Dynamic stress response kernels for dislocations and cracks: unified anisotropic Lagrangian formulation

Abstract

Elastodynamic cohesive-zone models for defects such as cracks or dislocations (such as the Geubelle-Rice model for cracks, or the Dynamic Peierls Equation for flat-core dislocations), feature the same stress-response convolution kernel in space and time. It accounts for in-plane elastic wave propagation, while its associated instantaneous radiative term accounts for radiative losses in the surrounding medium. These objects are well-known for isotropic elasticity, with their space-time representations involving generalized functions. For anisotropic elasticity they were unknown. The paper presents a derivation using the Stroh formalism. Their Fourier representation rests exclusively on the so-called prelogarithmic Lagrangian factor $L(v)$, while their space-time form involves its derivative $p(v)=L'(v)$, the prelogarithmic impulsion function. A straightforward consequence is the reformulation of the stress in the Weertman model of steadily-moving dislocations in terms of $L(v)$. Special care being paid to the causality constraint, the theory covers indifferently subsonic, intersonic and supersonic regimes of motion. The theory proposed is suitable to phase-field-type Fourier-based numerical codes for planar systems of defects in anisotropic elastodynamics.