Asymptotic rigidity of layered structures and its application in homogenization theory

TL;DR

This paper develops an asymptotic rigidity theorem for layered elastic structures, enabling the derivation of effective deformation properties in high-contrast composites and advancing homogenization theory.

Contribution

It introduces a new asymptotic rigidity result for layered structures and applies it to homogenize high-contrast bilayered materials using Gamma-convergence.

Findings

Global anisotropic constraints emerge as layer thickness vanishes.

Explicit homogenized limit model is derived with a cell formula.

Optimal scaling between layer thickness and stiffness is confirmed.

Abstract

In the context of elasticity theory, rigidity theorems allow to derive global properties of a deformation from local ones. This paper presents a new asymptotic version of rigidity, applicable to elastic bodies with sufficiently stiff components arranged into fine parallel layers. We show that strict global constraints of anisotropic nature occur in the limit of vanishing layer thickness, and give a characterization of the class of effective deformations. The optimality of the scaling relation between layer thickness and stiffness is confirmed by suitable bending constructions. Beyond its theoretical interest, this result constitutes a key ingredient for the homogenization of variational problems modeling high-contrast bilayered composite materials, where the common assumption of strict inclusion of one phase in the other is clearly not satisfied. We study a model inspired by hyperelasticity via Gamma-convergence, for which we are able to give an explicit representation of the homogenized limit problem. It turns out to be of integral form with its density corresponding to a cell formula.