Topology of the energy landscape of sheared amorphous solids and the irreversibility transition

Recent experiments and simulations of amorphous solids plastically deformed by oscillatory drive have foundsurprising behavior - for small strain amplitudes the dynamics can be reversible, which is contrary to the usual notion of plasticity as an irreversible form of deformation. This reversibility allows the system to reach limit-cycles in which plastic events repeat indefinitely under the oscillatory drive. Reaching reversible limit-cycles, can take a large number of driving cycles and it was surmised that the plastic events encountered during the transient period are not encountered again and are thus irreversible. Using a graph representation of the stable configurations of the system and the plastic events connecting them, we show that the notion of reversibility is more subtle. We find that reversible plastic events are abundant, and that a large portion of the plastic events encountered during the transient period are actually reversible, in the sense that they can be part of a reversible deformation path. We observe that the transition graph can be decomposed into clusters of configurations that are connected by reversible transitions. These clusters are the strongly connected components of the graph and their sizes turn out to be power-law distributed. The largest of these are grouped in regions of reversibility, which in turn are confined by regions of irreversibility whose number proliferates at larger strains. Our results provide an explanation for the irreversibility transition - the divergence of the transient period at a critical forcing amplitude. Long transients result from transition between clusters of reversibility in a search for a cluster large enough to contain a limit-cycle of a specific amplitude. For large enough amplitudes, the search time becomes very large, since sizes of the limit cycles become incompatible with the sizes of the regions of reversibility.