Self-organization in the localised failure regime: metastable attractors and their implications on force chain functionality

Abstract

In transitive metastable chaotic dynamical systems, there are no invariant neighbourhoods in the phase space. The best that one can do is look for metastable or almost-invariant (AI) regions as a means to decompose the system into its basic self-organising building blocks. Here we study the metastable dynamics of a dense granular material embodying strain localization in 3D from the perspective of its conformational landscape: the state space of all observed conformations as defined by the local topology of individual grains relative to their first ring of contacting neighbors. We determine the metastable AI sets that divide this conformational landscape, such that grain rearrangements from one conformation to another conformation in the same AI set occurs with high probability: by contrast, grain rearrangements involving conformational transitions between AI sets are unlikely. The great majority of conformational transitions are identity transitions: grains rearrange and exchange contacts to preserve those topological properties with the greatest influence on cluster stability, namely, the number of contacts and 3-cycles. Force chains show a clear preference for that AI set with the most number of accessible and highly connected conformations. Here force chains continually explore the conformational landscape, wandering from one rarely inhabited conformation to another. As force chains become overloaded and buckle, the energy released enables member grains to overcome the high dynamical barriers that separate metastable regions and subsequently escape one region to enter another in the conformational landscape. Thus, compared to grains locked in stable force chains, those in buckling force chains, confined to the shear band, show a greater propensity for not only non-identity transitions within each metastable region but also inter-transitions between metastable regions.