Nonlinear waves in adhesive strings

TL;DR

This paper investigates a 1D semilinear wave equation modeling an elastic string interacting with an adhesive layer, analyzing well-posedness, solution properties, and numerical evolution under discontinuous stress conditions.

Contribution

It introduces a mathematical model with a discontinuous source term for adhesive interactions and studies its well-posedness and solution behavior.

Findings

Well-posedness of the initial boundary value problem established.

Qualitative properties of solutions analyzed.

Numerical simulations illustrate solution evolution.

Abstract

We study a 1D semilinear wave equation modeling the dynamic of an elastic string interacting with a rigid substrate through an adhesive layer. The constitutive law of the adhesive material is assumed elastic up to a finite critical state, beyond such a value the stress discontinuously drops to zero. Therefore the semilinear equation is characterized by a source term presenting jump discontinuity. Well-posedness of the initial boundary value problem of Neumann type, as well as qualitative properties of the solutions are studied and the evolution of different initial conditions are numerically investigated.