The first spatial time derivative of a tensor t η defined using a Eulerian

description is simply indicated as ∂tη ( t x a ,t)

|t

∂t xa. Let us now calculate the material time derivative of a tensor t η defined

using an Eulerian description,

t t a···b

η = η c···d( t x a ,t) t g ...

a t t c t d

g g ... g

b

(2.18)

it is easy to derive the following relation (Truesdell & Noll 1965, Slattery

1972),

t ∂

˙η = tη ∂t + t v p ∂tη ∂t .

xp (2.19)

Using the spatial gradient of the tensor t η 3 :

∇ t η = t g p ∂t η

∂ t x p

=[ ∂t η a···b c···d

∂ t x p

+ t η k···b t a

c···d Γkp + ...+ t η a···k c···d t Γ b kp

− t η a···b k···d t Γ k pc − ...− t η a···b c···k t Γ k pd ]

we can rewrite Eq. (2.19) as,

2.5 Convected coordinates

(2.20a)

t p t

g ga ... t t c t d

g g ... g

b

t ˙η = ∂ t η

∂t + t v · (∇ t η) . (2.20b)

Let us consider a body B and define in its reference configuration a system of

curvilinear coordinates {θ i ,i=1, 2, 3}.

The curvilinear coordinate system {θ i } is a convected coordinate system

(Flügge 1972) if, when the body B undergoes a deformation process, for each

configuration, the triad {θ i } that defines a particle χ, isthesameasinthe

reference configuration.

2.6 The deformation gradient tensor

Let us consider the motion of the body B represented in Fig. 2.1. For the

reference configuration (t =0)we can write at the point (particle) χ:

3 See Appendix.

◦ dx = ◦ dx A ◦ gA

(2.21a)

14 **Nonlinear** continua

where the vector ◦ dx at the point χ in the reference configuration is called

a material line element or fiber (Ogden 1984).

Due to the motion t φ ,theabovedefined fiber is transformed into a fibre

in the spatial configuration,

t dx = t dx a t ga . (2.21b)

We now define a second-order tensor: t ◦X ,thedeformation gradient tensor

at χ,

From the above equation,

t dx = t ◦X · ◦ dx . (2.22)

t

◦X = ∂txa ∂◦xA t

ga

◦ A

g . (2.23)

Using the first of Eqs. (2.4) we get the following functional relation,

t

◦X a A = t ◦X a A( ◦ x B ,t) . (2.24)

t From Eq.(2.23), we see that the tensor ◦X has one base vector in the

reference configuration and the other in the spatial configuration. Hence, it is

a two-point tensor, (Marsden & Hughes 1983, Lubliner 1985).

It is important to note that t ◦X a A is a function of,

• the motion of the body ( t φ),

• the material coordinate system,

• the spatial coordinate system.

For a regular motion, the tensor that is the inverse of t ◦X at t x is,

◦ t

dx = ◦X −1 · t dx (2.25)

where,

t

◦X −1 = ∂◦x A

∂txa ◦

gA

t a

g . (2.26)

Using the second of Eqs. (2.4), we get the following functional relation,

( t ◦X −1 ) A a = ( t ◦X −1 ) A a ( t x b ,t) . (2.27)

We define the transpose of t ◦X at χ using the following relation (Marsden

& Hughes 1983, Strang 1980),

( t ◦X · ◦ dx) · t dx = ◦ dx · ( t ◦X T · t dx) , (2.28a)

if we now define t ◦X T = ( t ◦X T ) A a ◦ g A

t g a ,wegetfromEq.(2.28a),

hence,

2.6 The deformation gradient tensor 15

t

◦X a A ◦ dx A t d ◦ x b t gab = ◦ dx A ◦ gAB ( t ◦X T ) B b

t dx b

(2.28b)

( t ◦X T ) B b = t ◦X a A t gab ◦ g BA . (2.28c)

and therefore,

( t ◦X T ) B b = t ◦X B

b . (2.28d)

In the above equations, tgab = tg ·

a tg are the covariant components of the

b

metric tensor in the spatial configuration at { tx A } and ◦gAB = ◦gA · ◦gB are the contravariant components of the metric tensor in the reference configuration

at { ◦x a } 4 .

Referring the body B to a fixed Cartesian system, and using Eqs. (2.9a-

2.9b),

t

◦Xαβ = δαβ + ∂tuα ( t ◦X −1 )αβ = δαβ − ∂t u α

∂◦ , (2.29a)

β z

∂t , (2.29b)

zβ ( t ◦X T )αβ = t ◦Xβα , (2.29c)

where

δαβ: components of the Kronecker-delta.

It is important to remember that when a problem is referred to a Cartesian

system we do not need to make the distinction between covariant and

contravariant tensorial components (Green & Zerna 1968).

Example 2.1. JJJJJ

In a fixed Cartesian system, a rigid translation is represented by a deformation

gradient tensor with components t ◦Xαβ = δαβ. JJJJJ

Example 2.2. JJJJJ

In the rigid rotation represented in the following figure,

4 See Appendix.

16 **Nonlinear** continua

Rigid rotation

We can directly calculate the components t ◦Xαβ using Eq. (2.29a) but it may

be simpler to consider the following sequence:

Step 1: Transformation from Cartesian coordinates to cylindrical coordinates

in the reference configuration.

q

◦ 1

θ =

µ ◦z2

◦ 2 −1

θ =tan

◦ θ 3 = ◦ z 3 .

( ◦ z 1 ) 2 +( ◦ z 2 ) 2

◦ z 1

; (X1) l β = ∂◦ θ l

∂ ◦ z β

Without any motion, the change of the coordinate system in the reference

configuration produces a deformation gradient tensor.

Step 2: Rigid rotation.

t θ 1 = ◦ θ 1

t θ 2 = ◦ θ 2 + γ ; (X2) p

l = ∂t θ p

∂ ◦ θ l

t θ 3 = ◦ θ 3 .

Step 3: Transformation from cylindrical coordinates to Cartesian coordinates

in the spatial configuration.

t z 1 = t θ 1 cos t θ 2

t z 2 = t θ 1 sin t θ 2

t z 3 = t θ 3 .

; (X3) α

p = ∂t z α

∂ t θ p

Without any further motion, the change of the coordinate system in the spatial

configuration produces a deformation gradient tensor.

Using the chain rule,

t

◦Xαβ = ∂tz α

∂◦z β = ∂tz α

∂tθ p

t

◦Xαβ =(X3) α

p

2.6 The deformation gradient tensor 17

∂tθ p

∂◦θ l

∂◦θ l

∂◦z β

p

(X2) l

l

(X1) β

For step 1, the derivation of (X −1

1 )β l canbedonebyinspection.Thearray

of these components is,

inverting

For step 2,

For step 3,

Finally,

£ ¤ −1

X1 =

⎡

[X1] = ⎣

⎡

⎣ cos ◦θ 2 − ◦θ 1 sin ◦θ 2 ⎤

0

⎦ ,

sin ◦ θ 2 ◦ θ 1 cos ◦ θ 2 0

0 0 1

cos ◦θ 2

sin ◦θ 2

0

2 1

◦θ1 cos ◦θ 2 ⎤

0 ⎦ .

0 0 1

− 1

◦ θ 1 sin ◦ θ

⎡

[X2] = ⎣ 100

⎤

010⎦

.

001

⎡

[X3] = ⎣ cos tθ 2 − tθ 1 sin tθ 2 ⎤

0

⎦ .

sin t θ 2 t θ 1 cos t θ 2 0

0 0 1

£ t◦X ¤ ⎡

⎤

cos γ − sin γ 0

= ⎣ sin γ cosγ 0 ⎦ .

0 0 1

An important feature of the above matrix is that [ t ◦X] T

[ t ◦X] = [I] ,where

[I] is the unit matrix. That is to say, the matrix [ t ◦X] is orthogonal.JJJJJ

Example 2.3. JJJJJ

For the motion represented in the following figure, we can derive the deformation

gradient tensor directly by inspection,

18 **Nonlinear** continua

Simple deformation process

£ t◦X ¤ ⎡ ⎤

2.0 0.0 0.0

= ⎣ 0.0 0.5 0.0⎦

.

0.0 0.0 0.6

JJJJJ

Example 2.4. JJJJJ

For the motion represented in the following figure,

therefore,

Shear deformation

t z1 = ◦ z 1 + ◦ z 2 tan γ

t z 2 = ◦ z 2

t z 3 = ◦ z 3 ,

£ t◦X ¤ ⎡ ⎤

1tanγ0 = ⎣ 0 1 0⎦

.

0 0 1

JJJJJ

2.6 The deformation gradient tensor 19

When there is a sequence of motions (some of them can be just a change

of coordinate system) like the sequence depicted in Fig. 2.3, we can generalize

the result in Example 2.2,

Therefore,

Fig. 2.3. Sequence of motions

n∆t

◦ X a

P = ∂n∆txa ∂◦xP = ∂n∆txa ∂ (n−1)∆txb ······∂∆t xl ∂◦ . (2.30a)

P x

n∆t

◦ X = n∆t

(n−1)∆t

(n−1)∆t 2∆t

X · X ··· ∆t X · ∆t

◦ X . (2.30b)

(n−2)∆t

Example 2.5. JJJJJ

It is easy to show that when using a convected coordinate system

t

◦X a b = δ a b

where the δ a b are the mixed components of the Kronecker delta tensor.

JJJJJ

For a motion t φ we define at a point χ the Jacobian of the transformation

(Truesdell & Noll 1965),

t J(χ, t) =

t dV

◦ dV , (2.31)

where ◦ dV is a differential volume in the reference configuration and t dV

is the corresponding differential volume in the spatial configuration. Since in

a regular motion, a nonzero volume in the reference configuration cannot be

20 **Nonlinear** continua

collapsed into a point in the spatial configuration and vice versa (Aris 1962),

t J and t J −1 cannot be zero.

In a fixed Cartesian system we can define, for a motion t φ,

The vectors (Hildebrand 1976)

t tβ = ∂t z α

t z α = t z α ( ◦ z β ,t) . (2.32)

∂ ◦ z β

t eα (β =1, 2, 3) (2.33)

are the base vectors, in the spatial configuration, of a convected coordinate

system { ◦ z α } and

Therefore,

◦ dV = ◦ dz 1 ◦ dz 2 ◦ dz 3

(2.34a)

t dV = ◦ dz 1 ◦ dz 2 ◦ dz 3 [ t t1 · ( t t 2 × t t 3 )] . (2.34b)

t dV =det

⎡

⎢

⎣

∂ t z 1

∂ ◦ z 1

∂ t z 1

∂ ◦ z 2

∂ t z 1

∂ ◦ z 3

∂ t z 2

∂ ◦ z 1

∂ t z 2

∂ ◦ z 2

∂ t z 2

∂ ◦ z 3

∂ t z 3

∂ ◦ z 1

∂ t z 3

∂ ◦ z 2

∂ t z 3

∂ ◦ z 3

Since transpose matrices have the same determinant,

⎤

⎥ ◦

⎥ dV . (2.34c)

⎥

⎦

t dV = ¯ ¯t ◦X ¯ ¯ ◦ dV (2.34d)

where | t ◦X| =det[ t ◦X] .

When the motion of a body B is referred to a fixed Cartesian coordinate

system (Malvern 1969),

t J(χ, t) = ¯ ¯t ◦X ¯ ¯ . (2.34e)

When in the reference configuration we use a curvilinear system { ◦x A }

and in the spatial configuration a system { txa } , (Marsden & Hughes 1983):

∙ ¸ "

t α

t ∂ z ∂

J =det =det

tzα ∂txa #

. (2.34f)

∂ ◦ z β

∂ t x a

∂ ◦ x A

∂ ◦ x A

∂ ◦ z β

Following the reference (Green & Zerna 1968) and doing some algebra we

can show that:

¯

∙ ∙ ¸¸

¯ £ ¤ t α 2

¯ tgab ∂ z

=det = det

(2.34g)

and

¯t gab

| ◦ gAB| =det[ ◦ " "

gAB] = det

∂ t x a

∂ ◦ x A

∂ ◦ z β

## −2

. (2.34h)

Finally,

2.7 The polar decomposition 21

¯

t

J(χ, t) = ¯t

◦X ¯ s

¯

| tgab| | ◦ .

gAB|

(2.34i)

Example 2.6. JJJJJ

In a isocoric deformation (without change of volume) tJ =1, hence

¯

¯t

◦X ¯ p | tgab| = p | ◦gAB| .

◦ When in the problem we use a fixed Cartesian coordinate system, we have

¯

¯t ◦X ¯ ¯ =1.

◦ When we use convective coordinates

¯ ¯ ◦

= | gAB| .

¯t gab

2.7 The polar decomposition

JJJJJ

The polar decomposition theorem (Truesdell & Noll 1965, Truesdell 1966,

Malvern 1969, Marsden & Hughes 1983) is a fundamental step in the development

of the kinematic description of continuous body motions. It allows us to

locally (at a point χ) decompose any motion into a pure deformation motion

followed by a pure rotation motion or vice versa.

2.7.1 The Green deformation tensor

The Green deformation tensor is defined at a point χ as

t

◦C = t ◦X T · t ◦X . (2.35)

In some references, e.g. (Truesdell & Noll 1965, Truesdell 1966), the above

tensor is referred to as right Cauchy-Green deformation tensor .

Using Eq. (2.28c), we get

t

◦C =

and therefore,

h

t◦X a A t gab ◦ g AB ◦ t b

g g

B

i

·

h

t◦X d D t ◦ D

g g

d

i

, (2.36a)

22 **Nonlinear** continua

t

◦C = £ t

◦X a A t ◦X b D t gab ◦ g AB¤ ◦ ◦ D

gB g . (2.36b)

It is important to note that the Green deformation tensor is defined in the

reference configuration.

Using an equivalent definition of transposed tensor to that given in

Eq.(2.28a), we can write

( t ◦C · ◦ dx 1) · ◦ dx 2 = ◦ dx 1 · ( t ◦C T · ◦ dx 2) (2.37a)

where ◦ dx 1 and ◦ dx 2 are two arbitrary vectors definedinthereference

configuration at the point under study; after some algebra Eq. (2.37a) leads

to

t

◦C A B =( t ◦C T ) A B = t ◦C A

B . (2.37b)

The above equation indicates that the Green deformation tensor is symmetric.

For an arbitrary vector tdx definedinthespatialconfiguration we can

write the following equalities:

Hence,

t t a t

dx = dx ga = t ◦X a A ◦ dx A t g

a

(2.38a)

t t

dx = dxb t g b = t ◦X d B ◦ g BR ◦ dxR t gdb t g b . (2.38b)

t dx · t dx = t ◦X a A t ◦X d B t gda ◦ g BR ◦ dx A ◦ dxR . (2.38c)

Using Eq. (2.36b), we write

t dx · t dx = t ◦C R A ◦ dx A ◦ dxR = ◦ dx · t ◦C · ◦ dx . (2.38d)

Considering that:

• t dx · t dx ≥ 0 .

• t dx · t dx =0 ⇐⇒ | t dx| = 0 .

• If t φ is a regular motion, | t dx| = 0 ⇐⇒ | ◦ dx| = 0 we

conclude that t ◦C is a positive-definite tensor (Strang 1980).

2.7.2 The right polar decomposition

We define, in the reference configuration at the point under study, the right

stretch tensor as

t

◦U = £ t

◦C ¤ 1/2

(2.39)

and it follows immediately that the tensor t ◦U inherits from t ◦C the properties

of symmetry and positive-definiteness (Malvern 1969).

2.7 The polar decomposition 23

We define the right polar decomposition as a multiplicative decomposition

of the tensor t ◦X into a symmetric tensor ( t ◦U) premultiplied by a tensor

that we will show is orthogonal: the rotation tensor ( t ◦R) . Hence,

t

◦X = t ◦R · t ◦U . (2.40)

In order to show that t ◦R is an orthogonal tensor we have to show that:

(i) t ◦R T · t ◦R = ◦ g

(ii) t ◦R · t ◦R T = t g

where ◦g = δ A B ◦ ◦ B g g

A

is the unit tensor of the reference configuration

and tg = δ a b tg a

t b A

g is the unit tensor of the spatial configuration ( δ B

and δ a b are Kronecker deltas 5 ).

To prove the first equality (i), we start from Eq. (2.40) and get

t

◦R = t ◦X · t ◦U −1 = t ◦X a A ( t ◦U −1 ) A B t ◦ B

g g

a

. (2.41)

t The tensor ◦R defined by the above equation is, in the same sense as

t

t

◦X ,atwo-point tensor. The components of ◦R are:

and using an similar equation to (2.28c) we have

t

◦R a A = t ◦X a L ( t ◦U −1 ) L A , (2.42a)

( t ◦R T ) B b = t ◦X a L ( t ◦U −1 ) L A t gab ◦ g BA , (2.42b)

hence,

t

◦R T = t ◦X a L ( t ◦U −1 ) L A t gab ◦ g BA ◦ Considering that

g

B

t b

g . (2.42c)

and

t

◦X T = t ◦X d R t gdb ◦ g RB ◦ t b

g g , (2.42d)

B

t

◦U −T = ( t ◦U −1 ) L

D ◦ g D ◦ g , (2.42e)

L

t

◦U −T · t ◦X T =( t ◦U −1 ) L

D t ◦X d R δ R L t gdb ◦ g D t g b , (2.42f)

and since ◦gD = ◦gDB ◦g B

we write

t

◦U −T · t ◦X T =( t ◦U −1 ) L

D t ◦X d L t gdb ◦ g DB ◦ t b

g g .

B

(2.42g)

Comparing Eqs. (2.42g) and (2.42c) it is obvious that:

5 See Appendix.

24 **Nonlinear** continua

t

◦R T = t ◦U −T · t ◦X T = t ◦U −1 · t ◦X T t symmetry of ◦U).

Hence,

(the last equality follows from the

t

◦R T · t ◦R = t ◦U −1 · t ◦X T · t ◦X · t ◦U −1 . (2.42h)

Using also Eqs. (2.35) and (2.39), we obtain

t

◦R T

· t ◦R = t ◦U −1 · t ◦C · t ◦U −1 = ◦ g (2.42i)

and the first equality (i) is shown to be correct.

To prove the second equality (ii), we write (Marsden & Hughes 1983):

t

◦R · t ◦R T = t ◦R ·

³

t◦R T · t ´

◦R

· t ◦R T =

³ t◦R · t ◦R T ´ 2

, (2.43)

and since t ◦R cannot be a singular matrix, the second equality (ii) is shown

to be correct.

We will now show that the right polar decomposition is unique.

Assuming that it is not unique, we can have, together with Eq. (2.40),

another decomposition, for example:

t

◦X = t ◦ ˜ R · t ◦Ũ , (2.44a)

where t ◦ ˜ R is an orthogonal tensor and t ◦Ũ is a symmetric tensor.

We can write

t

◦C = t ◦X T · t ◦X = t ◦Ũ · t ◦Ũ , (2.44b)

and therefore,

t

◦Ũ = £ t

◦C ¤ 1/2

. (2.44c)

However, comparing the above with Eq. (2.39), we conclude that:

Then, from Eq. (2.44a),

t

◦ Ũ = t ◦U . (2.44d)

t

◦ ˜ R = t ◦X ·

t

◦U −1 , (2.44e)

and comparing the above with Eq. (2.41), we conclude that:

t

◦ ˜ R = t ◦R . (2.44f)

Equations (2.44d) and (2.44f) show that the right polar decomposition is

unique.

2.7.3 The Finger deformation tensor

2.7 The polar decomposition 25

The Finger deformation tensor, also known in the literature as the left

Cauchy-Green deformation tensor, isdefined at a point χ as:

t b = t ◦X · t ◦X T . (2.45a)

Using Eq. (2.28c), we have

t

b =

h

t◦X d D t ◦ D

g g

d

i h

·

t◦X a A t gab ◦ g BA ◦ g

B

t b

g i

and therefore,

(2.45b)

t t

b = ◦X d B t ◦X a A t gab ◦ g BA t t b

g g .

d

(2.45c)

It is important to note that the Finger deformation tensor is defined in

the spatial configuration.

Proceeding in the same way as in Sect. 2.7.1, we can show that:

• t b is a symmetric tensor.

• t b is a positive-definite tensor.

2.7.4 The left polar decomposition

We define the left polar decomposition as a multiplicative decomposition of the

tensor t ◦X into a symmetric tensor ( t ◦V) postmultiplied by the orthogonal

tensor t ◦R. Therefore,

t

◦X = t ◦V · t ◦R . (2.46)

From the above equation,

t

◦V = t ◦X · t ◦R T = t ◦R · t ◦U · t ◦R T

and taking into account that t ◦U is symmetric, we get

(2.47a)

t

◦V T = t ◦R · t ◦U · t ◦R T = t ◦V . (2.47b)

The above equation shows that the tensor t ◦V ,knownastheleft stretch

tensor, issymmetric.

From Eq. (2.45a), we get

t b = t ◦X · t ◦X T = t ◦V · t ◦V (2.48a)

and therefore,

t

◦V = £ ¤ t 1/2

b . (2.48b)

From the above equation we conclude that the left stretch tensor is defined

in the spatial configuration andthatitinheritsfrom t definiteness.

b the positive

Proceeding in the same way as in Sect. 2.7.2 we can show that the left

polar decomposition is unique.

26 **Nonlinear** continua

2.7.5 Physical interpretation of the tensors t ◦ R , t ◦ U and t ◦ V

In this Section, we will discuss a physical interpretation of the second-order

tensors introduced by the polar decomposition.

The rotation tensor

Assuming a motion in which t ◦U = ◦ g and therefore t ◦V = t g ,we get

t

◦X = t ◦R (2.49)

and considering in the reference configuration, at the point under analysis,

two arbitrary vectors ◦ dx 1 and ◦ dx 2 we have, in the spatial configuration:

hence, in the spatial configuration we can write

t dx1 = t ◦R · ◦ dx 1 (2.50a)

t dx2 = t ◦R · ◦ dx 2 (2.50b)

t dx1 · t dx 2 = t ◦R a A t ◦R b B t gab ◦ dx A 1

◦ dx B 2 . (2.50c)

Using Eqs. (2.42a) and (2.42b) we can rewrite the above equation as

t dx1 · t dx 2 = t ◦R a A ( t ◦R T ) C a ◦ dx A 1

and since t ◦R is an orthogonal tensor,

◦ dx2C , (2.50d)

t dx1 · t dx 2 = δ C A ◦ dx A 1 ◦ dx2C = ◦ dx 1 · ◦ dx 2 . (2.51)

The above equation shows that when t ◦X = t ◦R :

• The corresponding vectors in the spatial and reference configuration have

thesamemodulus.

• The angle between two vectors in the spatial configuration equals the angle

between the corresponding two vectors in the reference configuration.

Hence, the motion can be characterized, at the point under analysis, as a

rigid body rotation.

We can generalize Eqs.(2.50a-2.50d) for any vector Y that in the reference

configuration is associated to the point under analysis. For the particular

motion described by t ◦X = t ◦R ,weget

and since the rotation tensor is orthogonal,

where,

t y = t ◦R · Y , (2.52a)

Y = t ◦R T · t y . (2.52b)

If in the reference configuration there is a relation of the form:

Y = A · W , (2.53a)

2.7 The polar decomposition 27

Y , W : vectors defined in the reference configuration,

A : second order tensor defined in the reference configuration,

it is easy to show that:

t y =

h

t◦R

· A · t i

T

◦R

· t w . (2.53b)

In the above equation, the term between brackets is the result (in the

spatial configuration) of the rotation of the material tensor A.

If A is a symmetric second-order tensor we can write it using its eigenvalues

and eigenvectors,

A = λ AB Φ A Φ B , (2.54a)

where,

λ AB =0 if A 6= B,

and the set of vectors ΦA form an orthogonal basis in the reference configuration.

In the spatial configuration we get, from the rotation of A:

and therefore,

t a = t ◦R · A · t ◦R T , (2.54b)

t a = λ AB ( t ◦R · Φ A ) ( Φ B · t ◦R T ) . (2.54c)

We now define in the spatial configuration the set of vectors tϕ =

a t ◦R · ΦA ,

which obviously constitute an orthogonal basis; using Eq. (2.28a) we obtain

³

ΦB ·

t◦R T · t ´

ϕ

b

= ¡ ¢ t t

◦R · ΦB · ϕb , (2.54d)

and therefore,

t a = λ AB ( t ◦R · Φ A )( t ◦R · Φ B ) . (2.54e)

Comparing Eqs. (2.54a) and (2.54e), we conclude that:

• The material tensor A and the spatial tensor t a have the same eigenvalues.

• The eigenvectors of t a are obtained by rotating with t ◦R the eigenvectors

of A .

Since, according to Eq. (2.47a) t ◦V = t ◦R · t ◦U · t ◦R T , we can assess

that:

• The material tensor t ◦U and the spatial tensor t ◦V havethesameeigenvalues.

t • The eigenvectors of ◦V (and tb ) are obtained by rotating with t ◦R

t the eigenvectors of ◦U (and t ◦C ).

28 **Nonlinear** continua

The right stretch tensor

To study the physical interpretation of the right stretch tensor we consider, in

the reference configuration, at the point χ under analysis, two vectors ◦ dx 1

and ◦ dx 2 that in the spatial configuration are transformed into t dx 1 and

t dx2

After some algebra,

t dx1 = t ◦X · ◦ dx 1 (2.55a)

t dx2 = t ◦X · ◦ dx 2 . (2.55b)

t dx1 · t dx 2 = ◦ dx 1 · ( t ◦X T · t ◦X ) · ◦ dx 2 , (2.56a)

and using Eqs. (2.35) and (2.39), we write

t dx1 · t dx 2 = ◦ dx 1 · t ◦C · ◦ dx 2 = ◦ dx 1 · ( t ◦U · t ◦U ) · ◦ dx 2 . (2.56b)

It follows from the above equation that the changes in lengths and angles,

produced by the motion, are directly associated to the right stretch tensor

t

◦U . In the previous subsection we showed that these changes are nil when

t

◦U = ◦g .

The left stretch tensor

Starting from Eqs. (2.55a-2.55b) and using a left polar decomposition, we get

◦ dx1 · ◦ dx 2 = t dx 1 · ( t ◦V −1 · t ◦V −1 ) · t dx 2 . (2.57)

It is obvious from the above equation that changes in lengths and angles,

produced by the motion, are directly associated to the left stretch tensor t ◦V .

Above we showed that those changes are nil when t ◦V = t g .

2.7.6 Numerical algorithm for the polar decomposition

When analyzing finite element models of nonlinear solid mechanics problems,

we usually know the numerical value of the deformation gradient tensor at

a point and we need to use a numerical algorithm for performing the polar

decomposition.

In what follows, we present an algorithm that can be used for the right

polar decomposition when we refer the problem to a fixed Cartesian system.

√ t Starting from the matrix [ ◦X] that is a (3×3)-matrix in the general case,

we calculate the symmetric matrix,

£ t◦C ¤ = £ t ◦X ¤ T £ t◦X ¤ . (2.58a)

2.7 The polar decomposition 29

√ Using a numerical algorithm (Bathe 1996), we calculate the eigenvalues

λ 2 A and eigenvectors [ΦA] ;A =1, 2, 3 of the matrix [ t ◦C].

√ From the above step,

where

and

£ t◦C ¤ = [Ψ] [Λ] [Ψ] T

(2.58b)

[Ψ] = [[Φ1] [Φ2] [Φ3]] (2.58c)

⎡

(λ1)

[Λ] = ⎣

2

0 0

0 (λ2) 2

0

0 0 (λ3) 2

⎤

⎦ . (2.58d)

√

Using Eq. (2.39), we obtain (Strang 1980)

where

£ t◦U ¤ = [Ψ] [Λ] 1/2 [Ψ] T

[Λ] 1

2 =

√ Finally, using Eq. (2.40) we get

⎡

⎣ λ1 0 0

0 λ2 0

0 0 λ3

(2.58e)

⎤

⎦ . (2.58f)

£ t◦R ¤ = £ t ◦X ¤ £ t ◦U ¤ −1 . (2.58g)

And [ t ◦R] is now also completely determined.

Example 2.7. JJJJJ

For the case analyzed in Example 2.4, and for γ =15◦we can write

£ t◦X ¤ =

⎡

⎤

1.0 0.26795 0.0

⎣ 0.0 1.0 0.0⎦

.

0.0 0.0 1.0

Hence,

30 **Nonlinear** continua

£ t◦C ¤ = £ t ◦X T ¤ £ t ◦X ¤ =

⎡

⎤

1.0 0.26795 0.0

⎣ 0.26795 1.0718 0.0 ⎦ .

0.0 0.0 1.0

Solving for the eigenvalues and eigenvectors of [ t ◦C], weget

[Λ] =

⎡

⎤

0.76556 0.0 0.0

⎣ 0.0 1.30624 0.0 ⎦ ,

0.0 0.0 1.0

and,

[Ψ] =

Using Eq. (2.58e), we write

£ t◦U ¤ =[Ψ] [Λ] 1/2 [Ψ] T =

and finally, using Eq. (2.58g),

£ t◦R ¤ = £ t ◦X ¤ £ t ◦U ¤ −1 =

⎡

⎤

−0.75259 0.65849 0.0

⎣ 0.65849 0.75259 0.0 ⎦ .

0.0 0.0 1.0

⎡

⎤

0.99114 0.13279 0.0

⎣ 0.13279 1.02672 0.0 ⎦ ,

0.0 0.0 1.0

⎡

⎤

0.99114 0.13279 0.0

⎣ −0.13279 0.99114 0.0 ⎦ .

0.0 0.0 1.0

We urge the reader to verify that:

◦ [ t ◦R] T [ t ◦R] = [I]

◦ [ t ◦X] = [ t ◦R] [ t ◦U]

within the numerical accuracy used in the above calculations. JJJJJ

Example 2.8. JJJJJ

For the case analyzed in the previous example, we are now going to describe

the deformation of a material fiber that in the reference configuration contains

the (0, 0, 0) point and has the direction [ ◦ n] T = £ 0.0 1.0 0.0 ¤ . In the reference

configuration,

◦ dx = ◦ dS ◦ n

In the spatial configuration,

Using Eq. (2.56b),

◦ dx · ◦ dx = ( ◦ dS) 2

t dx = t dS t n ; k t nk = 1

t dx · t dx = ( t dS) 2 .

.

2.7 The polar decomposition 31

( t dS) 2 = ( ◦ dS) 2 £ ◦n · t ◦C · ◦ n ¤

and therefore, in the fixed Cartesian system that we are using,

t dS

◦ dS = £ [ ◦ n] T [ t ◦C] [ ◦ n] ¤1/2

and using the numerical values calculated in the previous example,

t dS

◦ dS = 1.03528 .

The reader can check the above numerical values using very simple geometrical

considerations.

◦ In the case of ◦ n = Φ i ,itiseasytoshowthat

t dS

◦ dS = λi .

◦ For [ ◦ m] T = £ 1.0 0.0 0.0 ¤ , we get

t dS

◦ dS = 1.0 .

◦ The vectors ◦ m and ◦ n, which are orthogonal in the reference configuration,

form an angle t θ in the spatial configuration,

t dxm = t dSm t m ; k t mk = 1

t dxn = t dSn t n ; k t nk = 1

t

dxm · t dxn = ( t dSm) ( t dSn) cos t θ =( ◦ dSm) ( ◦ dSn) [ ◦ m] T [ t ◦C] [ ◦ n]

µ tdS µ tdS

cos t θ =[ ◦ m] T [ t ◦C] [ ◦ n]

hence,

◦ dS

m

◦ dS

t θ = cos −1

n

"

[ ◦m] T [ t ◦C] [ ◦ #

n]

¢ ¢

¡ tdS

◦ dS

m

¡ tdS

◦ dS

and using the calculated numerical values, we get

t θ = 75 ◦ .

Once again, it is very simple for the reader to check the above numerical

result. JJJJJ

In the above examples we have numerically calculated the eigenvalues and

eigenvectors of the tensor t ◦C ; however, in some problems it is necessary to

n

32 **Nonlinear** continua

differentiate those eigenvalues and eigenvectors and it is therefore necessary

to use an analytical expression of them.

As is wellknown the eigenvalues of t ◦C are given by the roots of the following

polynomial (Strang 1980),

p(λ 2 i ) = − λ 6 i + I C 1 λ 4 i − I C 2 λ 2 i + I C 3 = 0 (i =1, 2, 3) (2.59a)

where, (McConnell 1957)

I C 1 = tr( t ◦C) (2.59b)

I C 2 = 1

2

£ C

(I1 ) 2 − tr( t ◦C 2 ) ¤

(2.59c)

I C 3 = det( t ◦C) , (2.59d)

and it is easy to verify the Serrin representation (Simo & Taylor 1991):

t t

ϕa ϕa = t λ 2 a

Φ A Φ A = t λ 2 A

t b − (I C 1 − λ 2 a) t g + I C 3 λ−2

a

2 λ 4 a − I C 1 λ2 a + I C 3 λ−2

a

t b −1

t

◦C − (IC 1 − λ 2 A) ◦g + IC 3 λ −2

A t ◦C−1 2 λ 4 A − I C 1 λ2 A + I C 3 λ−2

A

, (2.60a)

, (2.60b)

with no addition in “a”or “A”in the above equations. According to what we

showed in Sect. 2.7.5, λa = λA for a = A .

Example 2.9. JJJJJ

◦ To verify Eq. (2.60a)we start from,

For a =1

t 2 t t

b = λi ϕi ϕi

t b −1 = λ −2

i

t t

ϕi ϕi

I C 1 = λ 2 1 + λ 2 2 + λ 2 3

I C 2 = λ 2 1 λ 2 2 + λ 2 1 λ 2 3 + λ 2 2 λ 2 3

I C 3 = λ 2 1 λ 2 2 λ 2 3 .

t C

b − (I1 − λ 2 1) t g + I C 3 λ −2

1 t b −1 = 2λ41−I C 1 λ 2 1+I C 3 λ −2

1

λ 2 1

t t

ϕ1 ϕ1 .

The above verifies Eq. (2.60a) for the case a =1; the demonstrations for

a =2, 3 are identical.

◦ To verify Eq. (2.60b) we start from,

t

◦C = λI 2 ΦI ΦI t

◦C −1 = λI −2 ΦI ΦI 2.8 Strain measures 33

and proceed as before. JJJJJ

It is important to note that Eqs. (2.60a-2.60b) are only valid if the denominator

of the r.h.s. is not zero. The denominator is zero if we have repeated

eigenvalues.

2.8 Strain measures

Intheliteraturewecanfind a large number of strain measures that have been

proposed to characterize a deformation process. There are different approaches

for analyzing the deformation of continuum bodies and we usually find that,

for a given approach, one particular strain measure may be more suitable than

others.

In this section we will present a number of these strain measures without

making any claim of completeness.

2.8.1 The Green deformation tensor

We have already presented the Green deformation tensor in Sect. 2.7.1. It is

important to remember that this second-order tensor is defined in the reference

configuration and that for two vectors ◦ dx 1 and ◦ dx 2 defined at a point

χ in the reference configuration, the corresponding vectors in the spatial

configuration satisfy the relation,

t dx1 · t dx 2 = ◦ dx 1 · t ◦C · ◦ dx 2 . (2.61)

2.8.2 The Finger deformation tensor

We have already presented the Finger deformation tensor in Sect. 2.7.3. It is

important to remember that this second-order tensor is definedinthespatial

configuration.

Using Eq. (2.45a),

t b −1 = t ◦X −T · t ◦X −1 , (2.62)

and for two vectors t dx 1 and t dx 2 defined at a point t x in the spatial

configuration we can write,

34 **Nonlinear** continua

using Eq. (2.28a), we get

t dx1 · t b −1 · t dx 2 = t dx 1 · t ◦X −T · ◦ dx 2 (2.63a)

t dx1 · t b −1 · t dx 2 = ◦ dx 1 · ◦ dx 2 . (2.63b)

2.8.3 The Green-Lagrange deformation tensor

From Eq.(2.61),

t dx1 · t dx 2 − ◦ dx 1 · ◦ dx 2 = 2 ◦ dx 1 ·

∙

1

2 (t◦C − ◦ ¸

g) · ◦ dx2 . (2.64)

The Green-Lagrange strain tensor is defined in the reference configuration

as

t

◦ε = 1

2 (t◦C − ◦ g) . (2.65)

The second order tensor t ◦ε describes the deformation corresponding to the

t-configuration (spatial configuration) referred to the configuration at t =0

(reference configuration).

Example 2.10. JJJJJ

Considering a convected coordinate system © θ iª with covariant base vectors

t egi in the spatial configuration and ◦ eg i in the reference one, we can write,

and it is easy to show that,

t

◦ε = t ◦eεlm ◦eg l ◦eg m

t

◦eεlm = 1

h

tegl

·

2

teg −

m ◦eg l

·

i

◦egm .

JJJJJ

At the point χ under study, we now evolve from the t-configuration to a

t + ∆t-configuration by means of a rotation t+∆t

t R. Hence,

t+∆t

◦ X = t+∆t

t R · t ◦X , (2.66a)

t+∆t

◦ C = t ◦X T · t+∆t

t R T · t+∆t

t R · t ◦X , (2.66b)

and taking into account that the rotation tensor is orthogonal, we get

2.8 Strain measures 35

t+∆t

◦ C ≡ t ◦C (2.66c)

and therefore,

t+∆t

◦ ε ≡ t ◦ε . (2.66d)

From the above, we conclude that the Green deformation tensor and the

Green -Lagrange strain tensor are not affected by rigid body rotations: that

istosaytheyareindifferent to rotations.

2.8.4 The Almansi deformation tensor

From Eq. (2.63b), we get

t

dx1 · t dx2 − ◦ dx1 · ◦ dx2 = 2 t dx1 ∙

1

·

2 ( t g − t b −1 ¸

) · t dx2 . (2.67)

The Almansi strain tensor is defined in the spatial configuration as

t 1

e =

2 ( t g − t b −1 ) . (2.68)

At the point χ under study, we now evolve from the t-configuration to

the t + ∆t-configuration by means of a rotation t+∆t

t R. Hence,

and,

t+∆t

◦ X −1 = t ◦X −1 · t+∆t

t R T , (2.69a)

t+∆t

◦ X −T = t+∆t

t R · t ◦X −T , (2.69b)

using Eq. (2.62),

t+∆t −1

b

t+∆t

= t R · t b −1 · t+∆t

t R T , (2.69c)

and therefore,

t+∆t

e

t+∆t

= t R · t e · t+∆t

t R T . (2.69d)

Hence, the Finger and Almansi tensors are affectedbyrigid-bodyrotations.

2.8.5 The Hencky deformation tensor

The Hencky or logarithmic strain tensor is defined in the reference configuration

as

t

◦H = ln t ◦U . (2.70)

When the problem is referred to a fixed Cartesian system using Eq. (2.58e),

we get

[ t ⎡

◦H] = [Ψ] ⎣ ln λ1

⎤

0.0 0.0

0.0 lnλ20.0 ⎦ [Ψ]

0.0 0.0 lnλ3

T . (2.71)

Obviously, the Hencky deformation tensor is indifferent to rotations, since

from the polar decomposition, we can see that t ◦U does not incorporate the

effect of rigid-body rotations.

36 **Nonlinear** continua

2.9 Representation of spatial tensors in the reference

configuration (“pull-back”)

For the regular motion depicted in Fig. 2.1, we can define:

• An arbitrary curvilinear coordinate system { txa } in the spatial configuration.

At a point χ ( txa ,a =1, 2, 3) we can determine the covariant base

vectors tg and the contravariant base vectors

a tga •

.

An arbitrary curvilinear coordinate system { ◦x A } in the reference configuration.

At the point χ ( ◦xA ,A=1, 2, 3) we can determine the covariant

base vectors ◦g and the contravariant base vectors

A ◦gA •

.

A convected curvilinear coordinate system {θ i }.Atthepointχin the

spatial configuration we can determine the covariant base vectors t˜g and

a

the contravariant base vectors t˜g a , while in the reference configuration

we can determine the covariant base vectors ◦˜g a

base vectors

and the contravariant

◦˜g a

2.9.1 Pull-back of vector components

Let us consider in the spatial configuration at the point χ a vector,

t b = t b i t gi = t bi t g i = t˜ b i t egi = t˜ bi t eg i . (2.72)

We define in the reference configuration the following vectors (Dvorkin,

Goldschmit, Pantuso & Repetto 1994):

After some algebra,

t B = t˜ b i ◦ egi = [ t B ] A ◦ g A

(2.73a)

t B = t˜ bi ◦ eg i = [ t B ]A ◦ g A . (2.73b)

[ t B ] A = t b j ( t ◦X −1 ) A j (2.74a)

[ t B ]A = t bj t ◦X j

A . (2.74b)

Adopting the notation used in manifolds analysis (Lang 1972, Marsden &

Hughes 1983) we define the pull-back of the contravariant components t b j as

£ tφ ∗ ( t b j ) ¤ A = [ t B ] A

and the pull-back of the covariant components t bj as:

(2.75a)

£ tφ∗ t

( bj) ¤

A = [tB ]A . (2.75b)

We can therefore rewrite Eqs. (2.73a-2.73b) and (2.74a-2.74b),

2.9 Representation of spatial tensors in the reference configuration (“pull-back”) 37

t B = t b j ( t ◦X −1 ) A j ◦ g A = t b j ( t ◦X −1 ) A j ◦ gAC ◦ g C

t t

B = bj t ◦X j ◦ A t

A g = bj

t

◦X j

A

(2.76a)

◦ g AC ◦ gC . (2.76b)

For two vectors t b and t w ,defined in the spatial configuration at χ ,

using Eqs. (2.74a-2.74b), it is easy to show that:

t B · t W = t B · t W = t b · t w . (2.77)

Also, using Eqs.(2.74a-2.74b) and (2.36a-2.36b), we get

Using Eqs. (2.73a-2.73b), we can write

[ t B ] B t ◦CAB = [ t B ]A . (2.78)

( teg )

a = ◦eg a

(2.79a)

( teg a ) = ◦eg a . (2.79b)

Hence, we use the following notation (Moran, Ortiz & Shih 1990):

t φ ∗ ( t ega ) = ( t eg a ) = ◦ eg a

(2.79c)

t φ ∗ ( t eg a ) = ( t eg a ) = ◦ eg a . (2.79d)

From the above equations, we can get by inspection the geometrical interpretation

of the vectors t B and t B

:

• If tb,inthespatialconfiguration, is the tangent to a curve tc(ξ) at a

point χ, then tB

is the tangent, in the reference configuration, to the

curve C(ξ) = tφ −1 [ t •

c(ξ)] ,atthepointχ. In the transformation tB → tb the modulus of the original vector gets

stretched as the material fiber to which they are tangent.

In convected coordinates we have

that is to say,

t dx = dθ i t egi

◦ dx = dθ i ◦ egi

(2.80a)

(2.80b)

t dX = ◦ dx . (2.80c)

• For two vectors t b and t w that are orthogonal in the spatial configuration

it is obvious that:

38 **Nonlinear** continua

Fig. 2.4. Mappings

t B · t W = t B · t W = t b · t w = 0 . (2.81)

Hence, the orthogonality of t b and t w implies the orthogonality of

t B and t W and the orthogonality of t B and t W

in the reference

configuration.

It is important to take into account that in Eqs. (2.74a-2.74b) the terms

on the r.h.s. must be written as a function of the coordinates in the reference

configuration. We can indicate this using a more formal nomenclature (e.g.

Marsden & Hughes 1983),

[ t B ] A = £ ¤ t ∗ t a A

φ ( b ) =

£ t

( ◦X −1 ) A a ◦ t φ ¤ ( t b a ◦ t φ) (2.82a)

[ t B ]A = £ t ∗ t

φ ( ba) ¤

A = t ◦X a A ( t ba ◦ t φ) . (2.82b)

In order to understand the above equations we use Fig. 2.4 (Marsden &

Hughes 1983). In this figure we indicate with “f ◦ g” the composition of the

mapping “g” followed by the mapping “f”.

Example 2.11. JJJJJ

For a function f( t x a ) definedinthespatialconfiguration, we can write:

2.9 Representation of spatial tensors in the reference configuration (“pull-back”) 39

df = ∂f

∂ t x a

t dx a .

