DACIA ELECTRA - ingineria-automobilului.ro

Simulation of Elastic Properties of Composite Prepreges
with Applications in the Automotive Industry
Simularea proprietăţilor elastice ale prepreg-urilor compozite
cu aplicaţii în industria auto
Ingineria Automobilului
eterogene a dat o bună concordanţă între această
abordare teoretică și datele experimentale.
Cuvinte cheie: Prepreg-uri, Mediere,
Omogenizare, Limită superioară, Limită inferioară,
Proprietăţi elastice.
INTRODUCTION
Th e objectives of the paper is to compute the
upper and lower limits of the homogenized
elastic coeffi cients for a common 27% fi bers volume
fraction SMC based on a homogenization
method described by Ene and Pasa [1] as well as
ABSTRA CT
some averaging methods of the Young and shear
Th e paper presents some averaging methods to
moduli of various SMCs with diff erent fi bers
simulate the elastic properties of prepregs like
volume fractions. A Sheet Molding Compound
Sheet Molding Compounds (SMCs) used in the
(SMC) is a pre-impregnated material, chemi-
automotive industry. Th e upper and lower limits
cally thickened, manufactured as a continuous
of the homogenized coeffi cients for a 27% fi bers
mat of chopped glass fi bers, resin, fi ller and ad-
volume fraction SMC have been computed. A
ditives, from which blanks can be cut and placed
comparison between the upper and lower lim-
into a press for hot press molding [2-8]. Due
its of the homogenized elastic coeffi cients for a
to the signifi cant anisotropy and heterogeneity
27% fi bers volume fraction SMC and the experi-
of most pre-impregnated composite materials,
mental data is presented. Th e estimation model
their elastic properties may vary strongly in a
used as a homogenization method of these het-
quite large range. Th e aim is to obtain a computerogeneous
composite materials, gave emphasis
ing model, which takes into account the micro-
to a good agreement between this theoretical
structure and/or the local heterogeneity of these
approach and experimental data.
materials to simulate their elastic properties.
Key-words: Prepregs, Averaging,
A HOMOGENIZATION METHOD
Homogenization, Upper limit, Lower limit,
We consider a domain Ω from R
Elastic properties
REZUMAT
Lucrarea prezintă câteva metode de mediere
pentru simularea proprietăţilor prepreg-urilor
precum materialele compozite de tip Sheet
Molding Compound (SMC) utilizate în industria
auto. Au fost calculate limitele superioară și
inferioară ale coefi cienţilor omogenizaţi, pentru
un material de tip SMC cu 27% fracţiune volumică
a fi brelor. Se prezintă o comparaţie între
limitele superioară și inferioară a coefi cienţilor
elastici omogenizaţi ai materialului SMC cu
27% fracţiune volumică a fi brelor și datele experimentale.
Modelul estimat utilizat ca metodă
de omogenizare a acestor materiale compozite
3 riodic structure containing inclusions, a (x) is
ij
a function of x. If the period’s dimensions are
small in comparison with the dimensions of
the whole domain then the solution u of the
equation (1) can be equal with the solution
suitable for a homogenized material, where the
coeffi cients a are constants. In the R ij
space, in x co- i
ordinates, domain considered a SMC composite
material, in which a so-called replacement matrix
(resin and fi ller) represents the fi eld Y and 1
the reinforcement occupies the fi eld Y seen as
2
a bundle of glass fi bers. Let us consider the following
equation [1]:
∂ ⎡ ∂u
⎤
f ( x)
= − ⎢aij
( x)
⋅ ⎥;
aij
= a ji
(1)
∂xi
⎢⎣
∂x
j ⎥⎦
alternatively, writt en under the equivalent form:
∂pi
∂u
f = − ; pi
= aij
⋅ .
(2)
∂xi
∂x
j
In the case of SMC materials that present a pe-
3 space of
y coordinates, a parallelepiped with y i 0 sides is
i
considered, as well as parallelepipeds obtained
by translation n y i 0
i (n integer) in axes directions.
i
Th e functions can be defi ned [1]:
η ⎛ x ⎞
aij ( x)
= aij
⎜ ⎟
(3)
⎝η
⎠
where η is a real, positive parameter. Notice that
the functions a (x) are ηY-periodical in variable
ij
x (ηY being the parallelepiped with ηy0 i sides).
If the function f(x) is in Ω defi ned, the problem
at limit is [1]:
⎡
η
∂ u ⎤
η ∂
η
f ( x ) = − ⎢aij
( x ) ⋅ ⎥,
u = 0.
(4)
∂Ω
∂xi
⎢⎣
∂x
j ⎥⎦
η
p
Similar with equation (2), the vector
r
defi nes
the following elements [1]:
η
∂u
.
η η
pi
( x)
= aij
( x)
⋅
(5)
∂x
j
For the function uη (x), an asymptotic development
will be looking for, under the form [1]:
uη
( x ) = u0(
x,
y ) + η 1u1(
x,
y ) + η
(6)
u ( x,
y ) ...; y x ,
2 2 + η + =
η
where ui (x,y) are elements, Y-periodical in y variable.
Th e derivatives of the functions ui (x,y) behave
in the following manner [1]:
(7)
If the values of
⎛ x ⎞
u ⎜ x ⎟
⎝ η ⎠
i
Horaţiu TEODORESCU-DRĂGHICESCU
Transilvania University of Braşov,
Department of Mechanics
draghicescu.teodorescu@unitbv.ro
Khairul ALAM
Ohio University,
Russ College of Engineering and Technology,
Center for Advanced Materials Processing
alam@ohio.edu
Sorin VLASE
Transilvania University of Braşov,
Department of Mechanics
svlase@unitbv.ro
,
are compared in two homologous points P and 1
9