ON THE EFFECTIVE MECHANICAL-BEHAVIOR OF WEAKLY INHOMOGENEOUS NONLINEAR MATERIALS

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.