Topological study of persistent homology on complicated force chain network evolution in granular media

The force chain network is a crucial basis for characterizing the essential features of granular media microscopic mechanics and for studying the mechanical behavior of their macroscopic structures. Due to the complexity and multiformity of the force chain configuration, the topological data analysis (TDA) method provides a simple, effective, and manageable method to quantitatively describe the force network. Based on the theory of persistent homology in TDA, the analysis method of granules from contact network to force network to topological model is established. The two key factors affecting the structure of a force chain network are its connectivity and closure. β₀ reflects the number of particle clusters whose normal force is greater than the provided threshold. β₁ reflects the number of holes in the force chain cluster. Persistent homology, a mathematical method to calculate the topological features of structures in metric spaces with different resolutions, is applied to the structural analysis of force chain networks. Different from other methods that separately consider force threshold levels, the evolution of a force chain structure at the force threshold level is helpful in understanding the persistence characteristics of geometric structures at various force levels and describing the force chain network completely and deeply. For example, the top coal and overburden force chain evolution in top-coal caving mining is investigated through photoelastic experiments, and the TDA of the top-coal caving force network chain is performed. The research shows that when the particle stress is obtained as the threshold ε of persistent homology analysis in top-coal caving mining, the Betti number in persistent homology analysis can be used to evaluate periodic weighting. In the overlying strata and front of the working face, the curves of β₀ are parabolic, of which the peak values under the periodic weighting are higher than that under the initial condition. Furthermore, the difference between them is that the peak values in the overlying strata are located in the range of strong force chains, whereas those in front of the working face are located in the range of sub-strong force chains. The curves of β₁ show L-shaped, suggesting that within 0.3, 1, β₁ approaches 0; within 0, 0.2, β₁ is approximately inversely proportional to ε; and within 0.3, 1, β₁ trends to 0. Topological studies of persistent homology provide an effective method for quantitative analysis of the complicated force chain network evolution and macroscopic mechanical behavior in granular media.