A Note on a Frictional Unilateral Contact Problem in Nonlinear Elasticity

This note is devoted to a frictional unilateral contact problem in finite strain elasticity, where the Tresca friction model is adapted from linear elasticity. Here we focus to the issue of existence and aim at closing a gap in Gwinner (Philos Trans R Soc A 380(2236):Paper No. 20210356, 2022) between modeling and the existence proof using recession analysis together with the polyconvexity approach to nonlinear elasticity due to J. Ball (Arch Rational Mech Anal 63:337–403, 1977). As in Gwinner (Philos Trans R Soc A 380(2236):Paper No. 20210356, 2022) we include the delicate case where the elastic body is neither fixed nor a deformation is prescribed along some part of its boundary, but the body 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 such that the geometry of the obstacle prevents the body to escape. This new condition extends a similar condition of Ciarlet and Nečas (Arch Rational Mech Anal 87:319–338, 1985) from the frictionless case to the case of Tresca friction.