Calculus of Variations and Geometric Measure Theory

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


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


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.