Dislocation-induced linear-elastic strain dynamics by a Cahn–Hilliard-type equation
Nicolas Van Goethem · Mathematics and Mechanics of Complex Systems · 2016
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 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 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.L the one-dimensional Hausdorff measure concentrated in L, and b the Burgers vector, constant on the line.Moreover, inc is the incompatibility operator; i.e.,