Waves in flexural beams\with nonlinear adhesive interaction

TL;DR

This paper investigates the dynamic behavior of an elastic beam interacting with a substrate through a nonlinear, breakable adhesive force, analyzing existence, non-uniqueness, and long-term dynamics of solutions.

Contribution

It establishes existence of solutions, demonstrates non-uniqueness, and characterizes the influence of nonlinear adhesive interaction on the beam's dynamics.

Findings

Existence of solutions in energy space

Counterexamples to uniqueness

Long-term asymptotic behavior characterized

Abstract

The paper studies the initial boundary value problem related to the dynamic evolution of an elastic beam interacting with a substrate through an elastic-breakable forcing term. This discontinuous interaction is aimed to model the phenomenon of attachement-detachement of the beam occurring in adhesion phenomena. We prove existence of solutions in energy space and exhibit various counterexamples to uniqueness. Furthermore we characterize some relavant features of the solutions, ruling the main effectes of the nonlinearity due to the elasic-breakable term on the dynamical evolution, by proving the linearization property according to \cite{G96} and an asymtotic result pertaining the long time behavior.