Archive for Rational Mechanics and Analysis

, Volume 234, Issue 3, pp 1091–1120 | Cite as

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

  • Francesco MaddalenaEmail author
  • Danilo Percivale
  • Franco Tomarelli


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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Francesco Maddalena
    • 1
    Email author
  • Danilo Percivale
    • 2
  • Franco Tomarelli
    • 3
  1. 1.Dipartimento di Meccanica, Matematica, ManagementPolitecnico di BariBariItaly
  2. 2.Dipartimento di Ingegneria MeccanicaUniversità di GenovaGenovaItaly
  3. 3.Dipartimento di MatematicaPolitecnico di MilanoMilanoItaly

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