A multiple scales approach to crack front waves

TL;DR

This paper develops a multiple scales asymptotic method to analyze crack front waves, deriving dispersion relations and wave speeds, revealing dispersive properties for the first time in elastodynamics.

Contribution

It introduces a novel multiple scales approach using matched asymptotic expansions to analyze crack front waves in elastodynamics, providing new insights into wave dispersion.

Findings

Derived the dispersion relation for crack front waves.

Calculated wave speed as a function of nondimensional parameters.

Demonstrated dispersive properties of crack front waves for the first time.

Abstract

Perturbation of a propagating crack with a straight edge is solved using the method of matched asymptotic expansions (MAE). This provides a simplified analysis in which the inner and outer solutions are governed by distinct mechanics. The inner solution contains the explicit perturbation and is governed by a quasi-static equation. The outer solution determines the radiation of energy away from the tip, and requires solving dynamic equations in the unperturbed configuration. The outer and inner expansions are matched via the small parameter L/l defined by the disparate length scales: the crack perturbation length L and the outer length scale l associated with the loading. The method is first illustrated for a scalar crack model and then applied to the elastodynamic mode I problem. The dispersion relation for crack front waves is found by requiring that the energy release rate is unaltered under perturbation. The wave speed is calculated as a function of the nondimensional parameter kl where k is the crack front wavenumber, and dispersive properties of the crack front wave speed are described for the first time. The example problems considered here demonstrate that the potential of using MAE for moving boundary value problems with multiple scales.