On the plasticity of single crystals: free energy, microforces, plastic-strain gradients

Abstract

This study develops a general theory of crystalline plasticity based on classical crystalline kinematics; classical macroforces; microforces for each slip system consistent with a microforce balance; a mechanical version of the second law that includes, via the microforces, work performed during slip; a rate-independent constitutive theory that includes dependences on plastic strain-gradients. The microforce balances are shown to be equivalent to yield conditions for the individual slip systems, conditions that account for variations in free energy due to slip. When this energy is the sum of an elastic strain energy and a defect energy quadratic in the plastic-strain gradients, the resulting theory has a form identical to classical crystalline plasticity except that the yield conditions contain an additional term involving the Laplacian of the plastic strain. The field equations consist of a system of PDEs that represent the nonlocal yield conditions coupled to the classical PDE that represents the standard force balance. These are supplemented by classical macroscopic boundary conditions in conjunction with nonstandard boundary conditions associated with slip. A viscoplastic regularization of the basic equations that obviates the need to determine the active slip systems is developed. As a second aid to solution, a weak (virtual power) formulation of the nonlocal yield conditions is derived. As an application of the theory, the special case of single slip is discussed. Specific solutions are presented: one a single shear band connecting constant slip-states; one a periodic array of shear bands.