On the form for the plastic velocity gradient LP in crystal plasticity

It is usual to assume that the velocity gradient L-p associated with slip is Sigma (n)(kappa =1) gamma over dot(kappa)s(kappa) circle times n(kappa), where gamma over dot(kappa) is the rate of shear of the kappa th slip system defined by the slip direction s(kappa) and the normal to the plane n(kappa). The above expression, written directly as the linear superposition of instantaneous rates of shear over all the active slip systems, is motivated both by experiments and the single slip case. On the other hand, one might assume that the deformation gradient F-p has a multiplicative decomposition, but here the sequence of activation of the slip systems becomes important. These representations for F-p and L-p should be viewed as constraints, and they are consistent for linearized theories and proportional deformations but not for all deformations; that is, not all deformations for which L-p has the above form can one express F-p as a product of deformation gradients associated with each of the slip systems. In this paper, we discuss sufficient conditions under which the two constraints are equivalent, and we also provide a sufficient condition that guarantees that the form for the plastic deformation gradient is independent of the sequence of activation.