In the above equation, we use a formal analogy with vector calculus in which

(Marsden & Hughes 1983): df is a vector; ∂f

∂ t x a are its covariant components

(dfa) andd t x a are contravariant base vectors.

Hence we can do a pull-back operation,

£ tφ∗ (dfa) ¤

A = t ◦X a A dfa = ∂txa ∂◦xA ∂f

∂txa =

∂f

∂◦ . A x

Using a more formal nomenclature and the mappings in Fig. 2.7, we get

£ tφ∗ (dfa) ¤

A = ∂(f ◦ tφ) ∂◦ .

xA

JJJJJ

Example 2.12. JJJJJ

Equation (2.11b) defines, in the spatial configuration, the velocity of a material

point,

t t a t

v = v ga .

Using the expressions for the pull-back of contravariant components, we write

£ tφ ¤ ∗ t a A t

( v ) = ( ◦X −1 ) A a t v a .

Ifthecoordinatesystem{ ◦ x A },defined in the reference configuration, is a

convected system with covariant base vectors, t ˜gA ,inthespatialconfiguration,

we can write

◦ dx A t egA = t dx a t g a

hence,

and therefore,

t ga = ( t ◦X −1 ) A a t eg A

t v = t v a ( t ◦X −1 ) A a t eg A .

The components of the material velocity vector in the convected system { ◦ x A }

are,

t ˜v A = t v a ( t ◦X −1 ) A a

and therefore £ tφ ∗ ( t v a ) ¤ A = t ˜v A .

JJJJJ

40 **Nonlinear** continua

2.9.2 Pull-back of tensor components

Let us consider in the spatial configuration at the point χ ( t x a ,a =1, 2, 3)

a second-order tensor,

t t ij t = t t t g gj =

i

ttij tgi tgj = tt i

j t t j g g

i

= t˜t ij t eg tegj =

i

t˜tij teg i t j

eg = t˜t i

j t . (2.83)

eg t j

eg

i

We define in the reference configuration the following second-order tensors

(Dvorkin, Goldschmit, Pantuso & Repetto 1994):

t

T =

t˜t ij ◦ ◦ eg egj = [

i

t T ] AB ◦ g

A

◦ gB , (2.84a)

t

T =

t˜t i j ◦ ◦ j t A ◦ ◦ B eg eg = [ T ]

i

B gA g , (2.84b)

t T = t˜tij ◦ eg i ◦ eg j = [ t T ]AB ◦ g A ◦ g B . (2.84c)

After some algebra we get

We define:

[ t T ] AB = t t ab ( t ◦X −1 ) A a ( t ◦X −1 ) B b , (2.85a)

[ t T ] A B = t t a b (t◦ X−1 ) A a (t◦ X)bB , (2.85b)

[ t T ]AB = t tab ( t ◦X) a A ( t ◦X) b B . (2.85c)

• Pull-back of the contravariant components of t t ,

• Pull-back of the mixed components of t t ,

£ tφ ∗ ( t t ij ) ¤ AB = [ t T ] AB . (2.86a)

£ tφ∗ t i

( t j) ¤ A

B = [tT ] A B . (2.86b)

• t Pull-back of the covariant components of t ,

£ tφ∗ t

( tij) ¤

AB = [tT ]AB . (2.86c)

From Eqs. (2.84a-2.84c) and (2.85a-2.85c) we can write,

2.9 Representation of spatial tensors in the reference configuration (“pull-back”) 41

t T = t t ab ( t ◦X −1 ) A a ( t ◦X −1 ) B b ◦ g A

◦ gB

= t t ab ( t ◦X −1 ) A a ( t ◦X −1 ) B b ◦ gAL ◦ gBM ◦ g L ◦ g M

= t t ab ( t ◦X −1 ) A a ( t ◦X −1 ) B b ◦ gAL ◦ g L ◦ g B , (2.87a)

t T = t t a b ( t ◦X −1 ) A a ( t ◦X) b B ◦ g BL ◦ g A

◦ gL

= t t a b ( t ◦X −1 ) A a ( t ◦X) b B ◦ gAL ◦ g L ◦ g B

= t t a b ( t ◦X −1 ) A a ( t ◦X) b B ◦ ◦ B

g g , (2.87b)

A

t T = t tab ( t ◦X) a A ( t ◦X) b B ◦ g AL ◦ g BM ◦ g L

= t tab ( t ◦ X)a A (t ◦ X)b B ◦ g A ◦ g B

◦ gM

= t tab ( t ◦X) a A ( t ◦X) b B ◦ g BL ◦ g A ◦ g L . (2.87c)

For two tensors t t and t w definedinthespatialconfiguration at χ ( t x a ,a=

1, 2, 3) , using Eqs. (2.84a-2.84c), it is easy to show that:

t T :

t W = t T :

t W = t t :

and also, using Eqs.(2.87a-2.87c), we can show that

t w (2.88)

[ t T ]AB = [ t T ] P B t ◦CPA (2.89a)

[ t T ]AB = [ t T ] PQ t ◦CPA t ◦CQB . (2.89b)

If in the spatial configuration, at the point χ ( t x a ,a=1, 2, 3) ,twovectors

t b and t w are related by the second order tensor t t, via the equation,

t b = t t · t w (2.90)

in the reference configuration, we can easily verify that the following relations

are fulfilled:

t B = t T · t W

t B = t T · t W

(2.91a)

(2.91b)

t B = t T · t W . (2.91c)

42 **Nonlinear** continua

Example 2.13. JJJJJ

Instead of using the mixed components tti j , it is possible to use the compo-

, hence

nents t t i

j

t t = t t i

j t g j t gi =

t˜t i

j t ˜g j t ˜g i

wecanthendefine in the reference configuration the second order tensor,

after some algebra we then get,

t T ◦ = t˜t i

j ◦ ˜g j ◦ ˜g i = [ t T ◦ ] B

A ◦ g A ◦ g B

[ t T ◦ ] B

A = t t b

a (t ◦ X−1 ) B b (t ◦ X)a A .

Therefore,

[ t φ ∗ ( t t i

j )] B

A = [ t T ◦ ] B

A .

Starting from Eq.(2.90), we can also show that,

t B = t T ◦ · t W .

2.10 Tensors in the spatial configuration from

representations in the reference configuration

(“push-forward”)

JJJJJ

In the previous section we obtained representations in the reference configuration

of tensors definedinthespatialconfiguration (pull-back).

The inverse operation is named, in the manifold analysis literature (Lang

1972, Marsden & Hughes 1983), push-forward. For a second-order tensor, from

Eqs.(2.85a-2.85c), we can write:

t t ab =

t tab =

t t a b =

h tφ∗[ t T ] ABi ab

h tφ∗[ t T ]AB

h tφ∗[ t T ] A B

i

i a

= t ◦X a A t ◦X b B [ t T ] AB

(2.92a)

ab = (t ◦X −1 ) A a ( t ◦X −1 ) B b [ t T ]AB (2.92b)

b

= t ◦X a A ( t ◦X −1 ) B b [ t T ] A B . (2.92c)

2.11 Pull-back/push-forward relations between strain measures 43

Example 2.14.

Using Eqs. (2.92a-2.92c) we can show that,

JJJJJ

£ tφ∗(WA) ¤

a = (tb −1 )al

£ tφ∗(W A ) ¤l

.

2.11 Pull-back/push-forward relations between strain

measures

JJJJJ

In the two previous Sections we defined representations of the types known

as pull-back and push-forward. As we will see later on, these kinds of representations

are extremely useful in nonlinear continuum mechanics.

In Sect. 2.8 we defined a number of strain measures, some of them in the

spatial configuration (e.g. the Finger and Almansi tensors) and some of them

in the reference configuration (e.g. the Green, Green-Lagrange and Hencky

tensors).

In the present section we will establish relations of the pull-back/pushforward

type between those strain measures.

We define, as usual, at the material point χ:

◦ In the spatial configuration a coordinate system { txa } with covariant

base vectors tg .

a

◦ In the reference configuration a coordinate system { ◦x A } with covariant

base vectors ◦g .

A

We can therefore calculate at χ the deformation tensor t ◦X .

• The pull-backs of the spatial metric tensor are:

£ tφ ∗ ( t gab) ¤

AB

◦ g A ◦ g B = t g = t ◦C , (2.93a)

£ ¤AB tφ∗ t ab ◦gA ◦

( g ) gB = t g = t ◦C −1 . (2.93b)

• The pull-back of the Almansi strain tensor is:

£ tφ ∗ ( t eab) ¤

AB

◦ g A ◦ g B = t E = t ◦ε . (2.94a)

From Eqs. (2.77),(2.80a-2.80c) and (2.91a-2.91c), we get

t dx · t e · t dx = ◦ dx · t ◦ε · ◦ dx . (2.94b)

• The pull-back of the left stretch tensor is:

44 **Nonlinear** continua

t

◦V = t ◦U . (2.95)

• In many problems related to metallurgy (e.g. metal-forming problems) the

usual practice is to use a logarithmic strain measure (Hill 1978). Therefore,

in the spatial configuration the following strain measure is defined

Thepull-backofthetensor t h is:

t h = ln t ◦V . (2.96a)

t

◦H = t H = ln t ◦V = ln t ◦U . (2.96b)

The above tensor is known as the Hencky deformation tensor.

In following Chapters we will see the importance of the Hencky strain tensor

for the analysis of finite strain problems (Anand 1979). Some recent finite

element formulations, developed for the analysis of finite strain elastoplastic

problems use this strain measure (e.g. Rolph & Bathe 1984, Weber

& Anand 1990, Eterovic & Bathe 1990, Simo 1991, Dvorkin, Pantuso &

Repetto 1992\1993 \1994\1995, Dvorkin 1995a\1995b \1995c, Pèric, Owen

& Honnor 1992).

It is useful to note that we also get Eqs. (2.95) and (2.96a-2.96b) when

instead of searching for a representation in the reference configuration, we

search for a representation in a configuration rotated by t ◦R T from the

spatial configuration: corotational representation.

• The push-forward of the reference configuration metric tensor is:

£ tφ∗( ◦ gAB) ¤

ab

t g a t g b = ( t ◦X −1 ) A a ( t ◦X −1 ) B b ◦ gAB t g a t g b

= t b −1

(2.97a)

£ tφ∗( ◦ g AB ) ¤ab tga t

gb = t ◦X a A t ◦X b B ◦ g AB t t

g gb

a

= t b . (2.97b)

In Chap. 3 we will study relations of the pull-back/push-forward type

between stress measures.

2.12 Objectivity

The description of physical phenomena using objective formulations is a central

topic in continuum mechanics.

We will first present the classical concept of objectivity under rotations and

translations (isometries) (Truesdell & Noll 1965). Next, we will present the

concept of objectivity under general changes of the reference frame (Marsden

& Hughes 1983), we will use the word covariance to refer to this concept.

2.12.1 Reference frame and isometric transformations

2.12 Objectivity 45

We call an event the pair { t x,t} formed by a vector t x that defines a point

in the Euclidean space and a time t.

A reference or observation frame is a way of relating the physical world

to the points in an < 3 Euclidean space and a real axis of time (Truesdell &

Noll 1965).

Examples of reference frames:

- The system of fixedstarsandaclock.

-Thewallsofmyoffice and my watch.

- A system of coordinates drawn on a rotating platform and a clock.

An isometric transformation of reference frame is a mapping (< 3 ,t) → (< 3 ,t)

in which the distances between spatial points, the time intervals between

events and the time ordering of events are preserved. Obviously, isometric

transformations restrict us to Newtonian Mechanics.

To describe an isometric transformation of reference frame we define in

the spatial configuration:

• A fixed Cartesian system { t zα}.

• Another Cartesian system { t∗

z ∗ α} that rotates and translates.

For simultaneous events, in the fixed system we register a time t, andin

the moving system a time t∗ .

The base vectors of the fixed and moving Cartesian frames are teα and

t ∗

e∗ α , respectively; hence, we can write, at an instant t:

t

eα = Q(t) · t∗

e ∗ α . (2.98)

Obviously, the tensor Q(t) is orthogonal.

Let us call c(t) the vector that goes from tz = (0, 0, 0) (origin of the

fixed Cartesian frame) to t∗z∗

= (0, 0, 0) (origin of the moving Cartesian

frame).

An observer on the moving frame defines an event with the vector t∗z∗

and the time t∗ .

An observer on the fixed frame defines the same event with the vector tz and the time t.

Let us assume that the two observers are observing a lab experiment, e.g.

the measurement of the stretching of a spring when it is loaded with a weight

of 1 kg. Let us also assume that the lab is moving with the moving frame; for

the observer on this frame t∗z∗

is the position vector of the spring-end.

The observer on the fixed frame will see the spring-end moving:

t z(t) = c(t) + Q(t) · t ∗

z ∗ . (2.99a)

Since we are considering now isometric transformations, the following condition

holds:

46 **Nonlinear** continua

t = t ∗ − a, (2.99b)

where a is a constant. For simplicity, from here on, we will consider a =0(t =

t∗ ).

If we include the possibility that Q(t) represents not only rotations but

also reflections (transformations between left-handed and right-handed systems),

Eqs. (2.99a-2.99b) represent the most general isometric transformation

(Truesdell & Noll 1965, Truesdell 1966).

We should not confuse the concept of change of reference frame with the

concept of change of coordinates system, the latter being an instant concept

does not incorporate the frame velocity.

An isometric change of reference frame may induce changes in scalars,

vectors and tensors.

• Scalars do not change during an isometric transformation.

• Vectors definedinthespatialconfiguration can always be considered as

proportional to the difference between two position vectors in that configuration

(Truesdell & Noll 1965); hence, using Eq. (2.99a) we get

t v = Q(t) · t v ∗ . (2.100)

The above is the transformation law for spatial vectors under an isometric

transformation of reference frame.

• Second-order tensors defined in the spatial configuration relate vectors

defined in the same configuration (quotient rule) 6 . In the moving frame,

we can write for an arbitrary second-order tensor t s ∗

Using Eq.(2.100),

t v =

t v ∗ = t s ∗ · t w ∗ . (2.101a)

h

Q(t) · t s ∗ · Q T i

(t)

· t w . (2.101b)

Therefore, the transformation law for spatial second-order tensors under

an isometric transformation of reference frame is

t s = Q(t) · t s ∗ · Q T (t) . (2.101c)

• For spatial tensors of higher order, the transformation laws can be derived

in an identical way.

6 See Appendix.

2.12.2 Objectivity or material-frame indifference

2.12 Objectivity 47

Let us assume a deformation process taking place in the moving reference

frame. This deformation process can be referred to a fixed reference configuration.

There are general tensors (including scalars and vectors) defined in the

reference configuration (e.g. the Green-Lagrange strain tensor), they are called

material or Lagrangian tensors.

There are general tensors definedinthespatialconfiguration (e.g. the Almansistraintensor,thematerialvelocity),theyarecalledspatial

or Eulerian

tensors.

There are general tensors definedinbothconfigurations (e.g. the deformation

gradient tensor), they are called two-point tensors.

Following (Lubliner 1985) we define the following objectivity criteria under

isometric transformations of a reference frame (classical objectivity):

◦ A Lagrangian tensor is objective if it is not affected by changes of the

reference frame.

◦ An Eulerian tensor is objective if, under a change of reference frame, transforms

according to Eqs. (2.100) and (2.101c).

◦ A two points second order tensor is objective if, when operating on a Eulerian

objective vector, produces a Lagrangian objective vector.

Example 2.15. JJJJJ

If we differentiate Eq. (2.99a) with respect to time we get,

t ˙z(t) = t ˙c(t)+ ˙Q(t) · t z + Q(t) · t ˙z .

Hence, the velocity is not an objective vector. JJJJJ

Inthesamewaywecanshowthattheacceleration vector is not objective.

Therefore, even though forces are spatial objective tensors, Newton’s second

law ( t F = m t ¨x) is not objective and is only applicable in inertial frames

(Truesdell & Noll 1965).

In what follows, we will analyze the objectivity of some of the tensors

previously defined:

• The deformation gradient tensor, in the moving frame, satisfies the following

relation:

t dx ∗ = t ◦ X ∗ · ◦ dx . (2.102a)

48 **Nonlinear** continua

• The vector ◦ dx is an objective Lagrangian vector and t dx ∗ is an objective

Eulerian vector, hence t ◦X ∗ fulfills the objectivity definition.

In the fixed frame,

and finally,

Q T (t) · t dx = t ◦X ∗ · ◦ dx , (2.102b)

t

◦X = Q(t) · t ◦X ∗ . (2.102c)

Equation (2.102c) is the transformation law for objective two-point secondorder

tensors.

• Performing a right polar decomposition on both sides of Eq. (2.102c), we

get

t

◦R · t ◦U = Q(t) · t ◦R ∗ · t ◦U ∗ . (2.103a)

Taking into account that:

◦ The inner (dot) product of two orthogonal second-order tensors is an orthogonal

second order tensor.

◦ The polar decomposition is unique.

We get

t

◦R = Q(t) · t ◦R ∗

(2.103b)

t

◦U = t ◦U ∗ . (2.103c)

Hence, the rotation tensor (two-point tensor) and the right stretch tensor

(Lagrangian tensor) are objective under isometric transformations.

• From Eq. (2.47b), we can write

t

◦V = t ◦R · t ◦U · t ◦ R T . (2.104a)

and using Eqs. (2.103a-2.103c), we obtain

t

◦ V = Q(t) · t ◦ V∗ · Q T (t) . (2.104b)

The above equation shows that the left stretch tensor (Eulerian tensor) is

objective under isometric transformations.

• The Green-Lagrange strain tensor (Lagrangian tensor) is (Eq. (2.65))

t 1

h

t◦

◦ε = X

2

T · t ◦X − ◦ i

g

and using Eq. (2.102c), we get

2.12 Objectivity 49

(2.105a)

t

◦ε = t ◦ε

(2.105b)

The above equation depicts an objective behavior under isometric transformations.

• The Almansi strain tensor (Eulerian tensor) is (Eq. (2.68)):

t 1

e =

2

h

tg t

− ◦X −T · t ◦X −1i

and using Eq. (2.102c), we get

. (2.106a)

t e = Q(t) · t e ∗ · Q T (t) . (2.106b)

The above equation depicts an objective behavior under isometric transformations.

2.12.3 Covariance

Let us assume a body that remains undeformed, referred to a spatial coordinate

system that keeps changing (e.g. at a time t we have the spatial system

{ txa } and at a time ˆt the system { ˆt a ˆx }). For a spatial second order tensor

ta , using the usual tensor transformations, we can write (Truesdell & Noll

1965):

ˆt â ab = ∂ ˆt ˆx a

∂ t x l

ˆt âab = ∂t x l

∂ ˆt ˆx a

ˆt â a b = ∂ˆt ˆx a

∂ t x l

∂ ˆt ˆx b

∂ t x m

∂ t x m

∂ ˆt ˆx b

∂ t x m

∂ ˆt ˆx b

t a lm , (2.107a)

t alm , (2.107b)

t a l m . (2.107c)

Between the spatial coordinates of the material particles at t and the

spatial coordinates of the same particles at ˆt we can define a mapping ( ˆt tφ)

(see Eq. (2.4)). Hence, we can also define a deformation gradient tensor, ˆt tX ,

and we can rewrite Eqs. (2.107a-2.107c) as

50 **Nonlinear** continua

t a ab = ( ˆt tX −1 ) a l ( ˆt tX −1 ) b m

t aab = ˆt tX l a

ˆt

tX m b

t a a b = ( ˆt tX −1 ) a l

ˆt âlm =

ˆt

tX m b

ˆt â l m =

ˆt â lm =

h

ˆt

tφ ∗ ( ˆt

i

âlm )

h

ˆt

tφ ∗ ( ˆt

iab lm

â )

ab

h ˆt tφ ∗ ( ˆt â l m )

, (2.108a)

, (2.108b)

i a

b

. (2.108c)

If we consider now general transformations between the spatial configurations

at t and ˆt (not necessarily restricted to simple changes of coordinate

system) the spatial tensor t a is defined as covariant or objective (in a more

general way than the above defined objectivity under isometric transformations)

if under the mapping ( ˆt tφ) it transforms following Eqs.(2.108a-2.108c)

(Marsden & Hughes 1983).

For a two-point tensor, the covariance criterion is only applied to the

indices associated to the spatial basis (Lubliner 1985).

A Lagrangian tensor is always covariant if it is not affected by changes of

the reference frame. The reader can easily check that for the case of isometric

transformations the concept of covariance is coincident with the classical

concept of objectivity presented in the previous section.

A physical law is objective (either in the classical or in

the covariant sense) if all the tensors in its mathematical

formulation are objective.

2.13 Strain rates

In the previous sections we presented a static description of the kinematics

of continuous media: given a spatial and a reference configuration we developed

tools to relate both configurations (deformation gradient tensor, strain

measures, etc.)

In the present section we will study the kinematic evolution of the spatial

configuration. For this purpose we will introduce the time rates of the different

tensors described above.

2.13.1 The velocity gradient tensor

ThetimerateofEq.(2.23)is

t

◦ ˙X

∙ t a ∂ v

=

∂◦xA + t ◦X l A t Γ a

lb t v b

where t v a was defined in Eq. (2.10).

¸ tga

◦ g A , (2.109a)

2.13 Strain rates 51

It is important to remember that the functional dependence is

Using the chain rule in Eq. (2.109a),

t

◦ ˙ X a A = t ◦ ˙ X a A( ◦ x B ,t) . (2.109b)

t

◦ ˙ X a A = t v a |l t ◦X l A . (2.109c)

We define in the spatial configuration the velocity gradient tensor,

we can write the above as

Hence, we can write Eq.(2.109c) as

t t a

l = v |l t t l

g g , (2.110a)

a

t l = t v ∇ , (2.110b)

t l T = ∇ t v . (2.110c)

t

◦ ˙ X a

A = t l a l t ◦X l

A

(2.111a)

and therefore,

t

◦ ˙X = t l · t ◦X . (2.111b)

It is important to realize that the above is the material time derivate of

the deformation gradient tensor, t ◦ ˙X = D t ◦X Dt .

2.13.2 The Eulerian strain rate tensor and the spin (vorticity)

tensor

We can decompose the velocity gradient tensor into its symmetric and skewsymmetric

components:

t t t

l = d + ω (2.112a)

where,

t t T 1

d = d =

2 (tl + t l T ) (2.112b)

is the Eulerian strain rate tensor (defined in the spatial configuration) and,

t ω = − t ω T = 1

2 (t l − t l T ) (2.112c)

is the spin or vorticity tensor, alsodefined in the spatial configuration.

Let us assume a deformation process referred to a fixed Cartesian system.

The principal directions of t ◦U form, in the reference configuration, a Cartesian

system known as Lagrangian system. The principal directions of t ◦V form, in

the spatial configuration, a Cartesian system known as a Eulerian system (Hill

1978).

52 **Nonlinear** continua

Fig. 2.5. Rotations

We can go from one of the above-defined coordinate systems to another

one using the rotation tensors sketched in Fig. 2.5.

From Fig. 2.5, we get

For two consecutive rotations,

and therefore,

t RE = t ◦ R · t ◦ RL . (2.113)

t+∆t

◦ R = t+∆t

t R · t ◦R (2.114a)

t

◦ ˙R = lim ∆t → 0

We can define a rotation rate

" t+∆t

t

t ΩR = lim ∆t → 0

R − t #

g

·

∆t

t ◦R . (2.114b)

" t+∆t

t

and using it in Eq. (2.114b) (Hill 1978), we get

in the same way,

R − t #

g

∆t

(2.114c)

t

◦ ˙R = t Ω R · t ◦R (2.115a)

t

◦ ˙R L = t ◦Ω L · t ◦R L , (2.115b)

t ˙R E = t Ω E · t R E . (2.115c)

Since the rotation tensors are orthogonal we can write

2.13 Strain rates 53

t

◦R T · t ◦R = ◦ g , (2.116a)

taking the time derivative of the above equation and using Eq. (2.115a), we

have

t

ΩR + t Ω T

= 0 (2.116b)

R

in the same way,

t

◦Ω +

L t ◦Ω T

L

t ΩE + t Ω T

E

The above equations indicate that t t Ω ,

R ◦Ω

L

symmetric tensors.

2.13.3 Relations between different rate tensors

The time derivative of Eq. (2.113) leads to

= 0 , (2.116c)

= 0 . (2.116d)

and t Ω E are skew-

t ΩE · t R E = t Ω R · t ◦R · t ◦R L + t ◦R · t ◦Ω L · t ◦R L , (2.117a)

and therefore,

t

◦R T · ( t ΩE − t Ω ) ·

R t ◦R = t ◦Ω . (2.117b)

L

Using Eqs. (2.111b) and (2.40),

t l = t ◦ ˙R · t ◦R T + t ◦R · t ◦ ˙U · t ◦U −1 · t ◦R T , (2.118a)

splitting the above equation into its symmetric and skew-symmetric components,

we get

and

t

◦R T · t d · t ◦R = 1

2 (t◦ ˙U · t ◦U −1 + t ◦U −1 · t ◦ ˙U) (2.118b)

t

◦R T · ( t ω − t ΩR) · t ◦R = 1

2 (t◦ ˙U · t ◦U −1 − t ◦U −1 · t ◦ ˙U) . (2.118c)

It is very important to recognize that (Hill 1978):

t

◦U = ◦ g =⇒ t ω = t Ω R . (2.118d)

An example of the above situation is the beginning of the deformation

process (t = 0).

In the deformation process depicted in Fig. 2.6 (Truesdell & Noll 1965,

Malvern 1969) we can write, using Eqs. (2.30a-2.30b):

54 **Nonlinear** continua

Fig. 2.6. Relative deformation gradients

τ

◦X = τ t X · t ◦X .

For a fixed t-configuration, we can write

(2.119a)

d τ

dτ

◦X = d τ

t X ·

dτ

t ◦X . (2.119b)

Using Eq. (2.111b) and the polar decomposition in the above equation, we

get

τ τ

l · ◦X = ( d τ

t R ·

dτ

τ t U + τ t R · d τ

t U) ·

dτ

t ◦X , (2.119c)

for τ = t it is obvious that τ

t U = τ t R = ◦g , and since the above

equation has to hold for any t ◦X,

d τ

t R |τ=t +

dτ

d τ

t U |τ=t =

dτ

t l . (2.119d)

It is easy to show that the first tensor on the l.h.s. of the above equation

is skew-symmetric and the second one is symmetric; hence,

d τ

t R | τ=t =

dτ

1

2 (tl − t l T ) = t ω (2.120a)

d τ

t U | τ=t =

dτ

1

2 (tl + t l T ) = t d . (2.120b)

We can obtain an interesting picture of the deformation process by referring

the Lagrangian tensors to the Lagrangian coordinate system (principal

directions of t ◦U) and the Eulerian tensors to the Eulerian coordinate system

(principal directions of t ◦V). Hence (Hill 1978):

2.13 Strain rates 55

• In the Lagrangian system the components of t ◦U are tλr (weassumethem

to be different); the components of t ◦ ˙U are t ˙ λrs and the components of

t

◦Ω are

L t ◦ΩL •

rs.

In the Eulerian system the components of t ◦V are of course also tλr; the

components of td are tdrs; the components of tω are tωrs; the components

of tΩE are tΩE rs and the components of tΩR are tΩR rs.

From Eq. (2.117b) we have

t E

Ωrs − t Ω R rs = t ◦Ω L rs , (2.121a)

from Eq. (2.118b) we get

and from Eq. (2.118c) we get

t drs = t ˙ λrs

t ωrs − t Ω R rs = t ˙ λrs

t λr + t λs

2 t λr t λs

t λr − t λs

2 t λr t λs

(2.121b)

; (2.121c)

(in Eqs. (2.121b) and (2.121c) we do not use the summation convention).

In the fixed Cartesian system the components of t ◦U form the matrix [ t ◦U];

hence,

[ t ◦U] = [ t ◦RL] [ t Λ][ t ◦RL] T , (2.122a)

where,

[ t Λ] =

⎡

tλ1 0 0

⎣ 0 tλ2 0

0 0 t ⎤

⎦ .

λ3

(2.122b)

Taking the time derivative of Eq. (2.122a), we obtain

[ t ◦ ˙ U] = [ t ◦RL] [ t Λ] ˙ t

[ ◦RL] T + [ t ◦ΩL] [ t ◦RL] [ t Λ][ t ◦RL] T

hence,

− [ t ◦RL] [ t Λ][ t ◦RL] T [ t ◦ΩL] (2.122c)

[ t ◦RL] T [ t ◦ ˙ U][ t ◦RL] = [ t ˙

Λ] + [ t ◦RL] T [ t ◦ΩL] [ t ◦RL] [ t Λ]

The above equation shows that:

− [ t Λ][ t ◦ RL] T [ t ◦ ΩL][ t ◦ RL] . (2.122d)

tλrs ˙ = t ˙ tλrs ˙

λr

=(

(r = s)

tλs − tλr) t ◦ΩL rs

.

(r 6= s)

(2.123)

From Eq. (2.121b) for the case r = s (diagonal components), we get

t drr =

t ˙ λr

t λr

= d

dt (ln t λr) . (2.124)

56 **Nonlinear** continua

Example 2.16. JJJJJ

Using Eqs. (2.65), (2.35), (2.111b) and (2.112b) we can show that:

t

◦ ˙ε = 1 t

◦

2

˙C = t ◦X T · t d · t ◦X .

JJJJJ

Example 2.17. JJJJJ

The Hencky strain tensor components in the fixed Cartesian system are,

hence,

[ t ◦H] = [ t ◦RL] ln[ t Λ][ t ◦RL] T ,

[ t ◦ ˙ H] = [ t ◦RL] [ t Λ] −1 [ t ˙

Λ] [ t ◦RL] T +[ t ◦ΩL] [ t ◦RL] ln[ t Λ][ t ◦RL] T

− [ t ◦RL] ln[ t Λ][ t ◦RL] T [ t ◦ΩL]

2.14 The Lie derivative

.

JJJJJ

In the deformation process represented in Fig. 2.2 we can define, for a Eulerian

tensor tt,itsLie derivative associated to the flow of the spatial configuration

(Simo 1988, Marsden & Hughes 1983):

Ltv( t t) = t ½

d £

φ

tφ ¤ ∗ t

∗ ( t)

dt

¾

. (2.125)

As we already know (see Sect. 2.9) the operation of pull-back is not a

tensor operation since it operates on components. Hence, for calculating a Lie

derivative using Eq. (2.125) it is important to identify the components of t t

that we are using.

The Lie derivative of a scalar is

Ltvα = d ∂α

α =

dt ∂t

+ ∂α

∂ t x a

t v a . (2.126)

The covariant components of the Lie derivative of a spatial vector tw are

¡ ¢ t

Ltv w

i = (t ◦X −1 ) A i

½

d

dt [(t◦X) j

after some algebra,

¾

t

A wj] , (2.127a)

2.14 The Lie derivative 57

¡ ¢ t

Ltv w i = ∂twi ∂t + ∂twi ∂txa t a

v

∂

+ tva ∂txi t

wa . (2.127b)

Since,

t i t

w wi = t α (2.128a)

we can write

¡ ¢ t i twi

Ltv w + t w i ¡ ¢ t

Ltv w

i =

µ

d t i twi

w +

dt

t w i

µ

d t

wi

dt

(2.128b)

and from the above we get the contravariant components of the Lie derivative

of a spatial vector t w,

¡ ¢ t i ∂

Ltv w = twi ∂t + ∂twi ∂txa t v a − t w a ∂t v i

∂t . (2.128c)

xa Followingtheaboveprocedurewecanshowthatthemixedcomponents

of the Lie derivative of a general Eulerian tensor t t are

¡ Ltv( t t) ¢ a...b

c...d = ∂ tta...b c...d

∂t

− ∂ tva ∂txp + ∂ t v p

∂ t x c

+ ∂ tta...b c...d

∂txp t p

v

t t p...b c...d − ··· − ∂ t v b

∂ t x p

t t a...b p...d + ··· + ∂ t v p

∂ t x d

t t a...p c...d

t t a...b c...p

(2.129)

Example 2.18. JJJJJ

To calculate the Lie derivative of the spatial metric tensor tg directly use Eq. (2.125)

we can

³ ´

t

Ltv g

ij = t ∙

d

φ∗ dt

h i ¸

tφ∗ t

( g) ,

IJ

using now Eq. (2.93a), we get

³ ´

t

Ltv g

ij =

h

tφ∗( t ◦ ˙C)

i

ij .

Using the result in Example 2.16, we get

³ ´

t

Ltv g

ij = ¡ 2 t d ¢

ij .

JJJJJ

58 **Nonlinear** continua

Example 2.19. JJJJJ

To calculate the Lie derivative of the Almansi deformation tensor we use

Eq. (2.125) and get

¡ ¢ t

Ltv e

ij = t ∙ ¸

d £

φ

tφ ¤ ∗ t

∗ ( e)

dt

IJ

and resorting to Eq.(2.94a),

¡ Ltv t e ¢

ij = £ t φ∗( t ◦ ˙ε)¤

ij .

Taking into account the result obtained in Example 2.16 we can finally

write ¡ ¢ t

Ltv e

ij = ¡ ¢ t

d

ij .

JJJJJ

Example 2.20. JJJJJ

The Lie derivative of the Finger deformation tensor is

¡ ¢ t ij

Ltv b =

∙

t d

φ∗

dt

¸

£ tφ ¤ ∗ t IJ

( b) .

Using Eq. (2.97b) we get

and since

we get

£ tφ ∗ ( t b) ¤ IJ =

h tB i IJ

◦ ˙g = 0

¡ Ltv t b ¢ ij = 0 .

2.14.1 Objective rates and Lie derivatives

= ◦ g IJ

JJJJJ

In this Section we will show that the Lie derivative is the adequate mathematical

tool for deriving covariant (objective) rates from covariant (objective)

Eulerian tensors.

Let us consider the deformation processes schematized in Fig. 2.7. It is

obvious that

2.14 The Lie derivative 59

Fig. 2.7. Deformation processes between three configurations

ˆt

◦X = ˆt tX · t ◦X . (2.130a)

For a covariant Eulerian tensor t t, that without losing generality we take

as a second-order tensor:

¯ ˆt

◦φ ∗ ˆt

¯

îˆj ¯

t

ˆt t îˆj = ˆt tX î a

¯ AB

ˆt

tX ˆj t ab

b t , (2.130b)

= ( ˆt ◦X −1 ) Â ı ( ˆt ◦X −1 ) B ˆj

ˆt

tX î ˆt

a tX ˆj t ab

b t . (2.130c)

Using Eq. (2.130a) in the above, we get

¯ ˆt

◦φ ∗ ˆt

¯

îˆj ¯

t ¯ AB

= ( t ◦X −1 ) A l ( t ◦X −1 ) B k t t lk , (2.130d)

¯ ˆt

◦φ ∗ ˆt

¯

îˆj ¯

t = ¯ ¯t

◦φ ∗ t t lk¯¯ AB . (2.130e)

¯ AB

From the above and from Eq. (2.125) it follows that:

Lt ˆv ( ˆt t) â ˆ b = ˆt tX â l

ˆt

tX ˆ b m

h

Lt v ( t

ilm t)

=

½ h

ˆt ∗

tφ Ltv ( t i ¾

lm

âˆb t)

. (2.130f)

The above equality shows that the Lie derivative of a covariant Eulerian

tensor is also a covariant Eulerian tensor.

60 **Nonlinear** continua

Example 2.21. JJJJJ

Considering again the case of a moving Cartesian frame and a fixed one from

Example 2.15, we get

t v α = ˙c α + ˙ Q α γ ( t z ∗ ) γ + Q α γ ( t v ∗ ) γ ,

therefore taking into account that c , Q(t) and ˙Q(t), for the case under

consideration, are constant in space,

t l α β = ˙Q α γ

∂ ( t z ∗ ) γ

∂ t z β

+ Qα γ (t l ∗ ) γ

∂ ( t z ∗ )

∂ t z β

hence,

t α

l β = ˙ Q α γ (Q T ) γ

β + Qαγ ( t l ∗ ) γ (Q T ) β .

Comparing with Eq. (2.101c) it is obvious that tl is not an objective tensor.

Since the velocity gradient tensor is not objective in the classical sense we

know that it is not a covariant tensor. JJJJJ

Example 2.22. JJJJJ

From Example 2.19, we know that t d is the result of a Lie derivative; hence

we can assess that the Eulerian strain rate tensor is a covariant (objective)

tensor. JJJJJ

Example 2.23. JJJJJ

For a Eulerian tensor t t,wedefine in the reference configuration the tensor:

which can be written as

t T = ¡ t◦X −1¢ A

a

¡ t◦X −1¢ B

b

t T = t ◦X −1 · t t · t ◦X −T

Since, t ◦ X −1 · t ◦ X = ◦ g we can derive that,

and

we can write

d

³

tT dt

´

d

dt

d

dt

t ab ◦ ◦

t gA gB

¡ t◦X −1¢ = − t ◦X −1 · t l

³ ´

t◦X −T

= − t l T · t ◦X −T

= t ◦X −1 · t˙t · t ◦X −T − t ◦X −1 · t l · t t · t ◦X −T

− t ◦X −1 · t t · t l T · t ◦X −T

.

.

Also, from Eqs. (2.110c)

t l T = ∇ t v = t v p |n t g n t g p

2.15 Compatibility 61

Considering that the time derivative of the reference configuration base vectors

is zero and using the above together with the Lie derivative definition in Eq.

(2.125), we get

£

Ltv ( t t) ¤ ab t

= ˙t ab − t t nb t v a |n − t t al t v b |l .

The above equation is going to be used in Sect. 3.4 for deriving objective

stress rates. JJJJJ

2.15 Compatibility

In our previous description of the kinematics of continuous media we went

through the following path:

Assume the existence of a regular mapping t φ

⇓

Calculate the tensorial components of different deformation measures

If, instead of the above, we want to start by defining the tensorial components

of a given deformation measure, our freedom to define them is limited by

the fact that they should guarantee the existence of a regular mapping from

which they could be derived. The conditions that the tensorial components

of a deformation measure should fulfill in order to assure the existence of a

regular mapping are called their compatibility conditions.

In what follows we will derive the compatibility conditions for the Green

deformation tensor.

From Eqs. (2.61), (2.80c) and (2.93a) we have,

Eulerian tensor Spatial configuration Pull-back space

Length differential t dl t dl

Coordinate differential t dx a [( t dx a ) ] A = ◦ dx A

Hence,

Metric tensor t gab [( t gab) ]AB = t ◦CAB

t dx a t gab t dx b = ◦ dx A t ◦CAB ◦ dx B . (2.131)

From the above equation it is obvious that the covariant components of the

Green deformation tensor are the covariant components of the metric tensor

.

62 **Nonlinear** continua

of the pull-back space. Note that the pull-back space is by no means coincident

with the reference configuration, whose metric tensor has the covariant

components ◦ gAB.

Since we restrict our study of the kinematics of continuous media to the

Euclidean space, we can assess that the Riemann-Christoffel tensor is zero in

the spatial configuration (McConnell 1957). Hence,

t Rprsq = 0 . (2.132)

The above equation represents 81 compatibility conditions to be fulfilled in

the spatial configuration. However, the covariant components of the Riemann-

Christoffel tensor satisfy the following relations (Aris 1962) 7 :

t Rprsq = − t Rrpsq , (2.133a)

t Rprsq = − t Rprqs , (2.133b)

t Rprsq = t Rsqpr . (2.133c)

We must also consider that t Riiii = 0 ; t Riiij = 0 ; t Riijj = 0 can

be easily transformed into a trivial identity of the form 0=0. Hence we are

left with only 6 significant equations, namely:

t R1212 =0 ;

t R1313 =0 ;

t R1213 =0 ;

t R1323 =0 ;

t R1223 =0; (2.134)

t R2323 =0.

From Eqs. (A.79a-A.79e),

t

Rprsq = 1

2

µ 2 t ∂ gpq

∂txr ∂txs + ∂2 tgrs ∂txp ∂txq − ∂2 tgps ∂txr ∂txq − ∂2 tgrq ∂txp ∂txs

+ t g mn ¡ t

Γrsm t Γpqn − t Γrqm t ¢

Γpsn

(2.135a)

where the t Γijk are the Christoffel symbols of the first kind corresponding

to the coordinate system { t x a }.

Hence, using Eq. (A.79b)

t Rprsq = 1

2

+ t g mn

∙

1

− 1

4

4

µ 2 t ∂ gpq

∂txr ∂txs + ∂2 tgrs ∂txp ∂txq − ∂2 tgps ∂txr ∂txq − ∂2 tgrq ∂txp ∂txs

µ t ∂ gsm

∂txr + ∂ tgmr ∂txs − ∂ tgrs ∂txm µ t ∂ gqm

∂txr + ∂ tgmr ∂txq − ∂ tgrq ∂txm 7 See Appendix.

(2.135b)

µ t ∂ gqn

∂txp + ∂ tgnp ∂txq − ∂ tgpq ∂txn

µ t ∂ gsn

∂txp + ∂ tgnp ∂txs − ∂ tgps ∂txn ¸

=0.

2.15 Compatibility 63

Doing a pull-back operation on Eq. (2.132) we obtain,

£ tφ∗ ¡ ¢¤

t

Rprsq PRSQ = t ◦X p t

P ◦X r R t ◦X s S t ◦X q t

Q

Rprsq = 0 . (2.136a)

Example 2.24. JJJJJ

Equation (2.135b) represents the components of the tensorial equation

t R = 0 .

If in the spatial configuration we change from the { t x i } coordinate system to

the { t ˜x i } system, we write Eq. (2.135b) using,

t ˜gpq = t glm

and the equation would look like

t ˜ Rprsq = 1

2

µ 2 t ∂ ˜gpq

∂t ˜x r∂t + ···

˜x s

∂ t x l

∂ t ˜x p

t ˜g pq = t g lm ∂t ˜x p

∂ t x l

+ t ˜g mn

∂ t x m

∂ t ˜x q

∂ t ˜x q

∂ t x m

∙ µ t 1 ∂ ˜gsm

4 ∂t ¸

+ ···

˜x r

= 0 .

If we now want to do a pull-back of Eq.(2.135b) the algebra can get quite

lengthy, but we can use an analogy with the above tensor transformations:

£ tφ ¤ ∗ t t

( glm)PQ = glm

∂txl ∂◦xP ∂txm ∂◦xQ = t ◦CPQ

£ ¤PQ tφ∗ t lm

( g )

t lm

= g ∂◦x P

∂txl ∂◦x Q

∂txm = ¡ t

◦C −1¢ PQ

.

Using this formal analogy we can easily write:

£ tφ∗ t

( Rprsq) ¤

∙ 2 t 1 ∂ ◦CPQ

= PRSQ 2 ∂◦xR∂ ◦xS + ∂2 t ◦CRS

∂◦xP ∂◦xQ − ∂2 t ◦CPS

∂◦xR∂ ◦xQ − ∂2 t ◦CRQ

∂◦xP ∂◦xS ¸

+ ¡ t

◦C −1¢ MN

∙ µ t 1 ∂ ◦CSM

4 ∂◦xR + ∂ t ◦CMR

∂◦xS − ∂ t ◦CRS

∂◦xM µ t ∂ ◦CQN

∂◦xP + ∂ t ◦CNP

∂◦xQ − ∂ t ◦CPQ

∂◦xN

− 1

µ t ∂ ◦CQM

4 ∂◦xR + ∂ t ◦CMR

∂◦xQ − ∂ t ◦CRQ

∂◦xM µ t ∂ ◦CSN

∂◦xP + ∂ t ◦CNP

∂◦xS − ∂ t ◦CPS

∂◦xN ¸

JJJJJ

64 **Nonlinear** continua

1

2

Hence,

∙ 2 t ∂ ◦CPQ

∂◦xR∂ ◦xS + ∂2 t ◦CRS

∂◦xP ∂◦xQ − ∂2 t ◦CPS

∂◦xR∂ ◦xQ − ∂2 t ◦CRQ

∂◦xP ∂◦xS ¸

+ ¡ t

◦C −1¢ MN

(2.136b)

∙

1

4

µ t ∂ ◦CSM

∂◦xR + ∂ t ◦CMR

∂◦xS − ∂ t ◦CRS

∂◦xM µ t ∂ ◦CQN

∂◦xP + ∂ t ◦CNP

∂◦xQ − ∂ t ◦CPQ

∂◦xN

− 1

µ t ∂ ◦CQM

4 ∂◦xR + ∂ t ◦CMR

∂◦xQ − ∂ t ◦CRQ

∂◦xM µ t ∂ ◦CSN

∂◦xP + ∂ t ◦CNP

∂◦xS − ∂ t ◦CPS

∂◦xN ¸

=0.

