Coupled phase transformations and plasticity as a field theory of deformation incompatibility

Abstract

The duality between terminating discontinuities of fields and the incompatibilities of their gradients is used to define a coupled dynamics of the discontinuities of the elastic displacement field and its gradient. The theory goes beyond standard translational and rotational Volterra defects (dislocations and disclinations) by introducing and physically grounding the concept of generalized disclinations in solids without a fundamental rotational kinematic degree of freedom (e.g. directors). All considered incompatibilities have the geometric meaning of a density of lines carrying appropriate topological charge, and a conservation argument provides for natural physical laws for their dynamics. Thermodynamic guidance provides the driving forces conjugate to the kinematic objects characterizing the defect motions, as well as admissible constitutive relations for stress and couple stress. We show that even though higher-order kinematic objects are involved in the specific free energy, couple stresses may not be required in the mechanical description in particular cases. The resulting models are capable of addressing the evolution of defect microstructures under stress with the intent of understanding dislocation plasticity in the presence of phase transformation and grain boundary dynamics.