Nonlinear elasticity of composite materials

We investigate the elastic properties of model composites, consisting in a dispersion of nonlinear (spherical or cylindrical) inhomogeneities into a linear solid matrix. Both phases are considered isotropic. Under the simplifying hypotheses of small deformation for the material body and of small volume fraction of the embedded phase, we develop a homogenization procedure based on the Eshelby theory, aimed at describing nonlinear features. We obtain the bulk and shear moduli and Landau coefficients of the overall material in terms of the elastic behavior of the constituents and of their volume fractions. The mixing laws for the nonlinear properties describe a complex scenario where possible strong amplifications of the nonlinearities may arise in some given conditions