A theory of grain boundaries that accounts automatically for grain misorientation and grain-boundary orientation

This work is an attempt to answer the question:Is there a physically natural method of characterizing the possible interactions between the slip systems of two grains that meet at a grain boundary—a method that could form the basis for the formulation of grain-boundary conditions? Here we give a positive answer to this question based on the notion of a Burgers vector as described by a tensor field GG on the grain boundary [Gurtin, M.E., Needleman, A., 2005. Boundary conditions in small-deformation single-crystal plasticity that account for the Burgers vector. J. Mech. Phys. Solids 53, 1–31]. We show that the magnitude of GG can be expressed in terms of two types of moduli: inter-grain moduli that characterize slip-system interactions between the two grains; intra-grain moduli that for each grain characterize interactions between any two slip systems of that grain.We base the theory on microscopic force balances derived using the principle of virtual power, a version of the second law in the form of a free-energy imbalance, and thermodynamically compatible constitutive relations dependent on GG and its rate. The resulting microscopic force balances represent flow rules for the grain boundary; and what is most important, these flow rules account automatically—via the intra- and inter-grain moduli—for the relative misorientation of the grains and the orientation of the grain boundary relative to those grains.