Abstract
Pattern formation is often observed on the uniform (or macroscopically uniform) materials subjected to loading or deformation. In order to reveal the mechanism of pattern formation, we here investigate the mathematical structure of the bifurcation of a three-dimensional uniform domain with periodic boundaries by the group-theoretic bifurcation theory. First, we derive the concrete form of the bifurcation equations for three-dimensional uniform periodic materials by extending the results for two dimensional ones in a straightforward manner. Next, the symmetry of the kernel space of the bifurcation point and that of the bifurcation paths are classified by solving the bifurcation equations. Last, we conduct three-dimensional pattern simulation of the bifurcation phenomena of rocks and minerals in which the joint structure is observed.
