Large-Amplitude Elastic Free-Surface Waves: Geometric Nonlinearity and Peakons

TL;DR

This paper develops a nonlinear model for large-amplitude elastic free-surface waves, revealing phenomena like peakon formation and potential brittle failure, which are not captured by traditional linear theories.

Contribution

It introduces a numerical formulation that explicitly accounts for the unknown free surface location, capturing nonlinear effects and peakon formation in elastic wave propagation.

Findings

Linear theory underestimates elastic rebound strength.

Nonlinear solutions can form peakons in finite time.

Potential link between peakons and brittle failure initiation.

Abstract

An instantaneous sub-surface disturbance in a two-dimensional elastic half-space is considered. The disturbance propagates through the elastic material until it reaches the free surface, after which it propagates out along the surface. In conventional theory, the free-surface conditions on the unknown surface are projected onto the flat plane $y = 0$, so that a linear model may be used. Here, however, we present a formulation that takes explicit account of the fact that the location of the free surface is unknown {\it a priori}, and we show how to solve this more difficult problem numerically. This reveals that, while conventional linearized theory gives an accurate account of the decaying waves that travel outwards along the surface, it can under-estimate the strength of the elastic rebound above the location of the disturbance. In some circumstances, the non-linear solution fails in finite time, due to the formation of a ``peakon'' at the free surface. We suggest that brittle failure of the elastic material might in practice be initiated at those times and locations.