Dilatation of a one-dimensional nonlinear crack impacted by a periodic elastic wave

TL;DR

This paper investigates how a nonlinear crack in a one-dimensional elastic medium responds to periodic elastic waves, analyzing the nonlinear interactions, mean dilatation, and the effects of sinusoidal forcing through theoretical modeling.

Contribution

It provides a theoretical analysis of nonlinear crack response to periodic waves, extending previous numerical studies with bounds and solutions for a broader range of conditions.

Findings

Mean displacements are discontinuous across the crack.

Crack dilatation increases with forcing amplitude.

Bounds for solution extrema are established.

Abstract

The interactions between linear elastic waves and a nonlinear crack with finite compressibility are studied in the one-dimensional context. Numerical studies on a hyperbolic model of contact with sinusoidal forcing have shown that the mean values of the scattered elastic displacements are discontinuous across the crack. The mean dilatation of the crack also increases with the amplitude of the forcing levels. The aim of the present theoretical study is to analyse these nonlinear processes under a larger range of nonlinear jump conditions. For this purpose, the problem is reduced to a nonlinear differential equation. The dependence of the periodic solution on the forcing amplitude is quantified under sinusoidal forcing conditions. Bounds for the mean, maximum and minimum values of the solution are presented. Lastly, periodic forcing with a null mean value is addressed. In that case, a result about the mean dilatation of the crack is obtained.