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Analysis of an Inelastic Contact Problem for the Damped Wave Equation

Boris Muha, Srđan Trifunović

TL;DR

This work analyzes a one-dimensional viscoelastic string constrained by a rigid obstacle, governed by the damped wave equation with a singular, measure-valued contact force $F_{con}$ and dissipation $D_{con}$. The authors construct a global weak solution via a velocity-penalized approximation, establish energy dissipation that occurs only during contact and a weak vanishing of velocity on contact, and prove that the contact force acts reactively as a jump in stress at the contact boundary. A zero-trace framework is developed to rigorously describe velocity behavior on contact graphs, and the force is characterized through limit processes and renormalized momentum inequalities. Numerical experiments illustrate energy decay and the formation of contact sets, including multiple disconnected components, underscoring the model's ability to represent post-contact dynamics in a simplified fluid-structure-interaction context.

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 $\frac1\varepsilonχ_{\{η<0\}} (\partial_tη)^-$. 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.

Analysis of an Inelastic Contact Problem for the Damped Wave Equation

TL;DR

This work analyzes a one-dimensional viscoelastic string constrained by a rigid obstacle, governed by the damped wave equation with a singular, measure-valued contact force and dissipation . The authors construct a global weak solution via a velocity-penalized approximation, establish energy dissipation that occurs only during contact and a weak vanishing of velocity on contact, and prove that the contact force acts reactively as a jump in stress at the contact boundary. A zero-trace framework is developed to rigorously describe velocity behavior on contact graphs, and the force is characterized through limit processes and renormalized momentum inequalities. Numerical experiments illustrate energy decay and the formation of contact sets, including multiple disconnected components, underscoring the model's ability to represent post-contact dynamics in a simplified fluid-structure-interaction context.

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 . 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.

Paper Structure

This paper contains 18 sections, 5 theorems, 100 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem description
  3. Literature review
  4. Notation and preliminaries
  5. Weak formulation and main result
  6. Proof of Theorem \ref{['main1']}
  7. Approximate problem
  8. Approximate solutions and uniform estimates
  9. Passing to the limit $n\to\infty$
  10. Uniform estimates in $\varepsilon$
  11. Passing to the limit $\varepsilon\to 0$
  12. Proof of Theorem $\ref{['main2']}$
  13. Statement 1
  14. Statement 2
  15. Proof of Theorem $\ref{['main3']}$
  16. ...and 3 more sections

Key Result

Theorem 2.1

For a given $\eta_0\in H_0^1(0,l)$ with $\eta_0\geq c>0$ and $v_0\in L^2(0,l)$, there exists a global weak solution in the sense of Definition weak:sol.

Figures (5)

  • Figure 1: An illustrative example of the contact set with characterization of contact force.
  • Figure 2: Solution of example 1 at different times.
  • Figure 3: Contact set (left) and velocity field (right) for example 1.
  • Figure 4: Solution of example 1 at different times.
  • Figure 5: Contact set (left) and velocity field (right) for example 2.

Theorems & Definitions (12)

  • Definition 2.1
  • Remark 2.1
  • Theorem 2.1: Existence of a weak solution
  • Theorem 2.2: Characterization of the velocity upon contact
  • Remark 2.2
  • Theorem 2.3: Characterization of contact force
  • Definition 4.1
  • Remark 4.1
  • Lemma 4.1
  • proof
  • ...and 2 more