Mathematical Models of Interfacial Motion of Crystalline Materials

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Kobayashi had benn investigated the problem for the mathematical model of grain boundary in polycrystalline materials since the middle of 90's, and completed 2D model already. The main goal of this research is to extend Kobayashi-Warren-Carter model to 3D model, which includes essential difficulties because the orientation variable must be extended from SO(2)-valued to SO(3)-valued. Therefore we intended to formuLate the 3D model mathematically, study mathematical theories concerning to it, ・and develop the technique of numerical simulation.We first construct the SO(3)-valued singular diffusivity equation in one dimensional and two dimensional space, and simulation codes. Then we compare the 2 expressions for SO(3); local coordinates (Rodrigues, Vector) and imbedding to 9 dimensional Euclidean space R^{9}, and concluded that the imbedding method is the way we have to go. According to this decision we first developed a 3D numerial code in which the values of orientation variables are kept in SO(3) by means of the normal projection to SO(3) and some special penalty, function. By coupling this orientation equation with the phase field equation which describes the solid-liquid phase dynamics, we completed the model and 3D numerical code which can describe the while process from nucleation, solidification, formation of grain boundaries and growth of the grains accompanied by the motion of grain boundaries.Finally in order to include the symmetries of crystalline structure to our model, we took the orientation space as the homogeneous space (the space obtained by dividing SO(3) by symmetry group) and tried to imbed.it to some (higher dimensional) Euclidian space. However we found it is very difficult to obtain the concrete imbedding in the situation where we are interested in.Our research has interesting side effect, which is a construction of mathematical model of cleavage which can be considered (in some sense) to be a reverse process of coarsening of grains.