We can now define in the pull-back space, with metric t ◦CAB ,theChristof-

fel symbols of the first kind:

∗ ΓRSM = 1

2

µ ∂ t ◦CSM

∂ ◦ x R

+ ∂t ◦CMR

∂ ◦ x S

− ∂t ◦CRS

∂ ◦ x M

(2.136c)

therefore we obtain the following 6 compatibility conditions for the covariant

components of the Green deformation tensor:

φ ∗ ( t Rprsq) = 1

2

µ ∂ 2 t ◦CPQ

∂ ◦ x R ∂ ◦ x S + ∂2 t ◦CRS

∂ ◦ x P ∂ ◦ x Q − ∂2 t ◦CPS

∂ ◦ x R ∂ ◦ x Q − ∂2 t ◦CRQ

∂ ◦ x P ∂ ◦ x S

+ ¡ t ◦C −1¢ MN ( ∗ ΓRSM ∗ ΓPQN − ∗ ΓRQM ∗ ΓPSN ) = 0 . (2.136d)

The above equation indicates that t ◦CAB is a metric of the Euclidean space.

Taking into account the Bianchi identities in Eq. (A.82d) (Synge & Schild

1949) and Eqs.(2.135a), we get

t R1212|3 + t R 1213|1 + t R 1213|2 = 0 ,

t R1313|2 + t R 1323|1 − t R 1213|3 = 0 , (2.137)

t R2323|1 − t R 1323|2 + t R 1223|3 = 0 ,

andwereducethenumberofindependent compatibility conditions to 3.

Example 2.25. JJJJJ

Using as a metric the Green tensor, we can define an analog to Eq.(A.59):

GRADt ◦ C ( t A ) =

where for the Eulerian vector t a,

³ tA ´

∙ t ∂ AI ∂◦xA − t A D ∗ Γ D ¸

◦gI ◦ A

IA g

I = t ◦X i I ai ,

and,

∗ D

ΓIA = 1

2

Taking into account that

¡ t◦C −1¢ DK

it is easy to show that:

£ ∗

φ ¡ ∇ t a ¢

¡ t◦C −1¢ DK = ∂ ◦ x D

∙ t ∂◦CIK ∂◦xA + ∂t ◦CAK

∂◦xI t

◦CIK = ∂txr ∂◦xI ∂txs ∂◦xK t

grs ,

∂txr ∂◦x K

∂txs t rs

g ,

ia

¤

IA =

2.15 Compatibility 65

− ∂t ◦CIA

∂◦xK ¸

h

³

tA GRADt ◦C ´i

IA .

.

JJJJJ

Example 2.26. JJJJJ

In a Cartesian coordinate system, from Eqs. (2.9a-2.9b), (2.29a-2.29c), (2.65)

and (2.68) we get

t

◦εαβ = 1

2

t

eαβ = 1

2

µ ∂ t u α

∂◦z β + ∂tuβ ∂◦z α + ∂tuγ ∂◦z α

∂tuγ ∂◦z β

µ ∂ t u α

∂ t z β + ∂t u β

∂ t z α − ∂t u γ

∂ t z α

For linear kinematics,

∂tuα ∂◦ β z

66 **Nonlinear** continua

∂2 tεαβ ∂◦z γ ∂◦z δ + ∂2 tεγδ ∂◦z α ∂◦z β − ∂2 tεαδ ∂◦z γ ∂◦z β − ∂2 tεγβ ∂◦z α ∂◦ = 0 .

δ z

The above represents a set of 6 equations, that proceeding as in Eqs. (2.137),

can be reduced to 3 independent compatibility conditions.

The above result was obtained for a Cartesian system. Generalizing for an

arbitrary coordinate system we get

t εAB|CD + t ε CD|AB − t ε AD|BC − t ε BC|AD = 0 .

JJJJJ

3

Stress Tensor

To deform a continuous body the exterior medium has to produce a loading

on that body; therefore we get external forces acting on it. Also, during the

deformation of a continuous body, neighboring particles exert forces on each

other, they are the internal forces in the body.

The study of the internal forces in a body leads to the notion of stresses,

that we are going to develop in this chapter.

Some reference books for this chapter are: (Truesdell & Noll 1965, Truesdell

1966, Malvern 1969, Marsden & Hughes 1983).

3.1 External forces

When studying the deformation of a continuous body, to classify a force as

either external or internal, we have to carefully take into account our definition

of the body under consideration.

For example, in Fig. 3.1 a, we study the spatial configuration at t of the

body ( t ΩA ∪ t ΩB) with the external forces t F A and t F B andinFig.3.1b, to

study the same physical problem, we choose to consider t ΩA and t ΩB as

free bodies. Obviously both analyses will lead to the same conclusions but, it

is clear that in the first case t F AB and t F BA have to be considered as internal

forces and in the second case as external forces.

We can find different types of external forces, for example:

• External forces acting on the elements of volume or mass inside the body

(they are defined per unit volume or per unit mass). Theses forces are called

body forces and some examples are: gravitational forces, inertia forces, electromagnetic

forces, etc. (in general they are “action-at-a-distance” forces

(Malvern 1969).

Let us define in the t-configuration of a body a Cartesian system { t ˆz α ,

α =1, 2, 3} with base vectors t êα and an arbitrary curvilinear system

{ t x i ,i=1, 2, 3} with covariant base vectors t g i . We consider the vector

68 **Nonlinear** continua

Fig. 3.1. External and internal forces in a bar

field t b( t x) to define the forces per unit mass acting on the body at t .

We can write

t b = tˆ b α t êα = t b i t g i . (3.1)

The resultant of the forces per unit mass is:

Z

Z

t t t

ρ b dV =

t

ρ

tˆα t t

b êα dV =

t V

t V

Z

t V

t ρ t b i t gi

t dV (3.2)

where, t ρ :densityofthebodyint-configuration and t V : volume of the

body in the t-configuration.

• External forces acting on the elements of the body’s surface (they are

defined per unit surface). These forces are called surface forces and

some examples are: pressure forces, contact forces, friction forces, etc.

We consider the vector field t t( t x) to define the forces per unit surface

acting on the region t Sσ, a subset of the surface of the body at t .Wecan

write

The resultant of the forces per unit surface is:

Z

Z

t t

t dS =

tˆt α t Z

t

êα dS =

t Sσ

3.2 The Cauchy stress tensor 69

t t = tˆt α t ê α = t t i t g i . (3.3)

t Sσ

t Sσ

t t i t gi

t dS. (3.4)

It is important to point out that our description of the external forces

acting on a body excludes the possibility of considering distributed torques

per unit volume, mass or surface. Therefore, the moment with respect to a

point P of the considered external forces in the t-configuration is:

Z

Z

¡ t t

MP = ρ

tr ¢ t

× b

tdV ¡

+

tr ¢ t

× t

tdS

(3.5)

t V

where t r is the vector that in the t-configuration goes from the moment center

P toapointinsidethebody(firsttermonther.h.s.)ortoapointonthe

body surface (second term on the r.h.s.).

3.2 The Cauchy stress tensor

In Fig. 3.2 a we represent the spatial configuration of a body B corresponding

toatimet, and we identify a particle P .

t Sσ

Fig. 3.2. Internal forces at a point inside a continuum

The external forces acting on B per unit mass are given by the vector field

t b and the external forces per unit surface are given by the vector field t t .

70 **Nonlinear** continua

We now section the body B, inthet-configuration, with a surface tSc passing through P . The normal to the surface tSc at P is tn (see Fig. 3.2 b).

If we now analyze in Fig. 3.2 b the left part, tΩL ,asafreebody,wehave

to consider as external forces the internal forces at P in Fig. 3.2 a.

Considering on the surface tSc an area t∆S around P ,thesetofexternal

forces acting on t∆S can be reduced to a force t∆F through P and a moment

t∆MP .

When t∆S → 0:

t∆F lim =

t∆S→0 t∆S t t (3.6a)

t∆MP lim = 0 (3.6b)

t∆S→0 t∆S The vector t t is known in the literature as traction.

Equations(3.6a-3.6b) incorporate two fundamental hypotheses:

• The limit in Eq. (3.6a) exists. Therefore we exclude from the continuum

mechanics field the consideration of concentrated forces (concentrated

forces are also not physically possible).

• The condition in Eq. (3.6b) is a strong requirement in the classical formulation

of continuum mechanics. There are alternative formulations that

do not require the fulfillment of Eq. (3.6b) (e.g. the theory of polar media

(Truesdell & Noll 1965, Malvern 1969).

In this book we limit our study to the classical case of non-polar media.

It is interesting to note that from Eq. (3.6a) we can assess that, if we

consider different surfaces through P , which share the external normal t n

(tangent surfaces), we will arrive at the same traction vector t t (see Fig. 3.3).

We can define, in the t-configuration at the point P, a second-order tensor

t σ, the Cauchy stress tensor, via the following equation:

t t = t n · t σ . (3.7)

Since t t and t n are vectors, using the quotient rule (Sect. A.5), it is evident

that t σ is a second-order tensor.

We can consider Eq. (3.7) to be a condition of equivalence between external

forces and stresses inside a continuum.

3.2 The Cauchy stress tensor 71

Fig. 3.3. Tangent surfaces at P have the same traction vector

3.2.1 Symmetry of the Cauchy stress tensor (Cauchy Theorem)

In Sect. 4.4.2 we will prove that from the equilibrium equations of a nonpolar

continuum we get

that is to say, the Cauchy stress tensor is symmetric.

t σ = t σ T , (3.8)

Example 3.1. JJJJJ

In the following figure, at the point P , inside the t-configuration of a continuum

body, the stress tensor is t σ.

Cutting the continuum with the surface t S1, wegetatP the traction vector

t t1 = t n 1 · t σ ,

if we cut with t S2, the traction vector at P is:

t t2 = t n 2 · t σ .

72 **Nonlinear** continua

Secant surfaces at a point inside a continuum

In an arbitrary coordinate system { t xi} we can write

t

t1 · t n2 = t t ij t

n1i σ n2j ,

t

t2 · t n1 = t t ij t

n2i σ n1j ,

and since from Eq. (3.8) t σ ij = t σ ji we get

t t1 · t n 2 = t t 2 · t n 1 .

The above result, a direct consequence of the Cauchy Theorem, is known as the

projection theorem or reciprocal theorem of Cauchy (Malvern 1969).JJJJJ

3.3 Conjugate stress/strain rate measures

Let us assume, at an instant (load level) t, acontinuumbodyBin equilibrium

under the action of external body forces tb and external surface forces tt. Assuming a velocity field tv( tx) on B, thepowerprovided by the external

forces is:

Z

Z

t t t t t t t t

Pext = ρ b · v dV + t · v dS. (3.9a)

t V

Using Eq. (3.7) we can rewrite Eq.(3.9a) as

t S

t Pext =

Z

t V

3.3 Conjugate stress/strain rate measures 73

t t t t

ρ b · v dV +

Z

¡ tn ¢ t t t

· σ · v dS. (3.9b)

From the Divergence Theorem (Hildebrand 1976),

t

Pext =

Z

£ tρ t t

b · v +

¡

∇ ·

tσ ¢¤ t

· v

tdV

(3.9c)

t V

introducing Eq. (2.110a-2.110c) and after some algebra,

t

Pext =

Z

£ tσ

:

t

l +

¡ tρ ¢ ¤ t t t

b + ∇ · σ · v

tdV

. (3.9d)

t V

Since the t-configuration is an equilibrium configuration, the following

equation (to be proved in Chap. 4, Eq.(4.27b)) holds

t S

t ρ t b + ∇ · t σ = t ρ D t v

Dt

and an obvious result that we also need is:

D t v

Dt

· t v = D

Dt

∙

1 t t

v · v

2

¸

, (3.9e)

. (3.9f)

The kinetic energy of the body B, attheinstantt, isdefined as

Z t Z

t ρ t t t 1 t t

K = v · v dV = v · v dm . (3.9g)

tV 2

m 2

In the second integral of the above equation we integrate over the mass of

the body B. Since the mass of the body is invariant,

DtK Dt =

Z

D

m

tv ·

Dt

t v dm . (3.9h)

Finally, using the decomposition of the velocity gradient tensor in Eq.

(2.112a) and considering that since ( tσ) is a symmetric tensor and ( tω) is a

skew-symmetric one,

t t

σ : ω =0 (3.9i)

we get

We define

t Pext = Dt K

Dt +

t Pσ =

Z

t V

Z

t V

t σ :

t d t dV . (3.9j)

t σ : t d t dV (3.10)

as the stresses power. Obviously t Pσ is the fraction of t Pext that is not transformed

into kinetic energy and that is either stored in the body material or

dissipated by the body material, depending on its properties (see Chapter 5).

From Eq. (3.10) we define the spatial tensors t σ and t d to be energy

conjugate (Atluri 1984).

In what follows we will define other pairs of energy conjugate stress/strain

rate measures.

74 **Nonlinear** continua

3.3.1 The Kirchhoff stress tensor

From Eqs. (3.10) and (2.34d) and the mass-conservation principle (to be discussed

in Chapter 4, Eq.(4.20d)) we get

Z

Z

t

Pσ =

t

σ :

t t

d dV =

t

σ :

t ◦

d dV . (3.11)

t V

◦ V

The Kirchhoff stress tensor is defined as

t τ =

◦ ρ

t ρ

◦ ρ

t ρ

t σ (3.12)

where, ◦ ρ: density in the reference configuration and ◦ V : volume of the reference

configuration.

It is important to note that although the Kirchhoff stress tensor was introduced

by calculating t Pσ via an integral defined over the reference volume,

Eq. (3.12) clearly shows that t τ is definedinthesamespacewhere t σ is

defined: the spatial configuration. Hence, using in the t-configuration an arbitrary

curvilinear coordinate system { t x a } with covariant base vectors t g a we

obtain

and

t τ = t τ ab t ga

t σ = t σ ab t ga

t τ ab =

◦ ρ

t ρ

t gb

t gb

3.3.2 The first Piola-Kirchhoff stress tensor

◦ V

(3.13a)

(3.13b)

t σ ab . (3.13c)

From Eqs. (3.11) and (3.9i) we can write

Z

t t t ◦

Pσ = τ : l dV . (3.14)

In Chap. 2 we learned how to derive representations in the reference configuration

of tensors defined in the spatial configuration via pull-back operations.

We will now obtain a representation of the Kirchhoff stress tensor in the form

of a two-point tensor.

In the reference configuration we define an arbitrary coordinate system

{ ◦ x A } with covariant base vectors ◦ gA ;andinthespatialconfiguration a

system { t x a } with covariant base vectors t g a .Wealsodefine a convected

system {θ i } with covariant base vectors in the reference configuration ◦ eg i

and covariant base vectors in the spatial configuration t eg i .

In the spatial configuration we can write the Kirchhoff stress tensor as

3.3 Conjugate stress/strain rate measures 75

t τ = t ˜τ ij t egi

a pull-back of the above tensor to the reference configuration is:

t t ij ◦ ◦

T = ˜τ egi egj =

t egj

h tT i IJ

We define a two-point representation of t τ as

After some algebra,

t

◦P Ij = t ˜τ lm ∂◦x I

∂θ l

(3.15a)

◦ ◦

gI gJ . (3.15b)

t

◦P = t ˜τ ij ◦ t

eg egj =

i

£ t

◦P ¤ Ij ◦gI t

gj . (3.15c)

= t τ pj ¡ t ◦X −1¢ I

p

∂ t x j

∂θ m =

"

t τ pq ∂θl

∂ t x p

∂θ m

∂txq #

∂ ◦ x I

∂θ l

∂ t x j

∂θ m

. (3.15d)

The second-order two-point tensor t ◦P is the first Piola-Kirchhoff stress

tensor. ItisapparentfromEq.(3.15d)thatitisanon-symmetric tensor.

We can write, due to the symmetry of the Kirchhoff stress tensor:

Z

Pσ =

t ab t

τ lab ◦ Z

dV =

t ba t

τ lab ◦ dV . (3.16a)

◦ V

Hence, using Eq.(3.15d):

◦ V

Pσ =

Z

◦ V

Using Eq. (2.111a) we have

Z

Pσ =

t

◦P Ba t ◦ ˙ XaB ◦ Z

dV =

◦ V

t

◦P Ba t ◦X b B t lab ◦ dV . (3.16b)

◦ V

◦ V

t

◦P · ·

t

◦ ˙X ◦ dV . (3.17)

We can also write the above as (Malvern 1969):

Z

t

Pσ = ◦P T t

: ◦ ˙X ◦ dV . (3.18)

The above equation defines the two-point tensors t ◦P T and t ◦ ˙X as energy

conjugates.

We will not try to force a so-called “physical interpretation” of the first

Piola-Kirchhoff stress tensor; instead we will regard it only as a useful mathematical

tool.

76 **Nonlinear** continua

3.3.3 The second Piola-Kirchhoff stress tensor

The pull-back configuration of t τ to the reference configuration is:

t T =

h tτ ij ¡ t ◦X −1¢ I

i

¡ t◦X −1¢ i

J ◦gI ◦

gJ . (3.19)

j

The tensor t T , defined by the above equation, is the second Piola-

Kirchhoff stress tensor and it is a symmetric tensor.

Using Bathe’s notation (Bathe 1996), we identify the second Piola-Kirchhoff

stress tensor, corresponding to the t-configuration and referred to the configuration

in t =0as t ◦S.

Example 3.2. JJJJJ

From Eqs. (3.15d) and (3.19) we get

that is to say,

From Example 2.16:

and since from Eq. (3.19),

t

◦S IJ = ¡ t

◦X −1¢ J

j

using Eqs. (3.11) and (2.88), we get

Z

t

◦P Ij

t

◦S IJ = £ t φ ∗ ¡ t ◦P Ij¢¤IJ .

t

◦ ˙εAB = £ t ∗

φ ¡ ¢¤ t

dab AB

t

◦S AB = £ t ∗

φ ¡ t ab

τ ¢¤AB

t Pσ =

◦ V

t

◦S :

JJJJJ

(3.20a)

(3.20b)

t

◦ ˙ε ◦ dV . (3.20c)

From Eq. (3.20c) we define the tensors t ◦S and t ◦ ˙ε to be energy conjugate

(Atluri 1984).

Here, we will also not try to force a “physical interpretation” of the second

Piola-Kirchhoff stress tensor.

An important point to be analyzed is the transformation of t ◦S under rigidbody

rotations.

• Let us first consider the t-configuration of a certain body B. At an arbitrary

point P the Cauchy stress tensor is t σ.

3.3 Conjugate stress/strain rate measures 77

• Let us now assume that we evolve from the t-configuration to a (t + ∆t)configuration

imposing on B and on its external loads a rigid body rotation.

At the point P we can write:

and therefore,

t+∆t

t X = t+∆t

t R , (3.21a)

t+∆t t+∆t

dx = t R · t dx . (3.21b)

◦ For the external loads,

◦ For a velocity vector,

t+∆t t = t+∆t

t R · t t , (3.22a)

t+∆t b = t+∆t

t R · t b . (3.22b)

t+∆t t+∆t

v = t R · t v . (3.22c)

◦ For the external normal vector,

t+∆t n = t+∆t

t R · t n . (3.22d)

Example 3.3. JJJJJ

For an arbitrary force vector t f (it can be a force per unit surface, per unit

volume, etc.) and considering the evolution described above,

t+∆t t+∆t t+∆t

f · v = t R · t f · t+∆t v

and since the rotation tensor is orthogonal,

= t f · t+∆t

t R T · t+∆t v

= t f · t+∆t

t R T · t+∆t

t R · t v

t+∆t t+∆t t t

f · v = f · v .

The above equation states the intuitive notion that a rigid-body rotation

cannot affect the value of the deformation power performed by the external

forces. JJJJJ

78 **Nonlinear** continua

At t we can write

and at (t + ∆t),

Introducing Eq. (3.22d) in the above,

t t = t n · t σ (3.23a)

t+∆t t = t+∆t n · t+∆t σ . (3.23b)

t+∆t t = ¡ t+∆t

t R · t n ¢ · t+∆t σ . (3.23c)

And with Eq. (2.28a) and (3.22a), we finally have

t t

t = n ·

h

t+∆t

t R T · t+∆t σ · t+∆t

i

t R . (3.23d)

For deriving the above equation, we used that t+∆t

t R · tt = tt · t+∆t

t R T .

Hence,

t+∆t t+∆t

σ = t R · t σ · t+∆t

t R T . (3.23e)

The above equation indicates that the Cauchy stress tensor fulfills the criterion

for objectivity under isometric transformations, established for Eulerian

tensors in Sect. 2.12.2.

We define an arbitrary system { txa0} in the t-configuration and a system

{ t+∆txa } in the (t + ∆t)-configuration. Hence, from Eq. (3.23e),

and using Eq. (2.28c), we get

therefore,

t+∆t a

σ b = t+∆t

t R a c0 t+∆t σ a b = t+∆t

t R a c 0 t σ c0

d 0

t σ c 0

d 0

t+∆t σ al = t σ c 0 m 0

t+∆t

t R l m0 t+∆t

t R a c0 ¡ ¢d

0

t+∆t T

t R

b

t+∆t glb

t g m 0 d 0

(3.24a)

(3.24b)

t+∆t

t R l m0 . (3.24c)

It is easy to show that for the Kirchhoff stress tensor we can also write

From Eq. (3.19), we obtain

but since,

t+∆t t+∆t

τ = t R · t τ · t+∆t

t R T . (3.25)

t+∆t

◦ S IJ = t+∆t τ ij ¡ t+∆t

◦

X −1¢I

i

t+∆t

◦ X a A = t+∆t

t R a a0 t ◦X a0

¡ t+∆t

◦ X −1¢A

a = ¡ t

◦X −1¢ A

a0 using Eqs. (3.24c) and (3.26c) in Eq. (3.26a), we get

A

¡ t+∆t

◦ X −1¢J

j

¡ ¢a

0

t+∆t T

t R a

(3.26a)

(3.26b)

(3.26c)

3.3 Conjugate stress/strain rate measures 79

t+∆t

◦ S IJ = t ◦S IJ

(3.27a)

therefore,

t+∆t

◦ S = t ◦S . (3.27b)

The above equation indicates that the second Piola-Kirchhoff stress tensor

fulfills the criterion for objectivity under isometric transformations, established

for Lagrangian tensors in Sect. 2.12.2.

3.3.4 A stress tensor energy conjugate to the time derivative of

the Hencky strain tensor

In Sect. 2.8.5 we defined the logarithmic or Hencky strain tensor.

Let us now define, via a pull-back operation, the following stress tensor:

t Γ = t ◦R ∗ ( t τ ) . (3.28)

With the notation t ◦R∗ (·) we define the pull-back of the components of the

tensor (·) using the tensor t ◦R (Simo & Marsden 1984), that is to say, tΓ is

an unrotated representation of tτ .

From the symmetry of tτ ,theabovedefinition implies the symmetry of

tΓ .

We will now demonstrate, following (Atluri 1984), that for an isotropic

t t material Γ and ˙H ◦ are energy conjugate.

We can write Eq. (3.14) as

therefore,

t Pσ =

t Pσ =

From Eq. (2.28c),

Z

Z

◦ V

◦ V

t AB

Γ £ t

◦R ∗ ¡ ¢¤ t

dab AB

◦ dV (3.29a)

t Γ AB t ◦R a A t dab t ◦R b B ◦ dV . (3.29b)

t

◦R a A = ¡ ¢L t T

◦R l

t g al ◦ gAL

and using the above, the integrand in Eq. (3.29b) is:

t AB

Γ ¡ ¢L t T

◦R l

It is also easy to show that

t

◦R T · t d · t ◦R =

◦ V

h ¡t◦R

¢L T

l

t g al ◦ gAL t dab t ◦ Rb B

t g al ◦ gAL t dab t ◦R b B

(3.29c)

. (3.29d)

i ◦g A ◦ g B . (3.29e)

Hence, using Eq. (2.118b),

t

Pσ =

Z

1 t

Γ :

2

³

t◦ ˙U · t ◦U −1 + t ◦U −1 · t ◦ ˙U

´

◦dV

. (3.29f)

80 **Nonlinear** continua

In order to simplify the algebra, in what follows we will work in a Cartesian

system; [A] willbethematrixformedwiththeCartesiancomponentsofa

second-order tensor A.

From Eqs. (2.122a-2.122d), we get

[ t ◦ ˙ U][ t ◦U] −1 = [ t ◦RL] [ t Λ] ˙ t −1 t

[ Λ] [ ◦RL] T + [ t ◦ΩL] (3.30a)

− [ t ◦RL] [ t Λ][ t ◦RL] T [ t ◦ΩL] [ t ◦RL] [ t Λ] −1 [ t ◦RL] T

and

[ t ◦U] −1 [ t ◦ ˙ U] = [ t ◦RL] [ t Λ] −1 [ t ˙

Λ] [ t ◦RL] T

+[ t ◦RL] [ t Λ] −1 [ t ◦RL] T [ t ◦ΩL] [ t ◦RL] [ t Λ][ t ◦RL] T − [ t ◦ΩL]

it follows from the above two equations that

1

n

[

2

t ◦ ˙ U][ t ◦U] −1 + [ t ◦U] −1 [ t ◦ ˙ o

U] = [ t ◦RL] [ t Λ] −1 [ t Λ] ˙ t

[ ◦RL] T

+ 1

2 [t ◦RL] [ t Λ] −1 [ t ◦RL] T [ t ◦ΩL] [ t ◦RL] [ t Λ][ t ◦RL] T

− 1

2 [t◦ RL] [ t Λ][ t ◦RL] T [ t ◦ΩL] [ t ◦RL] [ t Λ] −1 [ t ◦RL] T ,

and using once again Eqs. (2.122a-2.122d), we get

1

2

n

[ t ◦ ˙U] [ t ◦ U]−1 + [ t ◦ U]−1 [ t ◦ ˙U]

o

= [ t ◦RL] [ t Λ] −1 [ t Λ] ˙ [ t T

◦RL] + 1

2 [t◦U] −1 [ t ◦ΩL] [ t ◦U] − 1

2 [t◦U] [ t ◦ΩL] [ t ◦U] −1 .

Using the result in Example 2.17, we can write

1

2

n

[ t ◦ ˙ U][ t ◦U] −1 + [ t ◦U] −1 [ t ◦ ˙ U]

o

(3.30b)

(3.30c)

(3.30d)

= [ t ◦ ˙ H] − [ t ◦ΩL] [ln t ◦U] (3.30e)

+[ln t ◦U] [ t ◦ΩL]+ 1

2 [t◦U] −1 [ t ◦ΩL] [ t ◦U] − 1

2 [t◦U] [ t ◦ΩL] [ t ◦U] −1 .

Using the above in Eq. (3.29f) and working with the matrix components,

Z

t

Pσ = [ t n

Γ ]αβ [ t ◦ ˙ H]αβ − [ t ◦ΩL]αγ [ln t ◦U]γβ (3.31a)

◦ V

+[ln t ◦U]αγ [ t ◦ΩL]γβ + 1

2 [t◦U −1 ]αγ [ t ◦ΩL]γδ [ t ◦U]δβ

− 1

2 [t◦U]αγ [ t ◦ΩL]γδ [ t ◦U −1 ¾

◦dV

]δβ .

Since [ t Γ ] and [ t ◦U] are symmetric and [ t ΩL] is skew-symmetric, we can

rewrite the above equation as

+ 1

2

t Pσ =

Z

◦ V

−

Z

◦ V

Z

◦ V

[ t Γ ]αβ [ t ◦ ˙ H]αβ ◦ dV

n £[ t Γ ][ t ◦H] ¤

n £[ t ◦U] −1 [ t Γ ][ t ◦U] ¤

βγ − £ [ t ◦H] [ t Γ ] ¤

3.4 Objective stress rates 81

βγ

o

γδ − £ [ t ◦U] [ t Γ ][ t ◦U] −1¤

γδ

[ t ◦ΩL]βγ ◦ dV

o

[ t ◦ΩL]γδ ◦ dV.

(3.31b)

We will show in Chap. 5 that for isotropic materials the Eulerian tensors

t t t σ ( τ ) and ◦V have coincident eigenvectors (they are coaxial).

Taking into account that

t

◦U = t ◦R ∗ ( t ◦V) (3.32)

and the definition of t Γ in Eq. (3.28) we conclude that the Lagrangian tensors

t Γ , t ◦U and t ◦H are also coaxial for isotropic materials.

Obviously, the coaxiality of t Γ and t ◦H implies that

[ t Γ ][ t ◦H] − [ t ◦H] [ t Γ ] = [0] (3.33a)

and the coaxiality of t Γ and t ◦U implies that

[ t ◦U] −1 [ t Γ ][ t ◦U] − [ t ◦U] [ t Γ ][ t ◦U] −1 = [0] . (3.33b)

Finally, for isotropic materials,

t P Isot.Mat.

σ

=

Z

◦ V

t Γ :

and therefore in this case t Γ and t ◦ ˙H are energy conjugates.

3.4 Objective stress rates

t

◦ ˙H ◦ dV (3.34)

In Sect. 2.14.1 we show that the adequate tool for deriving objective rates of

Eulerian tensors is the Lie derivative.

The Lie derivative of the Cauchy stress tensor is:

³ ´

t ◦ ab

σ = £ Ltv( t σ) ¤ ab t ab t cb t a

= ˙σ − σ l c − t σ ac t l b c (3.35)

the above stress rate is known as Oldroyd stress rate (Marsden & Hughes

1983).

82 **Nonlinear** continua

Example 3.4. JJJJJ

To derive the expression of Oldroyd’sstressratewe start from Eq. (2.129),

¡ ¢ t ab ∂

Ltv σ = tσab ∂t + ∂tσab ∂txp From Eq. (2.109a)

and Eq. (2.111b)

we get,

Then, from

we get,

t ˙σ =

t

◦ ˙X =

∙ t ab ∂ σ

∂t + ∂tσab ∂txl − ∂t v b

∂ ◦ x P

∙ ∂ t v a

¡ t◦X −1¢ P

p

t v p − ∂ t v a

t σ ap .

∂ ◦ x A + t ◦X p

A t Γ a pl t v l

t

◦ ˙X = t l · t ◦X ,

∂ ◦ x P

¸ tga

¡ t◦X −1¢ P

p

◦ g A ,

∂ t v a

∂ ◦ x A = t l a m t ◦X m A − t Γ a rs t v s t ◦X r A .

t σ = t σ ab t ga

t gb ,

t σ pb

¸

t l t mb t a t l t am t b t l tga t

v + σ Γml v + σ Γml v gb .

Replacing in the first equation, after some algebra, we finally get

¡ Ltv t σ ¢ ab = t ˙σ ab − t l a p t σ pb − t l b p t σ ap .

JJJJJ

• The Lie derivative of the Kirchhoff stress tensor is:

h i

t ◦ ab

τ = £ Ltv( t τ ) ¤ ab t ab

= ˙τ

t cb t a

− τ l c − t τ ac t l b c (3.36)

the above rate is known as the Truesdell stress rate (Marsden & Hughes 1983).

Example 3.5.

From Eq. (3.19),

JJJJJ

t ij

τ

t

= ◦S IJ t ◦X i I

t

◦X j

J ,

hence,

∂tτ ij

∂t + ∂tτ ij

∂txp t v p = t ◦ ˙ S IJ t ◦X i I

but h

tφ∗

( t ◦ ˙ S IJ iij )

Therefore,

h

tφ∗

( t ◦ ˙ S IJ iij )

+ t ◦S IJ t ◦X i I

3.4 Objective stress rates 83

t

◦X j

J + t ◦S IJ ∂tvi ∂ t v j

∂ ◦ x J

= t ◦ ˙ S IJ t ◦X i I

= ∂tτ ij

∂t + ∂tτ ij

∂txp t v p

− t τ lm ( t ◦X −1 ) I l ( t ◦X −1 ) J m

t

◦X j

J .

− t τ lm ( t ◦X −1 ) I l ( t ◦X −1 ) J m

∂ ◦ x I

∂tvi ∂◦xI ∂tvj ∂◦xJ t

◦X j

J

t

◦X j

J

t

◦X i I .

Using algebra along the lines of Example 3.4 and Eq.(3.36) we finally get,

t ◦ τ ij

=

h

tφ∗

( t ◦ ˙ S IJ iij ) .

JJJJJ

The above example shows that the necessary and sufficient condition for

the second Piola-Kirchhoff stress tensor to remain constant is that the Truesdell

stress rate is zero (Eringen 1967).

We can also perform pull-back and push-forward operations using the rotation

tensor t ◦R (Simo & Marsden 1984). Let us consider an arbitrary Eulerian

stress tensor tt (e.g. tσ or tτ ), and perform on its components a t ◦R-pullback:

£ t◦R ∗ ( t t) ¤ AB t ab t

= t ( ◦R T ) A a ( t ◦R T ) B b . (3.37a)

In order to simplify our calculations we will now work in a Cartesian

system; hence, £ t◦

R ∗ ( t t) ¤ = [ t ◦R]T [ t t][ t ◦R] . (3.37b)

Taking into account that d

©

T

dt [R] ª = © d

dt [R]ªT , we get

d £ t◦R

dt

∗ ( t t) ¤ = [ t ◦R] T [ t ˙t] [ t ◦R] + [ t ◦ ˙ R] T [ t t][ t ◦R] + [ t ◦R] T [ t t][ t ◦ ˙ R] (3.37c)

and using Eqs. (2.115a) and (2.116a) we get

d £ t◦R

dt

∗ ( t t) ¤ = [ t ◦R] T [ t ˙t] [ t ◦R] − [ t ◦R] T [ t ΩR] [ t t][ t ◦R]

+[ t ◦R] T [ t t][ t ΩR] [ t ◦R] . (3.37d)

84 **Nonlinear** continua

Since,

we finally arrive at

h

h

Lt ◦ R( t t)

Lt ◦ R( t t)

i

i

= [ t ◦R] d £ t◦R

dt

∗ ( t t) ¤ [ t ◦R] T

(3.37e)

= [ t ˙t] − [ t ΩR] [ t t] + [ t t][ t ΩR] (3.37f)

that in an arbitrary spatial coordinate system leads to

h

Lt

◦R( t iab t) = t ˙t ab − t Ω a

Rc t t cb + t t ac t Ω b

Rc. (3.38)

The above Lie derivative is the well-known Green-Naghdi stress rate (Dienes

1979, Marsden & Hughes 1983, Pinsky, Ortiz & Pister 1983, Simo &

Pister 1984, Cheng & Tsui 1990). From its derivation it is apparent that the

Green-Naghdi stress rate is objective under isometric transformations, therefore

it is known as a corotational stress rate.

In the case of the Kirchhoff stress tensor, its Green-Naghdi rate is the

t

◦R-push-forward of the rate of t Γ.

If as a reference configuration we use the t-configuration (Dienes 1979), we

will fulfill Eq. (2.118d), that is to say,

t ω = t ΩR . (3.39)

Using the above in Eq. (3.38) we obtain the Jaumann stress rate (Truesdell

& Noll 1965).

It is important to realize that (Dienes 1979):

• The Jaumann stress rate is only coincident with the corotational stress

rate when t ◦U ≈ ◦ g.

• When formulating a corotational constitutive relation, we will only be able

to use the Jaumann stress rate when the spatial and reference configurations

are coincident.

4

Balance principles

In this chapter we are going to present a set of basic equations (balance of

mass, momentum, angular momentum and energy) that govern the behavior

of the continuous media in the framework of Newtonian mechanics.

We are going to present these basic principles in an integral form and

also in the form of partial differential equations (localized form). On presenting

the basic principles we are going to use both, the Eulerian (spatial) and

Lagrangian (material) descriptions of motion.

Some reference books for this chapter are: (Truesdell & Toupin 1960, Fung

1965, Eringen 1967, Malvern 1969, Slaterry 1972, Oden & Reddy 1976, Marsden

& Hughes 1983, Panton 1984, Lubliner 1985, Fung & Tong 2001).

4.1 Reynolds’ transport theorem

We begin this chapter by presenting Reynolds’ transport theorem; which will

be used in what follows as a tool for calculating material derivatives of integrals

defined in a spatial domain.

Let us define in the spatial configuration of a continuum body B an arbitrary

coordinate system { txi ,i = 1, 2, 3} and let us assume a continuous

Eulerian tensor field tψ( txi ,t) to be a single-valued function of the coordinates

{ txi } and of time t. Also,wedefine in the reference configuration an

arbitrary coordinate system { ◦x I ,I = 1, 2, 3}.

We define tV as a volume in the spatial configuration and

D

Dt

Z

t V

t ψ ( t xi,t) t dV (4.1)

to be the material time derivative of a spatial volume integral; thatistosay,

Eq. (4.1) measures the rate of change of the total amount of the tensorial

property t ψ carried by the particles that at time t areinsidethevolume t V .

Using Eq.(2.31) we can write,

86 **Nonlinear** continua

D

Dt

Z

tV t t i t

ψ ( x ,t) dV =

d

dt

hence,

D

Dt

Z

t V

t ψ ( t x i ,t) t dV =

Z

◦ V

Z

◦ V

d t ψ ( ◦ x I ,t)

dt

t ψ ( ◦ x I ,t) t J ◦ dV (4.2a)

Z

t ◦

J dV +

◦ V

t ◦ I d

ψ ( x ,t) tJ ◦

dV .

dt

(4.2b)

Example 4.1. JJJJJ

Working in Cartesian coordinates we can write, from Eq. (2.34e) (Fung 1965):

t J = eαβγ t ◦Xα1 t ◦Xβ2 t ◦Xγ3 .

Using Eqs.(2.111a-2.111b) we can calculate the time rate of t J:

t ˙

J = eαβγ [ t lα t ◦X1 t ◦Xβ2 t ◦Xγ3 + t ◦Xα1 t lβ t ◦X2 t ◦Xγ3

+ t ◦Xα1 t ◦Xβ2 t lγ t ◦X3] .

After some algebra, the reader can easily verify that,

t ˙

J = t l eαβγ t ◦Xα1 t ◦Xβ2 t ◦Xγ3 .

Generalizing the above for any set of curvilinear coordinates in the Euclidean

spacewecanwrite,

t

J ˙

t t

= (∇ · v) J

where tv is the velocity vector at the t-configuration.

A proof of the above result in general curvilinear coordinates can be found in

(Marsden & Hughes 1983). JJJJJ

Using the result of Example 4.1 in Eq. (4.2b) we obtain,

D

Dt

Z

tV t t i t

ψ ( x ,t) dV =

Z "

d

(4.3a)

tψ ( ◦x I ,t)

dt

+ t ψ ( ◦ x I ,t)(∇ · t #

t ◦

v) J dV .

◦ V

Returningtothet-configuration we obtain,

D

Dt

Z

tV t t i t

ψ ( x ,t) dV =

Z ∙ t t i D ψ ( x ,t)

Dt

(4.3b)

+ t ψ ( t x i ,t)(∇ · t ¸

v)

tdV

.

t V

4.1 Reynolds’ transport theorem 87

Equation (4.3b) is one way of expressing Reynolds’ transport theorem. In

this Section, we will also discuss other expressions for this theorem.

Using Eq. (2.20b) we can rewrite Eq. (4.3b) as,

D

Dt

Z

t V

t ψ ( t x i ,t) t dV =

Z

[ ∂t ψ ( t x i ,t)

∂t

tV + t v · ∇ t ψ ( t x i ,t)+ t ψ ( t x i ,t)(∇ · t v)]

(4.3c)

t dV .

Example 4.2. JJJJJ

Let us assume a general tensor field:

t t a...b t

ψ = ψ c...d ga ... t t c t d

g g ... g

b

where the t g i are the covariant base vectors of the coordinate system { t x i }.

The velocity vector fieldcanbewrittenas,

t v = t v s t gs

therefore we can write, using Eq.(A.59),

t v · ¡ ∇ t ψ ¢ = t v s t gs · t ψ a...b c...d | n

Also, using Eq.(A.64),

= t v st ψ a...b c...d |s

t n t

g ga ... t t c t d

g g ... g

b

t

ga ... t t c t d

g g ... g .

b

t t t a...b t s

ψ(∇ · v) = ψ c...d v |s t g ...

a t t c t d

g g ... g .

b

Finally, using Eq. (A.62b),

∇ · ( t v t ³ ´

ψ)=

tvs t a...b

ψ c...d |s t g ...

a t t c t d

g g ... g

b

³

=

tvs t a...b

ψ c...d |s + t ψ a...b c...d t v s ´

tga

|s ... t t c t d

g g ... g .

b

From the above equations we get,

t v · (∇ t ψ) + t ψ (∇ · t v) = ∇ · ( t v t ψ) .

JJJJJ

Using the above result we can express Reynolds’ transport theorem as,

88 **Nonlinear** continua

D

Dt

Z

tV t t i t

ψ ( x ,t) dV =

Z

tV ∙ t t i ∂ ψ ( x ,t)

∂t

+ ∇ · ( t v t ψ( t x i ¸

,t))

tdV

.

(4.4)

The generalized Gauss’ theorem can be stated as (Malvern 1969):

Z

∇ · t ψ t Z

dV =

t t t

n · ψ dS (4.5)

t V

where t S is the closed surface that bounds the volume t V and t n is the surface’s

outer normal vector.

Using Gauss’ theorem in Eq. (4.4) and rearranging terms, we get the following

expression of Reynolds’ transport theorem:

⎧

⎪⎨

⎪⎩

Z

t V

∂ t ψ

∂t

t dV = D

Dt

Z

t V

t S

t ψ t dV −

Following (Malvern 1969) we can state:

Rate of increase

of the total amount

of tψ inside

avolume

t V

in the spatial

configuration

⎫

⎪⎬ ⎪⎨

=

⎪⎭

⎧

⎪⎩

Rate of increase

of the total amount

of tψ possessed

by the material

instantaneously

inside the volume

Z

⎫

⎪⎭ tV t S

⎪⎬ ⎪⎨

−

t n · t v t ψ t dS. (4.6)

⎧

⎪⎩

Net rate of outward

flux of tψ carried

by the material

transport through

the closed surface

⎫

⎪⎬

⎪⎭ tS The volume t V is usually called the control volume and the surface t S is

usually called the control surface.

4.1.1 Generalized Reynolds’ transport theorem

In the previous section, for deriving Reynolds’ transport theorem we considered

a material volume t V bounded by a material surface t S.Inthissection,

we are going to generalize the previous derivation considering in the spatial tconfiguration

an arbitrary volume Ω(t) bounded by a surface σ(t) that moves

with an arbitrary velocity field t w.

We can define inside the t-configuration of the body B asurface,

t f ( t x i ,t) = 0 . (4.7)

If the surface moves with the particles instantaneously on it, we say that

it is a material surface.

The Lagrange criterion states (Truesdell & Toupin 1960) that the necessary

and sufficient condition for the above-defined surface to be material is

that its material time derivative is zero; using Eq.(2.20b),

t ˙

f = ∂ t f

∂t + ∂ t f

∂ t x k

t v k = 0 . (4.8)

4.1 Reynolds’ transport theorem 89

Fig. 4.1. Body B with the surface t f ¡ t x i ,t ¢ =0

To prove the Lagrange criterion, let us consider the following general case,

In Fig. 4.1, at a point P on tf =0we define:

- The unit vector tn normal to the surface.

-Thematerialvelocitytvof the particle instantaneously at P .

-Thevelocitytwof the surface.

