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.

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

Tomarelli, Franco
2019-01-01

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.
2019
Pure Traction problems
Linear Elasticity
Nonlinear Elasticity
Finite Elasticity
Gamma-convergence
Asymptotic analysis
Nonlinear Neumann problems
Critical points
Calculus of Variations
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11311/1092889
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