P. Suquet et Pp. Castaneda, SMALL-CONTRAST PERTURBATION EXPANSIONS FO R THE EFFECTIVE PROPERTIES OF NONLINEAR COMPOSITES, Comptes rendus de l'Academie des sciences. Serie 2, Mecanique, physique, chimie, sciences de l'univers, sciences de la terre, 317(12), 1993, pp. 1515-1522
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.