The condition that the point P (not the particle instantaneously at P )

remains on tf =0when the surface moves is given by,

∂ tf ∂t + ∂ tf ∂ txk t k

w = 0 . (4.9a)

From geometrical considerations,

and we can write,

Hence,

t n =

t wn = t w · t n =

and using in the above Eq.(4.9a), we get

t wn = −

∂ t f

∂ txk t k g

q

∂ tf ∂ txl ∂ tf ∂ txm tglm (4.9b)

t w = t w p t gp . (4.9c)

q ∂ t f

∂ t x l

q ∂ t f

∂ t x l

∂ t f

∂t

t w k ∂ t f

∂ t x k

∂ t f

∂ t x m t g lm

∂ t f

∂ t x m t g lm

(4.9d)

. (4.9e)

90 **Nonlinear** continua

We can also define, using the material velocity of the particle instantaneously

at P ,

t vn = t v · t n =

q ∂ t f

∂ t x l

t v k ∂ t f

∂ t x k

∂ t f

∂ t x m t g lm

. (4.9f)

Using Eqs. (4.9e), (4.9f) and the first equality in Eq.(4.8), we obtain

r

t

f ˙

∂ tf =

∂ txl ∂ tf ∂ txm tglm ( t vn − t wn) . (4.9g)

If the particle instantaneously at P remains on the surface during the

motion, from obvious geometrical considerations

t vn = t wn , (4.9h)

and using the above in Eq.(4.9g), we have for a material surface

t ˙

f = 0 , (4.10)

which demonstrates the Lagrange criterion.

Equation (4.9h) indicates that if we consider in the spatial configuration

at time t fictitious particles (Truesdell & Toupin 1960) moving with a velocity

field t w,thevolumeΩ(t) and the surface σ(t) canbeconsideredmaterialand

we can write, using Eq.(4.6) for any Eulerian tensor field t ψ,

Dt w

Dt

Z

Ω(t)

t ψ t dV =

Z

Ω(t)

∂ t ψ

∂t

t dV +

Z

σ(t)

t n · t w t ψ t dS. (4.11)

With Dt w (·)/Dt we indicate that when taking the material time derivative,

the velocity field t w is considered.

Equation (4.11) is the expression of the generalized Reynolds’ transport

theorem. An application of this theorem is presented in Example 4.5.

4.1.2 The transport theorem and discontinuity surfaces

In Fig. 4.2 we represent a body B in its spatial configuration corresponding to

time t. We assume that the Eulerian tensor field t ψ has a jump discontinuity

across a surface t S12 inside the body and we let the material velocity field t v

to be also discontinuous across t S12 (Truesdell & Toupin 1960). At any point

on the discontinuity surface we define its normal ( t n 12 ) and its displacement

velocity ( t w), not necessarily coincident with the material velocity of the

particle instantaneously at that point ( t v).

Considering the region on the negative side of t n 12, wecandefine fictitious

particles with the following velocity field:

• t v on t S −

• t w on t S12

Fig. 4.2. Discontinuity surface

4.1 Reynolds’ transport theorem 91

and using the generalized Reynolds’ transport theorem we obtain,

Z

Dtw Dt tV −

t t

ψ dV =

Z

tV −

∂ t Z

ψ t

dV

∂t

+

Z

tS− t t t t

n · v ψ dS

+

t

n12 · t w t ψ − t dS. (4.12a)

t S12

Using the generalized Gauss’ theorem (Eq.(4.5)),

Z

∇ · ( t v t ψ) t Z

dV =

t t t t

n · v ψ dS

hence,

Dt w

Dt

t V −

Z

t V −

t ψ t dV =

Z

−

t V −

Z

+

t S12

t S −

Z

t S12

t n12 · t v − t ψ − t dS (4.12b)

∙ t ∂ ψ

∂t + ∇ · (tv t ¸

ψ)

tdV

(4.12c)

t ψ − t n12 · ( t v − − t w) t dS.

For the region on the positive side of tn12 , in the same way, we get

Dtw Dt

Z

tV +

t t

ψ dV =

Z ∙ t ∂ ψ

tV + ∂t + ∇ · (tv t Z

¸

ψ)

tdV

(4.12d)

+

t + t

ψ n12 · ( t v + − t w) t dS.

t S12

92 **Nonlinear** continua

The velocity field of the fictitious particles is coincident with the velocity

field of the actual particles everywhere except on t S12. Therefore:

D

Dt

Z

t V

t ψ t dV = Dt w

Dt

Z

t V −

t ψ t dV + Dt w

Dt

Z

t V +

t ψ t dV . (4.12e)

From Eqs.(4.12c) to (4.12e) we get,

D

Dt

Z

tV t t

ψ dV

Z ∙ t ∂ ψ

=

tV ∂t + ∇ · (tv t Z

¸

ψ)

tdV

+ [[ t ψ ( t vn − t wn)] t dS (4.13)

t S12

where,

[[ t ψ ( t vn − t wn)]] = t ψ + ( t v + n − t wn) − t ψ − ( t v − n − t wn) ,

t vn = t n 12 · t v ,

t wn = t n 12 · t w .

In order to obtain a localized version of Reynolds’ transport theorem at

the discontinuity surface we consider the arbitrary material volume enclosed

by the dashed line in Fig. 4.3.

Fig. 4.3. Derivation of the jump discontinuity condition t ν = t ν + ∪ t ν −

For the enclosed material volume, using Eq.(4.13) and the generalized

Gauss’ theorem, we write:

D

Dt

Z

tV t t

ψ dV

Z

=

tV ∂t Z

ψ

∂t

t

dV +

Z

tS− t t t t

n · v ψ dS

Z

(4.14)

+

t t t t

n · v ψ dS + [ t ψ ( t vn − t wn)] t dS,

t S +

t S12

4.2 Mass-conservation principle 93

when tdS + →t S12 and tdS − →t t + t −

S12 ; ν → 0 and ν → 0 we get from

Eq.(4.14):

Z

¡ t t

[[ ψ vn]] + [[ t ψ ( t vn − t wn)] ¢ t

dS = 0 . (4.15)

t S12

Therefore, in order for the above integral equation to be valid for any

arbitrary part of the discontinuity surface, we must fulfill

[ t ψ t vn]] + [ t ψ ( t vn − t wn)] = 0 (4.16a)

at every point on tS12. Equation (4.16a) is known as the jump discontinuity condition.

If we call tU = twn − tvn the discontinuity’s propagation speed, we can

write

[ t ψ t U]] = [ t ψ t vn]] . (4.16b)

The above equation is known as Kotchine’s theorem (Truesdell & Toupin

1960).

4.2 Mass-conservation principle

In Sect. 2.2, Eq.(2.6) introduced the concept of mass of a continuum body B.

In the study of continuum media, under the assumptions of Newtonian

mechanics, it is postulated that the mass of a continuum is conserved. Hence,

Z

D t t

ρ dV = 0 (4.17)

Dt tV where tρ = tρ ( txi ,t).

4.2.1 Eulerian (spatial) formulation of the mass-conservation

principle

Using in Eq.(4.17) the expression of Reynolds’ transport theorem given in

Eq.(4.4) we obtain:

Z

Z ∙ t D t t ∂ ρ

ρ dV =

Dt tV tV ∂t + ∇ · (tρ t ¸

v)

tdV

= 0 . (4.18)

Since the above equation has to be fulfilled for any control volume inside

the continuum, we can write for any point inside the spatial configuration:

∂tρ ∂t + ∇ · (tρ t v) = 0 . (4.19)

The above partial differential equation is the localized spatial form of the

mass-conservation principle in a Eulerian formulation anditiscalledthe

continuity equation.

94 **Nonlinear** continua

Example 4.3. JJJJJ

Using components in a general curvilinear spatial coordinate system, the continuity

equation is written as

∂ t ρ

∂t + t v a ∂t ρ

∂ t x a + t ρ t v a |a = 0 .

JJJJJ

Example 4.4. JJJJJ

For an incompressible material D t ρ

Dt = 0; hence the continuity equation is:

∇ · t v = 0 ,

or in components,

t v a |a = 0 .

JJJJJ

Example 4.5. JJJJJ

Let a fluid of density tρ = tρ ( txi ,t) have a velocity field tv. Let us consider in the spatial configuration a volume Ω(t) bounded by a

surface σ(t) that moves with an arbitrary velocity field tw. Following (Thorpe 1962), we first calculate the fluid mass instantaneously

inside the volume Ω(t):

Z

t t

M = ρ dV

Ω(t)

Ω(t)

and using the expression of the generalized Reynolds’ transport theorem in

Eq.(4.11), we get

dM

dt = Dt Z

Z

w t t ∂

ρ dV =

Dt

t Z

ρ t t t t t

dV + n · w ρ dS

∂t

Ω(t)

where tn is the external normal of the surface σ(t).

Using Eq.(4.5) (generalized Gauss’ theorem), we get

Z

∇ · ¡ Z

¢ t t

ρ v

tdV t t t t

= n · ( ρ v) dS

Ω(t)

σ(t)

σ(t)

and subtracting the above equation from the previous one,

dM

dt =

Z

Ω(t)

∙ t ∂ ρ

∂t + ∇ · (tρ t Z

¸

v)

tdV

+

t t t t t

ρ n · ( w − v) dS .

σ(t)

4.3 Balance of momentum principle (Equilibrium) 95

Using Eq.(4.19), we see that the first integral on the r.h.s. is zero; hence,

dM

dt =

Z

σ(t)

t t t t t

ρ n · ( w − v) dS.

The above equation is an integral equation of continuity for a control volume

in motion in the fluid velocity field (Thorpe 1962). JJJJJ

4.2.2 Lagrangian (material) formulation of the mass conservation

principle

Equation (4.17) implies that,

Z

◦ ◦ A ◦

ρ ( x ) dV =

◦ V

Z

t V

t ρ ( t x a ,t) t dV (4.20a)

where ( ◦ρ, ◦V ) correspond to the reference configuration and ( tρ, tV ) to the

spatial configuration. Using Eq.(2.31) in the r.h.s. of Eq.(4.20a) and changing

variables in the expression of tρ we obtain,

Z

Z

◦ ◦ A ◦

ρ ( x ) dV =

t ◦ A t ◦

ρ ( x ,t) J dV , (4.20b)

◦ V

hence, Z

◦ V

◦ V

( ◦ ρ − t ρ t J) ◦ dV = 0 . (4.20c)

Since the above equation has to be fulfilled for any control volume that we

define inside the continuum, we can write for any point inside the reference

configuration:

◦ t t

ρ = ρ J, (4.20d)

and therefore,

D

Dt (tρ t J) = 0 . (4.20e)

The above equation is the localized material form of the continuity equation.

4.3 Balance of momentum principle (Equilibrium)

The principle of balance of momentum is the expression of Newton’s Second

Law for continuum bodies. Quoting (Malvern 1969):

“The momentum principle for a collection of particles states that the time

rate of change of the total momentum of a given set of particles equals

the vector sum of all the external forces acting on the particles of the set,

provided Newton’s Third Law of action and reaction governs the internal

forces. The continuum form of this principle is a basic postulate of continuum

mechanics”.

96 **Nonlinear** continua

4.3.1 Eulerian (spatial) formulation of the balance of momentum

principle

For a body B in the t-configuration we define its momentum as,

Z

t t t

ρ v dV (4.21a)

t V

the resultant of the external forces acting on the elements of mass inside the

body are, from Eq.(3.2): Z

t t t

ρ b dV , (4.21b)

t V

and the resultant of the external forces acting on the elements of the body’s

surface are, from Eq.(3.4): Z

t t

t dS. (4.21c)

t S

Using Eqs.(4.21a-4.21c), we can state Newton’s Second Law for the body

B as,

Z

Z

Z

D t t t t t t t t

ρ v dV = ρ b dV + t dS. (4.22)

Dt

t V

t V

Using the condition of equivalence between external forces and Cauchy

stresses inside a continuum, defined in Eq.(3.7), we get:

Z

Z

Z

D t t t t t t t t t

ρ v dV = ρ b dV + n · σ dS. (4.23)

Dt

t V

t V

Using in the above the expression of Reynolds’ transport theorem given in

Eq.(4.4), we get

Z

tV ∙ t t ∂( ρ v)

∂t

+ ∇ · ( t ρ t v t ¸

v)

tdV

Z

=

tV Z

t t t

ρ b dV

+

t t t

n · σ dS.(4.24)

From Example 4.2, we obtain

∇ · ( t ρ t v t v) = t v · £ ∇ ( t ρ t v) ¤ + t ρ t v (∇ · t v) (4.25a)

also, from Eq.(2.20b), we get

D ( t ρ t v)

Dt

= ∂ (t ρ t v)

∂t

and, from Eq.(4.5) (Generalized Gauss’ Theorem), we get

t S

t S

t S

+ t v · £ ∇ ( t ρ t v) ¤ , (4.25b)

Z

t S

4.3 Balance of momentum principle (Equilibrium)

Z

97

t t t

n · σ dS = ∇ · ( t σ) t dV . (4.25c)

Using Eqs.(4.25a-4.25c) in Eq.(4.24) we arrive at the integral form of the

Eulerian formulation of the balance of momentum principle:

Z ∙

D

tV Dt (tρ t v) + t ρ t v (∇ · t ¸ Z

v)

tdV £

=

tρ ¤ t t

b + ∇ · σ

tdV

.

tV (4.26)

Since the above equation has to be fulfilled for any control volume that

we define inside the continuum, we can write for any point inside the spatial

configuration:

D

Dt (tρ t v) + t ρ t v (∇ · t v) = t ρ t b + ∇ · t σ (4.27a)

and using in the above the continuity equation, we have

t D

ρ tv Dt = t ρ t b + ∇ · t σ . (4.27b)

The above equation is the localized form of the balance of momentum

principle in an Eulerian formulation and it is known as the equilibrium equation.

Example 4.6. JJJJJ

Using Eq.(A.62b), in the general Eulerian curvilinear system { t x a },weget,

t V

∇ · t σ = t σ ab |a t g b

hence, using Eq.(A.55b),

∇ · t σ =

∙ t ab ∂ σ

∂txa + t σ sb t Γ a sa + t σ as t Γ b ¸

tgb

sa .

From the result in Example A.10, we can easily get

Therefore,

t Γ m il = 1

2

t g mj

∙ t ∂ gij

∂ txl + ∂ tgjl ∂ txi − ∂ tgli ∂ txj ¸

∇ · t ∙ t ab ∂ σ

σ =

∂t 1 ¡

+

tσsb t aj t as t bj

g + σ g

xa 2

¢

µ t ∂ gsj

∂ txa + ∂ tgja ∂ txs − ∂ tgas ∂ txj ¸

tgb

.

.

JJJJJ

98 **Nonlinear** continua

Example 4.7. JJJJJ

A perfect fluid is defined as a continuum in which, at every point, and for any

surface,

t t t t

n · σ = γ n ,

where tγ is a scalar (no shear stresses).

Since tσ is a symmetric second order tensor (to be shown in Sect. 4.4), its

eigenvalues are real and its eigenvectors are orthogonal (Appendix, A.4.1).

Referring the problem to the Cartesian system defined by the normalized

eigenvectors, têα we can write,

Then, for the perfect fluid,

t t

n = ˆnα t êα t

σ =

3X

t

ˆσββ t t

êβ êβ .

β=1

t ˆnβ t ˆσββ = t γ t ˆnβ (β =1, 2, 3)(no addition on β) .

The above set of equations is fulfilled only if the three eigenvalues t ˆσββ are

equal (hydrostatic stress tensor). Hence,

t σ = t p t êβ t êβ .

It is easy to show that as the three eigenvalues of t σ are equal, the above

equation is valid in any Cartesian system; hence we can write,

t σij = t pδij ,

where t p is the pressure. Generalizing the above to any arbitrary coordinate

system

t σ = t p t g .

Using Eq. (A.62b) and the result in Example A.11, the equilibrium equation,

Eq. (4.27b), can be written as,

t D

ρ tv Dt = t ρ t b + ∂p

∂ txi t ij t

g gj .

From Eq. (A.57) we identify the last term on the r.h.s. of the above equation

as ∇ t p, hence we can write the equilibrium equation for a perfect fluid as,

t D

ρ tv Dt = t ρ t b + ∇ t p.

The above equation is known as the Euler equation for perfect fluids. Many

authors get a minus sign for the second term on the r.h.s. because they define

t σij = − t pδij .

JJJJJ

4.3 Balance of momentum principle (Equilibrium) 99

Example 4.8. JJJJJ

Following with the topic discussed in Example 4.5 we consider a fluid, moving

with a velocity field t v, and a moving control volume, moving with a velocity

field t w. In this example, following (Thorpe 1962), we are going to analyze the

momentum balance inside the moving control volume. Using the generalized

Reynolds’ transport theorem (Eq.(4.11)) for the fluid momentum,

Dt w

Dt

Z

Ω(t)

t ρ t v t dV =

Z

+

Ω(t)

Z

∂( t ρ t v)

∂t

σ(t)

t dV

t ρ t v ( t n · t w) t dS.

From the generalized Gauss’ theorem (Eq.(4.5)),

Z

∇ · ( t ρ t v t v) t Z

dV =

t t t t t

n · ( ρ v v) dS.

Ω(t)

Subtracting the above equation from the previous one,

Dtw Dt

Z

Ω(t)

t t t

ρ v dV

Z

=

Ω(t)

∙ t t ∂( ρ v)

+ ∇ · (

∂t

t ρ t v t Z

+

¸

v)

tdV

£ t t

ρ v

tn ¤ t t

· ( w − v)

tdS

.

σ(t)

Using the result in Example 4.2 and Eq.(4.19) (continuity equation),

Dtw Dt

Z

Ω(t)

t t t

ρ v dV =

Z

Ω(t)

µ t

t ∂ v

ρ

∂t + t v · ∇ t Z

+

v

tdV

£ t t

ρ v

tn ¤ t t

· ( w − v)

tdS

.

σ(t)

Onther.h.s.oftheaboveequation,thetermbetweenthebracketsinthe

first integral is the fluid particles material acceleration. We can state, using

Newton’s second law, that the external force instantaneously acting on the

particles inside Ω(t) is,

Z

t t t t

F = ρ a dV .

Hence,

t F = Dw

Dt

Z

Ω(t)

Ω(t)

Z

t t t

ρ v dV +

σ(t)

σ(t)

t ρ t v £ tn · ¡ tv − t w ¢¤ tdS .

JJJJJ

100 **Nonlinear** continua

Example 4.9. JJJJJ

Let us consider the body B and the particle P on its external surface. We

define at P a convected coordinate system θ i with covariant base vectors t eg i

in the spatial configuration and ◦ eg i in the material configuration. The convected

system is definedsoastohave t eg 1 and t eg 2 in the plane tangent to t S

at t P ; and therefore ◦ eg 1 and ◦ eg 2 define the plane tangent to ◦ S at ◦ P .

Material and spatial normal vectors (Nanson’s formula)

The external unit normals at P are

t n =

t eg1 × t eg 2

| t eg 1 × t eg 2 | ,

and,

◦eg1 ×

◦

n =

◦eg 2

| ◦eg ×

1 ◦eg |

2 .

Also, the surface-area differentials are

If we define,

t dS t n = ( t eg1 × t eg 2 )dθ 1 dθ 2

◦ dS ◦ n = ( ◦ eg1 × ◦ eg 2 )dθ 1 dθ 2

t t1 = dθ 1 t eg 1 ,

t t2 = dθ 2 t eg 2 ,

◦ t1 = dθ 1 ◦ eg 1 ,

◦ t2 = dθ 2 ◦ eg 2 ,

(A) ,

(B) .

4.3 Balance of momentum principle (Equilibrium) 101

it is obvious from the results in Sect. 2.9.1 that

◦ t1 = t T

1

◦ t2 = t T

2 .

We can now define an arbitrary curvilinear system { t x i } in the spatial configuration

and another one { ◦ x I } in the material configuration, with covariant

base vectors t g i and ◦ g I respectively.

where,

t t1 = ( t t1) k t g k

t

t2 = ( t t2) k t g

k

t

T1 = (tt1) k ( t ◦X −1 ) K k ◦ g

K

t T

2 = (t t2) k ( t ◦X −1 ) K k ◦ g K

t n = t ni t g i

◦ n = ◦ nI ◦ g I

t

◦X i I = ∂txi ∂◦ I x

.

We write Eqs.(A) and (B) using the above as,

t dS t ni = t ijk ( t t1) j ( t t2) k

◦ dS ◦ nI = ◦ IJK ( t t1) j ( t t2) k ( t ◦X −1 ) J j ( t ◦X −1 ) K k .

Multiplying both sides of the above equation by ( t ◦X −1 ) I i ,weget

◦ dS ◦ nI ( t ◦X −1 ) I i = ◦ IJK ( t ◦X −1 ) I i ( t ◦X −1 ) J j ( t ◦X −1 ) K k ( t t1) j ( t t2) k

but, from Eqs.(A.37e) and (2.34g)

Hence, using Eq.(A.37c), we get

◦ IJK = eIJK

p | ◦ gAB| .

◦ dS ◦ nI ( t ◦X −1 ) I i = p | ◦ gAB| | t ◦X −1 | eijk ( t t1) j ( t t2) k

and again using Eq.(A.37e), we get

Therefore,

◦ dS ◦ nI ( t ◦X −1 ) I i = p | ◦ gAB| | t ◦X −1 |

t ijk

p | t gab| (t t1) j ( t t2) k .

102 **Nonlinear** continua

◦ dS ◦ nI ( t ◦X −1 ) I i = t dS t ni | t ◦X −1 |

and using Eq.(2.34i), we get

t t t ◦ t

n dS = J n · ◦X −1 ◦ dS.

s

| ◦ gAB|

| t gab|

The above equation is called Nanson’s formula (Bathe 1996). JJJJJ

Example 4.10. JJJJJ

ForanEulerianvector t a we define, using Eq.(2.76a),

t A =

h ¡t◦X −1¢ B

b

t a b i ◦ gB .

Using the generalized Gauss’ theorem (Eq.(4.5)),

Z

∇ · t a t Z

dV =

t t t

n · a dS.

t V

In the r.h.s. integral we introduce Nanson’s formula derived in Example 4.9;

hence,

Z

∇ · t a t Z

dV =

t ◦ t

J n · ◦X −1 · t a ◦ dS

t V

=

=

Z

◦ S

Z

◦ S

◦ S

t J ◦ n ·

◦ n ·

Using again the generalized Gauss’ theorem,

Z

∇ · t a t Z

dV = DIV

t V

◦ V

t S

h ¡t◦X −1¢ B

b

³ tJ t A ´ ◦ dS.

t a b i ◦ gB

³ tJ t A ´ ◦ dV .

With the notation DIV (·) we indicate a divergence in the reference configuration.

Using in the r.h.s. integral Eq.(2.31), we get

Z

Z

¡ ¢ ³

t t

J ∇ · a

◦dV

= DIV

tJ t

A ´ ◦

dV .

◦ V

The localized form of the above equation is known as the Piola Identity (Marsden

& Hughes 1983),

¡ ¢ ³

t t

J ∇ · a = DIV

tJ t

A ´

.

◦ V

◦ dS

JJJJJ

4.3 Balance of momentum principle (Equilibrium) 103

Example 4.11. JJJJJ

We can write the Piola Identity, derived in the above example, as:

t t b

J a |b = £ t t

J ( ◦X −1 ) B c t a c¤ Aftersomealgebra,weget

|B .

t t b

J a |b = £ t t

J ( ◦X −1 ) B ¤

c |B t a c + t J t a c |c .

Hence, £ tJ t

( ◦X −1 ) B ¤

c |B = 0

that is to say

DIV ¡ t J t ◦X −1¢ = 0 .

4.3.2 Lagrangian (material) formulation of the balance of

momentum principle

JJJJJ

In the previous section we derived, in the spatial configuration, the integral

and localized forms of the balance of momentum principle (equilibrium equations).

In this section we are going to derive, in the material configuration, the

integral and localized forms of the equilibrium equations.

We are going to refer Eq.(4.23) to volumes and surfaces definedinthe

material configuration, using Eq.(4.20d) and Nanson’s formula (Example 4.9).

D

Dt

Z

◦ V

◦ ρ t v ◦ dV =

Z

◦ V

◦ ρ t b ◦ dV +

Z

◦ S

◦ ρ

t ρ

◦ n · t ◦X −1 · t σ ◦ dS.

(4.28a)

In the above equation, all magnitudes are written as functions of ( ◦ x I ,t).

Using the generalized Gauss’ theorem together with the definition of the

first Piola-Kirchhoff stress tensor, we obtain

Z

D

Dt (◦ρ t v) ◦ dV =

Z

◦ t ◦

ρ b dV +

◦ V

◦ V

In the above (Malvern 1969),

Z

◦ V

DIV ( t ◦P) ◦ dV . (4.28b)

DIV ( t ◦P) = t ◦P Aa |A t g

a

∙ t ∂

=

◦P Aa

∂◦xA + t ◦P Da ◦ Γ A DA + t ◦P Ad t ◦X i A t Γ a

¸

tga

id .

Equation (4.28b) is an integral form of the equilibrium equations. It is important

to note that although the integrals are calculated on volumes defined

104 **Nonlinear** continua

in the reference configuration, the equilibrium is established in the spatial

configuration.

The corresponding localized form is,

◦ ρ D t v

Dt = ◦ ρ t b + DIV ( t ◦P) . (4.29)

Example 4.12. JJJJJ

From Eqs.(4.27b) and (4.29) we get,

◦ ρ

t ρ ∇ · t σ = DIV ( t ◦P) .

The above equation is a particular application of the Piola Identity. JJJJJ

In order to write the equilibrium equations in terms of fully material tensors

we have to pull-back Eq.(4.29).

For the material velocity field:

t V = £ tφ ∗ ( t v a ) ¤ A ◦gA . (4.30)

A“physical interpretation” of the pull-back of the material velocity contravariant

components was presented in Example 2.12.

In the same way, for the material acceleration,

and

h tA i A

t a = D t v

Dt = t a a t g a

= £ t φ ∗ ( t a a ) ¤ A = t ã A

(4.31)

(4.32)

the t ã A are the components of the material acceleration vector in the convected

system { ◦ x A } (convected acceleration (Simo & Marsden 1984)).

For the external loads per unit mass, we define

t B = £ tφ ∗ ( t b a ) ¤ A ◦gA . (4.33)

Since DIV ( t ◦P) is an Eulerian vector,

£ tφ∗ ¡ t

◦P Ia ¢¤A

|I

t

= ( ◦X −1 ) A a t ◦P Ia |I . (4.34)

Therefore, the pull-back of Eq.(4.29) is

◦ ρ t A = ◦ ρ t B + t ◦X −1 · DIV ( t ◦P) . (4.35)

In a Lagrangian formulation, the above equation is the localized form of

the equilibrium equations.

4.4 Balance of moment of momentum principle (Equilibrium) 105

4.4 Balance of moment of momentum principle

(Equilibrium)

Quoting (Malvern 1969) again:

“In a collection of particles whose interactions are equal, opposite, and

collinear forces, the time rate of change of the total moment of momentum for

a given collection of particles is equal to the vector sum of the moments of the

external forces acting on the system. In the absence of distributed couples, we

postulate the same principle for a continuum”.

The condition of no distributed couples (nonpolar media) was introduced

in Sect. 3.2.

4.4.1 Eulerian (spatial) formulation of the balance of moment of

momentum principle

For a body B in the t-configuration we define its moment of momentum with

respect to a given point O as,

Z

t t t t

ρ r × v dV , (4.36a)

t V

where

t t t

r = x − x◦ ;

and t t x ; x◦ are the position vectors of an arbitrary point P and of the

point O, respectively.

The resultant moment with respect to O of the external forces acting on

the elements of mass inside the body is,

Z

t t t t

ρ r × b dV , (4.36b)

t V

and the resultant moment with respect to O of the external forces acting on

the elements of the body’s surface is,

Z

t t t

r × t dS. (4.36c)

t S

Using Eqs.(4.36a-4.36c) we can state the balance of moment of momentum

principle for the continuum body B:

Z

Z

Z

D t t t t t t t t t t t

ρ r × v dV = ρ r × b dV + r × t dS. (4.37a)

Dt

t V

t V

With the condition of equivalence between external forces and Cauchy

stresses inside a continuum defined in Eq.(3.7), we get

t S

106 **Nonlinear** continua

Z

D

Dt

t t t t

ρ r × v dV =

t V

Z

t V

t ρ t r × t b t dV +

Z

t S

t t t t

r × ( n · σ) dS.

(4.37b)

Example 4.13. JJJJJ

The last integral on the r.h.s. of Eq.(4.37b) can be written as,

Z

t S

Z

¡ t

r ×

tn ¢ t

· σ

tdS

= −

Z

= −

t S

t V

¡ tn · t σ ¢ × t r t dS

∇ · ¡ t σ × t r ¢ tdV

where we used Eq. (4.5) (Generalized Gauss’ Theorem).

For any second-order tensor, t d , and for any vector, t c ,wecanwrite

Also,

t t t pq t r t t

d × c = d c gp gq × t g =

r t d pq t c r t qro t g

p

= − t d pq t c r t rqo t t o

g g .

p

t t T t l

c × d = c ¡ t T

d ¢mn tgl

× t t

g gn =

m

t c l t d nm t lmo t g o t g

n

and ³

tc t T

× d ´ T

= t c l t d nm t lmo t g

n

Hence,

Therefore,

Z

t S

³

t t

d × c = −

tc t T

× d ´ T

.

t g o .

t g o

t

r × ¡ Z

¢ t t

n · σ

tdS

= ∇ ·

tV ¡ t t T

r × σ ¢T tdV

.

JJJJJ

Using the above result,

D

Dt

Z

=

Z

t t t t

ρ r × v dV

tV t t t t

ρ r × b dV +

Z

∇ ·

(4.38a)

¡ t t T

r × σ ¢T tdV

.

t V

t V

4.4 Balance of moment of momentum principle (Equilibrium) 107

Using in the above equation the expression of Reynolds’ transport theorem

given in Eq.(4.4), we get

Z ∙

∂

tV ∂t (tρ t r × t v) + ∇ · ( t ρ t v t r × t ¸

v)

tdV

Z

Z

t t t t

= ρ r × b dV + ∇ · ¡ t t T

r × σ ¢T tdV

. (4.38b)

t V

With the result in Example 4.2, we get

∇ · ( t ρ t v t r × t v) = t v · £ ∇ ( t ρ t r × t v) ¤ +( t ρ t r × t v)(∇ · t v) (4.39)

using Eqs. (4.39) and (2.20b) we can write Eq.(4.38b) as,

Z

tV ∙ t t t D ( ρ r × v)

+

Dt

t ρ ¡ ¢ ¡ ¢ t t t

r × v ∇ · v ¸ t

dV

Z

Z

t t t t

= ρ r × b dV + ∇ · ¡ t t T

r × σ ¢T tdV

t V

and therefore,

Z ½

¡tr ¢ t

× v

tV ∙ Dtρ Dt + t ρ ¡ ∇ · t v ¢¸

+ t ρ D ¡ tr ¢ t

× v

Dt

¾ t

dV

Z

Z

t t t t

= ρ r × b dV + ∇ · ¡ t t T

r × σ ¢T tdV

. (4.40a)

t V

t V

Equation (4.19), the Eulerian continuity equation, can be written as,

Dtρ Dt + t ρ ¡ ∇ · t v ¢ = 0 (4.40b)

hence, introducing the above into Eq.(4.40a), we obtain

Z

t t t t

ρ r × a dV =

Z

t t t t

ρ r × b dV +

Z

∇ ·

tV tV tV ¡ t t T

r × σ ¢T tdV

.

(4.40c)

For the Eulerian formulation, the above is the integral expression of the

balanceofmomentofmomentumprinciple.

From it, we obtain the localized form:

t V

t ρ t r × t a = t ρ t r × t b + ∇ · ¡ tr × t σ T ¢T . (4.41)

4.4.2 Symmetry of Eulerian and Lagrangian stress measures

From the Eulerian localized form of the balance of moment of momentum

principle, we will first derive the symmetry of the Cauchy stress tensor.

t V

108 **Nonlinear** continua

Using an intermediate result that we got in Example 4.13, we can write

the localized form of the balance of moment of momentum as,

t r × t ρ D t v

Dt − t r × t ρ t b = − ∇ · ¡ t σ × t r ¢ . (4.42)

Example 4.14. JJJJJ

The divergence of the cross product between a second order tensor, t σ ,and

a vector, t r ,is

∇ · ¡ t σ × t r ¢ = ∂

∂ t x n

t g n · £ tσ × t r ¤

= ¡ ∇ · t σ ¢ × t r + t g n ·

= − t r × ¡ ∇ ·

= − t r × ¡ ∇ ·

t σ ¢ + t g n ·

t σ ¢ + t g n ·

= − t r × ¡ ∇ ·

h i

tσ t

× gn

h tσ ab t ga

h tσ ab t ga

t gb × t g n

t bno t g oi

t σ ¢ + t bno t σ nb t g o .

i

JJJJJ

Using the above result in Eq.(4.42), we get

t

r ×

∙

tρ Dtv Dt − t ρ t b − ∇ · t ¸

σ = − t bno t σ nb t g o . (4.43a)

Using the Eulerian localized equilibrium equation (Eq.(4.27b)),

t bno t σ nb t g o = 0 . (4.43b)

Introducing in the above Eq. (A.37e), we can write

¯

∂

¯

tzi ¯ t ebno t σ nb = 0 (o =1, 2, 3) (4.43c)

hence,

∂ t x j

Equations (4.43d) show that,

(o =1) t σ 23 − t σ 32 = 0 ,

(o =2) t σ 31 − t σ 13 = 0 , (4.43d)

(o =3) t σ 12 − t σ 21 = 0 .

4.5 Energy balance (First Law of Thermodynamics) 109

t σ = t σ T , (4.44)

that is to say, they show that the Cauchy stress tensor is symmetric.

The symmetry of the Cauchy stress tensor implies the symmetry of the

Kirchhoff stress tensor, the second Piola-Kirchhoff stress tensor and the stress

tensor defined in Eq.(3.28).

For the first Piola-Kirchhoff stress tensor, which is not symmetric, we can

define the following symmetry condition (see Eq.(3.15d)):

t

◦X i I t ◦P Ij = t ◦X j

I t ◦P Ii . (4.45)

4.5 Energy balance (First Law of Thermodynamics)

To study the conservation of energy in our framework of Newtonian continuum

mechanics, we will add to our set of variables a new one: the internal energy.

Quoting (Marsden & Hughes 1983) we can state that the internal energy

“represents energy stored internally in the body, which is a macroscopic reflection

of things like chemical binding energy, intermolecular energy and energy

of molecular vibrations”.

Examples of internal energy are:

• The energy stored in a deformed spring (elastic energy).

• The energy stored in a heated body (thermal energy).

• The energy stored in a bottle of oil (chemical energy).

In the next chapter, when we study the different material constitutive

relations we will present some phenomenological relations between different

forms of internal energy and the continuum state variables (stress, strain,

temperature, etc.).

In the following subsections, we will derive the Eulerian (spatial) and

Lagrangian (material) formulations of the First Law of Thermodynamics

(Malvern 1969).

4.5.1 Eulerian (spatial) formulation of the energy balance

For a body B in the t-configuration we define tu as its internal energy per

unit mass. BeingtKthe kinetic energy of B defined by Eq.(3.9g), the total

energy tE in the considered body, at the instant t, is

Z

t t t t t

E = K + ρ u dV . (4.46)

tV The external forces acting on the body provide a mechanical power input

tPext. Using Eq.(3.9j), we write

110 **Nonlinear** continua

t Pext = Dt K

Dt +

Z

t V

t σ :

t d t dV . (4.47)

We call “heat” ( t Q) the energy that flows due to a temperature gradient.

We will consider two types of heat:

• An outflowing heat flux through the body external surface ( t q:heatflux

vector). Examples are radiative and convective heat exchanges between

the body B and the external medium.

• An internal distributed heat source per unit mass ( t r). Examples are chemical

reactions, phase changes, etc.

The total heat input to B at the instant t is,

Z

Z

t

Qinput = −

t t t

q · n dS +

hence,

t S

Using the First Law of Thermodynamics, we can write

D

Dt

Z

t V

t ρ t u t dV =

t V

t ρ t r t dV . (4.48)

D t E

Dt = t Pext + t Qinput (4.49a)

Z

+

Z

t V

t V

t σ : t d t dV −

Z

t S

t q · t n t dS

t ρ t r t dV . (4.49b)

Using Gauss’ theorem (Eq.(4.5)), we get

D

Dt

Z

tV t t t

ρ u dV =

Z

tV t

σ :

t t

d dV −

Z

tV ∇ · t q t Z

dV

+

t t t

ρ r dV . (4.49c)

t V

Using in the above equation the expression of Reynolds’ transport theorem

given in Eq.(4.4), we get

Z

tV ∙ t t ∂ ( ρ u)

∂t

+ ∇ · ¡ ¢ t t t

v ρ u ¸ Z

Z

t

dV (4.49d)

=

t

σ :

t t

d dV − ∇ · t q t Z

dV +

t t t

ρ r dV .

t V

t V

Now, we introduce in the integral on the l.h.s. of the above equation the

result in Example 4.2, the Eulerian continuity equation in Eq.(4.19) and the

definition of material derivative in Eq.(4.49d); hence,

t V

Z

t V

4.5 Energy balance (First Law of Thermodynamics) 111

t D

ρ tu t

dV

Dt

=

Z

t t t

σ : d dV −

tV Z

tV ∇ · t q t Z

dV

+

t t t

ρ r dV . (4.49e)

t V

The above equation is the integral form of the Eulerian formulation for

the energy conservation principle.

Since the above equation has to be fulfilled for any control volume that

we define inside the continuum, we obtain the localized form of the energy

conservation principle in the Eulerian formulation,

t ρ D t u

Dt = t σ : t d − ∇ · t q + t ρ t r. (4.50)

Example 4.15. JJJJJ

Following from the topic discussed in Examples 4.5 and 4.8, we consider a

fluid moving with a velocity field t v and a moving control volume with a

velocity field t w. In this example, we are going to analyze the energy balance

inside the moving control volume. Using the generalized Reynolds’ transport

theorem (Eq. (4.11)) for the fluid internal energy,we get

Dt w

Dt

Z

Ω(t)

t ρ t u t dV =

Z

Ω(t)

∂ ( t ρ t u)

∂t

t dV +

Z

σ(t)

t ρ t u ¡ tn · t w ¢ tdS .

From the generalized Gauss’ theorem (Eq.(4.5)),

Z

∇ · ¡ ¢ t t t

ρ u v

tdV

=

Z

¡ t

n ·

tρ ¢ t t

u v

tdS

.

Ω(t)

Subtracting the above equation from the previous one,

Dtw Dt

Z

Ω(t)

t t t

ρ u dV

Z

=

Ω(t)

∙ t t ∂ ( ρ u)

+ ∇ ·

∂t

¡ ¢ t t t

ρ u v ¸ Z

+

t

dV

¡ t t t

ρ u n ·

tw ¢ t

− v

tdS

.

σ(t)

Using the result in Example 4.2 and Eq.(4.19) (continuity equation),

Dtw Dt

Z

Ω(t)

t t t

ρ u dV =

Z

Ω(t)

∙ t

t ∂ u

ρ

∂t + t v · ∇ ¡ ¢ t

u ¸ Z

+

t

dV

¡ t t t

ρ u n ·

tw ¢ t

− v

tdS

.

σ(t)

On the r.h.s. of the above equation, the term between brackets in the first

σ(t)

112 **Nonlinear** continua

integral is the material derivative of the fluid internal energy. Hence, using

Eq.(4.50),

Z

£ tσ ¤ t t t t

: d − ∇ · q + ρ r

tdV

Ω(t)

= Dtw Dt

Z

t Ω

t ρ t u t dV +

Z

σ(t)

t ρ t u t n · ¡ tv − t w ¢ tdS .

4.5.2 Lagrangian (material) formulation of the energy balance

JJJJJ

Let us call t U the internal energy per unit mass in the t-configuration but

referred to the reference configuration. Obviously,

From Eqs.(3.10) and (3.20c),

Z

t t t

σ : d dV =

t V

and using Eq.(2.76a), we define

t U ( ◦ x I ,t) = t u ( t x i ,t) . (4.51a)

Z

◦ V

t

◦S :

t

◦ ˙ε ◦ dV (4.51b)

t Q = ( t Q ) A ◦ gA = [( t ◦X −1 ) A a t q a ] ◦ g A . (4.51c)

For the heat sources per unit mass we can also define in the reference

configuration,

³ ´

t

R

◦xI ,t = t r ¡ ¢ t i

x ,t . (4.51d)

Using the Piola Identity (Example 4.10), we can write

Z

∇ · t q t dV =

Z

DIV

³

tJ t

Q ´ ◦

dV . (4.51e)

t V

◦ V

Hence, introducing into Eq.(4.49e) the results in Eq. (4.51a-4.51e), we get

the integral form of the Lagrangian formulation for the energy conservation

principle,

Z

◦ V

◦ ρ D t U

Dt

◦ dV =

Z

+

◦ V

Z

◦ V

t

◦S :

t

◦ ˙ε ◦ dV −

Z

◦ V

DIV

³ tJ t Q ´ ◦ dV

◦ ρ t R ◦ dV . (4.52)

Since the above equation has to be fulfilled for any volume that we define

in the reference configuration, we get the localized form of the energy

conservation principle in the Lagrangian formulation,

4.5 Energy balance (First Law of Thermodynamics) 113

◦ ρ D t U

Dt = t ◦S :

t

◦ ˙ε − DIV

³ tJ t Q ´

+ ◦ ρ t R. (4.53)

Example 4.16. JJJJJ

Using Eq.(3.18), we can write an alternative localized form of the energy

conservation principle,

◦ ρ D t U

Dt = t ◦P T :

t

◦ ˙X − DIV

³ tJ t Q ´

+ ◦ ρ t R .

JJJJJ

5

Constitutive relations

Following (Marsden & Hughes 1983) we can state that the constitutive relations

in a continuum are the functional forms that adopt the stress tensor,

thefreeenergyandtheheatflow as functions of the continuum deformation

and temperature.

Constitutive relations can be formulated using two different methodologies:

(i) Studying the phenomena that takes place on the atomic scale (deformation

of the atomic lattices, movement of dislocations, etc. (Dieter 1986)).

(ii) Using phenomenological mathematical models that can match laboratory

observations at the macroscopic scale.

The phenomenological constitutive relations are generally used in continuum

mechanics and we will concentrate on this approach in this chapter.

This chapter is intended as an introduction to a large number of different

constitutive models; hence, the recommended literature has to be classified

into different areas:

• For studying the fundamentals that have to be considered for formulating

constitutive relations some reference books are: (Truesdell & Noll 1965,

Malvern 1969, Marsden & Hughes 1983).

• For studying hyperelasticity: (Ogden 1984).

• For studying plasticity: (Hill 1950, Mendelson 1968, Johnson & Mellor

1973, Lubliner 1990, Simo & Hughes 1998, Kojić & Bathe 2005).

• For studying viscoplasticity: (Perzyna 1966, Kojić & Bathe 2005).

• For studying viscoelasticity: (Pipkin 1972).

• For studying damage mechanics: (Lamaitre & Chaboche 1990).

116 **Nonlinear** continua

5.1 Fundamentals for formulating constitutive relations

In this section, we will discuss some principles that shall be considered when

developing a constitutive relation for modeling the behavior of any material.

5.1.1 Principle of equipresence

This principle has been proposed in (Truesdell & Toupin 1960) and very generally

states that any independent variable that is included in the formulation

of any constitutive relation for a given material has to be included in the

formulation of all the other constitutive relations that are developed for the

same material unless it is shown that its inclusion is either not necessary or

violates some physical law.

Often, the constitutive models used by scientists and engineers are derived

considering a very simplified material behavior and some variables are not included

in the derived relation; e.g. usually when stating the relation between

heat flux and temperature the continuum deformations are not considered

(Fourier’s Law). The principle of equipresence requires that these simplifications

should not be made “by default” and that each of them should be

specifically analyzed. These analyses will also be helpful for evaluating and

understanding the limitations of the obtained results.

5.1.2 Principle of material-frame indifference

This principle states that the continuum constitutive relations shall be formulated

using objective physical laws (see Section 2.12).

