Mathematical concepts for the micromechanical modelling of dislocation dynamics with a phase-field approach

This contribution reviews the mathematical concepts of micromechanical modelling in the phase-field approach applied to dislocation dynamics. The intention is two-fold. On the one hand, modelling of dislocation dynamics is a very recent field of development in phase-field theory, in comparison to the simulation of diffusional phase transformation and related microstructure evolution problems in materials science. The reason is that modelling dislocation dynamics poses several challenges for phase-field concepts which go beyond purely diffusional problems in materials science such as, e.g. dendritic solidification, as we point out in Section 3. On the other hand, the modelling of dislocations has triggered further wide-ranging developments of phase-field based models for deformation problems. This is an important development, since a comprehensive model for deformation problems should include displacive as well as diffusional degrees of freedom from the atomic scale to the microscale. This is something phase-field theory is capable of, as discussed in this review article. We aim to give an overview of relevant mathematical concepts, and to stimulate further steps in this direction.