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 $\frac{1}{\varepsilon }{\chi }_{\{\eta <0\}}{\left({\partial }_t\eta \right)}^{-}$. The weak solution exhibits well-controlled energy dissipation, which occurs exclusively during contact and is concentrated on a set of zero measure, specifically 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.