When using Cartesian coordinate systems in the spatial and material configurations,

only classical objectivity is required. In more general cases, covariant

formulations should be used.

Quoting (Ogden 1984), we can describe classical material objectivity as:

“An important assumption in continuum mechanics is that two observers

in relative motion make equivalent (mathematical and physical) deductions

about the macroscopic properties of a material under test. In other words,

material properties are unaffected by a superposed rigid motion, and the relation

between the stress and the motion has the same form for all observers”.

5.1.3 Application to the case of a continuum theory restricted to

mechanical variables

This theory considers some measure of the continuum deformations as the

only independent variable and some measure of the continuum stresses as the

only dependent variable.

Since many of the physical problems usually analyzed by scientists and

engineers correspond to this category, it is an important case to be considered.

In (Truesdell & Noll 1965) the following principles are proposed for the

specific case of a continuum theory restricted to mechanical variables:

5.1 Fundamentals for formulating constitutive relations 117

• Stresses are deterministic functions of the continuum deformation history.

• Local action: the stresses acting on a material particle (material point)

are only a function of the strains at the same material particle and not of

the strains at neighboring particles. It is important to note that nowadays

nonlocal continuum theories are used for very specific problems (Pijaudier-

Cabot, Baˇzant & Tabbara 1988). Hence, we consider the principle of local

action as a convenient hypothesis that, although it does not represent a

physical law, provides a simplification to the constitutive relations that

agrees with physical observations for many materials.

Example 5.1. JJJJJ

Let us assume that for a problem in which only mechanical variables are

considered, we formulate the following constitutive relation:

¡ t t

σ = m

t◦X ¢

where t m is a tensorial function that maps the space of invertible two-point

tensors into the space of symmetric Eulerian tensors (Ogden 1984). It is important

to realize that t m depends on the selected reference configuration.

Since we have restricted ourselves to study a purely mechanical problem we

can accept that the proposed constitutive relation fulfills the principle of

equipresence.

To study the objectivity of the formulation (classical objectivity) we consider

in the spatial configuration two Cartesian coordinate systems: a fixed one

{ t zα} and a moving one { t z ∗ α}.

Since the Cauchy stress is an objective spatial tensor, from Eq. (2.101c),

t σ = Q(t) ·

t σ ∗ · Q T (t)

where Q(t) is an orthogonal tensor.

And since the deformation gradient tensor is a two-point objective tensor,

from Eq. (2.102c):

t

◦X = Q(t) ·

t

◦X ∗ .

Since the above relation is valid for any orthogonal Q, inparticularitisalso

valid for t ◦R (from the polar decomposition t ◦X = t t

◦R · ◦U ).

The observer in the moving frame writes, due to material objectivity,

t σ ∗ = t m ¡ t◦X ∗¢

and using variables measured in the stationary frame,

³

t ∗ t

σ = m

t◦R T t

· ◦R · t ´

◦U = t ◦R T · t σ · t ◦R .

118 **Nonlinear** continua

Hence,

t σ = Q(t) · t m ¡ t◦U ¢ · Q T (t) .

The principle of material-frame indifference imposes t m ¡ t◦U ¢ instead of

t m ¡ t◦X ¢ .

The requirements of determinism and local action are obviously fulfilled. JJJJJ

Example 5.2. JJJJJ

As an alternative formulation to the one presented in Example 5.1, we assume

that for a problem in which only mechanical variables are considered

t

◦PT = t ◦N ¡ t

◦X ¢

where t ◦N is a tensorial function that maps the space of invertible two-point

tensors into a space of general two-point tensors. The relation between tm (in

Example 5.1) and t ◦N can be derived using Eq. (3.15d).

For the fulfillment of the principle of equipresence, we make the same comment

as in the above example.

Again, to study the objectivity of the formulation, we consider in the spatial

configuration two Cartesian systems: a fixed one { tzα} and a moving one

{ tz∗ α}.

Since the first Piola-Kirchhoff stress tensor is an objective two-points tensor,

from Eq. ( 2.102c):

t

◦P T = Q (t) · t ◦P ∗T ,

where Q (t) is an orthogonal tensor.

Inthesameway

t

◦X = Q (t) · t ◦X ∗ .

The observer in the moving frame writes, due to material objectivity,

t

◦P ∗T = t ◦N ¡ t

◦X ∗¢ ,

and particularizing for Q (t) = t ◦R (from the polar decomposition

t

◦X = t ◦R · t ◦U ),

Hence,

t

◦P ∗T = t ³

t◦R

◦N

T · t ◦R · t ´

◦U

t

◦P T = Q(t) · t ◦N ¡ t ◦U ¢ .

= t ◦R T · t ◦P T .

The principle of material-frame indifference imposes the above form for the

constitutive relation. JJJJJ

5.1 Fundamentals for formulating constitutive relations 119

Example 5.3. JJJJJ

Another alternative formulation for a pure mechanical problem is,

t

◦S = t ◦M ¡ t

◦X ¢

where t ◦M is a tensorial function that maps the space of invertible two-point

tensors into the space of symmetric Lagrangian tensors. The relation between

t t m (in Example 5.1) and ◦M can be derived using Eq. (3.19).

For the fulfillment of the principle of equipresence, we make the same comment

as in Example 5.1. To study the objectivity of the formulation, as usual,

we consider two Cartesian systems in the spatial configuration: a fixed one

{ tzα} and a moving one { tz∗ α} .

Since the second Piola-Kirchhoff stress tensor is an objective Lagrangian tensor,

we have:

t

◦S = t ◦S ∗ .

From the objectivity of the deformation gradient tensor it follows that (Eq.

(2.102c)),

t

◦X = Q (t) · t ◦X ∗

where Q (t) is an orthogonal tensor.

Due to material objectivity, the observer in the moving frame writes:

t

◦S ∗ = t ◦M ¡ t

◦X ∗¢

and particularizing for Q (t) = t ◦R (from the polar decomposition

t

◦X = t ◦R · t ◦U ),

Hence,

and using

t

◦S ∗ = t ³

t◦R

◦M

T · t ◦R · t ´

◦U

t

◦S ∗ = t ◦M ¡ t ◦U ¢ ,

t

◦U = £ 2 t ◦ε + I ¤ 1/2

= t ◦S.

we get,

t

◦S = t ◦ c M ¡ t

◦ε ¢ .

The principle of material-frame indifference imposes the above form of the

constitutive relation. JJJJJ

120 **Nonlinear** continua

5.2 Constitutive relations in solid mechanics: purely

mechanical formulations

In this section we will analyze some of the constitutive relations that are used

to model the mechanical behavior of solids, neglecting the couplings with other

physical phenomena.

An elastic material model (also called Cauchy elastic material (Ogden

1984)) predicts a material behavior independent of the material history and

time,thatistosay,stressesareunivocally determined by strains and vice

versa.

Of course, an elastic material model cannot be used to model: permanent

deformation phenomena, damage of materials, creep effects, strain-rate effects,

etc.

In Chap. 3 we presented the definition of conjugate stress and strain rate

measures. If we define an arbitrary stress measure T (it can be either a Lagrangian,

Eulerian or two-point tensor) and its conjugate strain-rate measure

˙E , we can write the stress power per unit volume as,

for an elastic solid,

Pσ = T T : ˙E , (5.1a)

T = T(E) , (5.1b)

hence,

Pσ = T T (E) : ˙E . (5.1c)

In general (Ogden 1984), we cannot assess on the existence of a scalar

function, U(E) , such that,

˙U = T T (E) : ˙E , (5.1d)

that is to say, in general Pσ is not an exact differential.

When Pσ is an exact differential, we can write ( 1944)

˙U = ∂U

∂E : ˙E = T T (E) : ˙E , (5.1e)

and we say that the material model is hyperelastic (also called Green elastic

material model (Ogden 1984)).

For a hyperelastic material model, from Eq. (5.1e) taking into account

that ˙E is arbitrary,

T T (E) = ∂U

, (5.1f)

∂E

and U is called the elastic energy function per unit volume.

5.2 Constitutive relations in solid mechanics: purely mechanical formulations 121

Example 5.4.

For a hyperelastic material,

JJJJJ

T ij = ∂U

; T kl = ∂U

and since

we must have

∂Eij

∂ 2 U

∂Eij ∂Ekl

∂T ij

∂Ekl

=

∂ 2 U

∂Ekl ∂Eij

= ∂Tkl

∂Eij

.

∂Ekl

,

JJJJJ

The inelastic mechanical behavior of some materials can be described with

equations of the form,

which are the hypoelastic material models.

5.2.1 Hyperelastic material models

dT = C(T , E) : dE , (5.2)

For a hyperelastic material, the elastic energy in the spatial configuration per

unit volume of the reference configuration can be written as:

d t ◦U = t ◦S IJ d t ◦εIJ , (5.3a)

using the second Piola-Kirchhoff stress tensor and the Green-Lagrange strain

tensor.

If we use t U as the elastic energy in the spatial configuration per unit

mass, we get

d t ◦U dm

◦ ρ = d t U dm . (5.3b)

and

Hence,

d t U = 1 t

◦ρ ◦S IJ d t ◦εIJ , (5.3c)

t

◦S IJ = ◦ ρ ∂tU ∂t ◦εIJ

= 2 ◦ ρ ∂t U

∂ t ◦CIJ

. (5.3d)

Using the chain rule and considering Eq. (4.51a), we can write (Marsden

& Hughes 1983)

∂tu ∂t =

gab

∂tU ∂t ∂

◦CAB

t ◦CAB

∂t , (5.4a)

gab

122 **Nonlinear** continua

from Eq. (2.93a)

and therefore

∂ t u

∂ t gab

t

◦CAB = t gab t ◦X a A t ◦X b B , (5.4b)

= ∂t U

∂ t ◦ CAB

t

◦X a A t ◦X b B = t φ∗( ∂tU ∂ t ◦ CAB

Pushing-forward Eq. (5.3d) and using the above, we obtain

t τ ij = 2 ◦ ρ ∂ t u

∂ t gij

) . (5.4c)

, (5.4d)

and using Eq. (3.12) we obtain the Doyle-Ericksen formula (Simo & Marsden

1984):

t ij t ∂

σ = 2 ρ tu ∂t . (5.4e)

gij

It is important to realize that the above is the correct relation for deriving

the Cauchy stress tensor from the elastic energy per unit mass function, and

that (Marsden & Hughes 1983):

t ij

σ 6= 2 t ρ ∂tu ∂t . (5.5)

eij

Using the result in the Example 4.16 and remembering that for a hyperelastic

material the stress is only a function of the state variables, we also

get

t B

◦Pa = ◦ ∂

ρ

tU ∂t ◦X a .

B

5.2.2 A simple hyperelastic material model

In this section we will develop the simplest possible hyperelastic material

model: the isotropic, linear hyperelastic material model.

To use Eq. (5.3d), in the reference configuration we need to define the

strain energy per unit volume of the reference configuration

the simplest definition is:

t

◦U = t ◦A◦ + t ◦B IJ t ◦εIJ + 1

2

∧

t

◦U = t ◦U( t ◦εIJ) , (5.6a)

∧

t

◦

C IJKL t ◦εIJ t ◦ε KL , (5.6b)

where t t

◦A◦ ; ◦BIJ and t ◦C

IJKL are constants (independent of the deformation).

5.2 Constitutive relations in solid mechanics: purely mechanical formulations 123

• The value of t ◦A◦ is arbitrary since we are only interested in the derivatives

t of ◦U , therefore we set

t

◦A◦ = 0 . (5.6c)

• Using Eq. (5.1f) with Eq. (5.6b) we get,

t

◦S IJ = t ◦B IJ + 1

2

µ

∧

t◦

C IJKL + t ∧

◦C

KLIJ

t◦εKL

. (5.6d)

Since we are not considering an initial stress/strain state,

t

◦εIJ = 0 ⇐⇒ t ◦SIJ = 0 and therefore, we must have

t

◦B IJ = 0 . (5.6e)

Hence, for the simplest hyperelastic material model, we have the elastic

energy per unit volume of the reference configuration expressed as a quadratic

form of the Green-Lagrange strain tensor;

t

◦U = 1 t

◦

2

and a linear stress/strain relation,

where (Malvern 1969),

∧

C IJKL t ◦εIJ t ◦εKL

(5.7a)

t

◦S IJ = t ◦C IJKL t ◦εKL (5.7b)

t

◦C IJKL = 1

2

C IJKL + t ∧

◦C

KLIJ

. (5.7c)

µ

∧

t◦

Doing a push-forward of the above equation to the spatial configuration

we get,

t τ ij = t c ijkl t ekl (5.8a)

where the spatial elasticity tensor is defined by

t ijkl t

c = ◦C IJKL t ◦X i I t ◦X j t

J ◦X k K t ◦X l L . (5.8b)

Note that the obtained spatial elasticity tensor has components that are a

function of the deformation (not constant).

The Second Law of Thermodynamics indicates that for deforming a real

(stable) material we must spend an amount of work; hence,

t

◦U ≥ 0 (5.9)

and we can only have t ◦U = 0 when t ◦ε = 0 .

Therefore, t ◦U is a positive definite quadratic form.

Using Eqs. (5.7b) and (5.8a) together with the quotient rule (Sect. A.5) we

realize that t ◦C IJKL and t c ijkl are components of two fourth-order tensors,

t

◦C and t c .

124 **Nonlinear** continua

I Symmetries of t ◦C

In Eq. (5.7b) t ◦S IJ and t ◦ε IJ are components of symmetric tensors;

hence,

t

◦C IJKL = t ◦C JIKL

(5.10a)

t

◦C IJKL = t ◦C IJLK

. (5.10b)

Therefore the original 81 components of t ◦C are reduced to only 36 independent

ones.

Using for this case the result in Example 5.4 we arrive at,

t

◦C IJKL = t ◦C KLIJ . (5.11)

Therefore, we are left with only 21 independent constants.

It is important to note that while Eqs. (5.10a-5.10b) apply to any material,

Eq. (5.11) is only valid for a hyperelastic material.

Without introducing any symmetry inherent to a particular

material model, for the description of the most general linear

hyperelastic material model we have to use 21 material

constants.

Now we will consider materials with inherent symmetries, that is to say,

with planes of elastic symmetry. Letusfirst consider a material with one

plane of elastic symmetry; this means that if we define two coordinate systems

{ ◦ x I } and { ◦ ˜x I } with:

• ◦ x 1 and ◦ x 2 on the symmetry plane,

• ◦ x 3 normal to the symmetry plane,

• ◦ ˜x 1 = ◦ x 1 ; ◦ ˜x 2 = ◦ x 2 and ◦ ˜x 3 = − ◦ x 3 ,

we must have

t

◦C IJKL = t ◦ ˜ C IJKL

. (5.12)

Taking into account the tensor transformations

t

◦ ˜ C IJKL = t ◦C PQRS ∂◦ ˜x I

∂◦xP ∂ ◦ ˜x J

∂ ◦ x Q

∂ ◦ ˜x K

∂ ◦ x R

∂ ◦ ˜x L

∂ ◦ x S

(5.13)

it is obvious that the fulfillment of Eq. (5.12) imposes that the components

with an odd quantity of indices “3” have to be zero. Hence, a plane of elastic

symmetry reduces the material constants from 21 to 13.

Now we consider a further simplification imposed by the consideration of

more elastic symmetries: the orthotropic hyperelastic material model. In this

case we have to consider three mutually orthogonal planes of elastic symmetry.

The intersection of the three orthotropy planes determines a Cartesian

coordinate system { ◦ z α }.

5.2 Constitutive relations in solid mechanics: purely mechanical formulations 125

Besides the symmetry considerations with respect to the ( ◦z1 , ◦z2) plane, derived above, we have to consider a symmetry with respect to the

( ◦z2 , ◦z3) plane (Sokolnikoff 1956).

The fulfillment of Eqs. (5.12) and (5.13) imposes, for this second symmetry,

a further 4 zero material constants; hence, an orthotropic material model has

only 9 material constants.

Examples of materials that are adequately described using orthotropic

material models are wood, rolled steel plates, etc.

It is interesting to examine further the orthotropic constitutive relation

using the following arrays:

⎡ ⎤ ⎡

⎤ ⎡ ⎤

t

⎢◦S

⎢

⎣

11

t

◦S22 t

◦S33 t

◦S12 t

◦S23 ⎥ ⎢

⎥ ⎢

⎥ ⎢

⎥ ⎢

⎥ ⎢

⎥ ⎢

⎥ ⎢

⎥ = ⎢

⎥ ⎢

⎥ ⎢

⎥ ⎢

⎥ ⎢

⎥ ⎢

⎦ ⎣

sym

t

◦S31 t

◦C 1111 t ◦C 1122 t ◦C 1133 0 0 0

t

◦C 2222 t ◦C 2233 0 0 0

t

◦C 3333 0 0 0

t

◦C 1212 0 0

t

◦C 2323 0

t ◦C 3131

⎥ ⎢

⎥ ⎢

⎥ ⎢

⎥ ⎢

⎥ ⎢

⎥ ⎢

⎥ ⎢

⎥ ⎢

⎥ ⎢

⎥ ⎢

⎥ ⎢

⎥ ⎢

⎥ ⎢

⎥ ⎢

⎦ ⎣

t

◦ε11

t

◦ε22

t

◦ε33

2 t ◦ε12

2 t ◦ε23

2 t ◦ε31

⎥ . (5.14)

⎥

⎦

It is obvious from the above equation that if the orthotropy axes are coincident

with the principal axes of strain, they are also the principal axes of

the stress tensor (collinearity between the stress and strain tensor).

When for an orthotropic material the constitutive relation is written in a

coordinate system that is not coincident with the material orthotropy system,

the convenient form of Eq. (5.14) is lost.

I The isotropic hyperelastic material model

An important case of materials with inherent symmetry is analyzed in this

section: when every plane is a plane of elastic symmetry we have an isotropic

material; therefore t ◦C is an isotropic fourth-order tensor (Aris 1962).

It was shown in (Aris 1962) that the most general fourth-order isotropic

tensor has Cartesian components of the form,

t

◦Cαβγδ = λδαβ δγδ + µ ( δαγ δβδ + δαδ δβγ )+τ ( δαγ δβδ − δαδ δβγ ) .

(5.15)

Any of the two Eqs. (5.10a-5.10b), derived from the symmetry of the stress

and strain tensors, imposes the condition:

τ = 0 . (5.16)

Hence, it is obvious from Eqs. (5.15) and (5.16) that in the case of isotropic

materials there are only 2 independent constants; usually (Malvern 1969):

126 **Nonlinear** continua

• E : Young’s modulus

• ν : Poisson’s ratio.

For any Cartesian system we can write Eq. (5.14) as, (Bathe 1996):

⎡

t

⎢◦S

⎢

⎣

11

t

◦S22 t

◦S33 t

◦S12 t

◦S23 t

◦S31 ⎤

⎥ =

⎥

⎦

E (1−ν)

⎡

⎤ ⎡

⎢

(1+ν) (1−2ν) ⎢

⎣

1 ν

1−ν

1

ν

1−ν

ν

1−ν

1

0

0

0

1−2ν

2(1−ν)

0

0

0

0

1−2ν

2(1−ν)

0

0

0

0

0

t

⎥ ⎢ ◦ε11

⎥ ⎢

⎥ ⎢ t

⎥ ⎢ ◦ε22

⎥ ⎢

⎥ ⎢

⎥ ⎢ t

⎥ ⎢ ◦ε33

⎥ ⎢

⎥ ⎢

⎥ ⎢2

⎥ ⎢

⎥ ⎢

⎥ ⎢

⎦ ⎣

sym

1−2ν

2(1−ν)

t ◦ε12

2 t ◦ε23

2 t ⎤

⎥ ,

⎥

⎦

◦ε31

(5.17)

and the stress and strain tensors are always collinear.

It is important to realize that the numerical values of E and ν

usually found in the engineering literature refer to the relation [Cauchy

stresses/infinitesimal strains] rather than to the relation [second Piola-Kirchhoff

stresses/Green-Lagrange strains] that we use in this section.

We can define K , the volumetric modulus and G , the shear modulus

as:

K =

G =

E

3(1− 2ν)

E

2(1+ν)

(5.18a)

(5.18b)

and write for the simplest hyperelastic material model that we consider in this

section,

t

◦Sαβ = K t ◦εV δαβ + 2 G t ◦ε 0

αβ

(5.19a)

t

◦U = 1

2 K t 2

◦εV

t

+ G ◦ε 0 αβ t ◦ε 0 αβ (5.19b)

where, t ◦εV is the volumetric or hydrostatic component or the Green-Lagrange

t strain tensor, ◦εV = t ◦εαβ δαβ ,and t ◦ε0 αβ ] is the deviatoric component of

t the Green-Lagrange strain tensor, ◦ε0 αβ = t ◦εαβ − 1

3 δαβ t ◦εV .

From Eq. (5.19a-5.19b) we conclude that:

K ≥ 0 (5.20a)

G ≥ 0 . (5.20b)

5.2 Constitutive relations in solid mechanics: purely mechanical formulations 127

It is important to realize that t ◦εV is not a measure of the continuum

volume change.

The above implies,

E ≥ 0 (5.20c)

− 1 ≤ ν ≤ 0.5 . (5.20d)

We must remember that the symmetries discussed in the last two Sections

are directional properties and not positional properties. Even if a material

has a certain elastic symmetry at each point, the properties may vary from

point to point in a manner not possessing any symmetry with respect to the

shape of the analyzed body (Malvern 1969).

When considering an isotropic material t ◦S and t ◦ε are always collinear

tensors.

Example 5.5. JJJJJ

In the above equations we expressed the constitutive tensor for orthotropic

and isotropic material in Cartesian coordinates.

To express it in a general coordinate system { ◦ x I } we use:

t

◦ ˜ C IJKL = t ◦C αβγδ ¡ eα · ◦ g I¢¡ eβ · ◦ g J¢¡ eγ · ◦ g K¢¡ eδ · ◦ g L¢ .

Even though the material is still defined by the same number of material

constants, the convenient form of Eqs. (5.14) and (5.17) is lost. JJJJJ

Example 5.6. JJJJJ

A hyperelastic constitutive model cannot be formulated, in the spatial configuration,

as

t ij t ijkl t

σ = c ekl

(A)

where tc is a constant and isotropic fourth order tensor

If Eq. (A) is an acceptable constitutive relation, then taking Lie derivatives

on both sides of the equal sign, we get for a hyperelastic material,

t ◦ σ ij

= t c ijkl t dkl .

In Example 5.14, we will show that the above equation cannot represent a

hyperelastic material; hence, Eq. (A) cannot either.

It is important to point out that in (Simo & Pister 1984) it was also demonstrated

that the spatial elasticity tensor cannot be an isotropic and constant

fourth-order tensor. Simo and Pister developed their demonstration without

the need of going through an equivalent hypoelastic material; however, we

could not reproduce the demonstration in (Simo & Pister 1984). JJJJJ

128 **Nonlinear** continua

5.2.3 Other simple hyperelastic material models

In Sect. 5.2.1 we have formulated a simple hyperelastic material model using:

t

◦U = 1 t

2

◦ε :

t

◦C :

t

◦ε (5.21a)

t

◦S = ∂t◦U ∂t . (5.21b)

◦ε

We can use other conjugate stress/strain rate measures and get, using for

the elastic energy per unit volume in the reference configuration quadratic

forms with constant coefficients:

t

◦U = 1 t

2

◦H :

t Γ = ∂ t ◦U

∂ t ◦H

t

◦C : t ◦H (5.22a)

. (5.22b)

As we saw in Sect. 3.3.4, Eqs. (5.22a-5.22b) can only be used with isotropic

materials.

Even considering the above limitation, Eqs. (5.22a-5.22b) are very useful

for formulating practical constitutive relations because it was experimentally

shown by Anand that for metals, the values of E and ν that are used in

the usual engineering environment [Cauchy stresses/infinitesimal strains] can

be accurately used in Eqs. (5.22a-5.22b) for quite large strain values (Anand

1979).

Also, for the case of infinitesimal strains,

t 1 t t

◦U = ε : ◦C :

2

t ε , (5.23a)

t

σ =

∂t ◦U ∂ t ,

ε

(5.23b)

t

τ =

µ ◦ρ

tρ t ∂◦U ∂ t ,

ε

(5.23c)

where t ε is the infinitesimal strain tensor defined in Example 2.26. In this

case since t ρ ≈ ◦ ρ , t σ ≈ t τ .

The above equations are possible because

D t ε

Dt = t d . (5.24)

As we have seen above, in Eq. (5.5), it is not possible to use an alternative

hyperelastic formulation in terms of the Almansi strain tensor.

5.2 Constitutive relations in solid mechanics: purely mechanical formulations 129

5.2.4 Ogden hyperelastic material models

For an isotropic material we can write,

t

◦U = t ◦U(λ1,λ2,λ3) (5.25)

where the λi are the eigenvalues of the second order tensor, t ◦U, thatistosay,

the principal stretches defined in Eq. (2.58e).

Since

λi = λi(I C 1 ,I C 2 ,I C 3 ) (5.26)

where the values (IC 1 ,IC 2 ,IC 3 ) are the invariants of t ◦C defined in Eqs. (2.59b-

2.59d); we can write

t

◦U = t ◦U(I C 1 ,I C 2 ,I C 3 ) (5.27a)

I C 1 =(λ1) 2 +(λ2) 2 +(λ3) 2

(5.27b)

I C 2 =(λ2) 2 (λ3) 2 +(λ3) 2 (λ1) 2 +(λ1) 2 (λ2) 2

(5.27c)

I C 3 =(λ1) 2 (λ2) 2 (λ3) 2 . (5.27d)

Ogdenproposedtowrite t ◦U(I C 1 ,IC 2 ,IC ¡ 3 ) as an infinite series in powers of

C I1 − 3 ¢ , ¡ IC 2 − 3 ¢ , ¡ IC 3 − 1 ¢ (Ogden 1984).

Thus,

t

◦U(λ1,λ2,λ3) =

∞X

p,q,r=0

¡ C

Cpqr I1 − 3 ¢p ¡ C

I2 − 3 ¢q ¡ C

I3 − 1 ¢r

where the coefficients Cpqr are independent of the deformation.

For the unstrained configuration,

λ1 = λ2 = λ3 =1

I C 1 =3

I C 2 =3

I C 3 =1.

(5.28)

Therefore, in the unstrained configuration the strain energy per unit mass is

zero provided that C000 =0.

Also, the unstrained configuration has to be stress free; hence,

∂ t ◦U

| I C 1 =I C 2 =3;IC 3 =1 =0 for i =1, 2, 3 . (5.29a)

∂λi

The above equation leads using Eq. (5.28), to the condition,

C100 +2C010 + C001 =0. (5.29b)

130 **Nonlinear** continua

Example 5.7. JJJJJ

The third invariant, I C 3 , has an important physical interpretation. Considering

the eigendirections of t ◦U ,adifferential volume ◦ dV defined in the

reference configuration along those directions is transformed, in the spatial

configuration, in a differential volume t dV :

hence,

t dV = λ1λ2λ3 ◦ dV

t J =

t q

dV

= I ◦dV C 3 .

That is to say, the invariant IC 3 describes the volume change during the

continuum deformation.

Obviously, for incompressible deformations,

t J = I C 3 =1.

JJJJJ

Here Ogden introduces a simplificative hypothesis: the strain energy is

decoupled in two parts, a deviatoric part that is independent of the volume

change and a volumetric part that is only a function of the volume change

(Ogden 1984). The above hypothesis is introduced in Eq. (5.28) by imposing,

We can state,

t

◦U(λ1,λ2,λ3) =

+

∞X

p,q=0

∞X

r=1

g(I C 3 )=

¡ C

Cpq0 I1 − 3 ¢p ¡ C

I2 − 3 ¢q

¡ C

C00r I3 − 1 ¢r

. (5.30)

∞X

r=1

¡ C

C00r I3 − 1 ¢r

(5.31)

and retain only a few terms in Eq. (5.30); in this way we obtain specific

hyperelastic relations that have been successfully used for particular materials.

For example, for incompressible materials:

I C 3 =1 (5.32a)

g(I C 3 )=0 (5.32b)

t

◦U =

∞X ¡ C

Cpq0 I1 − 3 ¢p ¡ C

I2 − 3 ¢q

. (5.32c)

p,q=0

5.2 Constitutive relations in solid mechanics: purely mechanical formulations 131

• Retaining only two terms in Eq. (5.32c) we obtain the Mooney-Rivlin strain

energy function that has been successfully used for rubber-like materials,

¡ t

C

◦U = C100 I1 − 3 ¢ + C010

¡ C

I2 − 3 ¢ . (5.33)

• If C010 =0the above reduces to the neo-Hookean strain energy function

which has played an important role in the development of the theory and

applications of nonlinear elasticity,

¡ t

C

◦U = C100 I1 − 3 ¢ . (5.34)

The principal values of the collinear Green-Lagrange and second Piola-

Kirchhoff tensors are related by,

t

◦Si = ∂ t ◦U

∂εi

= ∂ t ◦U ∂λj

∂λj ∂εi

(5.35a)

taking into account that the εi are the eigenvalues of the Green-Lagrange

strain tensor, we can write

∂λj

=

∂εi

1

δji (no addition on j) . (5.35b)

λj

Hence,

t 1 ∂

◦Si =

λi

t ◦U

(no addition on i) . (5.35c)

∂λi

With a push-forward of the above, we get the eigenvalues of the Kirchhoff

stress tensor,

t ∂

τ i = λi

t ◦U

(no addition on i) . (5.35d)

∂λi

Hence, the eigenvalues of the Cauchy stress tensor are,

µ tρ

t ∂

σi = λi ◦ρ t ◦U

(no addition on i) . (5.35e)

∂λi

When the material is incompressible the three values of λi are not independent

since they have to fulfill the relation,

t J − 1=λ1λ2λ3 − 1=0. (5.36a)

In this case, Eq. (5.35a) determines the eigenvalues of the Cauchy stress

tensor except for the value of the hydrostatic pressure, 4P :

t ∂

σi = λi

t ◦U

+ 4P (no addition on i) . (5.36b)

∂λi

For determining the value of the hydrostatic pressure the equilibrium equations

shall be used.

132 **Nonlinear** continua

Example 5.8. JJJJJ

In the case of the biaxial tension of a square incompressible sheet (Ogden

1984),

therefore,

Also,

λ1λ2λ3 =1

λ1 = λ2 = ˆ λ

λ3 = 1

( ˆ .

λ) 2

t σ1 = t σ2 = t ˆσ

t σ3 =0.

Using the constitutive equation of the incompressible neo-Hookean material

we get,

t

◦U = C100(λ 2 1 + λ 2 2 + λ 2 3 − 3) .

Equations (5.36b) give,

Hence,

t ˆσ =2C100( ˆ λ) 2 + 4p

1

0=2C100

( ˆ + 4p .

λ) 4

t ˆσ =2C100

"

( ˆ λ) 2 − 1

( ˆ λ) 4

#

.

In plane Cauchy stresses in the biaxial tension of a square incompressible

sheet

JJJJJ

5.2 Constitutive relations in solid mechanics: purely mechanical formulations 133

Example 5.9. JJJJJ

For the case of an incompressible cube subjected to a uniform tension on its

faces, in (Marsden & Hughes 1983) a neo-Hookean material is considered with

a strain energy function of the form,

t

◦U = C100

£ λ 2 1 + λ 2 2 + λ 2 3 − 3 ¤ .

Incompressible cube subjected to a uniform tension on its faces

Using Eqs. (5.36b), we get

t σi =2C100(λi) 2 + 4p (i =1, 2, 3) .

From the equilibrium between internal and external forces, and taking into

account that due to incompressibility

we get,

where t = |t| .

λ1λ2λ3 =1

2C100(λ1) 2 + 4p = tλ1 ,

2C100(λ2) 2 + 4p = tλ2 ,

2C100(λ3) 2 + 4p = tλ3 ,

134 **Nonlinear** continua

Eliminating 4p gives,

In the intuitive case

[2C100 (λ1 + λ2) − t](λ1 − λ2) =0

[2C100 (λ2 + λ3) − t](λ2 − λ3) =0

[2C100 (λ3 + λ1) − t](λ3 − λ1) =0.

λ1 = λ2 = λ3

and the above equations are automatically fulfilled.

However, we will explore the possibility of other solutions.

• If we assume

λ1 6= λ2 6= λ3

we must have

t =2C100 (λ1 + λ2) =2C100 (λ2 + λ3) =2C100 (λ3 + λ1) .

Hence, we must have λ1 = λ2 = λ3. This contradiction shows that the assumed

solution is not possible.

• If we assume that two λis are equal and the third one is different, for

example:

λ1 = λ2 6= λ3 ,

we get

t =2C100 (λ1 + λ3)

and due to incompressibility we must have,

" #

− t =0.

2C100

λ1 + 1

(λ1) 2

In order to obtain λ1 = λ1(t) we must solve the equation

f(λ) =λ 3 1 − t

λ

2C100

2 1 +1=0.

Considering that only the positive roots are admisible, it is shown in (Marsden

& Hughes 1983) that:

¤ If t 0.

¤ If t =3 3√ 2 C100 =⇒ one root λ1 > 0.

¤ If t>3 3√ 2 C100 =⇒ two positive roots λ1 > 0.

Hence, t =3 3√ 2 C100 is a bifurcation load. JJJJJ

5.2 Constitutive relations in solid mechanics: purely mechanical formulations 135

5.2.5 Elastoplastic material model under infinitesimal strains

In this section we are going to present the elastoplastic material model, considering

only a purely mechanical formulation and infinitesimal strains.

In forthcoming sections we are going to present a thermoelastoplastic material

model and we are going to extend the model to consider finite strains.

The elastoplastic material model has to describe the following experimentally

observed phenomena:

• For loads below a certain limit loading condition, established via a yield

criterion, the material behavior is elastic and can be described using the

hyperelastic relations that we have discussed above.

• When the limit loading condition is reached there is an onset of permanent

or plastic deformations.

• The plastic deformations produce an evolution of the yield condition that

is described via a hardening law.

• When the limit loading condition is reached and then an unloading takes

place, elastic deformations are developed.

• The material behavior is rate independent, that is to say, the material

behavior is not a function of the loading or deformation rate.

• The material is stable: we have to spend mechanical work in order to

deform it.

I 1D case

The above observations easily fit into our experience of the 1D load/displacement

curve of a steel sample under tension.

In the 1D case it is obvious that (see Fig. 5.1):

Total 4L = Elastic 4L + Plastic 4L . (5.37a)

For the case of infinitesimal strains, dividing the above by ◦ L, we have

t ε = t ε E + t ε P

(5.37b)

where t ε : total axial strain in the sample, t ε E : elastic axial strain in the

sample and t ε P : plastic axial strain in the sample.

During the plastic loading we can also write, following Fig. 5.2, that

4ε = 4ε E + 4ε P . (5.37c)

Dividing Eq. (5.37c) by 4t and taking the limit for 4t → 0 we get for

thesimple1Dcase,

t ˙ε = t ˙ε E + t ˙ε P . (5.37d)

For the 3D case we generalize the above equation using Eq. (5.24) and get

the additive decomposition of the strain-rate tensor,

136 **Nonlinear** continua

Fig. 5.1. Tensile test of a steel sample

Fig. 5.2. Plastic loading (zoom)

t d = t d E + t d P . (5.38)

It is important to note that, for the moment, we are postulating the above

decomposition only for the case of infinitesimal strains, a very important one

for standard engineering applications.

In the 1D mechanical test described in Fig. 5.1 it is quite obvious that the

loading condition is,

and when loading:

if t P< ◦ Pyield

t ˙

P>0 , (5.39a)

5.2 Constitutive relations in solid mechanics: purely mechanical formulations 137

if t P > ◦ Pyield

4L E = 4P

AE

◦

L (5.39b)

4L P =0, (5.39c)

4L E = 4P

AE

◦

L (5.39d)

4L P > 0 , (5.39e)

where A is the sample transversal area, that due to the assumption of infinitesimal

strains we consider as being constant, and E is the steel Young’s

modulus.

As is shown in Fig. 5.1 when we reach a load t Pyield and unload

t ˙

P 0 and

then start loading again:

t

P>0 ˙

(5.41a)

the behavior will be elastic (Eqs. (5.39b) and (5.39c)) for

t P< t Pyield

and elastoplastic (Eqs. (5.39d) and (5.39e)) for

(5.41b)

t P > t Pyield . (5.41c)

Let us divide in Fig. 5.3 the P − values by A (assumed constant during

the deformation) and the 4L − values by ◦ L . We get the diagram shown in

Fig. 5.3 (Cauchy stress vs. infinitesimal strain).

For a material having a softening behavior, rather than a hardening behavior,

we would have the σ − ε diagram shown in Fig. 5.4 . It is evident in

this figure that when we go from point “1” to point “2” :

138 **Nonlinear** continua

and therefore

Fig. 5.3. σ-ε for a steel sample

Fig. 5.4. Softening in the σ − ε relation

4σ < 0 (5.42a)

4ε P > 0 (5.42b)

4σ 4ε P < 0 . (5.42c)

We will now show that the above stress/strain relation is incompatible

with the requirement of material stability.

5.2 Constitutive relations in solid mechanics: purely mechanical formulations 139

Let us now assume that the 1D sample is in equilibrium at “0” andan

external agent makes it describe the loading-unloading path 0 − 1 − 2 − 3,

described in Fig. 5.4.

The work performed by the external agent per unit volume of the sample

is,

Z 1

We.a. =

0

Z 1

−

0

σ ˙ε dt +

σ◦ ˙ε dt −

For a stable material it must be

Z 2

1

Z 2

1

σ ˙ε dt +

σ◦ ˙ε dt −

Z 3

2

Z 3

σ ˙ε dt

2

σ◦ ˙ε dt . (5.43)

We.a. > 0 . (5.44)

Taking into account that in the paths 0 − 1 and 2 − 3 weareinsidethe

elastic range, we can write

Z 1

We.a. =

0

Z 3

2

(σ − σ◦) ˙ε E dt +

(σ − σ◦) ˙ε E dt +

Z 2

1

Z 2

1

(σ − σ◦) ˙ε E dt +

(σ − σ◦) ˙ε P dt . (5.45)

The first three integrals on the r.h.s. of the above equation add up to

zero because they correspond to a loading-unloading elastic cycle and there is

neither energy dissipated nor generated in that cycle.

Z 2

We.a. =

1

When “1” and“2” areinfinitesimaly close,

(σ − σ◦) ˙ε P dt . (5.46)

d 2 We.a. =dσ dε P 6 0 . (5.47)

The points on the softening branch are in unstable equilibrium.

The material shown in Fig. 5.4, with a softening behavior

is not stable.

However, it is possible to obtain diagrams P − 4 with a softening region;

for example:

• In the 1D test of a steel sample after necking (localization). In this case, the

hypothesis of constant area is not valid and using at every configuration

the actual area for constructing the diagram in Fig. 5.4 (true stress), we

get a hardening behavior rather than a softening one.

140 **Nonlinear** continua

• In the 1D test of a concrete sample. During the descending branch of the

P − ∆ curve there are micro-cracks propagating inside the sample; hence,

again the consideration of a uniform stress across the transversal section

is not realistic (Ottosen 1986).

In what follows, we will generalize the intuitive concepts of this 1D example

to a general 3D formulation.

I The general formulation

For the general formulation of an elastoplastic material model we need the

following three ingredients:

• A yield surface that in the 3D stress space describes the locus of the points

where the plastic behavior is initiated.

• A flow rule that describes the evolution of the plastic deformations.

• A hardening law that describes the evolution of the yield surface during

the plastic deformation process.

The yield surface

In the stress space, for a given material we can define a yield surface,

t f( t σ , t qi i =1,n )=0 (5.48)

where t σ is the Cauchy stress tensor and t qi are internal variables to be defined

for every particular yield criterion.

The elastic range is described by the inequality

and the plastic range by the equality

t f0 is not possible in the elastoplastic framework.

Many different yield functions have been proposed over the years to

phenomenologically describe the behavior of different materials (Chen 1982,

Lubliner 1990). In this section, we concentrate on the yield function that is

generally used to describe the behavior of metals: the von Mises yield function.

In his experimental work, developed in the 1950s, Bridgman found that for

metals, it can be assumed that the yield function is not affected by the confining

hydrostatic pressure - at least for not very extreme hydrostatic pressures

(Hill 1950 and Johnson & Mellor 1973).

It is important at this point to introduce the following decomposition of

the Cauchy stress tensor:

5.2 Constitutive relations in solid mechanics: purely mechanical formulations 141

t t 1

³

σ = s +

tσ

:

3

´

t

g

tg

(5.50a)

which in Cartesian coordinates is written as,

t

σαβ = t sαβ + 1 ¡ tσγϕ

3

t ¢ tδαβ

δγϕ . (5.50b)

In the above equation ts is the deviatoric Cauchy stress tensor and

h ³ ´i

1 tσ t

3 : g is the hydrostatic or spherical component of the Cauchy stress

tensor.

Using the above decomposition together with Bridgman experimental observations

for the case of metals, we can write Eq. (5.48) as,

t f( t s , t qi i =1,n )=0. (5.51)

Taking into account that the first invariant or trace of the deviatoric

Cauchy stress tensor is,

t t

s : g =0 (5.52)

we can write Eq. (5.51) for an isotropic material whose behavior is symmetric

in the stress space as,

t f ¡ tJ2 , t J3 , t qi i =1,n ¢ =0 (5.53)

where t J2 and t J3 are the second and third invariant, respectively, of the

deviatoric Cauchy stress tensor.

By isotropic material we mean that the yield condition does not distinguish

orientations predefined in the material.

Experimental results indicate that when the yield surface is intersected in

the stress space with a plane that forms equal angles with the three principal

stress axes, a good approximation for the obtained curve is a circle.

Therefore, von Mises proposed

¡ t tJ2

f ,

t

qi i =1,n ¢ =0. (5.54)

Hence, the von Mises yield function is also known as the t J2 -yield function

(Simo & Hughes 1998).

More specifically the actual form of Eq. (5.54) can be written as,

or,

t 1 ¡

f =

ts ¢ ¡ t

− α :

ts ¢ t (

− α −

2

tσy) 2

=0 (5.55)

3

∙

t 3 ¡

f =

ts ¢ ¡ t

− α :

ts ¢ t

− α

2

¸1/2 − t σy =0. (5.56)

In the above we have introduced the following internal variables:

142 **Nonlinear** continua

Fig. 5.5. Von Mises yield surface

t σy : uniaxial yield stress at the t-configuration; that is to say corresponding

to a given plastic deformation. The evolution of t σy is going to be described

by the hardening law.

t α : back-stress tensor at the t-configuration. In many metals subjected to

cyclic loading it is experimentally observed that the center of the yield

surface experiences a motion in the direction of the plastic flow; the backstress

tensor describes this behavior (Mc Clintock & Argon 1966). The

evolution of t α is going to be described by the hardening law. We will

show that t α is a traceless tensor.

In Fig. 5.5 we represent von Mises yield surface in the space of princital

(over-)stresses.

The von Mises yield surface in the stress space is a circular cylinder whose

axis forms equal angles with each of the coordinates axes.

When, later, we discuss Drucker’s definition of stable materials, we will

show that for any stable material whose behavior is modeled using any yield

surface, at every time during the yield surface evolution, this surface has to

remain as a convex surface in the stress space.

The flow rule

As we stated at the beginning of the section, now we are focusing on the case of

infinitesimal strains; hence, the starting point for describing the deformation

of an elastoplastic solid during plastic loading is Eq. (5.38) which decomposes

the material flow into the addition of an elastic flow plus a plastic flow.

5.2 Constitutive relations in solid mechanics: purely mechanical formulations 143

During the material plastic flow there is a plastic dissipation per unit

volume of mechanical energy given by:

t t ij t P

D = σ dij (5.57)

(Hill 1950, Lubliner 1990, Simo & Hughes 1998).

t P t If d = 0 (no plastic loading) D =0, otherwise the Second Law of

Thermodynamics imposes that tD > 0.

For many materials, such as metals, the plastic flow is developed so as to

maximize the plastic dissipation.

