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

• Published in: Archive for rational mechanics and analysis Vol. 234; no. 3; pp. 1091 - 1120
• Main Authors: Maddalena, Francesco, Percivale, Danilo, Tomarelli, Franco
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2019; Springer Nature B.V
• Subjects: Classical Mechanics; Compatibility; Complex Systems; Elastic limit; Elasticity; Energy; Fluid- and Aerodynamics; Mathematical and Computational Physics; Physics; Physics and Astronomy; Theoretical; Traction
• ISSN: 0003-9527; 1432-0673
• DOI: 10.1007/s00205-019-01408-2

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.