Bifurcations to diversify geometrical patterns of shear bands on granular material.

The mechanism to diversify geometrical patterns on granular material was elucidated using a group-theoretic image analysis of patterned shear bands, with associated numerical bifurcation analysis. Pattern formation of granular materials took the course of the evolution of a diamondlike diffuse bifurcation breaking uniformity, followed by further bifurcation, mode jumping, and the formation and disappearance of shear bands through localization. A chaotic explosive increase of possible postbifurcation states was emphasized as a mechanism to diversify geometrical patterns.