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

TL;DR

This paper develops a distributional regularization method for the Green tensor in 2D elastodynamics, enabling better handling of hypersingular integrals related to nonuniformly moving sources across different motion regimes.

Contribution

It introduces a novel regularization of the Green tensor via complex time analytic continuation, facilitating practical computation of elastodynamic fields for moving dislocations.

Findings

Regularized Green tensor expressions valid for subsonic to supersonic motion.

Identification of Mach cone branch suppression at certain source orientations.

Enhanced mathematical framework for elastodynamic integral computations.

Abstract

The two-dimensional elastodynamic Green tensor is the primary building block of solutions of linear elasticity problems dealing with nonuniformly moving rectilinear line sources, such as dislocations. Elastodynamic solutions for these problems involve derivatives of this Green tensor, which stand as hypersingular kernels. These objects, well defined as distributions, prove cumbersome 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 is implemented by an analytic continuation of the Green tensor to complex times. It is applied to the computation of regularized forms of certain integrals of tensor character that involve the gradient of the Green tensor. These integrals are fundamental to the computation of the elastodynamic fields in the problem of nonuniformly moving dislocations. The obtained expressions indifferently cover cases of subsonic, transonic, or supersonic motion. We observe that for faster-than-wave motion, one of the two branches of the Mach cone(s) displayed by the Cartesian components of these tensor integrals is extinguished for some particular orientations of source velocity vector.