Stability of a dynamic frictionless contact problem for viscoelastic materials with rigid‐plastic foundation and long‐term memory

A mathematical model that describes a dynamic frictionless contact of a viscoelastic body with a rigid‐plastic foundation is considered. The constitutive law is assumed to be a nonlinear constitutive law with long memory. We derive a variational formulation of the model, and the global existence and uniqueness of a solution are proved. The proof is based on arguments of evolutionary variational inequalities, a classical existence and uniqueness result of nonlinear first‐order evolution inequalities, and Banach fixed point theorem. Finally, we apply the multiplier method to establish the exponential energy decay of the solution.