Continuum dislocation dynamics as a phase field theory with conserved order parameters: formulation and application to dislocation patterning

Abstract The dynamics of dislocations can be formulated in terms of the evolution of continuous variables representing dislocation densities (‘continuum dislocation dynamics’). We show for various variants of this approach that the resulting models can be envisaged in terms of the evolution of order-parameter-like variables that strive to minimize a free energy functional which incorporates interface energy-like terms, i.e. as a phase field theory. We show that dislocation density variables obey non-standard conservation laws. These lead, in conjunction with the externally supplied work, to evolution equations that go beyond the classical framework of Allen-Cahn vs. Cahn–Hilliard equations. The approach is applied to the evolution of dislocation patterns in materials with B1(NaCl) lattice structure and it is demonstrated that it gives access to the formation of cellular dislocation patterns, and the concomitant emergence of both incidental and geometrically necessary dislocation boundaries.