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.

Rapid formation of uniformly layered materials by coupling reaction-diffusion processes with mechanical responsiveness.

Straightforward manufacturing pathways toward large-scale, uniformly layered composites may enable the next generation of materials with advanced optical, thermal, and mechanical properties. Reaction-diffusion systems are attractive candidates to this aim, but while layered composites theoretically could spontaneously arise from reaction-diffusion, in practice randomly oriented patches separated by defects form, yielding nonuniformly patterned materials. A propagating reaction front can prevent such nonuniform patterning, as is the case for Liesegang processes, in which diffusion drives a reaction front to produce layered precipitation patterns. However, while diffusion is crucial to control patterning, it slows down transport of reactants to the front and results in a steady increase of the band spacing as the front advances. Here, we circumvent these diffusive limitations by embedding the Liesegang process in mechanically responsive hydrogels. The coupling between a moving reaction front and hydrogel contraction induces the formation of a self-regulated transport channel that ballistically carries reactants toward the area where patterning occurs. This ensures rapid and uniform patterning. Specifically, large-scale ([Formula: see text]5-cm) uniform banding patterns are produced with tunable band distance (d = 60 to 160 µm) of silver dichromate crystals inside responsive gelatin-alginate hydrogels. The generality and applicability of our mechanoreaction-diffusion strategy are demonstrated by forming patterns of precipitates in significantly smaller microscopic banding patterns (d = 10 to 30 µm) that act as self-organized diffraction gratings. By circumventing the inherent limitations of diffusion, our strategy unlocks the potential of reaction-diffusion processes for the manufacturing of uniformly layered materials.

Multi-step self-guided pathways for shape-changing metamaterials.

Multi-step pathways-which consist of a sequence of reconfigurations of a structure-are central to the functionality of various natural and artificial systems. Such pathways execute autonomously in self-guided processes such as protein folding1 and self-assembly2-5, but have previously required external control to execute in macroscale mechanical systems, provided by, for example, actuators in robotics6-9 or manual folding in origami8,10-12. Here we demonstrate shape-changing, macroscale mechanical metamaterials that undergo self-guided, multi-step reconfiguration in response to global uniform compression. We avoid the need for external control by using metamaterials that are made purely of passive components. The design of the metamaterials combines nonlinear mechanical elements with a multimodal architecture that enables a sequence of topological reconfigurations caused by the formation of internal self-contacts between the elements of the metamaterial. We realize the metamaterials by using computer-controlled water-jet cutting of flexible materials, and show that the multi-step pathway and final configuration can be controlled by rational design of the nonlinear mechanical elements. We also demonstrate that the self-contacts suppress errors in the pathway. Finally, we create hierarchical architectures to extend the number of distinct reconfiguration steps. Our work establishes general principles for designing mechanical pathways, opening up new avenues for self-folding media11,12, pluripotent materials9,13 and pliable devices14 in areas such as stretchable electronics and soft robotics15.

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.

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.

Machine Learning of Implicit Combinatorial Rules in Mechanical Metamaterials.

Combinatorial problems arising in puzzles, origami, and (meta)material design have rare sets of solutions, which define complex and sharply delineated boundaries in configuration space. These boundaries are difficult to capture with conventional statistical and numerical methods. Here we show that convolutional neural networks can learn to recognize these boundaries for combinatorial mechanical metamaterials, down to finest detail, despite using heavily undersampled training sets, and can successfully generalize. This suggests that the network infers the underlying combinatorial rules from the sparse training set, opening up new possibilities for complex design of (meta)materials.

Combinatorial design of textured mechanical metamaterials.

The structural complexity of metamaterials is limitless, but, in practice, most designs comprise periodic architectures that lead to materials with spatially homogeneous features. More advanced applications in soft robotics, prosthetics and wearable technology involve spatially textured mechanical functionality, which requires aperiodic architectures. However, a naive implementation of such structural complexity invariably leads to geometrical frustration (whereby local constraints cannot be satisfied everywhere), which prevents coherent operation and impedes functionality. Here we introduce a combinatorial strategy for the design of aperiodic, yet frustration-free, mechanical metamaterials that exhibit spatially textured functionalities. We implement this strategy using cubic building blocks-voxels-that deform anisotropically, a local stacking rule that allows cooperative shape changes by guaranteeing that deformed building blocks fit together as in a three-dimensional jigsaw puzzle, and three-dimensional printing. These aperiodic metamaterials exhibit long-range holographic order, whereby the two-dimensional pixelated surface texture dictates the three-dimensional interior voxel arrangement. They also act as programmable shape-shifters, morphing into spatially complex, but predictable and designable, shapes when uniaxially compressed. Finally, their mechanical response to compression by a textured surface reveals their ability to perform sensing and pattern analysis. Combinatorial design thus opens up a new avenue towards mechanical metamaterials with unusual order and machine-like functionalities.

