Three-dimensional continuum dislocation theory

TL;DR

This paper introduces a comprehensive three-dimensional continuum dislocation theory for single crystals with curved dislocations, deriving governing equations for equilibrium and motion, and providing simplified and asymptotic solutions.

Contribution

It presents a novel 3D continuum dislocation framework including equations for curved dislocations and accounts for dislocation resistance, extending existing theories.

Findings

Derived governing equations for dislocation equilibrium and motion.

Provided simplified small-strain theory.

Obtained asymptotic solutions for specific deformation problems.

Abstract

A three-dimensional continuum dislocation theory for single crystals containing curved dislocations is proposed. A set of governing equations and boundary conditions is derived for the true placement, plastic slips, and loop functions in equilibrium that minimize the free energy of crystal among all admissible functions, provided the resistance to the dislocation motion is negligible. For the non-vanishing resistance to dislocation motion the governing equations are derived from the variational equation that includes the dissipation function. A simplified theory for small strains is also provided. An asymptotic solution is found for the two-dimensional problem of a single crystal beam deforming in single slip and simple shear.