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Structural transitions are invariably affected by lattice distortions. If thebody is to remain crack-free, the strain field cannot be arbitrary but has tosatisfy the Saint-Venant compatibility constraint. Equivalently, anincompatibility constraint consistent with the actual dislocation network hasto be satisfied in media with dislocations. This constraint can be incorporatedinto strain-based free energy functionals to study the influence ofdislocations on phase stability. We provide a systematic analysis of thisconstraint in three dimensions and show how three incompatibility equationsaccommodate an arbitrary dislocation density. This approach allows the internalstress field to be calculated for an anisotropic material with spatiallyinhomogeneous microstructure and distribution of dislocations by minimizing thefree energy. This is illustrated by calculating the stress field of an edgedislocation and comparing it with that of an edge dislocation in an infiniteisotropic medium. We outline how this procedure can be utilized to study theinteraction of plasticity with polarization and magnetization.

On the incompatibility of strains and its application to mesoscopic studies of plasticity