On the commutability of homogenization and linearization in finite elasticity

TL;DR

This paper proves that, under certain conditions, homogenization and linearization processes in finite elasticity commute, and the homogenized energy retains a quadratic expansion derived from the linearized energy.

Contribution

It establishes the commutativity of homogenization and linearization for non-convex elastic energies with a quadratic Taylor expansion at identity.

Findings

Gamma-limits for homogenization and linearization commute.

Homogenized energy density admits a quadratic Taylor expansion.

Quadratic term of the homogenized energy is the homogenization of the linearized quadratic term.

Abstract

We study non-convex elastic energy functionals associated to (spatially) periodic, frame indifferent energy densities with a single non-degenerate energy well at SO(n). Under the assumption that the energy density admits a quadratic Taylor expansion at identity, we prove that the Gamma-limits associated to homogenization and linearization commute. Moreover, we show that the homogenized energy density, which is determined by a multi-cell homogenization formula, has a quadratic Taylor expansion with a quadratic term that is given by the homogenization of the quadratic term associated to the linearization of the initial energy density.