---
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.'
address: |
    Lehrstuhl für Mechanik - Materialtheorie, Ruhr-Universität Bochum,\
    D-44780 Bochum, Germany
author:
- 'K. C. Le[^1]'
title: 'Three-dimensional continuum dislocation theory'
---

dislocations (A) ,crystal plasticity (B) ,finite strain (B) ,variational calculus (C) .

Introduction {#sec:introduction}
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In view of a huge number of dislocations appearing in plastically deformed crystals (which typically lies in the range $10^8\div 10^{15}$ dislocations per square meter) the necessity of developing a physically meaningful continuum dislocation theory (CDT) to describe the evolution of dislocation network and predict the formation of microstructure in terms of mechanical and thermal 