Thermodynamic coarsening of dislocation mechanics and the size-dependent continuum crystal plasticity

Starting from the standard coarsening of dislocation kinematics, we derive the size-dependent continuum crystal plasticity by systematic thermodynamic coarsening of dislocation mechanics. First, we observe that the energies computed from different kinematic descriptions are different. Then, we consider systems without boundary dissipation (relaxation) and derive the continuum approximation for the free energy of elastic-plastic crystals. The key elements are: the two-dimensional nature of dislocation pile-ups at interfaces, the localized nature of the coarsening error in energy, and, the orthogonal decomposition theorem for compatible and incompatible elastic strain fields. Once the energy landscape is defined, the boundary dissipation is estimated from the height of energy barriers. The characteristic lengths are the average slip plane spacing for each slip system. They may evolve through the double-cross slip mechanism. The theory features the slip-dependent interface free energy and interface dissipation for penetrable interfaces. The main constitutive parameters are derived from elasticity. The exception is the dependence of interface energy on slip plane orientation, which is determined from numerical results. The theory requires no higher order boundary conditions. The only novel boundary conditions are kinematic, involving slip relaxation on the two sides of an interface. (C) 2009 Elsevier Ltd All rights reserved.