Multiple memory formation in glassy landscapes

Anomalous Softness in Amorphous Matter in the Reversible Plastic Regime

We study an elastoplastic model of an amorphous solid subject to athermal quasistatic cyclic shear strain. We focus on cycling amplitudes in the so-called reversible-plastic regime where, after a transient, the system locks into a hysteretic limit cycle and returns to the same microscopic configuration after one or more strain cycles. We show that the ground state energy of the terminal limit cycle decreases with increasing cycling amplitude. In analogy to an annealed alloy or an aged colloidal glass, one would expect the states with lower energy to be mechanically harder and to require larger stresses and strains to trigger microscopic rearrangements. However, we show the opposite result: the systems with lower energy cycled at higher strain amplitude are mechanically softer and begin to exhibit plastic rearrangements at smaller stresses and strains within the cycle. We explain this anomaly quantitatively in terms of Eshelby inclusion theory where an inclusion is subjected to a particular negative stress value after it undergoes a yielding event. These results point the way toward measurements to be conducted in experiments and particle-based computer simulations on cyclically sheared amorphous solids.

Emergent marginality in frustrated multistable networks

We study disordered networks of coupled bistable elastic elements, representing a coarse-grained view of amorphous solids. We find that such networks self-organize to a marginally stable state, in which the barrier for local activations becomes vanishingly small. The model provides unique access to both local and global properties associated with marginal stability. We directly measure pseudo-gaps in the spectrum of local excitations, as well as diverging fluctuations under shear. Crucially, the dynamics are dominated by a small population of bonds that are locally unstable, which give rise to quasi-localized, low-frequency vibrational modes and scale-free avalanches of instabilities. We propose a correction to the scaling between the pseudo-gap exponent and avalanche statistics based on diverging length fluctuations. Our model combines a coarse-grained view with a continuous, real-space implementation, providing novel insights to a wide class of amorphous solids.

Generalizing multiple memories from a single drive: The hysteron latch

Far-from-equilibrium systems can form memories of previous deformations or driving. In systems from sheared glassy materials to buckling beams to crumpled sheets, this behavior is dominated by return-point memory, in which revisiting a past extremum of driving restores the system to a previous state. Cyclic driving with both positive and negative strains forms multiple nested memories, as in a single-dial combination lock, while asymmetric driving (only positive strain) cannot. We study this case in a general model of hysteresis that considers discrete elements called hysterons. We show how two hysterons with a frustrated interaction can violate return-point memory, realizing multiple memories of asymmetric driving. This reveals a general principle for designing systems that store sequences of cyclic driving, whether symmetric or asymmetric. In disordered systems, asymmetric driving is a sensitive tool for the direct measurement of frustration.

Bifurcations of inflating balloons and interacting hysterons

While many materials exhibit a complex, hysteretic response to external driving, there has been a surge of interest in how the complex dynamics of internal materials states can be understood and designed to process and store information. We consider a system of connected rubber balloons that can be described by a Preisach model of noninteracting hysterons under pressure control but for which the hysterons become coupled under volume control. We study this system by exploring the possible transition graphs, as well as by introducing a configuration space approach which tracks the volumes of each balloon. Changes in the transition graphs turn out to be related to changes in the topology of the configuration space of the balloons, providing a particularly geometric view of how transition graphs can be designed, as well as additional information on the existence of hidden metastable states. This class of systems is more general than just balloons.

Controlled pathways and sequential information processing in serially coupled mechanical hysterons

The complex sequential response of frustrated materials results from the interactions between material bits called hysterons. Hence, a central challenge is to understand and control these interactions, so that materials with targeted pathways and functionalities can be realized. Here, we show that hysterons in serial configurations experience geometrically controllable antiferromagnetic-like interactions. We create hysteron-based metamaterials that leverage these interactions to realize targeted pathways, including those that break the return point memory property, characteristic of independent or weakly interacting hysterons. We uncover that the complex response to sequential driving of such strongly interacting hysteron-based materials can be described by finite state machines. We realize information processing operations such as string parsing in materia, and outline a general framework to uncover and characterize the FSMs for a given physical system. Our work provides a general strategy to understand and control hysteron interactions, and opens a broad avenue toward material-based information processing.

Emergent disorder and mechanical memory in periodic metamaterials

Ordered mechanical systems typically have one or only a few stable rest configurations, and hence are not considered useful for encoding memory. Multistable and history-dependent responses usually emerge from quenched disorder, for example in amorphous solids or crumpled sheets. In contrast, due to geometric frustration, periodic magnetic systems can create their own disorder and espouse an extensive manifold of quasi-degenerate configurations. Inspired by the topological structure of frustrated artificial spin ices, we introduce an approach to design ordered, periodic mechanical metamaterials that exhibit an extensive set of spatially disordered states. While our design exploits the correspondence between frustration in magnetism and incompatibility in meta-mechanics, our mechanical systems encompass continuous degrees of freedom, and thus generalize their magnetic counterparts. We show how such systems exhibit non-Abelian and history-dependent responses, as their state can depend on the order in which external manipulations were applied. We demonstrate how this richness of the dynamics enables to recognize, from a static measurement of the final state, the sequence of operations that an extended system underwent. Thus, multistability and potential to perform computation emerge from geometric frustration in ordered mechanical lattices that create their own disorder.

