We study a disordered network of bistable bonds subjected to periodic strain. The model is inspired by experiments on crumpled sheets, and it features behaviors associated with glasses, including a complex energy landscape, memories, and large avalanches. At small strain amplitudes, the system quickly converges to a limit cycle where it repeatedly cycles between a set of states. At large amplitudes, motion is erratic and does not converge to a limit cycle. The transition appears to be continuous, with diverging time scales. The nature of instabilities is different on both sides of the transition. At small strain amplitudes, instabilities are correlated only over a finite distance. Above the transition, instabilities are localized along diagonal bands. The distance between bands grows near the transition and appears to diverge. We propose a simple model that explains these observations. Below the transition, we propose a new "order parameter"-the polarization of the instabilities along the driving direction.

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.

Elastic structures can be designed to exhibit precise, complex, and exotic functions. While recent work has focused on the quasistatic limit governed by force balance, the mechanics at a finite driving rate are governed by Newton's equations. The goal of this work is to study the feasibility, constraints, and implications of creating disordered structures with exotic properties in the dynamic regime. The dynamical regime offers responses that cannot be realized in quasistatics, such as responses at an arbitrary phase, frequency-selective responses, and history-dependent responses. We employ backpropagation through time and gradient descent to design spatially specific steady states in disordered spring networks. We find that a broad range of steady states can be achieved with small alterations to the structure, operating both at small and large amplitudes. We study the effect of varying the damping, which interpolates between the underdamped and the overdamped regime, as well as the amplitude, frequency, and phase. We show that convergence depends on several competing effects, including chaos, large relaxation times, a gradient bias due to finite time simulations, and strong attenuation. By studying the eigenmodes of the linearized system, we show that the systems adapt very specifically to the task they were trained to perform. Our work demonstrates that within physical bounds, a broad array of exotic behaviors in the dynamic regime can be obtained, allowing for a richer range of possible applications.

Training materials through periodic drive allows to endow materials and structures with complex elastic functions. As a result of the driving, the system explores the high dimensional space of structures, ultimately converging to a structure with the desired response. However, increasing the complexity of the desired response results in ultra-slow convergence and degradation. Here, we show that by constraining the search space we are able to increase robustness, extend the maximal capacity, train responses that previously did not converge, and in some cases to accelerate convergence by many orders of magnitude. We identify the geometrical constraints that prevent the formation of spurious low-frequency modes, which are responsible for failure. We argue that these constraints are analogous to regularization used in machine learning. Our results present a unified understanding of the relations between complexity, degradation, convergence, and robustness.

We study networks of coupled bistable elastic elements, recently proposed as a model for crumpled thin sheets. The networks are poised on the verge of a localized instability, and the model allows 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. The networks also host 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. Crucially, the dynamics are dominated by a small population of bonds which are locally unstable. Our model combines a coarse-grained view with a continuous, real-space implementation, providing novel insights to a wide class of amorphous solids.

We introduce a training rule that enables a network composed of springs and dashpots to learn precise stress patterns. Our goal is to control the tensions on a fraction of "target" bonds, which are chosen randomly. The system is trained by applying stresses to the target bonds, causing the remaining bonds, which act as the learning degrees of freedom, to evolve. Different criteria for selecting the target bonds affects whether frustration is present. When there is at most a single target bond per node the error converges to computer precision. Additional targets on a single node may lead to slow convergence and failure. Nonetheless, training is successful even when approaching the limit predicted by the Maxwell Calladine theorem. We demonstrate the generality of these ideas by considering dashpots with yield stresses. We show that training converges, albeit with a slower, power-law decay of the error. Furthermore, dashpots with yielding stresses prevent the system from relaxing after training, enabling to encode permanent memories.

Material training is a method to endow materials with specific responses through external driving. We study the complexity of attainable responses, as expressed in the number of sites that are simultaneously controlled. With increased complexity, convergence to the desired response becomes very slow. The training error decays as a power law with an exponent that varies continuously and vanishes at a critical threshold, marking the limit of trainable responses. We study how the transition affects the vibrational properties. Approaching the critical threshold, low-frequency modes proliferate, approaching zero frequency. This implies that training causes material degradation and that training fails due to competing spurious low-frequency modes. We propose that the excess low-frequency spectrum is due to atypical local structures with bonds that nearly align. Our work explains how the presence of an exotic critical point affects the convergence of training, and could be relevant for understanding learning in physical systems.

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.

Traffic jams are an everyday hindrance to transport and typically arise when many vehicles have the same or a similar destination. We show, however, that even when uniformly distributed in space and uncorrelated, targets have a crucial effect on transport. At modest densities an instability arises leading to jams with emergent correlations between the targets. By considering limiting cases of the dynamics which map onto active Brownian particles, we argue that motility induced phase separation is the precursor to jams. That is, jams are MIPS seeds that undergo an extra instability due to target accumulation. This provides a quantitative prediction of the onset density for jamming, and suggests how jamming might be delayed or prevented. We study the transition between jammed and flowing phase, and find that transport is most efficient on the cusp of jamming.

Elastic metamaterials are often designed for a single permanent function. We explore the possibility of altering a material's function repeatedly through a self-organization,"training"process, controlled by applied strains. We show that the elastic function can be altered numerous times, though each new trained task imprints a memory. This ultimately leads to material degradation through the gradual reduction of the frequency gap in the density of states. We also show that retraining adapts previously trained low energy modes to a new function. As a result consecutive trained responses are realized similarly. We show how retraining can be exploited to attain a response that would otherwise be difficult.

