Analysis of an Inelastic Contact Problem for the Damped Wave Equation

TL;DR

This paper investigates the inelastic contact dynamics of a viscoelastic string with a rigid obstacle, providing a global weak solution that captures energy dissipation and velocity vanishing during contact, relevant for fluid-structure interactions.

Contribution

It introduces a novel approximation method to construct global weak solutions for inelastic contact problems in a one-dimensional viscoelastic string model.

Findings

Weak solutions exhibit energy dissipation only during contact.

Velocity vanishes after contact in a weak sense.

Model serves as a simplified framework for fluid-structure contact problems.

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 $\frac1\varepsilon\chi_{\{\eta<0\}} (\partial_t\eta)^-$. 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.