Calculus of Variations and Geometric Measure Theory
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F. Maddalena - D. Percivale - F. Tomarelli

The Gap Between Linear Elasticity and the Variational Limit of Finite Elasticity in Pure Traction Problems

created by tomarelli1 on 25 Feb 2019
modified on 28 Jun 2019

[BibTeX]

Accepted Paper

Inserted: 25 feb 2019
Last Updated: 28 jun 2019

Journal: Arch Rational Mech Anal (2019)
Pages: 1-30
Year: 2019
Links: to Journal

Abstract:

A limit elastic energy for the pure traction problem is derived from re-scaled nonlinear energies of a hyperelastic material body subject to an equilibrated force field. We prove that the strains of minimizing sequences associated to re-scaled nonlinear 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 the 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 unboundedness of strains and a limit energy which has infinitely many extra minimizers which are not minimizers of standard linear elastic energy. A consequence of this analysis is that a rigorous validation of linear elasticity fails for compressive force fields that infringe up on such a compatibility condition.

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