High contrast homogenisation in nonlinear elasticity under small loads

We study the homogenisation of geometrically nonlinear elastic composites with high contrast. The composites we analyse consist of a perforated matrix material, which we call the “stiff” material, and a “soft” material that fills the remaining pores. We assume that the pores are of size 0<ε≪10<ε≪1 and are periodically distributed with period ε. We also assume that the stiffness of the soft material degenerates with rate ε2γε2γ, γ>0γ>0, so that the contrast between the two materials becomes infinite as ε↓0ε↓0. We study the homogenisation limit ε↓0ε↓0 in a low energy regime, where the displacement of the stiff component is infinitesimally small. We derive an effective two-scale model, which, depending on the scaling of the energy, is either a quadratic functional or a partially quadratic functional that still allows for large strains in the soft inclusions. In the latter case, averaging out the small scale-term justifies a single-scale model for high-contrast materials, which features a non-linear and non-monotone effect describing a coupling between microscopic and the effective macroscopic displacements.