Causal Stroh formalism for uniformly-moving dislocations in anisotropic media: Somigliana dislocations and Mach cones

TL;DR

This paper extends Stroh's formalism to include causality for uniformly-moving dislocations in anisotropic media, enabling unified analysis of subsonic and supersonic regimes and revealing Mach cones through analytical and geometric methods.

Contribution

It introduces a causal modification to Stroh's formalism, allowing for comprehensive velocity-dependent dislocation field calculations including supersonic regimes.

Findings

Analytic expressions for dislocation fields valid at all velocities

Identification of Mach cones in supersonic dislocation motion

Explanation of backward Mach cones using slowness surfaces

Abstract

In this work, Stroh's formalism is endowed with causal properties on the basis of an analysis of the radiation condition in the Green tensor of the elastodynamic wave equation. The modified formalism is applied to dislocations moving uniformly in an anisotropic medium. In practice, accounting for causality amounts to a simple analytic continuation procedure whereby to the dislocation velocity is added an infinitesimal positive imaginary part. This device allows for a straightforward computation of velocity-dependent field expressions that are valid whatever the dislocation velocity ---including supersonic regimes--- without needing to consider subsonic and supersonic cases separately. As an illustration, the distortion field of a Somigliana dislocation of the Peierls-Nabarro-Eshelby-type with finite-width core is computed analytically, starting from the Green tensor of elastodynamics. To obtain the result in the form of a single compact expression, use of the modified Stroh formalism requires splitting the Green function into its reactive and radiative parts. In supersonic regimes, the solution obtained displays Mach cones, which are supported by Dirac measures in the Volterra limit. From these results, an explanation of Payton's `backward' Mach cones [R.G. Payton, Z. Angew. Math. Phys. 46, 282--288 (1995)] is given in terms of slowness surfaces, and a simple criterion for their existence is derived. The findings are illustrated by full-field calculations from analytical formulas for a dislocation of finite width in iron, and by Huygens-type geometric constructions of Mach cones from ray surfaces.