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On the Structure of Linear Dislocation Field Theory

Amit Acharya's picture

Amit Acharya          Robin J. Knops         Jeyabal Sivaloganathan

(In JMPS, 130 (2019), 216-244)

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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Amit Acharya's picture

I have uploaded, on imechanica, the final version of this paper and the publication information.

The imechanica version is marked up in blue in four places. I recommend this paper to students who want to learn about the big picture connections (with details) between plasticity theory and the theory of dislocations, including inertia, without getiing mired in intricate computations that might give the impression that the mathematical theory of dislocations has to do only with Green's function gymnastics (nothing wrong with the latter).

A theory is not a method, and a method is not a theory.

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