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

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

created by tomarelli1 on 25 Feb 2019



Inserted: 25 feb 2019
Last Updated: 25 feb 2019

Pages: 1-23
Year: 2018

ArXiv: 1810.01339 PDF
Links: WEB address on Arxiv


A limit elastic energy for pure traction problem is derived from re-scaled nonlinear energy of an hyperelastic material body with reference configuration in a Lipschitz domain Ω and subject to an equilibrated force field. We show that the strains of minimizing sequences associated to re-scaled non linear energies weakly converge in $L^2(\Omega, R^N)$, 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 nonconvex in 3D and even in 2D 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 unboundedness of strains and 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.

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