Abstract

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 ${ϵ}^0$, 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 ${ϵ}^0$, 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.

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Mathematical Subject Classification 2010

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