Phonons and elasticity in critically coordinated lattices

Fully Polarized Topological Isostatic Metamaterials in Three Dimensions

Topological surface states are unique to topological materials and are immune to disturbances. In isostatic lattices, mechanical topological floppy modes exhibit softness depending on the polarization relative to the terminating surface. However, in three dimensions, the polarization of topological floppy modes is disrupted by the ubiquitous mechanical Weyl lines. Here, we demonstrate, both theoretically and experimentally, the fully polarized topological mechanical phases free of Weyl lines. Floppy modes emerge exclusively on a particular surface of the three-dimensional isostatic structure, leading to the strongly asymmetric stiffness between opposing boundaries. Additionally, uniform soft strains can reversibly shift the lattice configuration to Weyl phases, switching the stiffness contrast to a trivially comparable level. Our work demonstrates the fully polarized topological mechanical phases in three dimensions, and paves the way towards engineering soft and adaptive metamaterials.

Integrating local energetics into Maxwell-Calladine constraint counting to design mechanical metamaterials

The Maxwell-Calladine index theorem plays a central role in our current understanding of the mechanical rigidity of discrete materials. By considering the geometric constraints each material component imposes on a set of underlying degrees of freedom, the theorem relates the emergence of rigidity to constraint counting arguments. However, the Maxwell-Calladine paradigm is significantly limited-its exclusive reliance on the geometric relationships between constraints and degrees of freedom completely neglects the actual energetic costs of deforming individual components. To address this limitation, we derive a generalization of the Maxwell-Calladine index theorem based on susceptibilities that naturally incorporate local energetic properties such as stiffness and prestress. Using this extended framework, we investigate how local energetics modify the classical constraint counting picture to capture the relationship between deformations and external forces. We then combine this formalism with group representation theory to design mechanical metamaterials where differences in symmetry between local energy costs and structural geometry are exploited to control responses to external forces.

Ultrahigh-quality-factor micro- and nanomechanical resonators using dissipation dilution

Mechanical resonators are widely used in sensors, transducers and optomechanical systems, where mechanical dissipation sets the ultimate limit to performance. Over the past 15 years, the quality factors in strained mechanical resonators have increased by four orders of magnitude, surpassing the previous state of the art achieved in bulk crystalline resonators at room temperature and liquid helium temperatures. In this Review, we describe how these advances were made by leveraging 'dissipation dilution'-where dissipation is reduced through a combination of static tensile strain and geometric nonlinearity in dynamic strain. We then review the state of the art in strained nanomechanical resonators and discuss the potential for even higher quality factors in crystalline materials. Finally, we detail current and future applications of dissipation-diluted mechanical resonators.

Nonreciprocity and odd viscosity in chiral active fluids

Odd viscosity couples stress to strain rate in a dissipationless way. It has been studied in plasmas under magnetic fields, superfluid [Formula: see text], quantum-Hall fluids, and recently in the context of chiral active matter. In most of these studies, odd terms in the viscosity obey Onsager reciprocal relations. Although this is expected in equilibrium systems, it is not obvious that Onsager relations hold in active materials. By directly coarse-graining the kinetic energy and independently using both the Poisson-bracket formalism and a kinetic theory derivation, we find that the appearance of a nonvanishing angular momentum density, which is a hallmark of chiral active materials, necessarily breaks Onsager reciprocal relations. This leads to a non-Hermitian dynamical matrix for the total hydrodynamic momentum and to the appearance of odd viscosity and other nondissipative contributions to the viscosity. Furthermore, by accounting for both the angular momentum density and interactions that lead to odd viscosity, we find regions in the parameter space in which 3D odd mechanical waves propagate and regions in which they are mechanically unstable. The lines separating these regions are continuous lines of exceptional points, suggesting a possible nonreciprocal phase transition.

