A small-deformation theory of strain-gradient plasticity for single crystals was developed. The theory was based upon: (i) a kinematic notion of a continuous distribution of edge and screw dislocations, (ii) a system of microscopic stresses consistent with a system of microscopic force balances, one balance for each slip system, (iii) a mechanical version of the second law that included, via the microscopic stresses, work performed during viscoplastic flow and (iv) a constitutive theory that permitted the free energy to depend upon densities of edge and screw dislocations and hence upon gradients of (plastic) slip; the microscopic stresses to depend upon slip-rate gradients. The microscopic force balanced when augmented by constitutive relations for the microscopic stresses resulted in a system of non-local flow rules in the form of second-order partial differential equations for the slips. When the free energy depended upon the dislocation densities the microscopic stresses were partially energetic, and this, in turn, led to back-stresses in the flow rules; on the other hand, a dependence of these stresses upon slip-rate gradients led to a strengthening. The flow rules, being non-local, require microscopic boundary conditions; as an aid to numerical solutions a weak (virtual power) formulation of the flow rule was derived.

Gradient Single-Crystal Plasticity with Free Energy Dependent on Dislocation Densities. M.E.Gurtin, L.Anand, S.P.Lele: Journal of the Mechanics and Physics of Solids, 2007, 55[9], 1853-78