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.

本文为半无限界面沿两种不同介质承受突然载荷的弹性动力学场提供了数学解决方案。本文为以下情况提供了解决方案：（1）界面打滑，即它不能将剪切应力从一种材料传递到另一种材料，这表示赫兹接触；（2）界面是焊接的，即所有应力分量都已转移，在这种情况下，它起裂缝的作用。我们通过Wiener-Hopf和Cagniard-de Hoop技术获得了沿滑动边界的界面场的完整的，明确的解析解。我们表明，由于剪切力没有在界面上传递，这样的界面在裂纹尖端处不会产生振荡奇点。焊接的界面裂纹导致维纳-霍夫矩阵问题，该问题无法简化为任何形式的形式，该形式将允许对所产生的散射核矩阵进行立即分析分解。在这种情况下，分解是通过散射核本身的连续Abrahams逼近来实现的，而不是通过弹性位移场的Williams展开来实现的。这导致了一种快速收敛的解决方案，该解决方案在近场和远场中都保留了渐近特性。通过研究散射矩阵，可以清楚地证明裂纹尖端处振荡奇异性的性质，该散射矩阵在近场中被证明可以简化为适合解析因式分解的Daniele-Khrapkov形式。本文介绍的解决方案是明确的，