Nonlinear effective properties of heterogeneous materials with ellipsoidal microstructure

The analysis and the synthesis of the nonlinear effective response of particulate composite materials are of great importance for developing new systems such as nonlinear elastic and electromagnetic metamaterials, nonlinear waveguides, nonlinear magnetoelectric devices and photonic or phononic crystals. Typically, classical homogenization schemes take into account the shape of the inhomogenieties but neglect the spatial correlation among them, a crucial feature for the above applications. In this paper we develop a nonlinear homogenization technique for dispersions of nonlinear particles in a linear matrix, which is able to take account of spatial correlation by means of the so-called ellipsoidal microstructure. While the linear result corresponds to the well known Ponte Castañeda-Willis estimate, we propose new formulae for the second and third order nonlinear behavior. We finally show applications to the nonlinear elastic Landau coefficients and to the nonlinear hypersusceptibility of transport processes.