The kinematics related to crystal plasticity starts with putting a triad of vectors to represent the coordinate axes on the crystalline axis is a basic assumption that was elucidated by Mandel . Once this is assumed, we can apply the deformation gradient in the same manner as we did in Chapter 2 for the macroscale formulation, where we assume a multiplicative decomposition of elastic and plastic deformation components after Lee and Liu  similar to Equation 2.26. Here we leave out the volumetric component due to damage (note: if we include the damage or porosity, it would relate to pores or damage growing within the crystal; otherwise, we can include it within the intergranular constraint because that is where many of the pore-inducing particles and grain boundaries are affected):
where Fp is the plastic or inelastic deformation gradient essentially related to crystallographic slip in the case presented in this chapter. Fp could also represent twinning, grain boundary sliding, or diffusion but typically has not been formulated to represent such inelastic behavior. Corresponding to the deformation gradient is the velocity gradient that is given by
where and . Now the plastic velocity gradient corresponding to crystallographic ...