The gap in Pure Traction Problems between Linear Elasticity and Variational Limit of Finite Elasticity

TL;DR

This paper investigates the connection between nonlinear hyperelastic energy and linear elasticity in pure traction problems, revealing conditions under which linear elasticity models are valid or fail.

Contribution

It derives a limit elastic energy from nonlinear energy, clarifies the role of compatibility conditions, and shows when linear elasticity approximations are valid or break down.

Findings

Weak convergence of strains under certain conditions

Compatibility condition ensures coincidence of minima

Violations lead to unbounded or multiple minimizers

Abstract

A limit elastic energy for pure traction problem is derived from re-scaled nonlinear energy of an hyperelastic material body subject to an equilibrated force field. We show that the strains of minimizing sequences associated to re-scaled non linear energies weakly converge, up to subsequences, to the strains of minimizers of a limit energy, provided an additional compatibility condition is fulfilled by the force field. The limit energy is different from classical energy of linear elasticity; nevertheless the compatibility condition entails the coincidence of related minima and minimizers. A strong violation of this condition provides a limit energy which is unbounded from below, while a mild violation may produce a limit energy with infinitely many extra minimizers which are not minimizers of standard linear elastic energy and whose strains are not uniformly bounded. A relevant consequence of this analysis is that a rigorous validation of linear elasticity fails for compressive force fields that do not fulfil such compatibility condition.