Group-theoretic bifurcation mechanism of pattern formation in three-dimensional uniform materials

An underlying mathematical mechanism for formation of periodic geometric patterns in uniform materials is investigated. Symmetry of a rectangular parallelepiped domain with periodic boundaries is modeled as an equivariance to a group O(2) × O(2) × O(2). The standard group-theoretic approach is used to investigate possible patterns of this domain that emerge through direct and some secondary bifurcations. This investigation clarifies the mechanism of successive symmetry-breaking bifurcation, which entails a variety of geometrical patterns in three-dimensional uniform materials. In particular, a few characteristic geometric patterns, such as oblique layer, column and diamond patterns, are identified and classified. Pattern simulations are conducted on geometrical patterns of joints in a calcite and folds in a stratum to reinforce pertinence of the pattern formation mechanism. Images of three-dimensional patterns of joints and folds are expanded into the triple Fourier series, and transient processes of bifurcation are reconstructed to arrive at possible courses of successive bifurcation. Qualitative information from this approach can offer insight into transient courses of deformation, which have been overlooked up to now.