Lipschitz estimates and existence of correctors for nonlinearly elastic, periodic composites subject to small strains

TL;DR

This paper proves Lipschitz estimates and the reduction of multi-cell to single-cell homogenization formulas for nonlinearly elastic periodic composites near rotations, under relaxed regularity assumptions, with applications to small-strain elasticity problems.

Contribution

It extends homogenization theory for nonlinear elasticity by relaxing regularity assumptions and establishing Lipschitz bounds for minimizers in periodic composites.

Findings

Multi-cell homogenization reduces to single-cell formula near rotations.

Lipschitz estimates for minimizers are uniform in small strains.

New Lipschitz estimate for systems with piecewise-constant coefficients.

Abstract

We consider periodic homogenization of nonlinearly elastic composite materials. Under suitable assumptions on the stored energy function (frame indifference; minimality, non-degeneracy and smoothness at identity; $p\geq d$-growth from below), and on the microgeometry of the composite (covering the case of smooth, periodically distributed inclusions with touching boundaries), we prove that in an open neighbourhood of the set of rotations, the multi-cell homogenization formula of non-convex homogenization reduces to a single-cell formula that can be represented with help of a corrector. This generalizes a recent result of the authors by significantly relaxing the spatial regularity assumptions on the stored energy function. As an application, we consider the nonlinear elasticity problem for $\varepsilon$-periodic composites, and prove that minimizers (subject to small loading and well-prepared boundary data) satisfy a Lipschitz estimate that is uniform in $0<\varepsilon\ll1$. A key ingredient of our analysis is a new Lipschitz estimate (under a smallness condition) for monotone systems with spatially piecewise-constant coefficients. The estimate only depends on the geometry of the coefficient's discontinuity-interfaces, but not on the distance between these interfaces.