High contrast homogenisation in nonlinear elasticity under small loads

TL;DR

This paper investigates the homogenisation of nonlinear elastic composites with high contrast, deriving effective models that account for large strains and the coupling between micro and macro displacements in the low energy regime.

Contribution

It introduces a two-scale homogenisation model for high-contrast nonlinear elastic composites, capturing large strains and non-monotone effects in the effective behavior.

Findings

Derived an effective two-scale model depending on energy scaling.

Identified conditions for quadratic and partially quadratic effective functionals.

Justified a single-scale model capturing micro-macro displacement coupling.

Abstract

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 pores. We assume that the pores are of size $0<\varepsilon\ll 1$ and are periodically distributed with period $\varepsilon$. We also assume that the stiffness of the soft material degenerates with rate $\varepsilon^{2\gamma},$ $\gamma>0$, so that the contrast between the two materials becomes infinite as $\varepsilon\to 0$. We study the homogenisation limit $\varepsilon\to 0$ 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.