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

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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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Authors and Affiliations

  1. Dipartimento di Meccanica, Matematica, Management, Politecnico di Bari, Via Re David 200, 70125, Bari, Italy

    Francesco Maddalena

  2. Dipartimento di Ingegneria Meccanica, Università di Genova, Piazzale Kennedy, Fiera del Mare, Padiglione D, 16129, Genova, Italy

    Danilo Percivale

  3. Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milano, Italy

    Franco Tomarelli

Authors
  1. Francesco Maddalena
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  2. Danilo Percivale
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  3. Franco Tomarelli
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Correspondence to Francesco Maddalena.

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Communicated by G. Dal Maso

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Research partially supported by C.N.R. INDAM Project 2018: G.N.A.M.P.A.–Problemi asintotici ed evolutivi con applicazioni a metamateriali e reti

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Maddalena, F., Percivale, D. & Tomarelli, F. The Gap Between Linear Elasticity and the Variational Limit of Finite Elasticity in Pure Traction Problems. Arch Rational Mech Anal 234, 1091–1120 (2019). https://doi.org/10.1007/s00205-019-01408-2

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