Analysis of an Inelastic Contact Problem for the Damped Wave Equation

2024/12/09 by Boris Muha, Muha, Boris, Srđan Trifunović +1
Computer Science · Engineering · #35L05 (Secondary) #74M15 (Primary) 74K05 #Analysis of PDEs (math.AP) #Contact Mechanics and Variational Inequalities #FOS: Mathematics #Mechanical stress and fatigue analysis #Vibration and Dynamic Analysis

paper · pdf · doi:10.48550/arxiv.2412.06185

openalex publication_date 2024/12/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \frac1εχ_\η<0\ (∂_tη)^-. 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.

Related

• ISS-Based Robustness to Various Neglected Damping Mechanisms for the 1-D Wave PDE
• Boundary controllability and source reconstruction in a viscoelastic string under external traction
• Exponential convergence to steady-states for trajectories of a damped dynamical system modelling adhesive strings
• At the end of a moving string
• A contact problem for viscoelastic bodies with inertial effects and unilateral boundary constraints