The elastic behavior of materials operating in the linear regime is constrained, by definition, to operations that are linear in the imposed deformation. Though the nonlinear regime holds promise for new functionality, the design in this regime is challenging. In this paper we demonstrate that a recent approach based on training [Hexner et al., PNAS 2020, 201922847] allows responses that are inherently non-linear. By applying designer strains, a disordered solids evolves through plastic deformations that alter its response. We show examples of elaborate nonlinear training paths that lead to the following functions: (1) Frequency conversion (2) Logic gate and (3) Expansion or contraction along one axis, depending on the sequence of imposed transverse compressions. We study the convergence rate and find that it depends on the trained function.

Materials and machines are often designed with particular goals in mind, so that they exhibit desired responses to given forces or constraints. Here we explore an alternative approach, namely physical coupled learning. In this paradigm, the system is not initially designed to accomplish a task, but physically adapts to applied forces to develop the ability to perform the task. Crucially, we require coupled learning to be facilitated by physically plausible learning rules, meaning that learning requires only local responses and no explicit information about the desired functionality. We show that such local learning rules can be derived for any physical network, whether in equilibrium or in steady state, with specific focus on two particular systems, namely disordered flow networks and elastic networks. By applying and adapting advances of statistical learning theory to the physical world, we demonstrate the plausibility of new classes of smart metamaterials capable of adapting to users' needs in situ.

We consider disordered solids in which the microscopic elements can deform plastically in response to stresses on them. We show that by driving the system periodically, this plasticity can be exploited to train in desired elastic properties, both in the global moduli and in local "allosteric" interactions. Periodic driving can couple an applied "source" strain to a "target" strain over a path in the energy landscape. This coupling allows control of the system's response, even at large strains well into the nonlinear regime, where it can be difficult to achieve control simply by design.

Disordered solids often change their elastic response as they slowly age. Using experiments and simulations, we study how aging disordered planar networks under an applied stress affects their nonlinear elastic response. We are able to modify dramatically the elastic properties of our systems in the nonlinear regime. Using simulations, we study two models for the microscopic evolution of properties of such a material; the first considers changes in the material strength, while the second considers distortions in the microscopic geometry. Both models capture different aspects of the experiments including the encoding of memories of the aging history of the system and the dramatic effects on the material's nonlinear elastic properties. Our results demonstrate how aging can be used to create complex elastic behavior in the nonlinear regime.

We consider zero temperature packings of soft spheres, that undergo a jamming to unjamming transition as a function of packing fraction. We compare differences in the structure, as measured from the contact statistics, of a finite subsystem of a large packing to a whole packing with periodic boundaries of an equivalent size and pressure. We find that the fluctuations of the ensemble of whole packings are smaller than those of the ensemble of subsystems. Convergence of these two quantities appears to occur at very large systems, which are usually not attainable in numerical simulations. Finding differences between packings in two dimensions and three dimensions, we also consider four dimensions and mean-field models, and find that they show similar system size dependence. Mean-field critical exponents appear to be consistent with the 3d and 4d packings, suggesting they are above the upper critical dimension. We also find that the convergence as a function of system size to the thermodynamic limit is characterized by two different length scales. We argue that this is the result of the system being above the upper critical dimension.

Disordered materials are often out of equilibrium and evolve very slowly in a rugged and tortuous energy landscape. This slow evolution, referred to as aging, is deemed undesirable as it often leads to material degradation. However, we show that aging also encodes a memory of the stresses imposed during preparation. Because of inhomogeneous local stresses, the material itself decides how to evolve by modifying stressed regions differently from those under less stress. Because material evolution occurs in response to stresses, aging can be "directed" to produce sought-after responses and unusual functionalities that do not inherently exist. Aging obeys a natural "greedy algorithm" as, at each instant, the material simply follows the path of most rapid and accessible relaxation. Our experiments and simulations illustrate directed aging in examples in which the material's elasticity transforms as desired because of an imposed deformation.

At densities higher than the jamming transition for athermal, frictionless repulsive spheres we find two distinct length scales, both of which diverge as a power law as the transition is approached. The first, ξ_{Z}, is associated with the two-point correlation function for the number of contacts on two particles as a function of the particle separation. The second, ξ_{f}, is associated with contact-number fluctuations in subsystems of different sizes. On scales below ξ_{f}, the fluctuations are highly suppressed, similar to the phenomenon of hyperuniformity usually associated with density fluctuations. The exponents for the divergence of ξ_{Z} and ξ_{f} are different and appear to be different in two and three dimensions.

We explore the range over which the elasticity of disordered spring networks can be manipulated by the removal of selected bonds. By taking into account the local response of a bond, we demonstrate that the effectiveness of pruning can be improved so that auxetic (i.e., negative Poisson's ratio) materials can be designed without the formation of cracks even while maintaining the global isotropy of the network. The bulk modulus and shear modulus scale with the number of bonds removed and we estimate the exponents characterizing these power laws. We also find that there are spatial correlation lengths in the change of bulk modulus and shear modulus upon removing different bonds that diverge as the network approaches the isostatic limit where the excess coordination number ΔZ → 0.