Pressure live loads and the variational derivation of linear elasticity

TL;DR

This paper extends the variational derivation of linear elasticity to include pressure live loads, analyzing the linearization for small pressures using Gamma-convergence and proving strong convergence of minimizers.

Contribution

It introduces the first rigorous derivation of linear elasticity under pressure live loads via Gamma-convergence, allowing for weakly coercive energy densities.

Findings

Linearization of pressure live loads obtained through Gamma-convergence.

Strong convergence of minimizers established.

Extension of classical elasticity derivations to include live loads.

Abstract

The rigorous derivation of linear elasticity from finite elasticity by means of Gamma-convergence is a well-known result, which has been extended to different models also beyond the elastic regime. However, in these results the applied forces are usually assumed to be dead loads, that is, their density in the reference configuration is independent of the actual deformation. In this paper we begin a study of the variational derivation of linear elasticity in the presence of live loads. We consider a pure traction problem for a nonlinearly elastic body subject to a pressure live load and we compute its linearization for small pressure by Gamma-convergence. We allow for a weakly coercive elastic energy density and we prove strong convergence of minimizers.