Reference configurations vs. optimal rotations: a derivation of linear elasticity from finite elasticity for all traction forces

TL;DR

This paper rigorously derives linear elasticity as a low energy limit of nonlinear elasticity without restrictive force assumptions, using $ ext{Gamma}$-convergence and optimal rotations to identify the reference configuration.

Contribution

It provides a full $ ext{Gamma}$-convergence derivation of linear elasticity from nonlinear elasticity, accounting for optimal rotations and fluctuations in reference configurations.

Findings

The $ ext{Gamma}$-limit is the standard linear elasticity model.

An additional penalty term accounts for fluctuations from optimal rotations.

On minimizers, the limit energy reduces to standard linear elasticity.

Abstract

We rigorously derive linear elasticity as a low energy limit of pure traction nonlinear elasticity. Unlike previous results, we do not impose any restrictive assumptions on the forces, and obtain a full $\Gamma$-convergence result. The analysis relies on identifying the correct reference configuration to linearize about, and studying its relation to the rotations preferred by the forces (optimal rotations). The $\Gamma$-limit is the standard linear elasticity model, plus a term that penalizes for fluctuations of the reference configurations from the optimal rotations. However, on minimizers this additional term is zero and the limit energy reduces to standard linear elasticity.