# Quantitative homogenization in nonlinear elasticity for small loads

## TL;DR

This paper investigates how nonlinear elastic materials with periodic microstructures can be effectively approximated by simpler models near rotations, showing that homogenization and linearization processes commute for small loads.

## Contribution

It demonstrates that under certain conditions, the multi-cell homogenization reduces to a single-cell formula near rotations, and establishes the commutation of homogenization and linearization in nonlinear elasticity.

## Key findings

- Homogenized integrand admits a quadratic Taylor expansion near rotations.
- Homogenization and linearization commute close to the rotations.
- Provides an estimate on homogenization error for small applied loads.

## Abstract

We study quantitative periodic homogenization of integral functionals in the context of non-linear elasticity. Under suitable assumptions on the energy densities (in particular frame indifference; minimality, non-degeneracy and smoothness at the identity; $p\geq d$-growth from below; and regularity of the microstructure), we show that in a neighborhood of the set of rotations, the multi-cell homogenization formula of non-convex homogenization reduces to a single-cell formula. The latter can be expressed with help of correctors. We prove that the homogenized integrand admits a quadratic Taylor expansion in an open neighborhood of the rotations -- a result that can be interpreted as the fact that homogenization and linearization commute close to the rotations. Moreover, for small applied loads, we provide an estimate on the homogenization error in terms of a quantitative two-scale expansion.