Tunable sequential pathways through spatial partitioning and frustration tuning in soft metamaterials

Elastic instabilities have been leveraged in soft metamaterials to attain novel functionalities such as mechanical memory and sequential pathways. Pathways have been realized in complex media or within a collection of hysteretic elements. However, much less has been explored in frustrated and partitioned soft metamaterials. In this work, we introduce spatial partitioning as a method to localize deformation in sub-regions of a large and soft metamaterial. The partitioning is achieved through the strategic arrangement of soft inclusions in a soft lattice, which form distinct regions behaving as mechanical units. We examine two partitions: an equally spaced layer partition with mechanical units connected in series, and a cross partition, represented by interconnected series of mechanical units in parallel. Sequential pathways are obtained by frustrating the partitioned metamaterial post-manufacture and are characterized by tracking the polarization change in each partition region. Through a combination of experiments and simulations, we demonstrate that partitioning enables tuning the pathway from longitudinal with weak interactions to a pathway exhibiting strong interactions rising from geometric incompatibility and central domain rotation. We show that tuning the level of uniform lateral pre-strain provides a wide range of tunability from disabling to modifying the sequential pathway. We also show that imposing a nonuniform confinement and altering the tilting of one or two of the domain edges enables to program the pathway, access a larger set of states, and tune the level of interaction between the regions.

Stress-activated friction in sheared suspensions probed with piezoelectric nanoparticles

A hallmark of concentrated suspensions is non-Newtonian behavior, whereby the viscosity increases dramatically once a characteristic shear rate or stress is exceeded. Such strong shear thickening is thought to originate from a network of frictional particle-particle contact forces, which forms under sufficiently large stress, evolves dynamically, and adapts to changing loads. While there is much evidence from simulations for the emergence of this network during shear thickening, experimental confirmation has been difficult. Here, we use suspensions of piezoelectric nanoparticles and exploit the strong local stress focusing within the network to activate charge generation. This charging can then be detected in the measured ac conductance and serve as a signature of frictional contact formation. The direct link between stress-activated frictional particle interactions and piezoelectric suspension response is further demonstrated by tracking the emergence of structural memory in the contact network under oscillatory shear and by showing how stress-activated friction can drive mechano-transduction of chemical reactions with nonlinear reaction kinetics. Taken together, this makes the ac conductance of piezoelectric suspensions a sensitive in-situ reporter of the micromechanics associated with frictional interactions.

Granular convergence as an iterated local map

Granular convergence is a property of a granular pack as it is repeatedly sheared in a cyclic, quasistatic fashion, as the packing configuration changes via discrete events. Under suitable conditions the set of microscopic configurations encountered converges to a periodic sequence after sufficient shear cycles. Prior work modeled this evolution as the iteration of a pre-determined, random map from a set of discrete configurations into itself. Iterating such a map from a random starting point leads to similar periodic repetition. This work explores the effect of restricting the randomness of such maps in order to account for the local nature of the discrete events. The number of cycles needed for convergence shows similar statistical behavior to that of numerical granular experiments. The number of cycles in a repeating period behaves only qualitatively like these granular studies.

Counting and Sequential Information Processing in Mechanical Metamaterials

Materials with an irreversible response to cyclic driving exhibit an evolving internal state which, in principle, encodes information on the driving history. Here we realize irreversible metamaterials that count mechanical driving cycles and store the result into easily interpretable internal states. We extend these designs to aperiodic metamaterials that are sensitive to the order of different driving magnitudes, and realize "lock and key" metamaterials that only reach a specific state for a given target driving sequence. Our metamaterials are robust, scalable, and extendable, give insight into the transient memories of complex media, and open new routes towards smart sensing, soft robotics, and mechanical information processing.

Competition between Energy and Dynamics in Memory Formation

Bistable objects that are pushed between states by an external field are often used as a simple model to study memory formation in disordered materials. Such systems, called hysterons, are typically treated quasistatically. Here, we generalize hysterons to explore the effect of dynamics in a simple spring system with tunable bistability and study how the system chooses a minimum. Changing the timescale of the forcing allows the system to transition between a situation where its fate is determined by following the local energy minimum to one where it is trapped in a shallow well determined by the path taken through configuration space. Oscillatory forcing can lead to transients lasting many cycles, a behavior not possible for a single quasistatic hysteron.

Dissipation Indicates Memory Formation in Driven Disordered Systems

Disordered and amorphous materials often retain memories of perturbations they have experienced since preparation. Studying such memories is a gateway to understanding this challenging class of systems. However, it often requires the ability to measure local structural changes in response to external drives. Here, we show that dissipation is a generic macroscopic indicator of the memory of the largest perturbation. Through experiments in crumpled sheets under cyclic drive, we show that dissipation transiently increases when first surpassing the largest perturbation due to irreversible structural changes with unique statistics. This finding is used to devise novel memory readout protocols based on global observables only. The general applicability of this approach is demonstrated by revealing a similar memory effect in a three-dimensional amorphous solid.

