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J Eng Math (2007) 59:399417 DOI 10.1007/s10665-007-9173-3

Summary: J Eng Math (2007) 59:399­417
DOI 10.1007/s10665-007-9173-3
A multiple-scales approach to crack-front waves
Andrew N. Norris · I. David Abrahams
Received: 25 June 2007 / Accepted: 5 July 2007 / Published online: 2 August 2007
© Springer Science+Business Media B.V. 2007
Abstract Perturbation of a steadily 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 gov-
erned 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 asso-
ciated with the loading. The method is illustrated for a scalar crack model and then applied to the elastodynamic
mode I problem. The crack-front wave-dispersion relation is found by requiring that the energy release rate is
unaltered under perturbation and dispersive properties of the crack-front wave speed are described for the first time.
The example problems considered demonstrate the potential of MAE for moving-boundary-value problems with
multiple scales.
Keywords Crack-front waves · Crack propagation · Dynamic fracture · Matched asymptotic expansions ·
Multiple scales · Wiener­Hopf
1 Introduction


Source: Abrahams, I. David - Department of Mathematics, University of Manchester


Collections: Mathematics