Publication 18-CNA-025

On the Structure of Linear Dislocation Field Theory

Amit Acharya
Dept. of Civil & Environmental Engineering
Center for Nonlinear Analysis
Carnegie Mellon University
Pittsburgh, PA 15213
acharyaamit@cmu.edu

R. J. Knopsy
The Maxwell Institute of Mathematical Sciences and
School of Mathematical and Computing Sciences
Heriot-Watt University
Edinburgh, EH14 4AS, Scotland, UK
r.j.knops@hw.ac.uk

J. Sivaloganathan
Department of Mathematical Sciences
University of Bath
Bath, BA2 7AY, UK
J.Sivaloganathan@bath.ac.uk

Abstract: Uniqueness of solutions in the linear theory of non-singular dislocations, studied as a special case of plasticity theory, is examined. The status of the classical, singular Volterra dislocation problem as a limit of plasticity problems is illustrated by a specific example that clarifies the use of the plasticity formulation in the study of classical dislocation theory. Stationary, quasi-static, and dynamical problems for continuous dislocation distributions are investigated subject not only to standard boundary and initial conditions, but also to prescribed dislocation density. In particular, the dislocation density field can represent a single dislocation line.

It is only in the static and quasi-static traction boundary value problems that such data are sufficient for the unique determination of stress. In other quasi-static boundary value problems and problems involving moving dislocations, the plastic and elastic distortion tensors, total displacement, and stress are in general non-unique for specified dislocation density. The conclusions are confirmed by the example of a single screw dislocation.

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