Fields of bounded deformation for mesoscopic dislocations

In this paper we discuss the consequences of the distributional approach to dislocations in terms of the mathematical properties of the auxiliary model fields such as displacement and displacement gradient which are obtained directly from the main model field here considered as the linear strain. We...

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Bibliographic Details
Published in:Mathematics and mechanics of solids 2014-07, Vol.19 (5), p.579-600
Main Author: Van Goethem, Nicolas
Format: Article
Language:English
Subjects:
Crystal growth
Crystals
Deformation
Dislocations
Displacement
Equivalence
Mathematical models
Strain
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Summary:In this paper we discuss the consequences of the distributional approach to dislocations in terms of the mathematical properties of the auxiliary model fields such as displacement and displacement gradient which are obtained directly from the main model field here considered as the linear strain. We show that these fields cannot be introduced rigourously without the introduction of gauge fields or, equivalently, without cuts in the Riemann foliation associated with the dislocated crystal. In a second step we show that the space of bounded deformations follows from the distributional approach in a natural way and discuss the reasons why it is adequate to model dislocations. The case of dislocation clusters is also addressed, as it represents an important issue in industrial crystal growth while from a mathematical point of view, peculiar phenomena might appear at the set of accumulation points. The elastic–plastic decomposition of the strain within this approach is also given a precise meaning.
ISSN:1081-2865
1741-3028
DOI:10.1177/1081286513479196