Linearization of elasticity models for incompressible materials

TL;DR

This paper demonstrates how linear elasticity models for incompressible materials can be derived as a limit of finite elasticity models using $ ext{Gamma}$-convergence, applicable to rubber-like materials.

Contribution

It establishes a rigorous mathematical derivation of linear elasticity as a limit of finite elasticity under incompressibility and boundary conditions for a broad class of energy densities.

Findings

Linear elasticity is obtained as a $ ext{Gamma}$-limit of finite elasticity models.

The result applies to a large class of energy densities for rubber-like materials.

The derivation is rigorous under Dirichlet boundary conditions.

Abstract

We obtain linear elasticity as $\Gamma$-limit of finite elasticity under incompressibility assumption and Dirichlet boundary conditions. The result is shown for a large class of energy densities for rubber-like materials.