Topological dynamics within the dissipative structure of granular materials at the critical state

Granular materials subjected to shear loading may reach a critical state where stress and volume remain constant while shear strain continues to develop. As a typical dissipative system, granular assemblies at the critical state undergo continuous particle rearrangements driven by contact sliding. In this study, DEM simulations of biaxial loading tests are performed on granular assemblies with different initial densities. Based on the microscopic energy dissipation, the shear band in the dense specimen and the whole loose specimen are identified as the dissipative structures, holding an identical dissipation rate at the critical state under the same confining pressure. The dissipative structures at the critical state are sustained by a unified topological dynamic mechanism. Specifically, particle displacement fields exhibit localized vortex structures that correlate with the spatial distribution of dissipative structures, suggesting that energy dissipation is mainly accommodated by locally self-organized particle motions. From a topological perspective, meso-loops continuously undergo generation and disappearance through dynamic reconfiguration, reaching a dynamic equilibrium with a unique reorganization rate within the dissipative structures at the critical state. Associating with this loop reconfiguration, the elastic energy storage and release processes within the dissipative structures achieve dynamic equilibrium, exhibiting a unique rate independent of initial density.