Signorini problem as a variational limit of obstacle problems in nonlinear elasticity

Abstract

An energy functional for the obstacle problem in linear elasticity is obtained as a variational limit of nonlinear elastic energy functionals describing a material body subject to pure traction load under a unilateral constraint representing the rigid obstacle. There exist 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 in linear elasticity. However, if the force field acting on the body fulfils an appropriate geometric admissibility condition, we can show coincidence of minima. The analysis developed here provides a rigorous variational justification of the Signorini problem in linear elasticity, together with an accurate analysis of the unilateral constraint.

Keywords

• calculus of variations
• signorini problem
• linear elasticity
• nonlinear elasticity
• finite elasticity
• gamma-convergence
• asymptotic analysis
• unilateral constraint