On the incompatibility of strains and its application to mesoscopic studies of plasticity

TL;DR

This paper analyzes the incompatibility constraints of strain fields in materials with dislocations, providing a systematic 3D approach to incorporate these constraints into free energy models for studying plasticity and microstructure interactions.

Contribution

It introduces a comprehensive 3D framework for the incompatibility equations accommodating dislocation densities in strain-based free energy models.

Findings

Derived three incompatibility equations for 3D dislocation networks.

Calculated internal stress fields for anisotropic materials with dislocations.

Demonstrated the application by comparing stress fields of edge dislocations.

Abstract

Structural transitions are invariably affected by lattice distortions. If the body is to remain crack-free, the strain field cannot be arbitrary but has to satisfy the Saint-Venant compatibility constraint. Equivalently, an incompatibility constraint consistent with the actual dislocation network has to be satisfied in media with dislocations. This constraint can be incorporated into strain-based free energy functionals to study the influence of dislocations on phase stability. We provide a systematic analysis of this constraint in three dimensions and show how three incompatibility equations accommodate an arbitrary dislocation density. This approach allows the internal stress field to be calculated for an anisotropic material with spatially inhomogeneous microstructure and distribution of dislocations by minimizing the free energy. This is illustrated by calculating the stress field of an edge dislocation and comparing it with that of an edge dislocation in an infinite isotropic medium. We outline how this procedure can be utilized to study the interaction of plasticity with polarization and magnetization.