On nonstationary von Karman variational inequalities

We deal with systems consisting of a nonlinear evolution variational inequality for the deflection and a nonlinear quasistationary equation for the Airy stress function. The systems describe moderately large deflections of thin viscoelastic plates with an inner obstacle. We distinguish two kinds of problems. Pseudoparabolic variational inequality for the quasistationary deflections and the hyperbolic inequality for the dynamic case. In both cases we transform the original problem to one canonical inequality in a Hilbert space of deflections. The pseudoparabolic problem is solved using a semidiscrete approximation transforming the problem into the sequence of stationary variational inequalities. The hyperbolic problem is solved by the penalization method.