In mathematical form we can say that for defining the plastic loading we

seek for the maximum of tD under the constraint

We define (Luenberger 1984),

t f =0. (5.58a)

t D = t D − t ˙ λ t f (5.58b)

where t ˙ λ is a Lagrange multiplier used to enforce the constraint in Eq. (5.58a).

The conditions for t D to attain a maximum under the constraint in Eq.

(5.58a) are,

∂tD ∂t =0

σij (5.59a)

∂tD ∂t ˙ =0.

λ

The first of the above equations leads to

(5.59b)

t P

dij = t λ˙ ∂tf ∂tσij and the second one to Eq. (5.58a).

Note that

t f0 (plastic loading) .

(5.60)

The above equations can be summarized using the Kuhn-Tucker conditions

for constrained optimization (Luenberger 1984),

t ˙ λ t f =0, (5.61a)

t ˙ λ > 0 , (5.61b)

t f 6 0 . (5.61c)

The Principle of Maximum Plastic Dissipation states that, for a given

plastic strain rate t d P , among the possible stresses t σ satisfying the yield

144 **Nonlinear** continua

criterion, the plastic dissipation attains its maximum for the actual stress

tensor (Simo & Hughes 1998).

The flow described by Eqs. (5.60-5.61c) is called associative or associated

plastic flow and presents many important properties that we will study in

what follows.

A general or nonassociated plastic flow is described by

Yield function :

t f =0 (5.62a)

Flow rule :

t P

dij = t λ˙ ∂tg ∂tσij (5.62b)

where tg is called the plastic potential and is different from the yield function.

Example 5.10. JJJJJ

For a material model developed using the von Mises yield criterion in Eq.

(5.55), we can write in a Cartesian coordinate system:

∂ t f

∂ t σαβ

= ∂ tf ∂ t ∂

sγδ

tsγδ ∂ tσαβ Hence, in the case of associated plastic flow

= ¡ t

sαβ − t ¢

ααβ

t P

dαβ = t ¡

λ ˙ tsαβ

− t ¢

ααβ

Since t s and t α are traceless tensors (see Eqs. (5.78) and (5.52)), it is obvious

that,

t d P αα =0,

which is the condition of incompressible plastic flow (see Example 4.4).

The above is, of course, a direct consequence of the fact that due to Bridgman

experimental observations the yield function does not include the trace

of t σ. JJJJJ

Stable materials - Drucker’s postulate

The principle of maximum plastic dissipation (associated plasticity) implies

that the plastic flow develops in the direction of the yield surface external normal,

Eq. (5.60). For an arbitrary yield function we schematize this normality

rule in Fig.5.6. Note that in the case of infinitesimal strains, it was shown in

Eq. (5.24) that t d = t . ε ; which is the nomenclature used in Fig. 5.6.

Let us assume an elasticplastic material that is initially at a stress state

t σ , inside the elastic range. Let us also assume that an “external agent”

(one that is independent of whatever has produced the current loads) slowly

.

.

5.2 Constitutive relations in solid mechanics: purely mechanical formulations 145

applies an incremental load resulting in a stress/strain increment and then

slowly removes it. For a stable material, the work performed by the external

agent in the course of the cycle consisting of the application and removal of

the external load is non-negative (Lubliner 1990).

Fig. 5.6. Normality rule in associated plasticity (maximum plastic dissipation)

The above is Drucker’s definition of a stable material, also known as

Drucker’s postulate.

In order to analyze the consequences of the above definition let us schematize

the above defined cycle in Fig. 5.7.

The total work per unit volume performed during the cycle 0-1-2-0 is,

Z 1

WTOTAL =

0

σ ij dεij +

Z 2

1

σ ij dεij +

Z 0

2

σ ij dεij

hence, the work performed by the external agent is,

Z 1

We.a =

¡ ij t ij

σ − σ ¢ dεij +

Z 2 ¡ ij t ij

σ − σ ¢ dεij

+

0

Z 0

2

1

(5.63a)

¡ σ ij − t σ ij ¢ dεij . (5.63b)

For the small strains case, using Eq. (5.38) we obtain,

146 **Nonlinear** continua

Fig. 5.7. Drucker’s postulate (work-hardening material)

Taking into account that

we get,

For a stable material

Trajectory dεij

Z 2

We.a =

1

0-1 dε E ij

1-2 dε E ij +dεP ij

2-0 dε E ij

I

σ ij dε E ij =0 (5.63c)

I

tσij E

dεij =0 (5.63d)

¡ σ ij − t σ ij ¢ dε P ij . (5.63e)

We.a ≥ 0 . (5.63f)

The above requirement implies that the integrand in Eq. (5.63e) has to be

non-negative, that is to say

5.2 Constitutive relations in solid mechanics: purely mechanical formulations 147

Fig. 5.8. Convexity of the yield surface as a consequence of Drucker’s postulate

¡ σ ij − t σ ij ¢ dε P ij ≥ 0 . (5.63g)

We now specialize the above equation for the case in which point “2”is

inside an infinitesimal neighborhood of “1”; hence,

dσ ij dε P ij ≥ 0 . (5.63h)

The above constrain has already been derived for the 1D case, see Eq.

(5.47).

Equations (5.63g) and (5.63h) are two mathematical constraints that a

stable material has to fulfill.

We can rewrite Eq. (5.63h) as,

t ˙σ ij t d P ij ≥ 0 . (5.64)

The above equation indicates that the plastic strain rate cannot oppose

the stress rate (Lubliner 1990).

Note that:

• While t ˙σ ij t d P ij > 0 indicates a work-hardening material, t ˙σ ij t d P ij =0

indicates a perfectly plastic material.

• Drucker’s postulate excludes from the range of stable materials the possibility

of strain-softening materials. However, Drucker’s postulate has been

obtained in the environment of stress - space plasticity, i.e. in our plasticity

theory the stress is the independent variable. Since the 1960s strain - space

plasticity formulations have been proposed even though their application

has not been widespread yet (Lubliner 1990).

148 **Nonlinear** continua

As a consequence of Drucker’s postulate, we can show that for a stable

material the yield surface has to be a convex surface in the stress space.

It is obvious from the two cases schematized in Fig. 5.8 that the nonconvex

one fails to fulfill Eq. (5.63g). Therefore, in the environment of stress-space

plasticity, a stable material has to have a convex yield surface in the stress

space.

Example 5.11. JJJJJ

In this example we are going to show that a frictional material cannot be

modeled using an associated plasticity formulation (Baˇzant 1979).

Let us assume the simplest frictional material represented in the following

figure: two rigid plates slide one on top of the other, and the sliding surface

hasaCoulombfrictioncoefficient µ.

We formulate a yield function in the stress space using T and N (the modulus

of T and N) as independent variables,

t f = T − µN ,

when t f

5.2 Constitutive relations in solid mechanics: purely mechanical formulations 149

Yield function and plastic deformation predicted using an associated

plasticity formulation (simplest frictional material)

The hypothesis of associated plasticity produces a nonphysical plastic displacement

component in the N-direction (remember that the plates were assumed

to be rigid).

Therefore, to model frictional materials, it is necessary to use nonassociated

plasticity formulations (Baˇzant 1979, Vermeer & de Borst 1984, Dvorkin,

Cuitiño & Gioia 1989). JJJJJ

Stress - strain relation

Let us consider an elastoplastic material during the process of plastic loading,

to relate the stress and strain increments. For the small-strains case in a

Cartesian coordinate system Eq. (5.38) can be written as

therefore, for a linear elastic behavior,

dεαβ =dε E αβ +dε P αβ , (5.65a)

dσαβ = C E αβγδ dε E γδ . (5.65b)

The fourth-order tensor C E is the material elastic constitutive tensor, it

is constant for linear elastic behavior and it is a function of the total elastic

strains for nonlinear elastic behavior. In this book we are not going to discuss

the influence of the plastic deformations on the material elastic properties

(damage theory (Lamaitre & Chaboche 1990)).

From Eq. (5.60), for an associated plasticity formulation,

dε P αβ =dλ ∂ t f

∂ t σαβ

=dλ ∂ tf ∂ t ∂

sγδ

tsγδ ∂ tσαβ (5.66a)

150 **Nonlinear** continua

and using Eq. (5.55), the von Mises yield criterion,

∂ tf = ¡ t

sγδ − t ¢

αγδ , (5.66b)

also, from Eq. (5.50b),

∂ t sγδ

∂ t sγδ

∂ t σαβ

= δγαδδβ − 1

3 δαβδγδ . (5.66c)

Taking into account that ts and tα are traceless tensors,

dε P αβ =dλ ¡ t

sαβ − t ¢

ααβ .

Using the above together with Eq. (5.65a) in Eq. (5.65b) we get

(5.67a)

dσαβ = C E £

αβγδ dεγδ − dλ ¡ t

sγδ − t ¢¤

αγδ . (5.67b)

During plastic loading dλ >0 ; hence, using Eq. (5.61a), we get the consistency

condition

df =0 (5.68a)

which leads to,

∂tf ∂t dσαβ +

σαβ

∂tf ∂tεP dε

αβ

P αβ =0.

Using the above, we get

(5.68b)

¡ tsαβ

− t ¢ E

ααβ Cαβγδ £

dεγδ − dλ ¡ t

sγδ − t ¢¤

αγδ

+ ∂tf ∂tεP dλ

αβ

¡ t

sαβ − t ¢

ααβ =0. (5.68c)

Hence,

dλ =

( t sαβ − t ααβ) C E αβγδ dεγδ

( tsεζ − tαεζ) CE εζηϑ ( tsηϑ − tαηϑ) − ∂tf ∂tεP (

εζ

tsεζ − tαεζ) Replacing in Eq. (5.67b) dλ with the above-derived value, we get

"

dσαβ =

. (5.68d)

C E αβγδ− (5.69)

( tsνµ − tανµ) CE αβνµ CE ϕξγδ ( tsϕξ − tαϕξ) ( tsρπ − t ⎤

⎦ dεγδ .

αρπ)

( t sρπ − t αρπ) C E ρπητ ( t sητ − t αητ) − ∂t f

∂ t ε P ρπ

The term between brackets, t C EP αβγδ , represents the Cartesian components

of the continuum tangential elastoplastic constitutive tensor. Therefore,

dσ = t C EP : dε .

It is important to note that the following symmetries are present:

5.2 Constitutive relations in solid mechanics: purely mechanical formulations 151

• t C EP αβγδ = t C EP βαγδ

• t C EP αβγδ = t C EP αβδγ

• t C EP αβγδ = t C EP γδαβ

It is important to realize that the last symmetry is lost in the case of

non-associated plastic models (see Eq. ( 5.62b)) (Baˇzant 1979, Vermeer & de

Borst 1984).

The hardening law

Following Hill we can say that “The yield law for a given state of the metal

must depend, in some complicated way, on the whole of the previous process

of plastic deformation since the last annealing”(Hill 1950).

In order to solve the equations that describe the elastoplastic deformation

process, we have to describe the yield surface evolution during plastic

deformation, that is to say, we have to describe the material hardening.

The simplest hardening models are :

• Isotropic hardening model.

• Kinematic hardening model.

While the first one does not describe the Bauschinger effect (Hill 1950),

the second one was developed to model the basic features of this effect.

In Fig. 5.9 we present a schematic description of the Bauschinger effect.

For an initially isotropic hardening material, after loading in tension to T σy ,

when we unload, the yield stress is

• T σy when we reload in tension,

• C σy when we load in compression.

Isotropic hardening

We assume that in Eq. (5.55)

and

t α = 0 (5.70a)

t σy = t σy( t W P ) . (5.70b)

Hence, the yield surface remains centered and the yield stress is a function

of the irreversible work performed on the solid (work hardening).

In the above equation t W P is the plastic work per unit volume performed

on the material. We can state that,

t D = D t W P

Dt

. (5.70c)

The total work per unit volume that has to be spent to deform a solid from

its unstrained configuration to a configuration defined by t εγδ is, assuming

infinitesimal strains,

152 **Nonlinear** continua

Using Eq. (5.65a), we get

t W =

Z t ε E γδ

0

Fig. 5.9. Bauschinger effect

t W =

Z t εγδ

0

σαβ dε E αβ +

σαβ dεαβ . (5.71a)

Z t ε P γδ

0

σαβ dε P αβ . (5.71b)

Note that we have not yet defined the upper limits of the above integrals.

The first integral on the r.h.s. of Eq. (5.71b) is the elastic (reversible) work

per unit volume while the second integral is the plastic (irreversible) work per

unit volume.

For an isotropic hardening von Mises material during plastic loading

t σ = t σy( t W P ) , (5.72a)

where t q

3

σ : equivalent stress = 2 ts : ts .

The rate of plastic work per unit volume is, considering that the plastic

flow is incompressible:

t

W˙ P t

= sαβ t d P αβ . (5.72b)

We define, for the isotropic hardening von Mises material the equivalent

plastic strain rate,

r

t P 2

d =

3 td P : td P . (5.73a)

5.2 Constitutive relations in solid mechanics: purely mechanical formulations 153

Hence,

r

t t P ¡ ¢

σ d = ts : ts ³ t P

d : t P

d ´

. (5.73b)

Using Eq. (5.60) we can see that t d P and t s are collinear tensors when

isotropic von Mises plasticity is considered; hence, we can write Eq. (5.73b)

as,

t σ t d P = t s : t d P = t ˙ W P

(5.73c)

therefore t σ and t d P are energy conjugate.

We define the equivalent plastic strain as

Hence,

t ε P =

Z t ε P

Considering Eqs. (5.72a) and (5.73c)

dσy = ∂t σy

∂ t W P

0

t d P dt. (5.74a)

dW P = t σy( t W P )dε P . (5.74b)

t

σy dε P = ∂tσy ∂tW P

∂tW P

∂tεP dεP = ∂tσy ∂tεP dεP . (5.74c)

Therefore, we can construct a curve,

¡ tεP ¢ . (5.75)

t σy = t σy

“The assumption that one universal stress - strain curve of the form of

Eq. (5.75) governs all possible combined - stress loadings of a given material

is obviously a very strong one” (Malvern 1969).

Example 5.12. JJJJJ

In a uniaxial tensile test, before the necking is localized, with the I-axis the

loading direction and the II and III - axes orthogonal ones, we can write

Hence,

t σI = σ ∗

t σII =0 ,

t σIII =0 ,

Also, due to incompressibility

,

σ = σ ∗ .

t sI = 2

3 σ∗ ,

t sII = − 1

3 σ∗ ,

t sIII = − 1

3 σ∗ .

154 **Nonlinear** continua

t P

εI = ε ∗ P

t P

εII = − 1

2 ε∗P t P

εIII = − 1

2 ε∗P .

Hence,

t P ∗

ε = εP .

Therefore, for an isotropic hardening von Mises material, the complete universal

stress - strain curve is determined with only an uniaxial test. JJJJJ

Kinematic hardening

In this hardening model, which was developed to simulate the basic features

of the Bauschinger effect, we assume:

t σy = ◦ σy = const. (5.76)

Kinematic hardening represents a translation of the yield surface in the

stress space by shifting its center. This is in fairly good agreement with the

Bauschinger effect for those materials whose stress-strain curve in the workhardening

range can be approximated by a straight line (“linear hardening”)

(Lubliner 1990).

Prager, in his kinematic hardening model, assumes a linear hardening

(Malvern 1969):

t

˙αij = c t d P ij

(5.77)

where c is a constant.

Since tdP αα =0(incompressibility), the back-stress tensor tα is also traceless;

that is to say

t α :

t g = 0 . (5.78)

In the nine-dimensional stress space, the yield surface is displaced in the

direction of its external normal at the load point.

Example 5.13. JJJJJ

We consider the 1D case of monotonic loading, also considered in the previous

example.

Considering the yield function in Eq. (5.55) under the constraints in Eqs.

(5.76) and (5.77); and imposing that during loading

we get,

t ˙

f =0,

5.2 Constitutive relations in solid mechanics: purely mechanical formulations 155

c =

·

¯σ

·

3

2¯ε

Therefore, the hardening parameter, c, is constant for linear hardening materials.

JJJJJ

When using the kinematic hardening model, as also discussed for the

isotropic hardening model, the complete universal stress-strain curve is determined

with one axial test.

It is important to realize that the results for monotonic loading are exactly

coincident when considering either the isotropic or the kinematic hardening

models.

5.2.6 Elastoplastic material model under finite strains

I Hypoelastic models

Following what has been done for the case of elastoplastic materials under infinitesimal

strains, we start this Section by assuming a hypoelastic description

of the deformation process.

Referring the problem to the equilibrium configuration at time t, we may

again use

t t E t P

d = d + d . (5.79)

t As we have shown in Example 2.22, d is an objective strain rate;

therefore, considering Eq. (5.79), both td E and td P have to be considered

as objective strain rates too. Hence, we can extend the constitutive laws used

for infinitesimal strains and write

t ◦ σ = t c E : t d E

.

(5.80a)

t d P = ·

λ t s . (5.80b)

In the above equation t ◦ σ is an objective stress rate such as the Oldroyd

stress rate (see Sect. 3.4) and tcE is the spatial elasticity tensor.

Equation (5.80a) is the hypoelastic expression of an elastic behavior and

we expect it to conserve energy; that is to say, it has to be the hypoelastic

expression of a hyperelastic material behavior.

156 **Nonlinear** continua

In the next example we show that only for the case of infinitesimal strains,

it is valid to assume that t c E is a constant and isotropic tensor.

Example 5.14. JJJJJ

Following (Simo & Pister 1984) we will show that the hypoelastic equation

t ◦ σ ij

= t c ijkl t dkl

fails to model a hyperelastic behavior when the fourth-order constitutive tensor

is constant and isotropic (see also Example 5.6).

(a) Calculation of t ˙σ ij

Using Eq. (3.35), we get

t ◦ σ = t ˙σ − t l · t σ − t σ · t l T .

Also, using the definition of the second Piola-Kirchhoff stress tensor, we can

write

t σ = t J −1 t ◦X · t ◦S · t ◦X T , (A)

where t J = ◦ ρ

tρ .

Using the result in Example 5.3, we can write

From Eq. (A)

Also,

t ˙σ = −

tJ˙ t J 2

t

◦S = t ◦S ¡ t ◦C ¢ .

t

◦ X · t ◦ S ¡ t ◦ C ¢ · t ◦ XT + t J −1 t l ·

+ t J −1 t ◦X · ∂t ◦S ¡ t ◦C ¢

∂ t ◦C

· t ◦

+ t J −1 t ◦ X · t ◦ S ¡ t ◦ C ¢ · t ◦ XT ·

·

C · t ◦X T

t l T .

t c kmpq = 2 J −1 t ◦X k K t ◦X m M t ◦X p

P t ◦X q

Q

Using Example 4.1 we can write

t

◦ X · t ◦ S ¡ t ◦ C ¢ · t ◦ XT

∂ t ◦S KM

∂ t ◦CPQ

t ·

J = t J (∇ · t v) = t J

. (5.81)

³ ´

td t : g

and taking into account that t g ab |i = 0 , after same algebra, we get, using

components, the equation (Truesdell & Noll 1965)

5.2 Constitutive relations in solid mechanics: purely mechanical formulations 157

t ˙σ km = t h kmpq t lpq

(B.1)

t h kmpq = − t σ km t g pq + t g kp t σ qm + t c kmpq + t σ kq t g mp . (B.2)

In the derivation of the above equation we made use of the symmetry condition

t c kmpq = t c kmqp .

(b) The Bernstein formula

Following (Truesdell & Noll 1965) we now present the conditions that t h kmpq

in Eqs. (B.1-B.2) needs to fulfill in order to model a hyperelastic material

behavior.

For a hyperelastic material behavior, we can write, using Eq. (5.4e),

t σ ij =2 t ρ ∂ t u

∂ t gij

where tu is the elastic energy per unit mass.

We can also write the following functional dependence,

t km km

σ = ϕ ¡ t

◦X a ¢

A . (C)

Hence,

t ˙σ km = ∂ϕ km

∂ t ◦X a A

,

t l a b t ◦X b A .

Therefore, using the above and Eq. (B.1), we get

and after some algebra

∂ϕ km

∂ t ◦X a A

t l a b t ◦X b A = t h kmpq t lpq ,

∂ϕkm ∂t ◦X p =

A

t h km p q ¡ t

◦X −1¢ A

q

. (D)

The above is written in (Truesdell & Noll 1965) as Eq. (100.24). The function

ϕ km has to fulfill the following condition to be a potential function,

∂ 2 ϕ km

∂ t ◦X p

A ∂t ◦X r B

=

∂ 2 ϕ km

∂ t ◦X r B ∂t ◦X p

A

. (E)

Using in Eq. (E) the equality (D) and considering that (Truesdell & Toupin

1960) the equation

158 **Nonlinear** continua

leads to

∂ ¡ t ◦ X −1¢ B

q

∂ t ◦X p

A

¡ t◦X −1¢ B

q

we finally get the Bernstein formula:

∂ t h kmpq

∂ t σ rs

t h rsjl − ∂ t h kmjl

∂ t σ rs

t

◦X q

M = δBM (5.82)

= − ¡ t ◦X −1¢ B

p

¡ t◦X −1¢ A

q

(5.83)

t h rspq − t h kmpl t g jq + t h kmjq t g pl =0. (F)

(c) The constant and isotropic constitutive tensor

Finally, we are going to show that when using a constant and isotropic constitutive

tensor in Eq. (5.80a), the Bernstein formula is not fulfilled and therefore,

a hyperelastic material behavior cannot be modeled.

A general constant isotropic tensor is written as (Aris 1962)

t c kmjl = λ t g km t g jl + µ ¡ tg kj t g ml + t g kl t g mj ¢

then we replace in Eq. (F) and we find that the equality can be satisfied only

for (λ + µ) =0. Since this constraint on the material properties does not

correspond to a physical acceptable material model, we conclude that it is not

possible to use a constant and isotropic constitutive tensor in Eq. (5.80a) to

model a hyperelastic material behavior. JJJJJ

I The multiplicative decomposition of the deformation gradient

The use of hypoelastic models to numerically analyze finite strains elastoplastic

problems leads to many difficulties:

• The strain- and stress-rate measures have to be both objective and incrementally

objective (i.e. ∆ε and ∆σ have to be objective tensors). As

we have shown in Sect. 3.4 the Jaumann stress-rate, which is an objective

stress-rate measure, is not suitable for producing an objective incremental

stress measure.

• The limitation discussed in Example 5.14 to represent an elastic behavior

using a hypoelastic constitutive model.

An alternative kinematic formulation to numerically analyze finite-strain

elastoplastic problems can be found in (Lee & Liu 1967, Lee 1969). Lee’s

kinematic formulation: the multiplicative decomposition of the deformation

gradient, is based on a micromechanical model of single-crystal metal plasticity

(Simo & Hughes 1998).

5.2 Constitutive relations in solid mechanics: purely mechanical formulations 159

Lee’s multiplicative decomposition of the deformation gradient has been

the basis of the hyperelastic model for finite-strain elastoplasticity developed

by Simo and Ortiz (Simo & Ortiz 1985, Simo 1988, Simo & Hughes 1998).

We will now present an “intuitive description” of Lee’s multiplicative decomposition

of the deformation gradient (Lubliner 1990).

Let us consider a reference configuration that we assume to be unstressed

and unstrained and the spatial configuration corresponding to a certain time

“t”. In this spatial configuration, we have reversible (elastic) deformations

and permanent (plastic) deformations.

If we now cut the spatial configuration into hexahedric infinitesimal volumes

and if we assume for each of those volumes an elastic unloading process,

we obtain a stress-free configuration that is called the intermediate configuration.

Obviously, the infinitesimal hexahedrals will not match together in the

stress free or intermediate configuration to form a continuum because compatibility

was lost when we divided the spatial continuum into infinitesimal

parts.

Therefore, the intermediate configuration is not a proper configuration

because there is not a bijective mapping between the material particles and

< 3 .

In Fig. 5.10 we schematize the three configurations: the reference one,

the intermediate one and the spatial one. We also indicate in this figure the

arbitrary coordinates defined on the reference and spatial configurations and

the corresponding deformation gradients.

From this figure

Notes:

t

◦X = t ◦X E · t ◦X P . (5.84)

• Since the intermediate configuration, due to its lack of compatibility is not

aproperconfiguration, the tensors t ◦X E and t ◦X P cannot be calculated

using the definition of a deformation gradient tensor (Eq. (2.23)).

• The mapping represented by t ◦X E is purely elastic and the stresses in the

spatial configuration are developed during this mapping.

The velocity gradient is defined in the spatial configuration and the following

relation holds

t l = t ◦ ˙X · t ◦X −1 . (5.85a)

Using the multiplicative decomposition in Eq. (5.84), we obtain

t l = t ◦ ˙X E ·

³ t◦X E´ −1

+ t ◦X E · t ◦ ˙X P ·

³ t◦X P ´ −1

·

³ t◦X E´ −1

. (5.85b)

160 **Nonlinear** continua

Fig. 5.10. Lee’s multiplicative decomposition of the deformation gradient

We call t¯P l = t ˙X ◦ P ·

³

t◦X P ´ −1

atensordefined in the intermediate

configuration, and we rewrite the above equation as:

t l = t ◦ ˙X E ·

³

t◦X E´ −1

+ t ◦Xe

³ ´

t¯l

P

. (5.85c)

In the above equation we used the notation t ◦Xe (.) to indicate the pushforward

of the components of the tensor (.) from the intermediate configuration

to the spatial configuration.

At the intermediate configuration, we can make the following additive

decomposition

t¯P l =

t¯P t P

d + ¯ω . (5.85d)

A standard hypothesis is that if the material under consideration has isotropic

elastic properties, we can impose

t ¯ω P ≡ 0 . (5.86)

The above hypothesis was used in (Weber & Anand 1990, Eterovic & Bathe

1990, Dvorkin, Pantuso & Repetto 1994).

By doing the polar decomposition of the elastic and plastic deformation

gradients, obtained in Eq. (5.84) via Lee’s multiplicative decomposition, we

get

5.2 Constitutive relations in solid mechanics: purely mechanical formulations 161

t

◦X E = t ◦R E · t ◦U E = t ◦V E · t ◦R E

(5.87a)

t

◦X P = t ◦R P · t ◦U P = t ◦V P · t ◦R P . (5.87b)

We can also define the elastic Hencky strain tensor as

t

◦H E ³

=ln

t◦U E´

. (5.88)

I Stresses and the yield criterion

Considering an elastically isotropic material, the energy conjugate of t ◦H E is

(see Sect. 3.3.4),

t t

Γ = ◦R e( t τ ) . (5.89)

Following the work by Lee (Lee & Liu 1967, Lee 1969) we formulate the

yield criterion in terms of Kirchhoff stresses. Since we are interested in the

modeling of elastoplastic deformation processes in metals, we use the von

Mises (J2) yield criteria combined with isotropic/kinematic hardening and

we have

t f =

∙

3

³

tτ

−

2 D t ´

α

In the above,

t τ D : deviatoric Kirchhoff stress tensor;

t α : back-stress tensor (traceless);

t σy: yield stress in the t-configuration.

:

³

tτ

−

D t ´

α

¸ 1

2

− t σy =0. (5.90)

Example 5.15. JJJJJ

In (Lee 1969), Lee noted that the rate of plastic work invested per unit volume

of the reference configuration is,

t

σ :

µ ◦ρ

t P

d tρ (A)

where ◦ ρ and t ρ are the densities of the reference and spatial configurations.

Since the plastic flow is incompressible, then only during the elastic deformation

can the densities change.

The plastic work in Eq. (A) is related to the material plastic hardening. Equation

(A) indicates a decrease in hardening when we increase the hydrostatic

pressure ( t ρ > ◦ ρ) . To avoid this coupling between the elastic and plastic behavior,

Lee suggests using, in the finite deformation elastoplastic formulation,

the Kirchhoff stress tensor rather than the Cauchy one in the yield criterion.

Obviously, for infinitesimal strains, both stress tensors are identical. JJJJJ

162 **Nonlinear** continua

The tensors defined by

t IJ t

B = ◦R ¡ tαij e

¢

t IJ

ΓD = t ◦R ³ ´

tτ ij

e D

are traceless.

By doing a t ◦Re-pull-back of Eq. (5.90), we get

t f =

∙

3

³

tΓD

−

2

t ´

B

:

(5.91a)

³

tΓD

− t ´

B

¸ 1

2

− t σy =0. (5.92)

We consider the following evolution equations (Eterovic & Bathe 1990,

Dvorkin, Pantuso & Repetto 1994):

t ˙σy = βh t ¯ d P

(5.93a)

t ˙B = 2

3 (1 − β) h t¯d P . (5.93b)

In the above, h = h ¡ t P ē ¢ is the hardening module, td¯ P is defined in

(5.73a) and β ∈ [0, 1] is the hardening ratio; β =0corresponds to purely

kinematic hardening and β =1corresponds to purely isotropic hardening.

We also define the equivalent plastic strain as:

I Energy dissipation

t ē P =

Z t

0

t ¯ d P dt. (5.94)

We now introduce t ψ as the free energy at the spatial configuration per

unit volume of the reference configuration.

Considering that the mechanical problem is uncoupled from the thermal

problem, we can write (Simo & Hughes 1998)

t τ :

t d − t ˙ ψ > 0 . (5.95a)

The above equation, known as the Clausius-Duhem inequality, is a restriction

imposed by the Second Law of Thermodynamics (Simo & Hughes

1998):

• For the elastic case the deformation is reversible and the equal sign holds.

• When there are plastic deformations the process is irreversible and the

greater-than sign holds.

5.2 Constitutive relations in solid mechanics: purely mechanical formulations 163

Therefore, using Lee’s multiplicative decomposition, we can write

t

◦X P αβ = δαβ −→ t τ :

t d − t ˙ ψ =0 (5.95b)

t

◦X P αβ 6= δαβ −→ t τ :

t t

d − ψ ˙ > 0 . (5.95c)

For the free energy of the purely mechanical problem, we can state the

following functional relation:

t ψ = t ψ

³

t◦H E , t σy, t ´

B

. (5.96a)

Following Simo 1988, we use the following uncoupled expression for the

free energy,

³

t t

ψ =

t◦H

ψe

E´

+ t ¡

ψ

tσy,

p

t B ¢ . (5.96b)

Considering a metal, t ψ e is the elastic free energy that we identify with

the atomic lattice deformation energy and t ψ p is the energy associated with

atomic lattice defects (e.g. dislocations) (Lubliner 1990).

From Eq. (5.85b),

Therefore,

t l = t l E + t l P

t l E = t ◦ ˙X E ·

³ t◦X E´ −1

t l P = t ◦X E · t¯ l P ·

t τ : t d = t τ :

Aftersomealgebra,wecanwrite

t τ :

t d P =

∙ ³t◦X E´ T

= t d E + t ω E

³ t◦X E´ −1

(5.97a)

(5.97b)

= t d P + t ω P . (5.97c)

³ td E + t d P ´

· t τ ·

³ t◦X E´ −T ¸

. (5.98)

:

t¯ l P . (5.99)

Since we restrict this formulation to the case of isotropic elastic properties,

we can write (see Sect. 3.3.4)

t τ :

t d E = t Γ :

t E

˙H ◦

and the Clausius-Duhem inequality takes the form,

(5.100)

Ã

t ∂

Γ − tψe ∂t ◦H E

!

: t ◦ ˙H E +

∙ ³t◦X E´ T

· t τ ·

³

t◦X E´ −T ¸

: t¯P l

t

− ψp ˙ > 0 .

(5.101)

Since the above must also be valid for the case of an elastic deformation,

we obtain

164 **Nonlinear** continua

t ∂

Γ = tψe ∂t E

◦H

We define as dissipation (Simo 1988):

. (5.102)

∙ ³t◦X

t

D =

E´ T

· t τ ·

³

t◦X E´ −T ¸

:

t¯P l

t

− ψp ˙ > 0 . (5.103)

Considering that:

• For elastically isotropic materials the tensors t Γ , t ◦H E and therefore

t

◦U E are collinear.

• The contraction of a symmetric tensor with a skew-symmetric one is zero.

We get,

t t

D = Γ :

t¯P t

d − ψp ˙ > 0 . (5.104)

t We search for the value of Γ that maximizes the dissipation under the

unilateral constraint tf 6 0 .

For the case tf

5.2 Constitutive relations in solid mechanics: purely mechanical formulations 165

where t+4t

◦ Π is the potential energy corresponding to the t + 4t configuration.

From Chap. 6, Eq. (6.51), we can write

Z

t+4t

◦ t+4t ◦ t+4t

◦ Π = ρ U dV + g (5.108a)

◦ V

where ◦ V : volume of the reference configuration; t+4t U : elastic energy per

unit mass stored in the (t + 4t)-configuration; and, t+4t g : potential of the

external loads acting on the (t + 4t)-configuration.

Z

UsingtheelasticfreeenergydefinedinEq.(5.96b),wecanwrite

t+4t

◦ Π =

Z

◦V t+4t ◦

ψe dV

t+4t

+ g. (5.108b)

Equation (5.107) together with the above leads to

◦ V

³

t+4t t+4t

Γ : δ ◦ H E´ ◦

dV =

Z

t+4tV t+4t

f b · δu t+4t Z

dV (5.108c)

+

t+4t

f s · δu t+4t dS.

t+4t S

In the above equation:

t+4t V : volume of the spatial configuration,

t+4t S : external surface of the spatial configuration,

t+4t f b : body forces per unit volume, acting on the (t + 4t)-configuration,

t+4t f s : surface forces acting on the (t + 4t)-configuration,

u : displacement from t to (t + 4t).

Also,

δ t+4t

◦

H E = ∂t+4t ◦ H E

∂ t+4t

◦ H

: δt+4t

◦ H . (5.109a)

∂ t+4t

◦ HE

∂ t+4t

◦ H

For the calculation of the fourth-order tensor

, we follow the

appendix in (Dvorkin, Pantuso & Repetto 1994) and we get

∂ t ◦H E IJ

∂ t ◦HKL

= ∂t ◦H E IJ

∂t ◦C E ∂

MN

t ◦C E MN

∂t ◦CRS

∂ t ◦CRS

∂ t ◦HKL

. (5.110a)

To calculate the derivative in the first factor on the r.h.s. of the above

equation, we use

t

◦HE q

=ln t◦ C E

(5.110b)

166 **Nonlinear** continua

being Φ T I =[αI,β I,γ I] (I =1, 2, 3) the 3 eigenvectors of t ◦C E and λI its

3 eigenvalues

∂t ◦H E IJ

∂t ◦C E MN

= ∂t ◦H E IJ

∂αK

+ ∂t ◦H E IJ

∂γ K

∂αK

∂ t ◦C E MN

∂γ K

∂ t ◦ CE MN

+ ∂t ◦H E IJ

∂β K

+ ∂t ◦H E IJ

∂λK

∂β K

∂ t ◦C E MN

∂λK

∂ t ◦ CE MN

. (5.110c)

To calculate the derivative in the second factor on the r.h.s. of Eq. (5.110a),

we use

t

◦C E =

³

t◦X P ´ −T

· t ◦C ·

³

t◦X P ´ −1

, (5.110d)

and to calculate the derivative in the third factor we use

q

t

◦H =ln t◦C , (5.110e)

and proceed as in Eq. (5.110c) but using the eigenvectors and eigenvalues of

t

◦C.

For the elastic deformation, we use

t+4t E t+4t

Γ = C : ◦ H E , (5.111)

where C E is a constant and isotropic elasticity tensor (Hooke’s law).

The main aspects of the finite strain elastoplastic formulation that we

presented in this Section are:

• We use Lee’s multiplicative decomposition of the deformation gradient.

• We use a hyperelastic constitutive relation for the elastic strains/stress

relation; that is to say, we relate total strains with total stresses.

• We describe the plastic flow using the maximum dissipation principle (associated

plasticity).

• We use the total Hencky strain as our strain measure. The reason for

this choice is that according to the experimental data reported in (Anand

1979), “the classical strain energy function of infinitesimal isotropic elasticity

is in good agreement with experiment for a wide class of materials

for moderately large deformations, provided the infinitesimal strain measure

occurring in the strain energy function is replaced by the Hencky or

logarithmic measure of finite strain”.

From a numerical viewpoint, an additional advantage in using the Hencky

strain measure is that the first invariant of the logarithmic strain tensor is the

logarithmic volume strain; therefore many techniques developed for handling

incompressibility in the infinitesimal strain problem can be carried over for

the finite strain problem.

We discussed the above formulation, the “total Lagrangian - Hencky formulation”anditsfinite

element implementation in (Dvorkin 1995a\1995b\1995c,

Dvorkin & Assanelli 2000, Dvorkin, Pantuso & Repetto 1992\1993\1994\1995).

5.3 Constitutive relations in solid mechanics: thermoelastoplastic formulations 167

5.3 Constitutive relations in solid mechanics:

thermoelastoplastic formulations

5.3.1 The isotropic thermoelastic constitutive model

Considering a hyperelastic solid under mechanical loads and thermal evolution,

the two point variables that define the state of the solid at any instant

are a strain measure and the temperature. Therefore, the stresses at any point

in the solid are a function of the strains and temperature at that point (local

action).

Using, for example, the Green-Lagrange strain tensor; and t T being the

temperature, we can write for any particle in the spatial configuration its

internal energy per unit mass (elastic energy + caloric energy) as

t U = t U ¡ t◦ε, t T ¢

(5.112a)

and considering a reversible process (Boley & Weiner 1960), we can write

t η = t η ¡ t◦ε, t T ¢

(5.112b)

where t η is the spatial entropy per unit mass.

The principle of energy conservation (First Law of Thermodynamics) can

be written as

D t U

Dt

= 1

◦ ρ

t

◦S :

t

◦ ˙ε + t T t ˙η. (5.113)

We can define Helmholtz’s free energy per unit mass, (Boley & Weiner

1960, Malvern 1969) as:

t ψ ¡ t◦ε, t T ¢ = t U ¡ t ◦ε, t T ¢ − t T t η. (5.114a)

Using the above and Eq. (5.113), we get

µ ∂ t ψ

∂ t ◦εIJ

− 1 t

◦ρ ◦S IJ

t◦

˙εIJ +

µ t ∂ ψ

∂tT + t

t

η T ˙ = 0 . (5.114b)

Hence considering the isothermal ( t ˙

T = 0 ) and isometric ( t ◦ ˙εIJ =0)

cases, we have

t

◦S IJ = ◦ ρ ∂tψ ∂t (5.114c)

◦εIJ

t ∂

η = − tψ ∂t . (5.114d)

T

In a hyperelastic material the stresses and entropy are a function of the

strain/temperature value at the point and not of its history; then, the above

equations are valid for any process.

168 **Nonlinear** continua

Since the free energy, t ψ , is invariant under changes of reference frame, for

an isotropic material, it can only depend on the invariants of t ◦ε ,whichfor

the t-configuration, and considering the strain tensor Cartesian components,

are

Hence,

t

◦I1 = t ◦εαα

t

◦I2 = 1

2

t

◦εαβ t ◦εαβ − 1

2

¡ ¢ t◦I1 2

(5.115a)

(5.115b)

t

◦I3 = 1

6 eαβγ eδζ t ◦εαδ t ◦εβ t ◦εγζ . (5.115c)

t ψ = t ψ ¡ t◦I1, t ◦I2, t ◦I3, t T ¢ . (5.116)

The most general expression for the free energy is

t ψ = a0 + a1 t ◦I1 + a2 t ◦I2 + a3 t ◦I3 + a4

+ a6 t ◦I1

¡ t◦I1

+ a9

¡ tT ¢

− TR + a7 t ◦I2

¡ tT ¢ ¡

− TR +

tT ¢ 2

a5 − TR

¡ tT ¢

− TR + a8 t ¡ tT ¢

◦I3 − TR

¢ 2 + ... . (5.117)

In the above equation TR is the reference temperature, usually the temperature

of the undeformed solid.

Using Eq. (5.114c), we get

where

∂t ◦I1

∂t ◦εαβ

∂t ◦I2

∂t ◦εαβ

∂t ◦I3

∂t ◦εαβ

t

◦Sαβ = ◦ ρ ∂tψ ∂t ◦Ii

∂t ◦Ii

∂t ◦εαβ

, (5.118a)

= δαβ , (5.118b)

= t ◦εαβ − t ◦I1 δαβ , (5.118c)

= 1

6 eαγδ eβζ t ◦εγ t ◦εδζ . (5.118d)

Physical restrictions used to determine the material constants:

t

◦εαβ =0

•

=⇒ tT = TR

t ◦Sαβ =0 ; hence, a1 =0.

• For the above condition, we set an arbitrary value for the free energy;

hence, we set a0 =0.

• Doing the same consideration for the entropy, we set a4 =0.

5.3 Constitutive relations in solid mechanics: thermoelastoplastic formulations 169

I Specialization for small strains and small temperature increments

In this case:

t

◦εαβ −→ tεαβ (infinitesimal strain tensor)

t

◦Sαβ −→ tσαβ (Cauchy stress tensor) .

Using Eq. (5.114c) and (5.117) and neglecting higher-order terms in tεαβ (

and

tT − TR) , we get, for an isotropic material (Malvern 1969),

t σαβ = λδαβ t εγγ + 2 G t εαβ −

Eα

(1 − 2ν)

¡ tT ¢

− TR δαβ

(5.119)

where E : Young’s modulus; ν : Poisson’s ratio; G : shear modulus, G = E

2(1+ν) ;

νE

λ = (1−2ν)(1+ν) ;and,α : linear coefficient of thermal expansion.

For an isotropic solid with constant and uniform material properties, in the

absence of volumetric heat generation or consumption, the heat conduction

equation is (Boley & Weiner 1960):

k ∇ 2 t T = ◦ ρ t T t ˙η (5.120)

where k is the heat conduction coefficient of the material (Fourier’s law).

The above equation together with Eq. (5.114d) leads to,

k ∇ 2 t T = − ◦ ρ t ∙

∂

T

2 t ¸

ψ

t

T˙

. (5.121)

∂ t T∂ t εαβ

t ˙εαβ + ∂2 t ψ

∂ t T 2

The specific heat of the material is defined as:

t ∂

c = tU ∂t (5.122)

T

and after some algebra,

t t ∂

c = T tη ∂t . (5.123)

T

Using Eqs. (5.114c) and (5.114d) in (5.121), we get

k ∇ 2t T = − ◦ ρ t T ∂

∂ t T

µ tσαβ

◦ ρ

t ˙εαβ + ◦ ρ t T ∂t η

∂ t T

And using Eqs. (5.119) and (5.123) in (5.124), we get

k ∇ 2t T =

t TEα

(1 − 2ν)

t ˙

T. (5.124)

t ˙εαα + ◦ ρ t c t ˙

T. (5.125)

Equations (5.119) and (5.125) show that in the most general case of thermoelastic

materials the heat transfer and stress analysis problems are coupled.

However, in most engineering applications the first term on the r.h.s. of

Eq. (5.125) can be neglected.

170 **Nonlinear** continua

In (Boley & Weiner 1960) numerical examples in aluminum and steel were

considered and it was shown that the coupling is negligible when

t ˙εαα

3 α

5.3.2 A thermoelastoplastic constitutive model

tT ˙

¿ 20 . (5.126)

In Sect. 5.2.5 we developed the constitutive relation for an elastoplastic material

under infinitesimal strains; in that section we made the assumption of

a purely mechanical formulation, now we shall remove that limitation incorporating

thermal effects. However, we will keep the other assumptions made

in that section: infinitesimal strains, limit loading condition (yield criterion),

rate-independent behavior, stable material, etc.

Sincewehavetotakeintoaccountthethermalstrains,werewriteEq.

(5.38) as:

t d = t d E + t d P + t d TH . (5.127)

For the yield condition, we rewrite Eq. (5.48) as:

t f ¡ tσ , t qi i =1,n , t T ¢ =0. (5.128)

Considering that we are focusing on the behavior of metals, we will keep

on using the von Mises yield criterion and therefore in the case of isotropic

hardening, we can rewrite Eq. (5.55) as:

t 1 t

f = s :

2

t s −

t σ 2 y

3

=0 (5.129a)

where tσy = t ¡

t¯ε

¢

P t

σy , T .

