We consider the slow and athermal deformations of amorphous solids and show how the ensuing sequence of discrete plastic rearrangements can be mapped onto a directed network. The network topology reveals a set of highly connected regions joined by occasional one-way transitions. The highly connected regions include hierarchically organized hysteresis cycles and subcycles. At small to moderate strains this organization leads to near-perfect return point memory. The transitions in the network can be traced back to localized particle rearrangements (soft spots) that interact via Eshelby-type deformation fields. By linking topology to dynamics, the network representations provide new insight into the mechanisms that lead to reversible and irreversible behavior in amorphous solids.

While memory effects emerge from systems of wildly varying lengthscales and timescales, the reduction of a complex system with many interacting elements into one simple enough to be understood without also losing the complex behavior continues to be a challenge. Here, we investigate how bistable cylindrical origamis provide such a reduction via tunably interactive memory behaviors. We base our investigation on folded sheets of Kresling patterns that function as two-state memory units. By linking several units, each with a selected activation energy, we construct a one-dimensional material that exhibits return-point memory. After a comprehensive experimental analysis of the relation between the geometry of the pattern and the mechanical response for a single bit, we study the memory of a bellows composed of four bits arranged in series. Since these bits are decoupled, the system reduces to the Preisach model, and we can drive the bellows to any of its 16 allowable states by following a prescribed sequence of compression and extension. We show how to reasonably discriminate between states by measuring the system's total height and stiffness near equilibrium. Furthermore, we establish the existence of geometrically disallowed defective stable configurations that expand the configuration space to 64 states with a more complex transition pattern. Using empirical considerations of the mechanics, we analyze the hierarchical structure of the corresponding diagram, which includes Garden of Eden states and subgraphs. We highlight two irreversible transformations, namely shifting and erasure of defects, leading to memory behaviors reminiscent of those observed with more complex glassy systems.

Recent experiments and simulations of amorphous solids plastically deformed by an oscillatory drive have found a surprising 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. It was also found that 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 in these systems 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. More specifically, 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 transition 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. The 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 the sizes of the limit cycles become incompatible with the sizes of the regions of reversibility.