The elastodynamic bimaterial interface under mode I and mode II loading

This article provides the mathematical solutions to the elastodynamic fields of a semi-infinite interface lying along two dissimilar media subjected to sudden loading. The article offers the solution to the cases when: (1) the interface slips, i.e., it cannot transfer shear stress from one material to the other, which represents a Hertzian contact; (2) the interface is welded, i.e., all stress components are transferred, in which case it acts as a crack. We obtain the full, explicit analytic solutions to the fields of the interface along the slipping boundary via the Wiener-Hopf and Cagniard-de Hoop techniques. We show that such interface does not entail an oscillatory singularity at the crack tip owing to the fact that shear forces are not transferred across the interface. The welded interface crack leads to a matricial Wiener-Hopf problem that is not reducible to any form that would allow an immediate analytic factorisation of the resulting scattering kernel matrix. The factorisation in this case is achieved via successive Abrahams approximations of the scattering kernel itself, rather than via a Williams expansion of the elastic displacement field. This leads to a quickly convergent solution that retains the asymptotic character in the near and in the far field. Explicit proof of the nature of the oscillatory singularity at the crack tip is provided by studying the scattering matrix, which in the near field is shown to reduce to a Daniele-Khrapkov form amenable to analytic factorisation. The solutions presented in this article are explicit, and will prove eminently useful in the modelling of fast fibre debonding in composite materials, and in the study of the scattering of seismic waves by cracks and faults in layered media