The critical state of granular media: convergence, stationarity and disorder

Discrete-element simulations are used to monitor several micro-scale characteristics within a granular material, demonstrating their convergence during loading toward the critical state, their stationarity at the critical state, and the evolution of their disorder toward the critical state. Convergence, stationarity and disorder are studied in the context of the Shannon entropy and two forms of Kullback-Leibler relative entropy. Probability distributions of 20 aspects of micro-scale configuration, force and movement are computed for three topological objects: particles, voids and contacts. The probability distributions of these aspects are determined at numerous stages during quasi-static biaxial compression and unloading. Not only do stress and density converge to the critical state, but convergence and stationarity are manifested in all of the micro-scale aspects. The statistical disorder (entropy) of micro-scale movements and strains generally increases during loading until the critical state is reached. When the loading direction is reversed, order is briefly restored, but continued loading induces greater disorder in movements and strains until the critical state is reached again.