A model of crystal plasticity containing a natural length scale

Abstract

A crystal plasticity model is developed which is embedded with a natural length scale. The model is developed within the framework of Coleman-Gurtin thermodynamics of internal state variables. The normal multiplicative decomposition of the deformation gradient utilized in crystal plasticity is expanded to add an extra degree of freedom. The internal state variables introduced in the free energy include the elastic strain associated with the statistically stored dislocations, and the curvature, which is derived from the continuum theory of dislocations. In this theory, the curvature is the curl of the elastic rotation associated with the polar decomposition of the elastic deformation gradient. This leads to an internal stress field that results from the presence of geometrically necessary dislocations and possesses an inherent length scale, in addition to the normal mechanical resistance that is proportional to the square root of the statistically stored dislocations. A crystal plasticity model incorporating these internal stresses is implemented into a FEM code. The results of the models prediction of the formation of misoriented cells, whose size is determined by the length scale of the model, as well as the prediction of the gradients of misorientation at the interface of the two cells is compared with bi-crystal experiments on Aluminum. During the compression of these bi-crystals, the dislocation patterns that develop encompass local lattice rotations or cells of misorientation that have a hierarchical of length scales. Therefore, the kinematics of the model are generalized to include an additional rotational degree of freedom, and additional tensor state variables are derived from the kinematics. This model has many similarities with the multipolar dislocation/disclination theory of Eringen and Claus and may be viewed as a finite deformation extension of that theory. The relationship of this model, which now has multiple length scales, to existing deformation theory type of strain gradient models is then discussed, and the length scales are related to actual physical mechanisms.