On the Derivation of Boundary Conditions for Continuum Dislocation Dynamics

Continuum dislocation dynamics (CDD) is a single crystal strain gradient plasticity theory based exclusively on the evolution of the dislocation state. Recently, we derived a constitutive theory for the average dislocation velocity in CDD in a phase field-type description for an infinite domain. In the current work, so-called rational thermodynamics is employed to obtain thermodynamically consistent boundary conditions for the dislocation density variables of CDD. We find that rational thermodynamics reproduces the bulk constitutive equations as obtained from irreversible thermodynamics. The boundary conditions we find display strong parallels to the microscopic traction conditions derived by Gurtin and Needleman (M.E. Gurtin and A. Needleman, J. Mech. Phys. Solids 53 (2005) 1–31) for strain gradient theories based on the Kröner–Nye tensor.