散体介质复杂力链网络演化持续同调拓扑研究

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

力链网络是表征颗粒物质和散体介质细观力学本质特征以及研究其宏观结构力学行为的重要基础.由于力链构型的复杂性和多重形态,使得力链网络难以分析处理.拓扑数据分析(TDA)为定量描述力链网络提供了一种简单、有效且易处理的方法.基于拓扑数据分析中的持续同调理论,建立了散体颗粒从接触网络到力链网络,再到拓扑空间模型的分析方法.以放顶煤开采为例,采用光弹实验方法,追踪放顶煤开采力链演化过程,对顶煤与覆岩力链网络进行持续同调拓扑分析.研究表明:将采场中散体颗粒应力作为持续同调分析的阈值ε时,Betti数可以作为衡量采场来压的一个量化指标,采场顶部覆岩和工作面前方煤岩力链网络β0曲线呈抛物线形,且来压时抛物线峰值高于初始状态下的峰值.顶部覆岩β0峰值位于强力链范围,而工作面前方煤岩β0峰值位于次强力链范围.β1曲线呈L形,在[0.3,1]区间内,β1趋近于0,在[0,0.2]区间内与ε近似呈反比.持续同调拓扑研究为定量分析颗粒介质复杂力链网络演化和宏观力学行为提供了有效方法.

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.