A three-dimensional misorientation axis- and inclination-dependent Kobayashi–Warren–Carter grain boundary model

The Kobayashi–Warren–Carter (KWC) phase-field model was originally conceived for two-dimensional systems to model grain boundary migration and grain rotation, which play a crucial role in nanocrystalline materials, and in phenomena such as superplasticity and recrystallization. Existing generalizations of the KWC model to three-dimensions construct the grain boundary energy as a function of misorientation angle between the grains, described as a scalar, and the inclination of the grain boundary. It is well-known that grain boundary energy is described on a five-dimensional space, where three dimensions describe misorientations and two describe inclinations. In this work, we generalize the KWC model by constructing a frame-invariant energy density that is sensitive to all the five-dimensions of misorientations and inclinations. In addition, we derive representations for energy density that result in different forms of anisotropies in inclination and misorientation. The developed framework enables us to introduce the effect of material symmetry on the inclination-dependence. We demonstrate the richness of the model using various three-dimensional numerical examples that simulate anisotropic grain coarsening and grain rotation.