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

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