Communications in Mathematical Sciences

Volume 15 (2017)

Number 7

Front migration for the dislocation strain in single crystals

Pages: 1843 – 1866



Nicolas van Goethem (Departamento de Matemática, Universidade de Lisboa, Portugal)


A single crystal is considered, i.e., a smooth elastic body $\Omega \subset \mathbb{R}^3$ containing a high density of point-defects and dislocations. In particular, we consider prismatic dislocation loops which result as the primary manifestation of irradiated or highly-deformed crystals. We consider linearized elasticity and identify the macroscopic dislocation-induced strain and its trace, directly related to the presence of dislocations, as the basic model variables. Further, we rely on a previously-introduced tensor version of a Cahn–Hilliard system in the context of incompatible linearized elasticity and consider the point-defects collapse into prismatic loops, yielding some well-formed microstructure. By means of a formal asymptotic analysis, we determine the front dynamics and obtain as a result a tensor version of Mullins–Sekerka dynamics. The associated gradient-flow formalism is also investigated.


dislocations, linear elasticity, incompatibility, Cahn–Hilliard system, evolution law, second principle

2010 Mathematics Subject Classification

35G31, 35J48, 35J50, 35K52, 35Q74

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The author was supported by the FCT Starting Grant “ Mathematical theory of dislocations: geometry, analysis, and modelling” (IF/00734/2013).

Received 14 July 2016

Accepted 28 April 2017

Published 16 October 2017