Thermodynamic forces in single crystals with dislocations

A simple model for the evolution of macroscopic dislocation regions in a single crystal is presented. This model relies on maximal dissipation principle within Kröner’s geometric description of the dislocated crystal. Mathematical methods and tools from shape optimization theory provide equilibrium relations at the dislocation front, similarly to previous work achieved on damage modelling (J Comput Phys 33(16):5010–5044, 2011). The deformation state variable is the incompatible strain as related to the dislocation density tensor by a relation involving the Ricci curvature of the crystal underlying elastic metric. The time evolution of the model variables follows from a novel interpretation of the Einstein–Hilbert flow in terms of dislocation microstructure energy. This flow is interpreted as the dissipation of non-conservative dislocations, due to the climb mechanism, modelled by an average effect of mesoscopic dislocations moving normal to their glide planes by adding or removing points defects. The model equations are a fourth-order tensor parabolic equation involving the operator “incompatibility,” here appearing as a tensorial counterpart of the scalar Laplacian. This work encompasses and generalizes results previously announced (C R Acad Sci Paris Ser I 349:923–927, 2011), with in addition a series of physical interpretations to give a meaning to the newly introduced concepts.