In the case of kinematic hardening we can rewrite Eq. (5.55) as (Snyder

1980):

t 1 ¡

f =

ts ¢ ¡ t

− α :

ts ¢ t

− α −

2

where tσy = tσy ( tT ); tαij = R t

◦

t σ 2 y

3

τ ˙αij dτ; and, t ˙αij = t c ( t T ) t d P ij

(5.129b)

.

For the unstressed state with no previously accumulated plastic strains

(Boley & Weiner 1960)

t t

f(0, 0, T ) < 0 . (5.130)

During plastic loading the consistency equation takes the form (Boley &

Weiner 1960),

f ˙ = ∂tf ∂tsij ˙s ij + ∂tf ∂tεP t P

dij +

ij

∂tf ∂t t

T ˙ =0. (5.131)

T

Following (Boley & Weiner 1960) we will study three possible cases:

5.3 Constitutive relations in solid mechanics: thermoelastoplastic formulations 171

(a)

∂tf ∂tsij t ij ∂

˙s + tf ∂t t

T ˙ =0. (5.132a)

T

Using the above equation together with Eq. (5.131), we see that it is possible

for the stress components and the temperature to change so that the

point remains on the yield surface but without further plasticity development.

Therefore this change is called neutral ¡ t P dij =0 ¢ .

(b)

∂ t f

∂ t s ij

t ˙s ij + ∂ t f

∂ t T

t ˙

T0 . (5.132c)

The above equation represents a condition of plastic loading and using

Eq.(5.131), we get t d P ij .

I Stress - strain - temperature relation

As we did above for the case of isothermal plasticity we consider that the

plastic flow maximizes the plastic dissipation (Lubliner 1985).

We again define the plastic dissipation via Eq. (5.57) and imposing during

plastic loading the condition tf = 0 we obtain, in a general curvilinear

system,

dσ ij t d P ij = t ∙ t

λ˙

∂ f

∂tσij dσij + ∂tf ∂tT dT

¸

. (5.133)

Considering Eq. (5.132c) and since tλ ˙ > 0, we get for plastic loading the

condition

t ij t P

˙σ dij > 0 , (5.134)

which is the same stability condition already obtained for the isothermal case

(Drucker’s postulate). Again, we obtain the convexity condition for the yield

surface in the stress space.

From Eq. (5.133),

∙

tdP ij − t λ˙ ∂tf ∂tσij ¸

dσ ij + ∂tf ∂t dT =0. (5.135)

T

Hence, the normality rule already developed for the isothermal case applies:

172 **Nonlinear** continua

t P

dij = t λ˙ ∂tf ∂t . (5.136)

σij Stress - strain relations for the case of isotropic hardening

Since we are considering the case of infinitesimal strains, we can write in

a Cartesian system,

dεαβ = dε E αβ + dε P αβ + dε TH

αβ

(5.137a)

and

t

σαβ = t C E αβγδ t ε E γδ . (5.137b)

We consider an isotropic linear elastic material with elastic constants function

of the temperature; therefore (Snyder 1980),

Hence,

dσαβ = t C E αβγδ dε E γδ + ∂t C E αβγδ

∂ t T

dσαβ = t C E αβγδ

£

dεγδ − dε P γδ − dε TH¤

∂

γδ + tC E αβγδ

∂tT For a von Mises material,

and for an isotropic thermal expansion,

t ε E γδ dT . (5.138)

t ε E γδ dT . (5.139a)

dε P γδ =dλ t sγδ (5.139b)

dε TH

γδ = t α dT δγδ . (5.139c)

During plastic loading df =0 and using Eq. (5.131),

∂ t f

∂ t σαβ

dσαβ + ∂t f

∂ t ¯ε P d¯εP + ∂t f

∂ t T

dT = 0 . (5.140)

Developing each of the terms in the above equation, we obtain

∂ t f

∂ t σαβ

dσαβ = t sαβ [ t C E αβγδ

+ ∂t C E αβγδ

∂ t T

∂tf ∂t¯ε P d¯εP = − 4

9

∂tf ∂t dT

T

2

= −

3

therefore (Snyder 1980),

t σ 2 y

t σy

¡

dεγδ − dλ t sγδ − t ¢

α dT δγδ

t ε E γδ dT ] (5.141a)

∂ t σy

∂t dλ (5.141b)

¯ε P

∂ t σy

∂ t T

dT, (5.141c)

5.3 Constitutive relations in solid mechanics: thermoelastoplastic formulations 173

dλ =

t sαβ [ t C E αβγδ (dεγδ − t α dT δγδ)+ ∂t C E αβγδ

∂ t T

t sηθ t C E ηθµξ t sµξ + 4

9 t σ 2 y

t ε E γδ dT ] − 2

∂ t σy

∂ t ¯ε P

3 tσy ∂tσy ∂tT dT

(5.142)

Hence, we introduce the above in Eqs. (5.139a-5.139c) and we can inmediately

relate increments in strains/temperature with stress increments.

In order ³ to be able to evaluate the terms in Eq. (5.142) it is necessary to

t

∂ σy

relate ∂t¯ε P

´ ³ t

∂ σy

and ∂t ´

T to the actual material behavior (Snyder 1980).

From the data obtained in isothermal tensile tests of virgin samples, we

can develop the idealized bilinear stress-strain curves shown in Fig. 5.11.

Fig. 5.11. Stress-strain curves at different temperatures, Ti

For a constant temperature curve, we can write

t

σy =( ◦ σy) T +

∙

tε

−

µ ◦σy

E

¸

Therefore,

t ε = t ε P +

t σy

(E) T

t σy =( ◦ σy) T + t ε P (E Et) T

(E − Et) T

T

(Et) T

(5.143a)

. (5.143b)

. (5.143c)

.

174 **Nonlinear** continua

Using, as in the isothermal case, the concept of a universal stress-strain

curve that is valid for any multiaxial stress-strain state, we can use Eq.

(5.143c) for any stress - strain state, provided that tεP is replaced by t¯ε P

(Eq. (5.74a)). Hence, in Eq. (5.142), we have

∂tσy ∂t =

¯ε P

∂ t σy

∂ t T =

µ

EEt

E − Et

T

+ t ¯ε P

µ ∂ ◦ σy

∂ t T

Hence, we can rewrite Eq. (5.142) as:

dλ =

−

t sαβ

∙

2

3 t σy

T

∙

∂

∂t µ

EEt

¸

T E − Et T

t E Cαβγδ (dεγδ − tα dT δγδ)+ ∂tC E αβγδ

∂tT tsηθ tC E ηθµξ tsµξ + 4

³ ´

E Et

9 tσ2 y E − Et T

h³ ◦

∂ σy

∂t ´

T

T + t¯ε P

³

∂

∂t ³ ´´ i

E Et

T E − Et T

tsηθ tC E ηθµξ tsµξ + 4

9 tσ2 ³ ´

E Et

y E − Et

T

(5.144a)

. (5.144b)

t ε E γδ dT

dT

¸

. (5.145)

In the above equation, we consider a linear isotropic elastic model; therefore

using Eqs. (5.15) and (5.16), we get

t C E αβγδ =(λ) T δαβδγδ +(G) T (δαγδβδ + δαδδβγ) (5.146)

and taking into account that t sαα =0,weget

t sαβ t C E αβγδ =2 (G) T

t sγδ . (5.147)

Taking into account that (Snyder 1980)

h ¡ ¢−1

i ³

t E ν

´

C = − δαβδγδ +

αβγδ E T

1

(δαγδβδ + δαδδβγ)

4(G) T

we can show that

(5.148)

t

sαβ

∂tC E αβγδ

∂tT ∙ µ

t E 1 ∂G

εγδ =

G ∂t ¸

T T

t

sαβ t σαβ . (5.149)

Hence,

dλ = 2(G) T tsγδ dεγδ + ¡ ¢

1 ∂G t

G ∂T

sαβ T

tsαβ dT

4

3 (G) T (tσy) 2 + 4

9 tσ2 ³ ´

E Et

y E−Et T

h³ ´ 0

2 t ∂ σy

3

σy ∂T

T −

+ tεP ³

∂

∂t ³ ´´ i

E Et

T

dT

E−Et T

4

3 (G) T (tσy) 2 + 4

9 tσ2 ³ ´

E Et

y E−Et T

. (5.150)

5.3 Constitutive relations in solid mechanics: thermoelastoplastic formulations 175

Stress - strain relations for the case of kinematic hardening

We use the kinematic hardening results in Section 5.2.5 and adapt them for

the case of nonisothermal processes,

t 1 ¡ tsγδ

f =

2

− t ¢¡ tsγδ

αγδ − t ¢ 1

αγδ −

3

¡ tT ¢

t σy = t σy

t αγδ =

Zt

0

t σ 2 y =0 (5.151a)

(5.151b)

t ˙αγδ dt (5.151c)

t ˙αγδ = t c t d P γδ (5.151d)

t c = t c ¡ tT ¢ . (5.151e)

During the plastic loading

∂tf ∂t dσγδ +

σγδ

∂tf ∂t dαγδ +

αγδ

∂tf ∂t dσy =0. (5.152)

σy

Developing each of the terms in the above equation, we obtain

∂ t f

∂ t σγδ

(

dσγδ = ¡ t

sγδ − t ¢

αγδ

t C E γδϕξ

∂ t f

∂ t αγδ

∂ t f

∂ t σy

£

dεϕξ − dλ ¡ t

sϕξ − t ¢ ¤ ∂ t

αϕξ − α dT δϕξ + tC E γδϕξ

∂T

(5.153a)

t ε E ϕξ dT

dαγδ = − ¡ t

sγδ − t ¢ tc ¡ tsγδ

αγδ dλ − t ¢ 2 t t 2

αγδ = − c dλ σy 3

(5.153b)

dσy = − 2 t

σy

3

∂ t σy

∂ t T

dT (5.153c)

therefore (Snyder 1980),

(

dλ =

tsγδ − t ∙

t E

αγδ) Cγδϕξ (dεϕξ − tα dT δϕξ)+ ∂tC E ¸

γδϕξ t E

∂T εϕξ dT

( tsηθ − tαηθ) tC E ηθµψ ( tsµψ − tαµψ)+ 2

3 tc tσ2 y

−

2

3

tσy ∂tσy ∂tT dT

( tsηθ − tαηθ) tC E ηθµψ ( tsµψ − tαµψ)+ 2

3 tc tσ2 y

)

(5.154)

176 **Nonlinear** continua

Again, as in the case of isotropic hardening, we relate the above expression

to the actual material behavior using the information contained in the

isothermal uniaxial stress-strain curves.

For an isothermal loading in a bi-linear material, we can use the result in

Example 5.13 and obtain,

µ

t 2 EEt

c(T )= . (5.155)

3 E − Et

5.4 Viscoplasticity

In Sects. 5.2 and 5.3, we discussed constitutive relations that have a common

feature: the response of the solids is instantaneous; thatistosay,whenaload

is applied, either a mechanical or a thermal load, the solid instantaneously

develops the corresponding displacements and strains.

We know, from our experience, that this is not the case in many situations;

e.g. a metallic structure under elevated temperature increases its deformation

with time; a concrete structure in the first few months after it has been cast

increases its deformation with time, etc.

There is also an important experimental observation related to the response

of materials, in particular metals, to rapid loads: the apparent yield

stress increases with the deformation velocity. In the previous sections, when

considering instantaneous plasticity, we represented the strain hardening of

metals with equations of the form:

T

σy = σy(ε, T ) . (5.156)

To take into account the above commented experimental observation, the yield

stress has to present the following functional dependence (Backofen 1972):

σy = σy(ε, ˙ε, T ) . (5.157)

We can say that the strain-rate effect shown in Eq.(5.157) is a viscous

effect. There are basically two ways in which a viscous effect can enter a

solid’s constitutive relation:

• In the viscoelastic constitutive relations, the elastic part of the solid deformation

presents viscous effects. In this book, we are not going to discuss

this kind of constitutive relations and we refer the readers to (Pipkin 1972)

for a detailed discussion.

• In the viscoplastic constitutive relations (Perzyna 1966), the permanent

deformation presents viscous effects. The examples we discussed above are

described using viscoplastic constitutive relations and also, other important

problems like metal-forming processes are very well described using

this constitutive theory (Zienkiewicz, Jain & Oñate 1977, Kobayashi, Oh

& Altan 1989).

5.4 Viscoplasticity 177

As in the case of elastoplasticity, we can divide the total strain rate into

its elastic and viscoplastic parts; hence, we get an equation equivalent to Eq.

(5.38), but now for an elastoviscoplastic solid:

t d = t d E + t d VP

(5.158)

where, t d VP is the viscoplastic strain rate tensor.

In some cases, for example when modeling bulk metal-forming processes

(Zienkiewicz, Jain & Oñate 1977), t d E 0 (5.161a)

hai =0 if a ≤ 0 . (5.161b)

178 **Nonlinear** continua

An important difference between the flow rate for the viscoplastic constitutive

model (Eq. (5.160)) and the flow rate for the plastic constitutive model

(Eq. (5.60)) is that in the present case, γ the fluidity parameter is a material

constant, while in the plasticity theory t ˙ λ is a flow constant, derived by

imposing the consistency condition during the plastic loading.

Obviously, the correct value of γ and the correct expression for φ ( t f) are

derived from experimental observations.

In what follows we will concentrate on the details of a rigid-viscoplastic

relation suited for describing the behavior of metals with isotropic hardening,

φ( t f)=

" µ1

2

t sαβ t sαβ

1 2

−

t σy

√3

#δ

· (5.162)

IntheaboveequationthetermbetweenbracketsisthevonMisesyield

function.

Using the definition of the second invariant of the deviatoric Cauchy

stresses we get,

¯

∂f ¯

∂σαβ

¯ =

t

1

2 √ t

sαβ

(5.163)

tJ2 hence, using Eq.(5.160), we get

t VP

dαβ = γ

2 √ t tf δ

sαβ

tJ2 ® · (5.164)

The above equation indicates that with the selected yield function the resulting

viscoplastic flow is incompressible; a result that matches the experimental

observations performed on the viscoplastic flow of metals.

Using the definition of equivalent viscoplastic strain associated to the von

Mises yield function, Eq. (5.73a), we have

t ˙εVP = γ

√ tf δ

3

® · (5.165)

Therefore, for tf ≥ 0

√

¡ tf ¢ δ 3 t

= ˙εVP· (5.166)

γ

Formulating, for a rigid-viscoplastic material model, the relation among

deviatoric stresses and strains as,

t sαβ =2 t µ t d VP

αβ

and using the above equations we get, for t f ≥ 0

t µ =

∙

t √3

σy

√3 t + γ

·

¸ 1

δ

εVP

√

3 t ˙εVP

(5.167)

· (5.168)

5.4 Viscoplasticity 179

From Eqs. (5.167) and (5.168) we see that a rigid-viscoplastic material

behavesasanon-Newtonianfluid. It comes as no surprise that the solid behaves

in a “fluid way”, since we have neglected the solid elastic behavior and

therefore its memory; the material memory is the main difference between the

behavior of solids and fluids.

In the limit, when γ →∞Eq. (5.168) describes the behavior of a rigidplastic

material (inviscid), in this case,

t

µ =

3

t σy

t ˙εVP

· (5.169)

Example 5.16. JJJJJ

An important experimentally observed effect, that the viscoplastic material

model explains, is the increase in the apparent yield stress of metals when the

strain rate is increased (Malvern 1969) (strain-rate effect).

Let us assume a uniaxial test in a rigid-viscoplastic bar,

Therefore,

σ11 = bσ

σ22 = σ33 =0·

s11 = 2

3 bσ

s22 = s33 = − 1

bσ ·

3

Also, for the viscoplastic strain rates we can write,

d VP

11 = · ε

d VP

22 = d VP

33 = − 1

˙ε ·

2

Hence, the equivalent viscoplastic strain rate is,

·

εVP = ˙ε .

Using Eqs. (5.167) and (5.168) together with the above we get,

bσ = σy + √ 3

Ã√

3

γ ˙ε

! 1/δ

In the above equation, σy is the bar yield stress obtained with a quasistatic

test and bσ is the apparent yield stress obtained with a dynamic test.

When γ →∞(inviscid plasticity), the strain-rate effect vanishes.

Using other functions in Eq. (5.162) more complicated strain-rate dependences

can be explained (Backofen 1972). JJJJJ

.

180 **Nonlinear** continua

In (Zienkiewicz, Jain & Oñate 1977) a finite element methodology, based

on a rigid-viscoplastic constitutive relation was developed, for analyzing bulk

metal-forming processes. This methodology known as the flow formulation has

been widely used since then for analyzing many industrial processes (Dvorkin,

Cavaliere & Goldschmit 2003, Cavaliere, Goldschmit & Dvorkin 2001a\2001b,

Dvorkin 2001, Dvorkin, Cavaliere & Goldschmit 1995\1997\1998, Dvorkin &

Petöcz 1993).

5.5 Newtonian fluids

We define as an ideal or Newtonian fluid flow a viscous and incompressible

one.

The first property of a Newtonian fluid is the lack of memory: Newtonian

fluids do not present an elastic behavior and they do not store elastic energy.

Regarding the incompressible behavior we can write the continuity equation,

using the result of Example 4.4 as,

∇ · t v =0· (5.170)

It is important to remark that even though there are some fluids that can

be considered as incompressible, most of the cases of interest in engineering

practice are flows where Eq. (5.170) is valid even though the fluids are not

necessarily incompressible in all situations (e.g. isothermal air flow at low

Mach numbers) (Panton 1984).

The constitutive relation for the Newtonian fluids can be written in the

spatial configuration as,

t σ = t p t g +2µ t d . (5.171)

In the above equation, t σ is the Cauchy stress tensor, t p is its first invariant

also called the mechanical pressure, t d is the strain-rate tensor and µ is the

fluid viscosity that we assume to be constant (it is usually called “molecular

viscosity”).

Note that for an incompressible flow t dii =0and therefore t d = t d D .

Taking into account the incompressibility constraint in Eq. (5.170), it is

important to realize that the pressure cannot not be associated to its energy

conjugate: the volume strain rate, because it is zero; hence, the pressure will

have to be determined from the equilibrium equations on the fluid-flow domain

boundaries. Therefore it is not possible to solve an incompressible fluid flow

in which all the boundary conditions are imposed velocities, at least at one

boundary point we need to prescribe the tractions acting on it.

Many industrially important fluids, like polymers, do not obey Newton’s

constitutive equation. They are generally called non-Newtonian fluids. When

5.5 Newtonian fluids 181

bulk metal forming processes are described neglecting the material elastic behavior

(i.e. neglecting the material memory) the resulting constitutive equation

is usually a non-Newtonian one (Zienkiewicz, Jain & Oñate 1977).

5.5.1 The no-slip condition

When solving a fluid flow usually two kinematic assumptions are made:

• At the interface between the fluid and the surrounding solid walls the

velocity of the fluid normal to the walls is zero.

• At the interface between the fluid and the surrounding solid walls the

velocity of the fluid tangential to the walls is zero.

The first of the above assumptions is quite obvious when referring to nonporous

walls: the fluid cannot penetrate the walls.

The second of the above assumptions is not so obvious and, as a matter

of fact, it has been historically the subject of much controversy; our faith in

it is only pragmatic: it seems to work (Panton 1984).

6

Variational methods

In this chapter we will assume that the reader is familiar with the fundamentals

of variational calculus. The topic can be studied from a number of

references, among them (Fung 1965, Lanczos 1986, Segel 1987, Fung & Tong

2001).

The most natural way for starting the presentation of the theory of mechanics

is by accepting the Principle of Momentum Conservation as a law of

Nature and then stepping forward to demonstrate the Principle of Virtual

Work as a consequence of the momentum conservation; this is perhaps the

most direct way for developing the mechanical concepts because the Principle

of Momentum Conservation is quite intuitive to the reader with a background

in basic mechanics.

An alternative route for developing the theory of mechanics is by accepting

the Principle of Virtual Work as a law of Nature and then stepping forward to

demonstrate the Principle of Momentum Conservation. This route is perhaps

not as intuitive as the first one but equally valid from a formal point of view.

However, more important than deciding which formulation is aesthetically

more rewarding, an important fact for the scientist or engineer interested in

solving advanced problems in mechanics is that the Principle of Virtual Work,

and the other variational methods that can be derived from it, are the bases

for the development of approximate solutions to problems for which analytical

solutions cannot be found (Washizu 1982, Fung & Tong 2001).

6.1 The Principle of Virtual Work

We have represented in Fig. 6.1 the spatial configuration of a continuum body

t B; its external surface t S can be subdivided into:

t Su : on this surface the displacements are prescribed as boundary conditions,

t Sσ : on this surface the external loads are prescribed as boundary conditions.

184 **Nonlinear** continua

Fig. 6.1. Spatial configuration of a continuum

It is important to realize that a given point can pertain to t Su in one

direction and to t Sσ in another direction, but at one point, the displacement

and the external load corresponding to the same direction cannot be simultaneously

specified. Taking this into account we realize that the surfaces t S, t Su

and t Sσ have to be defined as the addition of the surfaces corresponding to

each of the three space directions. Also,

t S = t Su ∪ t Sσ ,

t Su ∩ t Sσ = ∅ .

The external forces acting on the body tB in the spatial configuration are:

t t t : external loads per unit surface acting on Sσ,

tb : external loads per unit mass.

We refer the continuum body to a spatial Cartesian coordinate system

{ tzα }.

To each point in the t−configuration, which is an equilibrium configuration

of the continuum body, we can associate a displacement vector tu. We can also define an admissible displacements field as (Fung 1965),

t eu( t z α )= t u( t z α )+δ t u( t z α ) . (6.1)

In the above δ t u is the variation of the displacements field, called the

virtual displacements. The virtual displacements have to satisfy the boundary

condition δ t u ≡ 0 on t Su and they are arbitrary on t Sσ, (Fung 1965).

6.1 The Principle of Virtual Work 185

Assuming that when the continuum evolves from tu to teu, theexternal

loads remain constant, the work performed by them is the virtual work of the

external loads ( δ t Wext),

δ t Z

Z

Wext =

t t t t

b · δ u ρ dV +

t t t

t · δ u dS. (6.2)

t V

Using in the above Eq.(3.7), which is an equilibrium equation for the particles

on the body surface, we get

Z

Z

t t t

t · δ u dS =

¡ tσαβ

δ

tSσ tSσ t ¢ tnα

uβ

t dS (6.3a)

Z

¡ tσαβ

= δ

tS t ¢ tnα

uβ

t dS (6.3b)

Z

¡ tσαβ

= δ t ¢

uβ ,α t dV . (6.3c)

t V

In the above, for deriving the last line we have used Gauss’ theorem.

Hence,

Z

Z

t t t

t · δ u dS =

t

σαβ,α δ t uβ t Z

dV +

t

σαβ δ( t uβ,α) t dV. (6.4)

t Sσ

t V

In the last integral we have used the equality (Fung 1965)

µ t ∂ uβ

δ

∂t

=

zα

∂

∂t ¡ ¢ t

δ uβ . (6.5)

zα

Using in Eq. (6.4) the momentum conservation equation (Eq.(4.27b)),

Z

Z

t t t

t · δ u dS = −

t

bα δ t uα t ρ t Z

dV +

t

σαβ δ ¡ ¢ t tdV

uβ,α .(6.6)

t Sσ

t V

In the derivation of the above equation we have assumed in Eq. (4.27b)

that Dtv Dt =0; however, dynamic problems can also be considered by including

the inertia forces among the external loads per unit mass (Crandall 1956).

It is easy to show that,

t σαβ δ t uβ,α = t σαβ δ t εαβ

δ t εαβ = 1

2

t Sσ

t V

t V

(6.7a)

" ¡ ¢ t

∂ δ uα

+ ∂ ¡ δ t ¢ #

uβ

. (6.7b)

∂ t zβ

∂ t zα

In the above equations the terms δ t εαβ are the infinitesimal strain components

developed by the virtual displacements; hence, we refer to them as

virtual strain components.

186 **Nonlinear** continua

Note that the actual strains in the t-configuration

are arbitrary, only the virtual strains are infinitesimal.

Replacing with Eq. (6.7a) in Eq. (6.6),

Z

Z

Z

t t t t

b · δ u ρ dV +

t t t

t · δ u dS =

t V

t Sσ

t V

t σ : δ t ε t dV . (6.8)

The above equation is the mathematical statement of the Principle of

Virtual Work and it states that for a continuum body in equilibrium, the

virtual work of the external loads equals the virtual work of the stresses.

Notes:

• No assumption was made on the material, i.e. on its stress - strain relation;

hence, the Principle of Virtual Work holds for any constitutive relation.

• No assumption was made on the actual strains in the spatial configuration;

hence, the Principle of Virtual Work holds for finite or infinitesimal strains.

• No assumption was made on the external loads; hence, the Principle of

Virtual Work holds for conservative and nonconservative loads (Crandall

1956).

• The integrals in Eq. (6.8) are calculated on the spatial configuration of the

body.

• The Principle of Virtual Work was derived from momentum conservation

and not from energy conservation.

As we see the Principle of Virtual Work is a very general statement, holding

for any type of nonlinearities that may be present in the spatial configuration

(material and geometrical nonlinearities).

It is also possible to go through the inverse route, that is to say, starting

from the Principle of Virtual Work to demonstrate the equations of momentum

conservation. For this demonstration we refer the reader to (Fung 1965,

Fung & Tong 2001).

6.2 The Principle of Virtual Work in geometrically

nonlinear problems

In Eq. (6.8) the integrals are calculated in the spatial configuration of the

continuum, which is normally one of the problem unknowns; however, for geometrically

linear problems ( t uα,β

6.2 The Principle of Virtual Work in geometrically nonlinear problems 187

In the case of geometrically nonlinear problems it is convenient to calculate

the integrals in Eq. (6.8) using the reference configuration.

The coordinates t zα remain constant during the virtural displacement;

hence,

dδ t εαβ

dt

µ t

1 ∂δ vα

=

2 ∂t +

zβ

∂δtvβ ∂t

zα

(6.10)

where δ t v is the virtual velocity vector.

Equation (3.11) is valid for any velocity field, in particular when we use

the virtual velocity field, we get using Eq. (6.10),

Z

Z

t t t

σ : δ ε dV =

t t ◦

τ : δ ε dV. (6.11)

t V

Replacing in Eq. (6.8),

Z

t V

t b · δ t u t ρ t dV +

Z

t Sσ

◦ V

Z

t t t

t · δ u dS =

◦ V

t τ : δ t ε ◦ dV. (6.12)

Also, using another pair of energy conjugate measures,

Z

Z

Z

t t t t

b · δ u ρ dV +

t t t

t · δ u dS =

t

oS : δ t oε ◦ dV. (6.13)

t V

t Sσ

In the above,

t

oS: second Piola-Kirchhoff stress tensor,

t

oε: Green-Lagrange strain tensor.

We can define a general load, either a load per unit mass or per unit surface

as:

t l m

f = λ ϕ . (6.14)

In the above, l λ is a scalar proportional to the load modulus and (Schweizerhof

& Ramm 1984):

• l =0implies that the load modulus is a function of the reference coordinates

(body-attached loads);

• l = t implies that the load modulus is a function of the spatial coordinates

(space-attached loads).

The unitary vector m ϕ is a direction and (Schweizerhof & Ramm 1984):

• m =0implies a constant direction load;

• m = t implies a follower load (the direction is a function of the body

displacements).

◦ V

188 **Nonlinear** continua

Example 6.1. JJJJJ

Buckling of a circular ring.

In (Brush & Almroth 1975) we find that the elastic buckling pressure acting

on a circular ring depends on the type of load that we consider:

Load Buckling pressure

Hydrostatic pressure loading pcr =3 EI

a 3

Centrally directed pressure loading pcr =4.5 EI

a 3

Both are cases of follower loads but, the description of the load as a function

of the displacements is different. JJJJJ

Using the mass conservation principle in Eq. (4.20d) we can write,

Z

Z

t t t t

b · δ u ρ dV =

t t o ◦

b · δ u ρ dV. (6.15)

t V

At each point on the surface bounding the continua we can calculate,

hence, Z

t Sσ

◦ V

t dS = t JS ◦ dS, (6.16)

Z

t t t

t · δ u dS =

◦ Sσ

t t · δ t u t JS ◦ dS. (6.17)

Therefore, we can write the principle of Virtual Work calculating the integrals

over the reference configuration as,

Z

Z

t t o ◦

b · δ u ρ dV +

t t t

t · δ u JS ◦ Z

dS =

t

oS : δ t oε ◦ dV. (6.18)

◦ V

◦ Sσ

Example 6.2. JJJJJ

Let us consider the work of the external loads per unit surface of the spatial

configuration for the case of a typical follower load: the hydrostatic fluid

pressure. In this case,

t t = t p t n

where tn is the surface external normal. For this case,

δ t W S Z

t t t t

ext = pδu · n dS.

t Sσ

◦ V

6.2 The Principle of Virtual Work in geometrically nonlinear problems 189

Using Nanson’s formula (Example 4.9) we get,

δ t W S Z

t t t ◦ t

ext = pδu · J n · oX −1 ◦ dS .

◦ Sσ

For a case with infinitesimal strains,

and therefore,

δW S Z

ext =

◦ Sσ

t J ≈ 1

t

oX −1 ≈ t oR T

h

toR T · δ t i

u · £ ¤ t ◦

p n

◦dS

.

In the above equation, the first bracket inside the integral is the displacements

variation rotated back from the spatial configuration to the material one; and

the second bracket is the load per unit surface, but in the reference configuration.

Hence, it is very important to realize that for the case of fluid-pressure loads

and infinitesimal strains we can easily calculate the external surface loads

virtual work in the reference configuration. JJJJJ

Using other alternative energy conjugate stress/strain measures,

Z

Z

t t o ◦

b·δ u ρ dV +

t t t

t·δ u JS ◦ Z

dS =

t

oP T : δ t oX ◦ dV,(6.19)

◦ V

◦ Sσ

and, for an isotropic constitutive relation

Z

Z

t t o ◦

b · δ u ρ dV +

t t t

t · δ u JS ◦ Z

dS =

◦ V

◦ Sσ

6.2.1 Incremental Formulations

◦ V

◦ V

t Γ : δ t oH ◦ dV.(6.20)

We have represented in Fig. 6.2 a typical Lagrangian analysis where the configuration

at t =0is known and the configuration at t = t1 is sought.

Normally, we perform an incremental analysis; that is to say, we determine

the sequence of equilibrium configurations at t =0, ..., t, t + ∆t, ...t1.

In this incremental analysis, the basic link to be analyzed is the generic

step t → t + ∆t; that is to say, knowing the equilibrium configuration at t we

seek the one at t + ∆t. Of course, once this generic step can be solved, then

the complete incremental analysis can be performed.

In what follows, to describe the step t → t + ∆t we follow the presentation

in (Bathe 1996).

190 **Nonlinear** continua

Fig. 6.2. Lagragian incremental analysis

First, we have to recognize that for describing the t + ∆t-configuration

we can use as a reference configuration either the one at t =0or any of the

intermediate ones, already known. In what follows we will specifically analyze

two particular cases:

• The total Lagrangian formulation, whereweuseasthereferenceconfiguration

the one at t =0(usually the undeformed configuration)

• The updated Lagrangian formulation, where we use as the reference configuration

the previous one (t).

The total Lagrangian formulation

Using the principle of Virtual Work to define the t + ∆t equilibrium configuration,

we write:

Z

t+∆t

o S IJ δ t+∆t

o εIJ ◦ dV = δ t+∆t Wext . (6.21)

◦ V

In the above equation,

t+∆t

o SIJ : components of the second Piola-Kirchhoff stress tensor, correspondingtothet

+ ∆t configuration and referred to the configuration at t =0.

t+∆t

o εIJ: components of the Green-Lagrange strain tensor, corresponding to

the t + ∆t configuration and referred to the configuration at t =0.

n o

A general curvilinear coordinate system ◦xI ,I =1, 2, 3 is used in the

reference configuration, with covariant base vectors ◦g (L =1, 2, 3).

L

Thevolumeofthereferenceconfiguration is ◦V and the virtual work of

the external loads acting at time t + ∆t is δ t+∆t Wext (see the previous

section for a discussion on the calculation of this term).

We can write,

t+∆t

o S IJ = t oS IJ + oS IJ

(6.22)

where oSIJ are the components of the incremental second Piola-Kirchhoff

stress tensor. It is important to recognize that the three tensors in Eq. (6.22)

6.2 The Principle of Virtual Work in geometrically nonlinear problems 191

Fig. 6.3. Total Lagrangian formulation

are referred to the same reference configuration (t = 0). For the Green-

Lagrange strain tensor we can also write an incremental equation,

t+∆t

o εIJ = t oεIJ + oεIJ , (6.23)

again, in the above equation the three tensors are referred to the same reference

configuration (t =0).

Replacing with Eqs. (6.22) and (6.23) in Eq. (6.21) and taking into account

that for the step that we are investigating, t oεIJ is data and therefore, δ t oεIJ =

0, weget Z

( t oS IJ + oS IJ ) δoεIJ ◦ dV = δ t+∆t Wext . (6.24)

◦ V

We write an incremental constitutive equation of the form,

and get,

Z

◦ V

oS IJ = oC IJKL oεKL

(6.25)

( t oS IJ + oC IJKL oεKL) δoεIJ ◦ dV = δ t+∆t Wext . (6.26)

Refering the problem to a fixed Cartesian system, we can write for a generic

particle P , as shown in Fig. 6.3:

We can show that (Bathe 1996),

t P t P ◦ P

u = x − x (6.27a)

u P = t+∆t u P − t u P . (6.27b)

192 **Nonlinear** continua

oεαβ = 1 ¡

ouα,β + ouβ,α +

2

t ouγ,α ouγ,β + t ¢

ouγ,β ouγ,α + ouγ,α ouγ,β .

(6.28)

In the above equation, ouα,β = ∂uα

∂ ◦ z β and t ouα,β = ∂t uα

∂ ◦ z β .

Hence, it is possible to rewrite Eq. (6.28) as,

oεαβ = oeαβ + oηαβ (6.29a)

oeαβ = 1 ¡

ouα,β + ouβ,α +

2

t ouγ,α ouγ,β + t ¢

ouγ,β ouγ,α (6.29b)

oηαβ = 1

2 ouγ,α ouγ,β . (6.29c)

The term in Eq. (6.29b) is linear in the unknown incremental displacements,

u, while the term in Eq. (6.29c) is nonlinear.

We introduce Eq. (6.29a) in Eq. (6.26) and obtain,

Z

[ t oSαβ + oCαβγδ (oeγδ+ oηγδ)] δ(oeαβ +oη αβ) ◦ dV = δ t+∆t Wext . (6.30)

◦ V

The above is the momentum balance equation at time t + ∆t; whichisa

nonlinear equation in the incremental displacement vector. In order to solve

it we use an iterative technique, in which the first step is the linearization of

Eq. (6.30) (Bathe 1996). Keeping only up to the linear terms in u, weobtain

the linearized momentum balance equation:

Z

[oCαβγδ oeγδ δoeαβ +

◦V t oSαβ δoηαβ] ◦ dV (6.31)

Z

= δ t+∆t Wext −

◦ V

t

oSαβ δoeαβ ◦ dV .

The updated Lagrangian formulation

Using the principle of Virtual Work to define the t + ∆t equilibrium configuration,

we write:

Z

t+∆t

t S IJ δ t+∆t

t εIJ t dV = δ t+∆t Wext . (6.32)

t V

In the above equation,

t+∆t

t SIJ : components of the second Piola-Kirchhoff stress tensor, correspondingtothet

+ ∆t configuration and referred to the configuration at t.

t+∆t

t εIJ: components of the Green-Lagrange strain tensor, corresponding to

the t + ∆t configuration and referred to the configuration at t.

We can write,

t+∆t

t S IJ = t tS IJ + tS IJ

(6.33)

6.2 The Principle of Virtual Work in geometrically nonlinear problems 193

where t tS IJ = t σ IJ and tS IJ are the components of the incremental second

Piola-Kirchhoff stress tensor; it is important to recognize that the three tensors

in Eq. (6.33) are referred to the spatial configuration at time t.

Also,

t+∆t

t εIJ = tεIJ . (6.34)

because t tεIJ =0.

Replacing with Eqs. (6.33) and (6.34) in Eq. (6.32), we get

Z

¡ tσIJ + tS IJ¢ δ (tεIJ) t dV = δ t+∆t Wext , (6.35)

t V

we can write an incremental constitutive equation referred to the t−configuration,

and get,

Z

t V

tS IJ = tC IJKL tεKL , (6.36)

¡ tσIJ + tC IJKL ¢

tεKL δ (tεIJ) t dV = δ t+∆t Wext . (6.37)

In a fixed Cartesian system we can show that (Bathe 1996),

tεαβ = 1

2 (tuα,β + tuβ,α + tuγ,α tuγ,β) (6.38)

where tuα,β = ∂uα

∂tzβ .

We can decompose the strain increment into a linear and a nonlinear

increment in the unknown incremental displacement; that is to say,

tεαβ = teαβ + tηαβ teαβ = 1

2 (tuα,β + tuβ,α) (6.39)

tηαβ = 1

2 (tuγ,α tuγ,β)

Hence we can write Eq. (6.37) as,

.

Z

tV £ tσαβ

+ tCαβγδ

¡

teγδ

¢¤ ¡

+ tηγδ δ teαβ

¢

+

tdV

tηαβ t+∆t

= δ Wext .

(6.40)

The above is the momentum balance equation at time t + ∆t; whichis

a nonlinear equation in the incremental displacement vector. Proceeding in

the same way as in the total Lagrangian formulation we obtain the linearized

momentum balance equation (Bathe 1996):

Z

tV tCαβγδ teγδ δteαβ t Z

dV +

tV t t

σαβ δtηαβ dV (6.41)

= δ t+∆t Z

Wext −

t

σαβ δteαβ t dV.

t V

194 **Nonlinear** continua

It is easy to show that

oS IJ =

o ρ

t ρ tS ij ¡ t oX −1¢ I

i

¡ toX −1¢ J

j

(6.42)

oεIJ = tεij t oX i I t oX j

J

(6.43)

and therefore if the same material is considered in both formulations the

incremental constitutive tensors should be related,

◦C IJKL =

o ρ

t ρ tC mnpq ¡ t oX −1¢ I

m

¡ toX −1¢ J

n

¡ toX −1¢ K

p

¡ toX −1¢ L

q

. (6.44)

Any problem can be alternatively solved using either the total or the updated

Lagrangian formulations and the results should be identical (Bathe

1996).

For solving finite-strain elastoplastic problems, in Sect. 5.2.6 we introduced

an adhoc incremental formulation, the total Lagrangian-Hencky formulation.

6.3 The Principle of Virtual Power

There are formulations where the primary unknowns are the material velocities

rather than the material displacements (e.g. fluid problems, metal-forming

Eulerian (Dvorkin, Cavaliere & Goldschmit 1995, Dvorkin & Petöcz 1993) or

ALE formulations (Belytschko, Liu & Moran 2000), etc.). For these cases the

momentum conservation leads to,

Z

Z

Z

t t t t t t t t t t

b · δ v ρ dV + t · δ v dS = σ : δ d dV . (6.45)

t V

t Sσ

In the above equation tv is the material velocity at a point and td is the

strain-rate tensor.

Of course, we can use, for formulating the Principle of Virtual Power, other

energy conjugated stress/strain rate measures, for example:

Z

Z

t t o ◦ t t t

b · δ v ρ dV + t · δ v JS ◦ Z

t t ◦

dS = τ : δ d dV, (6.46a)

Z

Z

◦ V

◦ V

Z

◦ V

◦ V

◦Sσ Z t

t t o ◦

b · δ v ρ dV +

Z

t t o ◦

b · δ v ρ dV +

◦ Sσ

Z

t t o ◦

b · δ v ρ dV +

◦Sσ t V

◦ V

t t t

t · δ v JS ◦ Z

t

dS = oS : δ

◦V t ·

oε

◦ dV, (6.46b)

t t t

t · δ v JS ◦ Z

t

dS = oP

◦V T : δ t ·

oX

◦ dV,

(6.46c)

t t t

t · δ v JS ◦ Z

·

t t

dS = Γ : δoH ◦ dV, (6.46d)

◦ Sσ

◦ V

6.4 The Principle of Stationary Potential Energy 195

the last one only being valid for isotropic constitutive relations.

6.4 The Principle of Stationary Potential Energy

As we remarked above, the Principle of Virtual Work can be used for any

material constitutive relation, for any type of loading and for any nonlinearity

inthecasetobeanalyzed.

In the present section we will specialize the Principle of Virtual Work for:

• Hyperelastic materials.

• Conservative external loads.

For a hyperelastic material we have seen in Chap. 5 (Eq. (5.3d) that,

t

oS IJ = o ρ ∂ tU( t oε)

∂ t oεIJ

. (6.47)

The external conservative loads are the external loads that can be derived

from a potential. Hence, a load field is said to be conservative in a region if

the net work done around any closed path in that region is zero (Crandall

1956).

A typical conservative load system can be represented as,

t f = ◦ λ ◦ ϕ . (6.48)

Following the definitions introduced above, the load system in Eq. (6.48)

is a body attached load system with constant direction.

For conservative loads per unit mass, we write

t ∂

b = − tG ( tu) (6.49)

∂u

and for conservative surface loads

t ∂

t = − tg ( tu) . (6.50)

∂u

Note that the above-defined surface loads are defined as loads per unit

reference surface; therefore, its resultant at time t is R t ◦

◦ t dS .

Sσ

We now define a functional of the function tu called the potential energy

functional:

Z

t

oΠ =

◦V ◦ ρ ¡ tU + t G ¢ ◦dV +

Z

◦ Sσ

t g ◦ dS (6.51)

196 **Nonlinear** continua

Therefore,

δ t Z "

◦Π =

◦ V

◦ ρ ∂ t U

∂ t ◦ε : δt ◦ε + ◦ ρ ∂t G

∂u · δt u

# Z

◦

dV +

◦ Sσ

∂ t g

∂u ·δt u ◦ dS. (6.52)

In the above, δ t u are admissible variations ¡ δ t u = 0 on ◦Su see Fig. 6.1 ¢

and δ t ◦ε is derived from the displacement variations. Therefore,

δ t Z

◦Π =

h

t◦S:

δ t ◦ε − ◦ ρ t b · δ t i Z

u

◦dV

−

t t ◦

t · δ u dS. (6.53)

◦ V

Hence, for a hyperelastic material under a conservative load system, the

principle of virtual work, in Eq. (6.18), can be written as

◦ Sσ

δ t ◦Π =0. (6.54)

The above equation states that when the t−configuration is in equilibrium

the potential energy functional reaches a stationary value; i.e. it fulfills the

necessary requirements for being an extreme (Fung 1965).

In what follows we show that in the case of infinitesimal strains the potential

energy not only is stationary at the equilibrium configuration but it

actually attains there a minimum.

Using the nomenclature introduced in Eq. (6.1) we write the potential

energy functional for an admissible configuration close to the equilibrium one

as

t

oΠ 0

Z

=

◦V −

Z

◦ Sσ

Using a Taylor expansion,

t t

U( ◦ε + δ t ◦ε) = t U( t ◦ε)+ ∂tU ∂t ¯

◦ε ¯

t◦ ε

Hence,

t

oΠ 0

− t oΠ = δ t Z

◦Π +

◦V ◦ ρ £ tU( t ◦ε + δ t ◦ε) − t b · ¡ t u + δ t u ¢¤ ◦dV (6.55)

t t · ¡ tu + δ t u ¢ ◦dS .

1 ◦ t

ρδ

2

◦ε :

: δ t ◦ε + 1

2 δt ◦ε :

∂2tU ∂t ◦ε ∂t ¯

◦ε ¯

t◦ ε

∂2tU ∂t ◦ε ∂t ¯

◦ε ¯

t◦ ε

: δ t ◦ε + ··· .

(6.56)

: δ t ◦ε ◦ dV + ··· . (6.57)

Since at equilibrium δ t ◦Π = 0, the sign of the l.h.s. is the sign of the

integrand on the r.h.s..

