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.

Training precise stress patterns.

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.

Periodic training of creeping solids.

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.

Training nonlinear elastic functions: nonmonotonic, sequence dependent and bifurcating.

The elastic behavior of materials operating in the linear regime is constrained, by definition, to operations that are linear in the imposed deformation. Although 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 solid 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.

Enhanced hyperuniformity from random reorganization.

Diffusion relaxes density fluctuations toward a uniform random state whose variance in regions of volume [Formula: see text] scales as [Formula: see text] Systems whose fluctuations decay faster, [Formula: see text] with [Formula: see text], are called hyperuniform. The larger [Formula: see text], the more uniform, with systems like crystals achieving the maximum value: [Formula: see text] Although finite temperature equilibrium dynamics will not yield hyperuniform states, driven, nonequilibrium dynamics may. Such is the case, for example, in a simple model where overlapping particles are each given a small random displacement. Above a critical particle density [Formula: see text], the system evolves forever, never finding a configuration where no particles overlap. Below [Formula: see text], however, it eventually finds such a state, and stops evolving. This "absorbing state" is hyperuniform up to a length scale [Formula: see text], which diverges at [Formula: see text] An important question is whether hyperuniformity survives noise and thermal fluctuations. We find that hyperuniformity of the absorbing state is not only robust against noise, diffusion, or activity, but that such perturbations reduce fluctuations toward their limiting behavior, [Formula: see text], a uniformity similar to random close packing and early universe fluctuations, but with arbitrary controllable density.

Can a Large Packing be Assembled from Smaller Ones?

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.

Two Diverging Length Scales in the Structure of Jammed Packings.

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.

Role of local response in manipulating the elastic properties of disordered solids by bond removal.

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.

Noise, Diffusion, and Hyperuniformity.

We consider driven many-particle models which have a phase transition between an active and an absorbing phase. Like previously studied models, we have particle conservation, but here we introduce an additional symmetry-when two particles interact, we give them stochastic kicks which conserve the center of mass. We find that the density fluctuations in the active phase decay in the fastest manner possible for a disordered isotropic system, and we present arguments that the large scale fluctuations are determined by a competition between a noise term which generates fluctuations, and a deterministic term which reduces them. Our results may be relevant to shear experiments and may further the understanding of hyperuniformity which occurs at the critical point.

Reply to the 'Comment on "Spatial structure of states of self stress in jammed systems"' by E. Lerner, Soft Matter, 2017, 13, DOI.

Lerner's theoretical analysis neglects a normalizing factor which distinguishes states of self stress from the stress response to a unit dipolar force along a bond. This factor leads to different spatial profiles upon ensemble averaging.

Hyperuniformity of critical absorbing states.

The properties of the absorbing states of nonequilibrium models belonging to the conserved directed percolation universality class are studied. We find that, at the critical point, the absorbing states are hyperuniform, exhibiting anomalously small density fluctuations. The exponent characterizing the fluctuations is measured numerically, a scaling relation to other known exponents is suggested, and a new correlation length relating to this ordering is proposed. These results may have relevance to photonic band-gap materials.

Entropic commensurate-incommensurate transition.

The equilibrium properties of a minimal tiling model are investigated. The model has extensive ground state entropy, with each ground state having a quasiperiodic sequence of rows. It is found that the transition from the ground state to the high temperature disordered phase proceeds through a sequence of periodic arrangements of rows, in analogy with the commensurate-incommensurate transition. We show that the effective free energy of the model resembles the Frenkel-Kontorova Hamiltonian, but with temperature playing the role of the strength of the substrate potential, and with the competing lengths not explicitly present in the basic interactions.

Self-propulsion and self-navigation: Activity is a precursor to jamming.

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.

Tug of war in motility assay experiments.

The dynamics of two groups of molecular motors pulling in opposite directions on a rigid filament is studied theoretically. To this end we first consider the behavior of one set of motors pulling in a single direction against an external force using a new mean-field approach. Based on these results we analyze a similar setup with two sets of motors pulling in opposite directions in a tug of war in the presence of an external force. In both cases we find that the interplay of fluid friction and protein friction leads to a complex phase diagram where the force-velocity relations can exhibit regions of bistability and spontaneous symmetry breaking. Finally, motivated by recent work, we turn to the case of motility assay experiments where motors bound to a surface push on a bundle of filaments. We find that, depending on the absence or the presence of bistability in the force-velocity curve at zero force, the bundle exhibits anomalous or biased diffusion on long-time and large-length scales.

Directed aging, memory, and nature's greed.

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.

Linking microscopic and macroscopic response in disordered solids.

The modulus of a rigid network of harmonic springs depends on the sum of the energies in each of the bonds due to an applied distortion such as compression in the case of the bulk modulus or shear in the case of the shear modulus. However, the distortion need not be global. Here we introduce a local modulus, L_{i}, associated with changing the equilibrium length of a single bond, i, in the network. We show that L_{i} is useful for understanding many aspects of the mechanical response of the entire system. It allows an efficient computation of how the removal of any bond changes the global properties such as the bulk and shear moduli. Furthermore, it allows a prediction of the distribution of these changes and clarifies why the changes of these two moduli due to removal of a bond are uncorrelated; these are the essential ingredients necessary for the efficient manipulation of network properties by bond removal.