Vol. 4, No. 2, 2016

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Dislocation-induced linear-elastic strain dynamics by a Cahn–Hilliard-type equation

Nicolas Van Goethem

Vol. 4 (2016), No. 2, 169–195
DOI: 10.2140/memocs.2016.4.169

In a single crystal containing dislocations, the elastic strain defined by a linear constitutive law from the stress tensor can be written as the sum of a symmetric gradient and a solenoidal tensor ϵ0, called the dislocation strain. This latter part of the elastic strain is related to dislocations since its incompatibility equals the curl of the contortion. The aim of this paper is to derive a time-evolution law for the internal thermodynamic variable ϵ0, arising from the second law of thermodynamics, and to discuss its mathematical setting. This encompasses a discussion on the functional space used and about the equation’s well-posedness. A fourth-order time-dependent nonlinear PDE involving the incompatibility operator is found, which is similar in form to the Cahn–Hilliard equation, and represents in this respect a tensor generalization for solenoidal fields.

dislocations, linear elasticity, incompatibility, Cahn–Hilliard, evolution law, second principle
Mathematical Subject Classification 2010
Primary: 35J48, 35J50, 35G31, 35K52, 35Q74
Received: 25 January 2016
Revised: 23 May 2016
Accepted: 30 June 2016
Published: 21 November 2016

Communicated by Francesco dell'Isola
Nicolas Van Goethem
Departamento de Matemática
Universidade de Lisboa
Alameda da Universidade, C6
1749-016 Lisboa