Matched asymptotics of Rayleigh-wave fields near cuspidal ridges and gorges

TL;DR

This paper develops a matched asymptotic framework to analyze Rayleigh wave fields near cuspidal surface features, revealing how different geometries influence stress behavior and wave fields.

Contribution

It introduces a local asymptotic description for elastic fields near cuspidal ridges and gorges, highlighting the effects of cusp exponents on stress and wave behavior.

Findings

Stress near gorges behaves as r^{-1/2}

Ridge fields are asymptotically rigid with bounded stress

Finite-element simulations support theoretical stress laws

Abstract

We construct a local matched-asymptotic description of time-harmonic elastic fields generated by Rayleigh waves near cuspidal elements of a traction-free surface. The free surface is represented locally by a cusp graph with exponent $0<\alpha<1$, or equivalently by a vanishing-width horn $b(s)=B s^m$, $m=1/\alpha>1$. A cuspidal gorge is a zero-opening re-entrant notch: its leading field is the Williams crack-tip field, and the stresses behave as $r^{-1/2}$. The cusp exponent affects the gorge through lower-order corrections and through the stress cut-off produced by rounding the bottom. In contrast, a cuspidal ridge behaves as an elastic horn with vanishing width. The leading admissible free-tip field is asymptotically rigid (bounded stress), distinct from the high-energy branch supported by a finite tip truncation, where stresses grow as $\sigma \sim \rho^{-m}$. Finite-element calculations for the local static Lam\'e problems support these predictions: the free-tip ridge test confirms the absence of crack-like growth, the truncated ridge recovers the high-energy law, and the gorge stress slope is found to be close to $-1/2$.