Analysis of an Inelastic Contact Problem for the Damped Wave Equation

Abstract

In this paper, we examine the dynamic behavior of a viscoelastic string oscillating above a rigid obstacle in a one-dimensional setting, accounting for inelastic contact between the string and the obstacle. We construct a global-in-time weak solution to this problem by using an approximation method that incorporates a penalizing repulsive force of the form $\frac{1}{\epsilon }{\chi }_{\{\eta <0\}}({\partial }_t\eta {)}^{-}$. The weak solution exhibits well-controlled energy dissipation, occurring only during contact on a set of zero measure and exclusively when the string moves downward. Furthermore, the velocity is shown to vanish after contact in a specific weak sense. This model serves as a simplified framework for studying contact problems in fluid-structure interaction contexts.