Pure traction and Signorini problem between linear and finite elasticity

Abstract: A limit elastic energy for pure traction problem is derived from re-scaled nonlinear energies of a hyper-elastic material body subject to an equilibrated force field. We show 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. The limit energy functional exhibits a gap that makes it different from the classical linear elasticity functional. Nevertheless a suitable compatibility condition entails the coincidence of related minima and minimizers, hence the rigorous validation of linear elasticity. 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. As a consequence, a rigorous validation of linear elasticity fails for compressive force fields that infringe such a compatibility condition. This phenomenon has relevant consequences also on the validation of the Signorini problem in linear elasticity: there are loads pushing the body against the obstacle, but unfit for the geometry of the whole system body-obstacle, so that the corresponding variational limit turns out to be different from the classical Signorini problem. However, if the force field acting on the body fulfils an appropriate geometric admissibility condition, we can show coincidence of related solutions. The asymptotic analysis provides a sharp and rigorous variational justification of the unilateral problem in linear elasticity, together with an accurate analysis of the unilateral constraint. These results were obtained in a joint research with Francesco Maddalena (Politecnico di Bari) and Danilo Percivale (Università di Genova).

MathematicsPhysics

Audience: researchers in the topic

Nečas Seminar on Continuum Mechanics

Series comments: This seminar was founded on December 14, 1966.

Faculty of Mathematics and Physics, Charles University, Sokolovská 83, Prague 8. If not written otherwise, we will meet on Mondays at 15:40 in lecture hall K3 (2nd floor)