Pp. Castaneda et P. Suquet, ON THE EFFECTIVE MECHANICAL-BEHAVIOR OF WEAKLY INHOMOGENEOUS NONLINEAR MATERIALS, European journal of mechanics. A, Solids, 14(2), 1995, pp. 205-236
This paper is concerned with the prediction of the effective propertie
s of nonlinear composite materials with local constitutive behavior co
ntrolled by a convex potential w depending on a small parameter t. The
case where t serves to characterize the amplitude of the local variat
ions in the properties of the composites is studied in detail. Express
ions are generated for each of the first four terms in a perturbation
series expansion for the effective potential of the composites in the
contrast t; their derivation being reduced to the solution of standard
linear-elasticity problems for homogeneous anisotropic media with bod
y force distributions determined by the relevant polarization tensors.
The first two terms in such an expansion are found to add up to the v
olume average of the potentials of the N phases. For the case where th
e microstructure of the composite is periodic, explicit expressions ar
e derived for the second- and third-order terms, in terms of Fourier s
eries involving the characteristic functions of the constituent phases
in the unit cell. For the case of random microstructures, the express
ions generated for the second- and third-order terms depend on up to t
wo- and three-point correlation functions, respectively. When the micr
ostructure is further assumed to be statistically homogeneous and isot
ropic, the second-order term can be shown to depend only on the volume
fractions and properties of the phases. The results are specialized t
o composites with N isotropic phases, and explicit expressions are gen
erated for statistically isotropic and transversely isotropic fiber-re
inforced and laminated materials. It is found that the results for the
statistically isotropic composites have explicit dependence on the de
terminant of the applied strain, as expected from general symmetry con
siderations. In addition, it is found that the perturbation expansions
diverge in the limit of rigid/perfectly plastic behavior for the cons
tituent phases, under certain special types of loading conditions, whi
ch may be associated with the possible development of shear bands in t
he composites.