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.

Design of Pseudo-Mechanisms and Multistable Units for Mechanical Metamaterials.

Mechanisms-collections of rigid elements coupled by perfect hinges which exhibit a zero-energy motion-motivate the design of a variety of mechanical metamaterials. We enlarge this design space by considering pseudo-mechanisms, collections of elastically coupled elements that exhibit motions with very low energy costs. We show that their geometric design generally is distinct from those of true mechanisms, thus opening up a large and virtually unexplored design space. We further extend this space by designing building blocks with bistable and tristable energy landscapes, realize these by 3D printing, and show how these form unit cells for multistable metamaterials.

From Bouncing to Floating: The Leidenfrost Effect with Hydrogel Spheres.

The Leidenfrost effect occurs when a liquid or stiff sublimable solid near a hot surface creates enough vapor beneath it to lift itself up and float. In contrast, vaporizable soft solids, e.g., hydrogels, have been shown to exhibit persistent bouncing-the elastic Leidenfrost effect. By carefully lowering hydrogel spheres towards a hot surface, we discover that they are also capable of floating. The bounce-to-float transition is controlled by the approach velocity and temperature, analogously to the "dynamic Leidenfrost effect." For the floating regime, we measure power-law scalings for the gap geometry, which we explain with a model that couples the vaporization rate to the spherical shape. Our results reveal that hydrogels are a promising pathway for controlling floating Leidenfrost objects through shape.

Softening and yielding of soft glassy materials.

Solids deform and fluids flow, but soft glassy materials, such as emulsions, foams, suspensions, and pastes, exhibit an intricate mix of solid- and liquid-like behavior. While much progress has been made to understand their elastic (small strain) and flow (infinite strain) properties, such understanding is lacking for the softening and yielding phenomena that connect these asymptotic regimes. Here we present a comprehensive framework for softening and yielding of soft glassy materials, based on extensive numerical simulations of oscillatory rheological tests, and show that two distinct scenarios unfold depending on the material's packing density. For dense systems, there is a single, pressure-independent strain where the elastic modulus drops and the particle motion becomes diffusive. In contrast, for weakly jammed systems, a two-step process arises: at an intermediate softening strain, the elastic and loss moduli both drop down and then reach a new plateau value, whereas the particle motion becomes diffusive at the distinctly larger yield strain. We show that softening is associated with an extensive number of microscopic contact changes leading to a non-analytic rheological signature. Moreover, the scaling of the softening strain with pressure suggest the existence of a novel pressure scale above which softening and yielding coincide, and we verify the existence of this crossover scale numerically. Our findings thus evidence the existence of two distinct classes of soft glassy materials - jamming dominated and dense - and show how these can be distinguished by their rheological fingerprint.

Criticality in Vibrated Frictional Flows at a Finite Strain Rate.

We evidence critical fluctuations in the strain rate of granular flows that are weakly vibrated. Strikingly, the critical point arises at finite values of the mean strain rate and vibration strength, far from the yielding critical point at a zero flow rate. We show that the global rheology, as well as the amplitude and correlation time of the fluctuations, are consistent with a mean-field, Landau-like description, where the strain rate and the stress act as conjugated variables. We introduce a general model which captures the observed phenomenology and argue that this type of critical behavior generically arises when self-fluidization competes with friction.

Programmable mechanical metamaterials: the role of geometry.

We experimentally and numerically study the role of geometry for the mechanics of biholar metamaterials, which are quasi-2D slabs of rubber patterned by circular holes of two alternating sizes. We recently showed how the response to uniaxial compression of these metamaterials can be programmed by lateral confinement. In particular, there is a range of confining strains εx for which the resistance to compression becomes non-trivial-non-monotonic or hysteretic-in a range of compressive strains εy. Here we show how the dimensionless geometrical parameters t and χ, which characterize the wall thickness and size ratio of the holes that pattern these metamaterials, can significantly tune these ranges over a wide range. We study the behavior for the limiting cases where the wall thickness t and the size ratio χ become large, and discuss the new physics that arises there. Away from these extreme limits, the variation of the strain ranges of interest is smooth with porosity, but the variation with size ratio evidences a cross-over at low χ from biholar to monoholar (equal sized holes) behavior, related to the elastic instabilities in purely monoholar metamaterials. Our study provides precise guidelines for the rational design of programmable biholar metamaterials, tailored to specific applications, and indicates that the widest range of programmability arises for moderate values of both t and χ.

