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The two-dimensional elastodynamic Green tensor is the primary building blockof solutions of linear elasticity problems dealing with nonuniformly movingrectilinear line sources, such as dislocations. Elastodynamic solutions forthese problems involve derivatives of this Green tensor, which stand ashypersingular kernels. These objects, well defined as distributions, provecumbersome to handle in practice. This paper, restricted to isotropic media,examines some of their representations in the framework of distribution theory.A particularly convenient regularization of the Green tensor is introduced,that amounts to considering line sources of finite width. Technically, it isimplemented by an analytic continuation of the Green tensor to complex times.It is applied to the computation of regularized forms of certain integrals oftensor character that involve the gradient of the Green tensor. These integralsare fundamental to the computation of the elastodynamic fields in the problemof nonuniformly moving dislocations. The obtained expressions indifferentlycover cases of subsonic, transonic, or supersonic motion. We observe that forfaster-than-wave motion, one of the two branches of the Mach cone(s) displayedby the Cartesian components of these tensor integrals is extinguished for someparticular orientations of source velocity vector.

wave motion

On the gradient of the Green tensor in two-dimensional elastodynamic  problems, and related integrals: Distributional approach and regularization,  with application to nonuniformly moving sources