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Willis and Movchan [Willis, J.R., Movchan, A.B., 1995. Dynamic weight functions for a moving crack I. Mode I loading. J. Mech. Phys. Solids 43, 319.] devised weight functions for a dynamic mode I fracture, within the singular crack model, using a ®rst order perturbation of in-plane crack motion from the 2D results. Ramanathan and Fisher [Ramanathan, S., Fisher, D.S., 1997. Dynamics and instabilities of planar tensile cracks in heterogeneous media. Phys. Rev. Lettr. 79, 877.] reformulated the Willis-Movchan's result in terms of crack growth at constant fracture energy, thereby con®rming the existence of a crack front wave. Such a wave, as a propagating mode local to the moving crack front, was seen in the non-perturbative numerical simulations based on a cohesive zone fracture model, equivalent to growth at constant fracture energy. In this paper, the result of Ramanathan and Fisher, given in the wavenumber±frequency domain, is recast in the wavenumber±time domain to analyze fracture propagation within ®rst-order perturbations for the singular crack model. This allows application of a spectral numerical methodology and is shown to be consistent with the known 2D results. Through analysis of a single spatial mode of crack shape, the propagating crack front wave and its resonance are demonstrated. Crack propagation through a randomly heterogeneous zone, and growth of disorder with propagation distance, are also examined. 7

SUMMARYThis paper summarizes the formulation and numerical implementation of a spectral scheme specially designed for dynamic anti-plane shear (mode III) fracture problems. The scheme allows for a wide variety of simulations ranging from the dynamic loading of stationary cracks to the spontaneous propagation of faults. The method is based on a spectral form of the elastodynamic relation between the shearing tractions acting on the fracture surface and the resulting slip velocity response for a planar two-dimensional crack in an infinite linearly elastic medium. The formulation is expressed in the Fourier domain and involves a convolution over the past slip or slip rate history. Conversion between spectral and real domains is performed through the fast Fourier transform algorithm. The time-integration scheme is explicit and a variety of constitutive laws can be used to express the strength of the material on the fault plane. The stability and accuracy of the numerical scheme are discussed through comparison with existing analytical solutions involving non-propagating and propagating cracks. The extraction of the dynamic stress intensity factor from the computed slip history is described.1997 by John Wiley & Sons, Ltd.

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