On hyperbolic contact problems

Abstract

We deal with hyperbolic variational inequalities modeling
 vibrations of two-dimensional structures with an obstacle. We focus
 on the plates with moderately large deflections. The nonlinear
 strain-displacements relations imply nonlinear elliptic parts of
 differential operators in considered problems. We distinguish two types of problems.
 In the first case only the deflections are considered with
 accelerations and the plane displacements are expressed using the Airy
 stress function. In the case of plane accelerations the full von
 K\'arm\'an system consisting of two equations and one variational
 inequality is considered. The existence of solutions is derived
 using the penalization method.