6.4 The Principle of Stationary Potential Energy 197

In the case of infinitesimal strains case we can assume that t ◦ε ≈ 0 and we

have tU(0) =0(convention) and t ◦S ¯ =

0 ◦ρ ∂tU ∂t ◦ε ¯ = 0; hence, from Eq. (6.56)

0

t U(δ t ◦ε) = 1

2 δt ◦ε :

∂2tU ∂t ◦ε ∂t ¯

◦ε ¯

0

: δ t ◦ε + ··· . (6.58)

Since, in a stable material the value of the elastic strain energy is positive

for any strain tensor (the elastic strain energy is a positive-definite function)

we conclude that,

t

oΠ 0

− t oΠ>0 , (6.59)

and the potential energy is a local minimum at the equilibrium configuration.

In the infinitesimal strains case we call it the minimun potential energy

principle (Washizu 1982).

Example 6.3. JJJJJ

Conservative and nonconservative loading.

(a) Conservative loading

Let us consider a linear elastic, cantilever beam under the conservative endload

shown in the figure,

Conservative load

The elastic energy stored in the beam is,

Z L

t

U =

0

EI

2

µ 2 t d u2

dtz 2

2

1

d t z1

where E is Young’s modulus and I is the beam cross section moment of inertia.

The Principle of Virtual Work states,

198 **Nonlinear** continua

δ t U = t Pδu2

δ ¡ ¢ t t t

U− P u2 =0

where t Π = t U − t P t u2 is the potential energy of the system.

(b) Nonconservative loading

We now consider the same linear elastic cantilever beam but under a follower

load, as shown in the figure

The principle of virtual work states,

Body-attached follower load

δ t U = − t P sin ¡ t θ ¢ δu1 + t P cos ¡ t θ ¢ δu2 .

For small displacement derivatives we can approximate

sin ¡ µ

¢ t

t t d u2

θ ≈ θ ≈

hence,

Since

t P

δ t U = t P

∙ µ t d u2

−

cos ¡ t θ ¢ ≈ 1

d t z1

∙ µ t d u2

−

d t z1

δ

L

t u1 + δ t u2

d t z1

L

δ

L

t u1 + δ t u2

¸

¸

6= − ∂t G

∂u · δt u

the load is nonconservative and the principle of stationary potential energy

cannot be used. JJJJJ

.

6.4 The Principle of Stationary Potential Energy 199

Example 6.4. JJJJJ

Stability of the equilibrium configuration (buckling) (Hoff 1956).

Let us consider the system shown in the following figure, in equilibrium at

time t, inthestraightconfiguration:

t P : axial conservative load ; L : length of the rigid bar ; k : stiffness of

the linear spring; kT : stiffness of the torsional spring.

Assume that the equilibrium configuration is perturbed with a rotation

θ

200 **Nonlinear** continua

For θ

Pcr defines the bifurcation or buckling load.

The equilibrium path is

6.4 The Principle of Stationary Potential Energy 201

Since in the above derivation the terms higher than θ 2 were neglected, we

cannot assess anything about the branching equilibrium path. JJJJJ

Example 6.5. JJJJJ

Postbuckling behavior.

We repeat the previous example derivation keeping terms higher than θ 2 .By

doing this, we get

µ 2

θ

∆P = − L (1 − cos θ) ≈−L

Hence

t

oΠ = 1

2 kL2

t Uk = 1

2 k (L sin θ)2 ≈ 1

2 kL2

− θ4

2 4!

t t

G = − P∆P≈ t µ 2

θ θ4

PL −

2 4!

µ

θ − θ3

3!

t UT = 1

2 kT θ 2 .

µ

θ 2 − θ4

θ6

+ +

3 36

1

2 kT θ 2 + t PL θ2

2 − t PL θ4

24 .

For the equilibrium configuration δ t ◦Π =0and therefore,

∙

kL 2

2

µ

2 θ3 θ5

θ − + + kT θ +

3 12

t PLθ− t PL θ3

¸

6

δθ =0.

202 **Nonlinear** continua

Since δθ is arbitrary, we get, neglecting terms higher than θ 3 ,

∙

kL 2

µ

1 −

2 θ2

+ kT +

3

t PL

µ

1 − θ2

¸

6

θ =0

which has again two possible solutions:

(i) θ =0the straight solution

(ii) tP = − kL

³ ´

2 θ2 1− 3 + kT L

1− θ2

6

In the second solution, for θ =0, tP = Pcr = − ¡ kL+ kT

¢

L .

The bifurcation point is the same as the one calculated in the previous example;

however, now θ is defined.

We see that for θ>0 Eq. (ii) provides tP = tP (θ).

If we examine the case with kT =0,weget

− t ³ ´

2 θ2

kL 1 − 3

P =

and we can represent

If we examine the case with k =0,weget

and we can represent

− t P =

³

L

1 − θ2

6

kT

1 − θ2

6

´

6.4 The Principle of Stationary Potential Energy 203

It is clear that the above cases represent two very different behaviors from a

structural point of view.

For the case kT = 0, the buckling is catastrophic because for θ > 0 the

load-carrying capacity of the structure keeps dimishing: unstable postbuckling

behavior.

For the case k =0, the load-carrying capacity of the structure increases after

buckling: stable postbuckling behavior. JJJJJ

Example 6.6. JJJJJ

Natural boundary conditions.

Let us study the following linear elastic cantilever beam under conservative

loads:

E : Young’s modulus; I : moment of inertia of the beam cross section.

Assume tu2 = t u2( tz1) is the beam transversal displacement and using linear

beam theory (Hoff 1956)

204 **Nonlinear** continua

t Π =

Z L

0

− t PL

EI

2

¢

¡ tu2

µ 2 t d u2

dtz1 2

2

L − t ML

d t Z L

z1 −

0

µ t d u2

.

d t z1

For the equilibrium configuration δ t Π =0; hence

Z L

0

EI

µ 2 t d u2

dtz1 2

δ

− t PL δ ¡ ¢ t

u2

µ 2 t d u2

dtz1 2

In the first integral we use (Fung 1965) δ

grating by parts twice, we get

Z L

0

EI

µ 2 t d u2

dtz1 2

2 d

dtz1 2

∙

− EI d3 tu2 dtz1 3 δt ¸L

u2

0

L

d t Z L

z1 −

0

∙µ t d u2

t q t u2 d t z1

¡ ¢ t tu2 t

qδ d z1

¸

((A))

L − t ML δ

dtz1 L

=0.

³ 2 t

d u2

dtz1 2

´

= d2

dtz1 2

¡ ¢ t

δ u2 and inte-

∙

¡ ¢ t t

δ u2 d z1 = EI d2 tu2 Z L

+

0

d t z1 2

d

d t z1

EI d4 t u2

d t z1 4 δt u2 d t z1 .

At tz1 = 0 we have as boundary conditions tu2 = dtu2 dtz1 ¡ ¢ t

δ u2 tz1=0 =

h ³ t

d u2 δ dt ´i

=0.

z1 tz1=0 Replacing in (A), we get

Z L

0

dtz1 4 − t ¸

q

∙

EI d4 t u2

+

∙µ

EI d2 t u2

d t z1 2

δ t u2 d t ∙µ

z1 − EI d3 t

u2

L

− t ML

¸

d t z1 3

µ t d u2

δ

d t z1

L

L

+ t PL

=0.

¡ ¢ t

δ u2

¸L

0

= 0, hence,

¸

δ ¡ ¢ t

u2 L

Since δ t u2 is arbitrary at every point 0 6 t z1 6 L we must fulfill the

following differential equation

t q = EI d 4 t u2

d t z1 4

which is the well-known equation of beam theory.

At t z1 = L we get

6.4 The Principle of Stationary Potential Energy 205

Essential (rigid) boundary conditions or Natural boundary condition

Either ( t u2) L is fixed and δ ( t u2) L =0 or t PL = − EI

Either

³ t

d u2

dt ´

is fixed and δ

z1 L

³ t

d u2

dt ´

z1 L =0 or tML = EI

³ 3 t

d u2

dtz1 3

´

L

³ 2 t

d u2

dtz1 2

´

L

. JJJJJ

Example 6.7. JJJJJ

The Rayleigh-Ritz method.

In the previous example, from the principle of stationary potential energy we

derived the differential equation that governs the deformation of a cantilever

beam.

Usually, we need to work in the opposite direction: we know the differential

equations that govern the deformation of a continuum but we cannot integrate

them and we resort to the principle of stationary potential energy to derive

an approximate solution. One method for deriving approximate solutions is

the Rayleigh-Ritz method (Hoff 1956).

Let us consider again the linear case analyzed in the previous example and

letusassumethatwewanttoderiveanapproximatesolutionforthecase

t q = t PL =0.Forthiscase

t Π =

Z L

0

EI

2

µ 2 t d u2

dtz1 2

2

d t z1 − t ML

µ t d u2

d t z1

To derive an approximate solution we consider trial functions that fulfill the

geometrical or essential boundary conditions,

µ t

t d u2

u2(0) = =0.

For example

dtz1 0

t ª

u 2 ( t µ

z1) =a 1 − cos π t

z1

2L

where the parameter a will be determined by imposing the minimization condition

on

t Π ª = t Π ª (a) .

Using the adopted trial function, we get

t Π ª = EIπ 4 a 2

64 L 3

−

t ML aπ

2 L

.

L

.

206 **Nonlinear** continua

The minimum value that can attain the above functional is, within the considered

set of trial functions, our best approximation to the equilibrium configuration.

Imposing

∂ t Π ª

∂a =0,

we get

EIπ3 .

Therefore our approximate solution is

t ª

u 2 ( t z1) = 16 tML L2 EIπ3 µ

1 − cos πt

z1

2L

a = 16 t ML L 2

t ª 4

Π = − tM 2 L L

EIπ2 .

For the case we are analyzing the exact solution is

¡ ¢ t

t exact tz1 ML (

u2 =

tz1) 2

2 EI

t M 2 L L

t exact

Π = −

2 EI

It is obvious that t Π ª > t Π exact .

If we want to improve our approximate solution we enrich the trial function

set using, for example

µ

¡ ¢ t ⊕ tz1

u2 = a 1 − cos πt µ

z1

+ b 1 − cos

2L

πt

z1

L

It is important to note that the above defined trial function:

◦ Fulfills the essential (rigid) boundary conditions.

◦ Contains the previous one, tu ª

2 ( tz1), asaparticularcase(b =0).

Since we will determine the values of both constants by imposing on t Π ⊕ the

necessary conditions for attaining a minimum, it is obvious that

t Π ⊕ 6 t Π ª .

That is to say, we will either find the same solution as before (b =0and

t Π ⊕ = t Π ª )orabetterone(b 6= 0 and t Π ⊕ < t Π ª ). We cannot

deteriorate the solution by adding more terms in the trial function.

Using tu ⊕

2 ( tz1) , we get

t ⊕ EIπ

Π = 3

2 L3 µ

ab π

+

3 32 a2 + π

2 b2

tML aπ

− .

2 L

Imposing ∂t Π ⊕

∂a = ∂t Π ⊕

∂b =0,weget

.

.

a =0.6294

b = −0.0668

t Π ⊕ = −0.4940

t ML L 2

EI

6.5 Kinematic constraints 207

t ML L 2

EI

t M 2 L L

EI

It is clear from the derived values that, as expected: t Π exact < t Π ⊕ < t Π ª ,

and therefore t u ⊕

2

( t z1) isa“better”approximationthan t u ª

2

.

( t z1). JJJJJ

In the previous example we have introduced three relevant topics that we

want to highlight:

1. When obtaining approximate solutions using the Rayleigh-Ritz method,

based on the minimum potential energy principle (infinitesimal strains),

we can only rank the merit of different solution using their potential energy

value; that is to say, if t Π A < t Π B then the A-solution is a better

approximation than the B-solution.

2. The trial functions have to exactly satisfy the rigid boundary conditions

(admissible functions) but not the natural boundary conditions.

3. Approximate solutions do not need to fulfill exactly either the equilibrium

equation inside the dominium or the natural boundary conditions

(equilibrium equations on the boundary).

6.5 Kinematic constraints

IntheexampleshowninFig.6.4,wheretPis a conservative load, the potential

energy is,

t 1

Π =

2 k t u 2 − t P t v. (6.60)

For the inextensible string there is a kinematic constraint given by,

2 t v − t u =0. (6.61)

Hence, we have to minimize the functional of the potential energy given

in Eq.(6.60) under the constraint expressed in Eq. (6.61). Using the Lagrange

multipliers technique (Fung 1965, Fung & Tong 2001), we define a new functional

( t Π ∗ ) and we perform on it an unconstrained minimization:

t ∗ 1

Π =

2 k t u 2 − t P t v + t λ ¡ 2 t v − t u ¢

where tλ is the Lagrange multiplier.

(6.62)

208 **Nonlinear** continua

Fig. 6.4. Kinematics constraints

We need to determine the set ( t λ, t u, t v) that satisfies δ t Π ∗ =0.The

necessary conditions are,

From the above we get,

∂ t Π ∗

∂ t λ

∂ t Π ∗

∂ t u

∂ t Π ∗

∂ t v

=0 (6.63a)

=0 (6.63b)

=0. (6.63c)

t

t P

u =

2 k

(6.64a)

t

t P

v =

4 k

(6.64b)

t

t t P

λ = k u =

2

. (6.64c)

Example 6.8. JJJJJ

Physical interpretation of the Lagrange multiplier in the above example (Crandall

1956).

Let us assume that the string, instead of being inextensible has a stiffness ks;

hence, with tλ the tensile load on the spring we have,

¡ ¢ t t

2 v − u .

t λ = ks

6.6 Veubeke-Hu-Washizu variational principles 209

For this system with no constraints we can define the potential energy

t Πks

and δ t Πks =0leads to,

Hence,

Also,

= 1

2 k t u 2 + 1

2 ks

k t u − ks

2 ks

t u =

¡ 2 t v − t u ¢ 2 − t P t v

¡ 2 t v − t u ¢ = 0

¡ 2 t v − t u ¢ − t P =0.

t P

2 k

2 t v − t tλ u =

;

ks

t λ =

→

ks→∞

t P

2 .

Therefore, t λ is independent of the displacement (conservative load) and we

can use Eq. (6.62) for writing the potential energy, and identify the Lagrangian

multiplier with the string tensile load. JJJJJ

In the above example, it is important to realize that the Lagrangian multiplier

is the energy conjugate of the physical magnitude that represents the

constraint equation: the string elongation.

0 .

6.6 Veubeke-Hu-Washizu variational principles

In the previous section we analyzed a simple mechanical system and we developed

the imposition, using the Lagrange multipliers technique, of a kinematic

constraint on the stationarization of the potential energy functional.

In the present section we are going to generalize the above technique to

impose different constraints on the potential energy functional (Fraeijs de

Veubeke 1965).

6.6.1 Kinematic constraints via the V-H-W principles

Let us assume that we relax the compatibility conditions (see Sect. 2.15)

when formulating the potential energy functional; that is to say we consider

in Eq. (6.51) a strain tensor t oε that can not necessarily be derived from the

displacements field.

We now define the functional:

Z

Z

t

oΠ ∗ =

◦ V

o ρ £ tU ¡ toε ¢ + t G( t u) ¤ ◦ dV +

Z

+

◦V ◦ Sσ

t λ :( t oε ¡ t u ¢ − t o ε) ◦ dV

t g ¡ tu ¢ ◦dS (6.65)

210 **Nonlinear** continua

and we search for the equilibrium configuration imposing

δ t ◦Π ∗ =0 (6.66)

we have as independent variables: tu, t oε and tλ;witht oS = oρ ∂tU ∂t oε we get

t

oS : δ t oε ◦ Z

dV −

◦V Z

o t t ◦

ρ b · δ u dV −

◦Sσ Z

t t ◦

t · δ u dS (6.67)

◦V Z

+ δ

◦V t λ : ( t oε − t oε) ◦ Z

dV +

◦V t t

λ :(δoε − δ t oε) ◦ dV =0.

Considering that the variations are arbitrary,

Z

t t

λ : δ oε ◦ Z

Z

dV =

o t t ◦

ρ b · δ u dV +

t t ◦

t · δ u dS (6.68)

◦ V

Z

Z

◦ V

◦ V

The last equations impose the conditions

◦ V

◦ Sσ

¡ toS − t λ ¢ : δ t oε ◦ dV = 0 (6.69)

δ t λ :( t oε − t oε) ◦ dV =0 . (6.70)

t

oS = t λ (6.71)

t

oε = t oε (6.72)

which are always fulfilled for the continuum problem but not necessarily in

finite element approximations.

Adding Eqs. (6.68) and (6.69), we get

Z

t

oS : δ

◦V t oε ◦ Z

dV +

◦V t t

λ :(δoε − δ t oε) ◦ Z

dV

Z

(6.73)

=

o t t ◦

ρ b · δ u dV +

t t ◦

t · δ u dS .

◦ V

From the above, we can state the Principle of Virtual Work as,

Z

t

oS : δ t oε ◦ Z

Z

dV =

o t t ◦

ρ b · δ u dV +

t t ◦

t · δ u dS (6.74)

◦ V

◦ V

as long as we fulfill the condition of variational consistency (Simo & Hughes

1986), Z

t t

λ :(δoε − δ t oε) ◦ dV =0, (6.75)

◦ V

◦ Sσ

◦ Sσ

which is obviously fulfilled for the continuum problem.

6.6 Veubeke-Hu-Washizu variational principles 211

6.6.2 Constitutive constraints via the V-H-W principles

Let us now assume that we consider in Eq. (6.51) a stress tensor t oS that is

not necessarily derived from the kinematically consistent strain field t oε.

We can write

Therefore

Z

t

oΠ ∗ =

◦ V

t

oS = o ρ ∂tU ∂t oε

(6.76a)

t

oS = o ρ ∂tU ∂t . (6.76b)

oε

£ o

ρ

tU ¡ toε ¢ + t G( t u) ¤ Z

¡ ◦ t

dV + g

tu ¢ ◦dS

◦Sσ Z

(6.77)

+

t t

λ :( oS − t oS) ◦ dV .

◦ V

We search for the equilibrium configuration imposing

δ t ◦Π ∗ =0, (6.78)

and considering the independent variables tu, t λ and ¡ t

oε , t oS¢ :

Z

◦V t

oS : δ t oε ◦ Z

dV −

◦V t t

λ : δoS ◦ Z

dV =0 (6.79)

t t

λ : δoS ◦V ◦ Z

Z

dV −

o t t ◦

ρ b · δ u dV −

t t ◦

t · δ u dS =0 (6.80)

◦V Z

◦Sσ δ t λ :( t oS − t oS) ◦ dV =0 . (6.81)

◦ V

The last equation imposes the condition

t

oS = t oS (6.82)

whichisalwaysfulfilled for the continuum problem but not necessarilly in the

finite element approximations.

Adding Eqs. (6.79) and (6.80), we get

Z

◦V t

oS : δ t oε ◦ Z

dV +

t t

λ :(δ oS − δ

◦V t oS) ◦ Z

dV

Z

=

o t t ◦

ρ b · δ u dV +

t t ◦

t · δ u dS. (6.83)

◦ V

From the above, we can state the Principle of Virtual Work as,

◦ Sσ

212 **Nonlinear** continua

Z

t

oS : δ t oε ◦ Z

dV =

◦ V

◦ V

o ρ t b · δ t u ◦ dV +

Z

◦ Sσ

t t · δ t u ◦ dS (6.84)

as long as we fulfill the condition,

Z

t t

λ :(δ oS − δ t oS) ◦ dV =0, (6.85)

◦ V

which is obviously fulfilled for the continuum problem.

Other constraints can also be considered and they constitute the basis of

different finite element applications.

When using variational principles of the Veubeke-Hu-Washizu type for

developing finite element formulations, the interpolation functions selected

for the different interpolated fields should fulfill orthogonality conditions of

the form of Eqs. (6.75) or (6.85) (Simo & Hughes 1986).

Different forms of the Veubeke-Hu-Washizu variational principles have

been used to develop mixed and hybrid finite element formulation some of

them can be read from the classical paper (Pian & Tong 1969) and also from

(Dvorkin & Bathe 1984, Bathe & Dvorkin 1985\1986, Dvorkin & Vassolo 1989,

Fung & Tong 2001), etc.

A

Introduction to tensor analysis

In this Appendix, assuming that the reader is acquainted with vector analysis,

we present a short introduction to tensor analysis. However, since tensor

analysis is a fundamental tool for understanding continuum mechanics, we

strongly recommend a deeper study of this subject.

Some of the books that can be used for that purpose are: (Synge & Schild

1949, McConnell 1957, Santaló 1961, Aris 1962, Sokolnikoff 1964, Fung 1965,

Green & Zerna 1968, Flügge 1972, Chapelle & Bathe 2003).

A.1 Coordinates transformation

Let us assume that in a three-dimensional space (< 3 ) we can define a system

of Cartesian coordinates: we call this space the Euclidean space.

In this Appendix we will restrict our presentation to the case of the Euclidean

space.

In the < 3 spacewedefine a system of Cartesian coordinates { z α , α =

1, 2, 3 }, and an arbitrary system of curvilinear coordinates { θ i ,i=1, 2, 3 } .

The following relations hold:

θ i = θ i (z α ,α=1, 2, 3) , i =1, 2, 3 . (A.1)

The above functions are single-valued, continuous and with continuous

first derivatives.

We call J the Jacobian of the coordinates transformation defined by Eq.

(A.1). Hence

"

∂θ

J =

i

∂zα #

. (A.2)

An admissible transformation is one in which det J 6= 0,thatistosay,

a transformation where a region of nonzero volume in one system does not

collapse into a point in the other system and vice versa.

214 **Nonlinear** continua

0.

A proper transformation is an admissible transformation in which det J>

A.1.1 Contravariant transformation rule

From Eq. (A.1) we obtain

dθ i = ∂θi

∂zα dzα . (A.3)

When the coordinates system is changed, the mathematical entities ai at a certain point of < 3 that transform following the same rule as does the

coordinate differentials (Eq. (A.3)) are said to transform according to a contravariant

transformation rule. We indicate these mathematical entities using

upper indices.

Now we consider two systems of curvilinear coordinates © θ iª and { ˆθ i

},

related by the following equations:

ˆθ i

= ˆ θ i

(θ j ,j=1, 2, 3) , i =1, 2, 3 (A.4a)

and

θ k = θ k ( ˆθ l

,l=1, 2, 3) , k =1, 2, 3 . (A.4b)

We can write the coordinate differentials as:

dθ i = ∂θi

∂ ˆ θ j dˆ θ j

(A.4c)

dˆθ i

= ∂ˆθ i

∂θ j dθj . (A.4d)

In the same way, a contravariant mathematical entity can be defined in

either of the two systems

a i = a i (θ j ,j=1, 2, 3) , i =1, 2, 3 (A.4e)

â i = â i ( ˆθ j

,j=1, 2, 3) , i =1, 2, 3 (A.4f)

and we transform it from one curvilinear system to the other following a transformation

rule similar to the transformation rule followed by the coordinate

differentials:

a i = ∂θi

∂ ˆ θ

â i = ∂ˆ θ i

j âj

aj

j

∂θ

, i =1, 2, 3 (A.4g)

, i =1, 2, 3 . (A.4h)

A.2 Vectors 215

Although the contravariant transformation rule applies to

dθ i and not to θ i , using a notation abuse, we follow the

convention of using upper indices for the coordinates.

A.1.2 Covariant transformation rule

Given an arbitrary continuous and differentiable function f(θ 1 ,θ 2 ,θ 3 ) and

using the chain rule, we write

∂f

∂ˆ i

θ

=

∂f

∂θ j

∂θ j

∂ˆθ i , i =1, 2, 3 . (A.5a)

We define

aj = ∂f

∂θ j , j =1, 2, 3 . (A.5b)

In the { ˆθ i

} coordinate system we define

âj = ∂f

âi = aj

∂ ˆ θ j , j =1, 2, 3 (A.5c)

∂θ j

∂ ˆ θ i , i =1, 2, 3 (A.5d)

∂

ai = âj

ˆθ j

∂θ i , i =1, 2, 3 . (A.5e)

When the coordinates system is changed, the mathematical entities ai

at a certain point of < 3that transform following the same rule as does the

derivatives of a scalar function (Eqs. (A.5d) and (A.5e)) are said to transform

accordingtoacovariant transformation rule. We indicate those mathematical

entities using lower indices.

A.2 Vectors

There are some physical properties like mass, temperature, concentration of a

given substance, etc., whose values do not change when the coordinate system

used to describe the problem is changed. These variables are referred to as

scalars.

On the other hand, there are other physical variables like velocity, acceleration,

force, etc. that do not change their intensity and direction when the

coordinate system used to describe the problem is changed. They are called

vectors.

In what follows, we will make use of the above intuitive definition of scalars

and vectors. However, in Sect. A.4 we will see that they represent two particular

kinds of tensors (order 0 and 1, respectively).

216 **Nonlinear** continua

A.2.1 Base vectors

Asetofnlinearly independent vectors is a basis of the space < n and any

othervectorin< n can be constructed as a linear combination of those base

vectors.

Let us consider the three linearly independent vectors g (i =1, 2, 3) in

i

< 3 . Any vector v inthesamespacecanbewrittenas:

v = v i g i . (A.6)

The mathematical entities v i (i =1, 2, 3) are the components of v in the

basis g i (i =1, 2, 3).

Example A.1. JJJJJ

In a Cartesian system {z α , α =1, 2, 3} thebasevectorsare

e1 = (1, 0, 0) ,

e2 = (0, 1, 0) ,

e3 = (0, 0, 1) ,

wherewehaveindicatedtheprojectionofthebasevectorsontheCartesian

axes.

The position vector r of a point P in < 3 is

Hence,

but also,

Therefore, we get

A.2.2 Covariant base vectors

r = z α e α .

dr = dz α e α ,

dr = ∂r

∂z α dzα .

e α = ∂r

∂z α , α =1, 2, 3 .

JJJJJ

In the arbitrary curvilinear system {θ i ,i=1, 2, 3} we can write, at any point

P of the space,

dr = ∂r

∂θ i dθi . (A.7a)

Since

we obtain

Fig. A.1. Covariant base vectors at a point P

dr = dθ i g i

A.2 Vectors 217

(A.7b)

g i = ∂r

∂θ i , α =1, 2, 3 . (A.7c)

The vectors g i ,defined with the above equation, are the covariant base

vectors of the curvilinear coordinate system { θ i } at the point P .

From its definition, the vector g 1 is tangent to the line, θ2 = θ2 (P ) and

θ3 = θ3 (P ) .

Similar conclusions can be reached for the covariant base vectors g 2 and

g 3 .

In a Cartesian system, we can write Eq. (A.7c) as:

g i = ∂zα

∂θ i e α , i =1, 2, 3 . (A.8)

In a second curvilinear system { ˆθ i

,i=1, 2, 3 },

218 **Nonlinear** continua

Hence, we have

dr = dˆθ i

ˆg = dθ

i i ∂ˆθ j

∂θ i ˆg . (A.9a)

j

g =

i ∂ˆθ j

∂θ i ˆg , i =1, 2, 3 . (A.9b)

j

Due to the similarity between Eqs. (A.9b) and Eq. (A.5e) the base vectors

g are called covariant base vectors.

i

A.2.3 Contravariant base vectors

In an arbitrary curvilinear coordinate system { θ i ,i=1, 2, 3 } we define the

contravariant base vectors (dual basis) (g i ,i=1, 2, 3) with the equation

g i

· g j = δ i j , (A.10)

where the dot indicates a scalar product (“dot product”) between two vectors

and δ i j is the Kronecker delta (δ i j = 1 for i = j and δ i j = 0 for i 6= j ).

Defining in < 3 two curvilinear systems {θ i } and { ˆθ i

} and using Eq. (A.9b),

we obtain

ˆg i

· ˆg = ˆg

j i

· ∂θm

∂ˆθ j g m = δij . (A.11a)

Hence, using Eq. (A.10), we obtain

If we define

g i

from Eq. (A.11b), we obtain

· g j = ˆg i

ˆg i = ∂ˆ θ i

g i

gl

l

∂θ

· g j = ∂ˆ θ i

∂θ l

· ∂θm

∂ˆθ j g . (A.11b)

m

, i =1, 2, 3 (A.11c)

∂θ m

∂ ˆ θ

j gl

· g m

(A.11d)

and

g i

· g =

j ∂ˆθ i

∂θ l

∂θ m

∂ˆθ j δlm = ∂ˆθ i

∂ˆθ j = δij , (A.11e)

where we can see that the relation (A.11a) is satisfied.

Therefore, Eq.(A.11c) can be considered the transformation rule for the

contravariant base vectors. Due to the similarity between Eqs. (A.11c) and

(A.4h) the base vectors gi are called contravariant base vectors.

A.3 Metric of a coordinates system 219

Fig. A.2. Covariant and contravariant base vectors

InFig.A.2werepresentinaspace< 2 ,atapointP ,thecovariantand

contravariant base vectors, in order to provide the reader with a useful geometrical

insight.

It is important to note that in a Cartesian system the

covariant and contravariant base vectors are coincident.

A.3 Metric of a coordinates system

If a position vector r defines a point P in < 3 and a position vector (r +dr)

defines a neighboring point P 0 , the distance between these points is given by

A.3.1 Cartesian coordinates

In a Cartesian system in < 3 ,wehave

ds = p dr · dr . (A.12)

ds 2 = dz α dz β (e α · e β) . (A.13a)

220 **Nonlinear** continua

If we call e α · e β = δαβ , then

ds 2 = dz α dz β δαβ . (A.13b)

We call the nine number δαβ the Cartesian components of the metric tensor

at P (this notation will be clarified in Sect. A.4.3).

From its definition, it is obvious that δαβ = 1 if α = β and δαβ = 0

if α 6= β .

A.3.2 Curvilinear coordinates. Covariant metric components

In an arbitrary curvilinear system { θ i , i =1, 2, 3 } the distance between P

and P 0 is given by

ds 2 = dr · dr = dθ i dθ j (g i

· g j ) . (A.14a)

We call

gij = gji = g · g (A.14b)

i j

the covariant components of the metric tensor at P in the curvilinear system

{θ i } (this notation will be clarified in Section A.4.3.).

Using Eq. (A.8), we get

gij = ∂zα

∂θ i

∂z β

∂θ j δαβ . (A.15a)

Defining a second curvilinear system { ˆθi ,i=1, 2, 3 } and using Eq. (A.9b),

we get

∂ˆθ m

gij = ∂ˆ θ l

∂θ i

Equations (A.15a-A.15b) are the reason for using the name “covariant”

for the metric tensor components defined in Eq. (A.14b).

∂θ j ˆglm . (A.15b)

A.3.3 Curvilinear coordinates. Contravariant metric components

The scalars defined in Eq. (A.14b) by the dot product of the covariant base

vectors were named covariant components of the metric tensor. In the same

way, we define

g ij = g ji = g i

· g j

(A.16a)

the contravariant components of the metric tensor at P .

Using Eq. (A.11c), we get

g ij = ∂θi

∂ ˆ θ l

∂θ j

∂ ˆ θ m ˆg lm . (A.16b)

A.3 Metric of a coordinates system 221

The above equation is the reason for using the name “contravariant” for the

metric tensor components defined in Eq. (A.16a).

It is obvious that in the Cartesian system, we have

When { ˆ θ i

} is a Cartesian system, Eq. (A.16b) is

δ αβ = δαβ . (A.17a)

g ij = ∂θi

∂z α

∂θ j

∂z β δαβ . (A.17b)

A.3.4 Curvilinear coordinates. Mixed metric components

In any curvilinear coordinate system {θ i , i =1, 2, 3 } we can define the

mixed components of the metric tensor as

g i j = g i

· g j = δ i j (A.18a)

g i

j = g j · g i = δ i j . (A.18b)

Example A.2. JJJJJ

Any vector in < 3 can be written as a linear combination of the covariant base

vectors; hence, we can write

g i = α ij g j

When we postmultiply by g k on both sides, we obtain

Thus, we have

α ik = g i

.

· g k = g ik

g i = g ij g j

.

.

JJJJJ

Example A.3. JJJJJ

Proceeding as we did in the previous examples, the reader can easily show

that:

g

i

= gij g j

.

JJJJJ

222 **Nonlinear** continua

A.4 Tensors

We show in Sect. A.2 that in the space < 3 we can define two sets of linearly

independent vectors: the covariant and the contravariant base vectors. Hence,

any arbitrary vector in < 3 can be written as:

v = v i g i = vi g i . (A.19)

We are now going to show that for the vector v to remain invariant under

coordinate changes, the components vi should transform following a contravariant

rule and the components vi should transform following a covariant

rule.

When we go from the system { θ i , i =1, 2, 3 } to the system { ˆθ i

, i =

1, 2, 3 }, using Eqs. (A.9b), we obtain

Hence,

v = v i g i = v i ∂ˆ θ j

∂θ i ˆg j = ˆvj ˆg j . (A.20a)

ˆv j = v i ∂ˆθ j

∂θ i ; j =1, 2, 3 . (A.20b)

We see from the above that when the coordinate system is changed, the

components vi transform following a contravariant rule.

Using Eq. (A.11c), we can write

v = ˆvj ˆg j = ˆvj

∂ ˆ θ j

∂θ i gi = vi g i . (A.21a)

Hence,

vi = ˆvj

∂ˆθ j

∂θ i ; i =1, 2, 3 . (A.21b)

We see from the above that when the coordinate system is changed, the

components vi transform following a covariant rule.

As a conclusion to this section, we can state that the invariance of

under coordinate transformation requires the use of:

v

• covariant components + contravariant base vectors

or

• contravariant components + covariant base vectors.

A.4.1 Second-order tensors

A.4 Tensors 223

Generalizing the concept of vectors that we presented above, we define as

second-order tensors the following mathematical entities:

a = aij g i g j = a ij g i g j = a i j g i g j = a j

i gi g j

(A.22)

that remain invariant under coordinate transformations.

In the above equation, we used tensorial or dyadic products between vectors

(g g ; g

i j i gj ; gi g ; etc.) that we are going to formally define in this Section.

j

For the transformation {θ i } → { ˆθ i

} using Eq. (A.9b), we get

a = a ij g i g j = a ij ∂ˆ θ k

Thus, we have

∂θ i

∂ ˆ θ l

∂θ j ˆg k ˆg l = âkl ˆg k ˆg l . (A.23a)

â kl = a ij ∂ˆθ k

∂θ i

∂ˆθ l

∂θ j ; k, l =1, 2, 3 . (A.23b)

That is to say, the components aij transform following a double contravariant

rule.

In the same way, we can show that

∂θ

âkl = aij

i

∂ˆθ k

∂θ j

∂ˆθ l ; k, l =1, 2, 3 . (A.23c)

That is to say, the components aij transform following a double covariant

rule.

We can also show that:

â k l = a i j

â k

l

= a i

j

∂ ˆ θ k

∂θ i

∂θ j

∂ ˆ θ l

∂θ j

∂ ˆ θ l ; k, l =1, 2, 3 (A.23d)

∂ ˆ θ k

That is to say, the components a i j

From Eq. (A.22), we have

∂θ i ; k, l =1, 2, 3 . (A.23e)

and a j

i

transform following mixed rules.

akl g k g l = a ij g i g j . (A.24a)

When we postmultiply by g r on both sides (the formal definition of this

operation is given in what follows), we get

akl g k δ l r = a ij gjr g i

(A.24b)

224 **Nonlinear** continua

and if we now postmultiply (inner product) both sides by g s ,weobtain

akl δ l r δ k s = a ij gis gjr . (A.24c)

Finally,

asr = a ij gis gjr ; r, s =1, 2, 3 . (A.24d)

In the same way, we can show the following relations for k, l =1, 2, 3:

akl = a j

k gjl (A.24e)

a kl = aij g ik g jl = a k j gjl

(A.24f)

a k l = a kj gjl = ajl g jk . (A.24g)

It is evident that we can raise and lower indices using the proper components

of the metric tensor.

In Eq. (A.22) and the ones that followed, we wrote dyads of the type

g g or g

i j

i gj or gi g or g g

j i j the tensorial product of two vectors.

that define an operation known as

To define the tensorial product of two vectors a and b (ab in our

notation or a ⊗ b in the notation used by other authors), we will define

the properties of this new operation:

• Given a scalar α,

• Givenathirdvectorc,

and

• In general,

α (ab) = (α a)b = a(α b) = αab . (A.25a)

(ab)c = a(bc) = abc (A.25b)

a(b + c) = ab + ac 6= ba + ca = (b + c) a . (A.25c)

ab 6= ba . (A.25d)

• The scalar product of a vector c with the dyad ab is a vector,

where (c · a) is a scalar. Also,

c · (ab) = (c · a)b , (A.25e)

A.4 Tensors 225

(ab) · c = a(b · c) 6= c · (ab) = (c · a)b . (A.25f)

It should be notd that (ab) · c is a vector with the direction of the vector

a , while c · (ab) is a vector with the direction of the vector b .

• The scalar or inner product between two dyads is another dyad:

(ab) · (cd) = (b · c) ad . (A.25g)

• The double scalar or inner product between two dyads is a scalar:

Besides (Malvern 1969),

which is a scalar too.

(ab) : (cd) = (a · c) (b · d) . (A.25h)

(ab) ··(cd) = (a · d) (b · c) (A.25i)

Using the above definition, we can perform the scalar product of the

second-order tensor a defined by Eq. (A.22) and the vector v defined by Eq.

(A.19),

a · v = aik v k g i = a ik vk g i = a i k v k g i = a k

i vk g i

then, we obtain a vector b = a · v with:

◦ covariant components: bi = aik vk = a k

i vk

◦ contravariant components: bi = aik vk = ai k vk .

It is easy to show that (v · a) is also a vector and that in general

(A.26a)

v · a 6= a · v . (A.26b)

Eigenvalues and eigenvectors of second-order tensors

We say that a vector v is an eigenvector of a second-order tensor a if

a · v = λ v (A.27)

and we call λ the eigenvalue associated to the eigendirection v.

It is easy to show that the following relation holds

( aij − λgij ) v j = 0 . (A.28)

Equation (A.28) represents a system of 3 homogeneous equations (j =

1, 2, 3) with 3 unknowns (v 1 ,v 2 ,v 3 ) . To obtain a solution different from the

trivial one, we must have

226 **Nonlinear** continua

| aij − λgij | = 0 . (A.29)

The above is a cubic equation in λ that leads to 3 eigenvalues and therefore

3 associated eigendirections. It is obvious that if a pair (λ, v) satisfies

Eq. (A.27), the pair (λ, α v) will also do so. Hence, the modulus of the

eigenvectors remains undefined.

The following properties can be derived:

• If a is symmetric, the eigenvalues and eigenvectors are real.

Proof. (Green & Zerna 1968)

Assume λ is not real, then

λ = α + iβ (A.30a)

v j = η j + iµ j . (A.30b)

From Eq. (A.28), equating real and imaginary parts,

(aij − αgij) η j + βgij µ j = 0 (A.30c)

(aij − αgij) µ j − βgij η j = 0 . (A.30d)

After some algebra, from the two above equations, we get

β (gij η i η j + gij µ i µ j ) = 0 (A.30e)

for aij = aji.

Since all the η i and µ i cannot be zero and the terms (gij η i η j ) and

(gij µ i µ j ) are always positive (see Eq. (A.14a)), then

β = 0 . (A.30f)

Therefore, the eigenvalues are always real and to satisfy Eq. (A.28) the

eigenvectors shall also be real.

• If a is symmetric, the eigenvectors form an orthogonal set.

Proof. (Crandall 1956)

For two pairs (λ1, v 1) and (λ2, v 2) from Eq. (A.27)

A.4 Tensors 227

v 1 · a · v 2 − v 2 · a · v 1 = (λ2 − λ1) v 1 · v 2 . (A.31)

For a symmetric tensor, the l.h.s. of the above equation is zero. Hence,

◦ If λ1 6= λ2 then v 1 · v 2 = 0.Thatistosay,v 1 and v 2 are orthogonal.

◦ If λ1 = λ2 there are infinite vectors v 1 , v 2 that satisfy the above equation.

Among them we can select a pair of orthogonal vectors.

Hence, in general we assess that for symmetric second-order tensors, the

eigenvectors are orthogonal.

Example A.4. JJJJJ

As a is a symmetric second order tensor, with eigenvalues λI and eigenvectors

v I (I =1, 2, 3) with |v I| = 1 ,thecanonical form of a is:

a = λ1 v 1 v 1 + λ2 v 2 v 2 + λ3 v 3 v 3

also known as the diagonalized form. JJJJJ

A.4.2 n-order tensors

Inthesamewaythatwedefined the tensorial products of two vectors (dyad),

we can define the tensorial product of n vectors (n-poliad). Therefore, we can

define mathematical entities of the type:

t = t ij...n g i g j ··· g n = tij...n g i g j ··· g n

= t ij...k lm...n g i g j ··· g k g l g m ··· g n

(A.32)

which we call tensors of order n and we associate to them the property of

remaining invariant when coordinate transformations are performed.

When we go from the curvilinear system { θ i , i =1, 2, 3 } to the curvilinear

system { ˆθ i

,i=1, 2, 3 }, due to the invariance property, weget

t = t ij...k lm...n g i g j ··· g k g l g m ··· g n

= ˆt ab...c de...f ˆg a ˆg b ··· ˆg c ˆg d ˆg e ··· ˆg f .

Hence, the following relations can be derived:

(A.33a)

228 **Nonlinear** continua

ˆt ab...c de...f = t ij...k lm...n (ˆg a · g i )(ˆg b · g j ) ··· (ˆg c · g k )

(ˆg d · g l )(ˆg e · g m ) ··· (ˆg f · g n ) (A.33b)

and using Eqs. (A.9b) and (A.11c), we obtain

ˆt ab...c de...f = tij...k lm...n

A.4.3 The metric tensor

∂ ˆ θ a

∂θ i

∂ˆθ b

∂θ j ··· ∂ˆθ c

∂θ k

∂θ l

∂ ˆ θ d

∂θ m

∂ˆθ e ··· ∂θn

∂ˆ . (A.33c)

f

θ

As a particular but important example of second-order tensors, we will refer

in this section to the metric tensor, g,

g = gij g i g j = g ij g i g j = δ i j g i g j = δ j

i gi g j . (A.34)

In Sects. A.3.1 - A.3.3, we introduced the covariant, contravariant and

mixed components of this tensor.

We can rewrite Eq.(A.14a) as:

ds 2 ³

= dr · g · dr = dθ i ´

g ·

i

¡ gkl g k g l¢ ³

· dθ j ´

g (A.35a)

j

and therefore,

ds 2 = dθ i dθ j gkl δ k i δ l j = dθ i dθ j gij . (A.35b)

Going back to Eq. (A.34), we post-multiply both sides by the vector gp and

gij g i g j · g p = g kl g g · g

k l

p . (A.36a)

Operating, we get

gij g jp g i = g kl δ p

l g . (A.36b)

k

Using Eqs. (A.15a) and (A.17b), we arrive at

gij g jp = ∂zα

∂θ i

∂z β

∂θ

j δαβ

∂θ j

∂z γ

∂θ p

∂z δ δγδ = ∂zα

∂θ i

Rearranging,

gij g jp = ∂θp

∂zα ∂zα i

∂θ

∂θp

=

∂θ

Using the above in Eq. (A.36b), we finally obtain

∂z α

∂z γ

= δp

i i

∂θ p

. (A.36c)

∂zγ . (A.36d)

g p = g pk g k . (A.36e)

The above equality was also derived in Example A.2. In an identical way,

we can also derive the result of the Example A.3.

A.4.4 The Levi-Civita tensor

A.4 Tensors 229

The Cartesian components of the Levi-Civita or permutation tensor are defined

as:

eαβγ = e αβγ ⎧

⎨ 0 when two of the indices are equal

= 1 when the indices are arranged as 1,2,3

⎩

−1 when the indices are arranged as 1,3,2

⎫

⎬

⎭ .

By using the tensorial components transformation rules in an arbitrary

curvilinear system { θ i

get

,i =1, 2, 3 } and for the covariant components, we

ijk = ∂zα

∂θ i

∂zβ ∂θ j

∂zγ ∂θ k eαβγ . (A.37a)

Taking into account that the determinant of a (3×3) matrix can be written

as: ¯

¯a i ¯