Strain energy-based homogenization of nonlinear elastic particulate composites

The macroscopic constitutive law for a heterogeneous solid containing two dissimilar nonlinear elastic phases undergoing finite deformation is obtained. Attention is restricted to the case of spherical symmetry such that only the materials consisting of an irregular suspension of perfectly spherical particles experiencing all-round uniform loading are considered which leads to a one-dimensional modeling. For the homogenization procedure, a strain–energy based scheme which utilizes Hashin’s composite sphere is employed to obtain the macroscopic stress–deformation relation added by the initial volume fraction of the particles. As applications of the procedure, the closed-form macroscopic stress expression for a generalized Carroll composite material is derived. Then, by choosing carbon black-filled rubbers, unknown bulk modulus of the carbon black particles is calculated. Finally, the particle-reinforced flexible polyurethane foam is studied using the Ritz method. It is shown that the analytical outcome for composites filled by compressible inclusions is applicable for porous materials with the same matrix.