Stress-free states of continuum dislocation fields: Rotations, grain boundaries, and the Nye dislocation density tensor

TL;DR

This paper develops a theoretical framework linking grain boundaries, rotational deformations, and stress-free dislocation states in continuum dislocation theory, providing explicit forms and interpretations for stress-free configurations.

Contribution

It derives general relations for stress-free dislocation fields, showing they can be expressed as superpositions of Frank walls and linked to rotational deformations.

Findings

Stress-free states can be explicitly constructed as superpositions of Frank walls.

Grain boundaries satisfy Frank's formula and have vanishing stress in the continuum limit.

A least-squares method for determining the rotation field of dislocation densities is proposed.

Abstract

We derive general relations between grain boundaries, rotational deformations, and stress-free states for the mesoscale continuum Nye dislocation density tensor. Dislocations generally are associated with long-range stress fields. We provide the general form for dislocation density fields whose stress fields vanish. We explain that a grain boundary (a dislocation wall satisfying Frank's formula) has vanishing stress in the continuum limit. We show that the general stress-free state can be written explicitly as a (perhaps continuous) superposition of flat Frank walls. We show that the stress-free states are also naturally interpreted as configurations generated by a general spatially-dependent rotational deformation. Finally, we propose a least-squares definition for the spatially-dependent rotation field of a general (stressful) dislocation density field.