Mechanical annealing and memories in a disordered solid

Shearing a disordered or amorphous solid for many cycles with a constant strain amplitude can anneal it, relaxing a sample to a steady state that encodes a memory of that amplitude. This steady state also features a remarkable stability to amplitude variations that allows one to read the memory. Here, we shed light on both annealing and memory by considering how to mechanically anneal a sample to have as little memory content as possible. In experiments, we show that a "ring-down" protocol reaches a comparable steady state but with no discernible memories and minimal structural anisotropy. We introduce a method to characterize the population of rearrangements within a sample and show how it connects with the response to amplitude variation and the size of annealing steps. These techniques can be generalized to other forms of glassy matter and a wide array of disordered solids, especially those that yield by flowing homogeneously.

Memory from coupled instabilities in unfolded crumpled sheets

Crumpling an ordinary thin sheet transforms it into a structure with unusual mechanical behaviors, such as enhanced rigidity, emission of crackling noise, slow relaxations, and memory retention. A central challenge in explaining these behaviors lies in understanding the contribution of the complex geometry of the sheet. Here we combine cyclic driving protocols and three-dimensional (3D) imaging to correlate the global mechanical response and the underlying geometric transformations in unfolded crumpled sheets. We find that their response to cyclic strain is intermittent, hysteretic, and encodes a memory of the largest applied compression. Using 3D imaging we show that these behaviors emerge due to an interplay between localized and interacting geometric instabilities in the sheet. A simple model confirms that these minimal ingredients are sufficient to explain the observed behaviors. Finally, we show that after training, multiple memories can be encoded, a phenomenon known as return point memory. Our study lays the foundation for understanding the complex mechanics of crumpled sheets and presents an experimental and theoretical framework for the study of memory formation in systems of interacting instabilities.

Sequential snapping and pathways in a mechanical metamaterial

Materials that feature bistable elements, hysterons, exhibit memory effects. Often, these hysterons are difficult to observe or control directly. Here, we introduce a mechanical metamaterial in which slender elements, interacting with pushers, act as mechanical hysterons. We show how we can tune the hysteron properties and pathways under cyclic compression by the geometric design of these elements and how we can tune the pathways of a given sample by tilting one of the boundaries. Furthermore, we investigate the effect of the coupling of a global shear mode to the hysterons as an example of the interactions between hysteron and non-hysteron degrees of freedom. We hope our work will inspire further studies on designer matter with targeted pathways.

Cooperative effects driving the multi-periodic dynamics of cyclically sheared amorphous solids

When subject to cyclic forcing, amorphous solids can reach periodic, repetitive states, where the system behaves plastically, but the particles return to their initial positions after one or more forcing cycles, where the latter response is called multi-periodic. It is known that plasticity in amorphous materials is mediated by local rearrangements called ``soft spots' or ``shear transformation zones'.Experiments and simulations indicate that soft spots can be modeled as hysteretic two-state entities interacting via quadrupolar displacement fields generated when they switch states and that these interactions can give rise to multi-periodic behavior. However, how interactions facilitate multi-periodicity is unknown. Here we show, using a model of random interacting two-state systems and molecular dynamics simulations, that multi-periodicity arises from oscillations in the magnitudes of the switching field of soft spots which cause soft spots to be active during some forcing cycles and idle during others. We demonstrate that these oscillations result from cooperative effects facilitated by the frustrated interactions between the soft spots. The presence of such mechanisms has implications for manipulating memory in frustrated hysteretic systems.

Profusion of transition pathways for interacting hysterons

The response, pathways, and memory effects of cyclically driven complex media can be captured by hysteretic elements called hysterons. Here we demonstrate the profound impact of hysteron interactions on pathways and memory. Specifically, while the Preisach model of independent hysterons features a restricted class of pathways which always satisfy return point memory, we show that three interacting hysterons generate more than 15 000 transition graphs, with most violating return point memory and having features completely distinct from the Preisach model. Exploring these opens a route to designer pathways and information processing in complex matter.

Complex pathways and memory in compressed corrugated sheets

The nonlinear response of driven complex materials-disordered magnets, amorphous media, and crumpled sheets-features intricate transition pathways where the system repeatedly hops between metastable states. Such pathways encode memory effects and may allow information processing, yet tools are lacking to experimentally observe and control these pathways, and their full breadth has not been explored. Here we introduce compression of corrugated elastic sheets to precisely observe and manipulate their full, multistep pathways, which are reproducible, robust, and controlled by geometry. We show how manipulation of the boundaries allows us to elicit multiple targeted pathways from a single sample. In all cases, each state in the pathway can be encoded by the binary state of material bits called hysterons, and the strength of their interactions plays a crucial role. In particular, as function of increasing interaction strength, we observe Preisach pathways, expected in systems of independently switching hysterons; scrambled pathways that evidence hitherto unexplored interactions between these material bits; and accumulator pathways which leverage these interactions to perform an elementary computation. Our work opens a route to probe, manipulate, and understand complex pathways, impacting future applications in soft robotics and information processing in materials.