Modeling Material Microstructure - Continuum Mechanics of Anisotropic Materials - page 177

7.3 Effective Material Parameters

One of the prime objectives in the discipline of composite materials, a discipline
that has developed over the last half-century, is to evaluate the effective material
parameters of a composite in terms of the material parameters and configurational
geometries of its constituent components or phases. The purpose of this section is to
show that the effective material properties may be expressed in terms of integrals
over the surface of the RVE. The conceptual strategy is to average the heteroge-
neous properties of a material volume and to conceptually replace that material
volume with an equivalent homogenous material that will provide exactly the same
property volume averages as the real heterogeneous material, allowing the calcula-
tion of the material properties of equivalent homogeneous material. The material
volume selected for averaging is the RVE and the material properties of equivalent
homogeneous material are then called the effective properties of the RVE. The
calculational objective is to compute the effective material properties in terms of an
average of the real constituent properties. This is accomplished by requiring that the
integrals of the material parameters over the bounding surface of the RVE for the
real heterogeneous material equal those same integrals obtained when the RVE
consists of the equivalent homogeneous material. Thus we seek to express the
physical fields of interest for a particular RVE in terms of RVE boundary integrals.
In order to construct an effective anisotropic Hooke's law it is necessary to
represent the global or macro stress and strain tensors,

,respectivelyas
integrals over the boundary of the RVE. To accomplish this for the stress tensor
we begin by noting the easily verified identity (see the Appendix, especially
problem A.3.3)

h E i and h T i

@ x i

T

ðr x Þ ¼ 1 ;

x j e i

e j ¼ d ij e i e j : (7.2)

@

Using the identity ( 7.2 ) a second identity involving the stress is constructed,

T T

T ¼ 1 T ¼ðr x Þ T ¼frð T x Þg : (7.3)

The derivation of the last equality in ( 7.3 ) employs the fact that the divergence of
the stress tensor is assumed to be zero, 0. This restriction on the stress tensor

follows from the stress equation of motion (3.37) when the acceleration and action-
at-a-distance forces are zero. The term

D T ¼

in ( 7.3 ) is written with the T first
in the parenthesis to indicate that the divergence operator is to be applied to
T and not to x ; the latter case would be written as

rð T x Þ

rð x T Þ . The expansion of

rð T x Þ is

rð T x Þ¼r T x þ T ðr x Þ¼ T (7.4a)

since r T ¼ 0and r x ¼ 1 . In the indicial notation this development is written