SMALL-CONTRAST PERTURBATION EXPANSIONS FO R THE EFFECTIVE PROPERTIES OF NONLINEAR COMPOSITES

This Note deals with nonlinear composite materials with local constitu tive behaviour controlled by a convex potential w, which varies slight ly from point to point in the composite, as determined by a small para meter t. The effective behaviour of the composite is in turn controlle d by a macroscopic potential W, which is assumed to depend smoothly on the contrast t. Exact expressions are obtained for the first three te rms in a perturbation expansion of W about t = 0; their derivation bei ng reduced to the solution of a standard linear elasticity problem for a homogeneous anisotropic body with body forces determined by the rel evant polarization tensors. An explicit expansion, exact to second ord er in the contrast, and depending only on the volume fractions for the N phases, is obtained for the special class of power-law incompressib le composites with a statistically homogeneous and isotropic distribut ion of the phases. One interesting feature of the result is the explic it dependence of the second-order term on the third invariant of the a pplied strain.