Origami multistability: from single vertices to metasheets.

We show that the simplest building blocks of origami-based materials-rigid, degree-four vertices-are generically multistable. The existence of two distinct branches of folding motion emerging from the flat state suggests at least bistability, but we show how nonlinearities in the folding motions allow generic vertex geometries to have as many as five stable states. In special geometries with collinear folds and symmetry, more branches emerge leading to as many as six stable states. Tuning the fold energy parameters, we show how monostability is also possible. Finally, we show how to program the stability features of a single vertex into a periodic fold tessellation. The resulting metasheets provide a previously unanticipated functionality-tunable and switchable shape and size via multistability.

Rigidity loss in disordered systems: three scenarios.

We reveal significant qualitative differences in the rigidity transition of three types of disordered network materials: randomly diluted spring networks, jammed sphere packings, and stress-relieved networks that are diluted using a protocol that avoids the appearance of floppy regions. The marginal state of jammed and stress-relieved networks are globally isostatic, while marginal randomly diluted networks show both overconstrained and underconstrained regions. When a single bond is added to or removed from these isostatic systems, jammed networks become globally overconstrained or floppy, whereas the effect on stress-relieved networks is more local and limited. These differences are also reflected in the linear elastic properties and point to the highly effective and unusual role of global self-organization in jammed sphere packings.

Discontinuous Buckling of Wide Beams and Metabeams.

We uncover how nonlinearities dramatically alter the buckling of elastic beams. First, we show experimentally that sufficiently wide ordinary elastic beams and specifically designed metabeams-beams made from a mechanical metamaterial-exhibit discontinuous buckling, an unstable form of buckling where the postbuckling stiffness is negative. Then we use simulations to uncover the crucial role of nonlinearities, and show that beams made from increasingly nonlinear materials exhibit an increasingly negative postbuckling slope. Finally, we demonstrate that for sufficiently strong nonlinearity, we can observe discontinuous buckling for metabeams as slender as 1% numerically and 5% experimentally.

Fluctuations in flows near jamming.

Bubbles, droplets or particles in flowing complex media such as foams, emulsions or suspensions follow highly complex paths, with the relative motion of the constituents setting the energy dissipation rate. What is their dynamics, and how is this connected to the global rheology? To address these questions, we probe the statistics and spatio-temporal organization of the local particle motion and energy dissipation in a model for sheared disordered materials. We find that the fluctuations in the local dissipation vary from nearly Gaussian and homogeneous at low densities and fast flows, to strongly intermittent for large densities and slow flows. The higher order moments of the relative particle velocities reveal strong evidence for a qualitative difference between two distinct regimes which are nevertheless connected by a smooth crossover. In the critical regime, the higher order moments are related by novel multiscaling relations. In the plastic regime the relations between these moments take on a different form, with higher moments diverging rapidly when the flow rate vanishes. As these velocity differences govern the energy dissipation, we can distinguish two qualitatively different types of flow: an intermediate density, critical regime related to jamming, and a large density, plastic regime.

Heaping, secondary flows and broken symmetry in flows of elongated granular particles.

In this paper we report experiments where we shear granular rods in split-bottom geometries, and find that a significant heap of height of least 40% of the filling height can form at the particle surface. We show that heaping is caused by a significant secondary flow, absent for spherical particles. Flow reversal transiently reverses the secondary flow, leading to a quick collapse and slower regeneration of the heap. We present a symmetry argument and experimental data that show that the generation of the secondary flow is driven by a misalignment of the mean particle orientation with the streamlines of the flow. This general mechanism is expected to be important in all flows of sufficiently anisometric grains.

Programmable mechanical metamaterials.

We create mechanical metamaterials whose response to uniaxial compression can be programmed by lateral confinement, allowing monotonic, nonmonotonic, and hysteretic behavior. These functionalities arise from a broken rotational symmetry which causes highly nonlinear coupling of deformations along the two primary axes of these metamaterials. We introduce a soft mechanism model which captures the programmable mechanics, and outline a general design strategy for confined mechanical metamaterials. Finally, we show how inhomogeneous confinement can be explored to create multistability and giant hysteresis.

Shock waves in weakly compressed granular media.

We experimentally probe nonlinear wave propagation in weakly compressed granular media and observe a crossover from quasilinear sound waves at low impact to shock waves at high impact. We show that this crossover impact grows with the confining pressure P0, whereas the shock wave speed is independent of P0-two hallmarks of granular shocks predicted recently. The shocks exhibit surprising power law attenuation, which we model with a logarithmic law implying that shock dissipation is weak and qualitatively different from other granular dissipation mechanisms. We show that elastic and potential energy balance in the leading part of the shocks.