On a frictional unilateral contact problem in nonlinear elasticity—existence and smoothing procedure

Abstract

This note is devoted to a novel frictional unilateral contact problem in finite-strain elasticity. Here, we adopt the Tresca friction model from linear elasticity. Our analysis relies on the polyconvexity approach to nonlinear elasticity due to J. Ball. We include the delicate case where the elastic body neither is fixed nor has a deformation prescribed along some part of its boundary, but rests on a rigid foundation with a free boundary and is only submitted to forces and loads acting in the interior (like gravity) and at the boundary, respectively. This leads to a loss of coercivity and necessitates an extra condition that prevents the body from escaping by the geometry of the obstacle. This new condition extends a similar condition of Ciarlet and Nečas from the frictionless case to the case of Tresca friction. In addition, as a first step towards a numerical treatment of such nonlinear problems, we present a smoothing procedure that tackles the non-smooth term from Tresca friction and provide a convergence result for the novel smoothing method. This article is part of the theme issue ‘Non-smooth variational problems and applications’.