Gradient crystal plasticity models with a natural length scale in the hardening law

A class of crystal plasticity models based on the concept of microforces conjugate to slip-rate gradients is examined in the small strain framework. As an extension of the usual formulation, slip-rate gradients are introduced here into the incremental hardening law, including in this way a natural internal length scale derived recently in a closed form from relationships of the physically-based dislocation theory of plasticity. The condition for plastic flow on a crystallographic slip system involves other length scales, associated with the second-order gradients of slip and slip rate in energetic and dissipative terms, respectively. The interplay between the length-scales of physically different origin is illustrated by the examples of monotonic and cyclic deformation of one- and two-dimensional models of Cu single crystals with boundary constraints imposed on plastic slips. It is shown that selected earlier results are reproduced accurately if one or another length scale ceases to play an essential role. For cyclic deformations, the effects of the energetic length scale in the flow condition and of the natural length scale in the incremental hardening law can both be significant at the micron scale.