GENERALIZED CONVEXITY IN NONLINEAR ELASTICITY WITH APPLICATIONS TO UNILATERAL CONTACT

Abstract

This research/survey paper firstly gives an overview of generalized convexity in calculus of variations and nonlinear elasticity, centered at the notions of quasiconvexity, polyconvexity, and rank-one-convexity. Then ${\cal A}$-convexity based on Young measures and relaxation are discussed. In this context a general version of the Jensen’s inequality for ${\cal A}$-convex functions is given that extends the classical Jensen’s inequality for convex functions.

Secondly new results for the unilateral contact problem in nonlinear elasticity are presented. In particular existence results are derived for the pure contact-traction problem under an appropriate recession condition for quasiconvex as well as for nonquasiconvex energy densities, using in the latter case the Young measure approach.