Micromechanics and dislocation theory in anisotropic elasticity

TL;DR

This paper develops dislocation master-equations for anisotropic materials using micromechanics, deriving key tensor functions and applying them to 3D crack modeling and spherical inclusions.

Contribution

It introduces new dislocation master-equations in anisotropic elasticity and specifies them for Somigliana dislocations, with applications to crack and inclusion modeling.

Findings

Derived the second derivative of the anisotropic Green tensor as a sum of a $1/R^3$-term and a Dirac delta-term.

Specified dislocation master-equations for Somigliana dislocations.

Derived the interior Eshelby tensor for a spherical inclusion in anisotropic materials.

Abstract

In this work, dislocation master-equations valid for anisotropic materials are derived in terms of kernel functions using the framework of micromechanics. The second derivative of the anisotropic Green tensor is calculated in the sense of generalized functions and decomposed into a sum of a $1/R^3$-term plus a Dirac $\delta$-term. The first term is the so-called "Barnett-term" and the latter is important for the definition of the Green tensor as fundamental solution of the Navier equation. In addition, all dislocation master-equations are specified for Somigliana dislocations with application to 3D crack modeling. Also the interior Eshelby tensor for a spherical inclusion in an anisotropic material is derived as line integral over the unit circle.