Zero modes activation to reconcile floppiness, rigidity, and multistability into an all-in-one class of reprogrammable metamaterials

Existing mechanical metamaterials are typically designed to either withstand loads as a stiff structure, shape morph as a floppy mechanism, or trap energy as a multistable matter, distinct behaviours that correspond to three primary classes of macroscopic solids. Their stiffness and stability are sealed permanently into their architecture, mostly remaining immutable post-fabrication due to the invariance of zero modes. Here, we introduce an all-in-one reprogrammable class of Kagome metamaterials that enable the in-situ reprogramming of zero modes to access the apparently conflicting properties of all classes. Through the selective activation of metahinges via self-contact, their architecture can be switched to acquire on-demand rigidity, floppiness, or global multistability, bridging the seemingly uncrossable gap between structures, mechanisms, and multistable matters. We showcase the versatile generalizations of the metahinge and remarkable reprogrammability of zero modes for a range of properties including stiffness, mechanical signal guiding, buckling modes, phonon spectra, and auxeticity, opening a plethora of opportunities for all-in-one materials and devices.

Transformable Superisostatic Crystals Self-Assembled from Segment Colloidal Rods

Transformable mechanical structures can switch between distinct mechanical states. Whether this kind of structure can be self-assembled from simple building blocks at microscale is a question to be answered. In this work, we propose a self-assembly strategy for these structures based on a nematic monolayer of segmented colloidal rods with lateral cutting. By using Monte Carlo simulation, we find that rods with different cutting degrees can self-assemble into different crystals characterized by bond coordination z that varies from 3 to 6. Among these, we identify a transformable superisostatic structure with pgg symmetry and redundant bonds (z = 5). We show that this structure can support either soft bulk modes or soft edge modes depending on its Poisson's ratio, which can be tuned from positive to negative through a uniform soft deformation. We also prove that the bulk soft modes are associated with states of self-stress along the direction of zero strain during uniform soft deformation. The self-assembled transformable structures may act as mechanical metamaterials with potential applications in micromechanical engineering.

Soft and Stiff Normal Modes in Floppy Colloidal Square Lattices

Floppy microscale spring networks are widely studied in theory and simulations, but no well-controlled experimental system currently exists. Here, we show that square lattices consisting of colloid-supported lipid bilayers functionalized with DNA linkers act as microscale floppy spring networks. We extract their normal modes by inverting the particle displacement correlation matrix, showing the emergence of a spectrum of soft modes with low effective stiffness in addition to stiff modes that derive from linker interactions. Evaluation of the softest mode, a uniform shear mode, reveals that shear stiffness decreases with lattice size. Experiments match well with Brownian particle simulations, and we develop a theoretical description based on mapping interactions onto a linear response model to describe the modes. Our results reveal the importance of entropic steric effects and can be used for developing reconfigurable materials at the colloidal length scale.

Self-driven configurational dynamics in frustrated spring-mass systems

Various physical systems relax mechanical frustration through configurational rearrangements. We examine such rearrangements via Hamiltonian dynamics of simple internally stressed harmonic four-mass systems. We demonstrate theoretically and numerically how mechanical frustration controls the underlying potential energy landscape. Then, we examine the harmonic four-mass systems' Hamiltonian dynamics and relate the onset of chaotic motion to self-driven rearrangements. We show such configurational dynamics may occur without strong precursors, rendering such dynamics seemingly spontaneous.

Jamming on convex deformable surfaces

Jamming is a fundamental transition that governs the behavior of particulate media, including sand, foams and dense suspensions. Upon compression, such media change from freely flowing to a disordered, marginally stable solid that exhibits non-Hookean elasticity. While the jamming process is well established for fixed geometries, the nature and dynamics of jamming for a diverse class of soft materials and deformable substrates, including emulsions and biological matter, remains unknown. Here we propose a new scenario, metric jamming, where rigidification occurs on a surface that has been deformed from its ground state. Unlike classical jamming processes that exhibit discrete mechanical transitions, surprisingly we find that metric jammed states possess mechanical properties continuously tunable between those of classically jammed and conventional elastic media. The compact and curved geometry significantly alters the vibrational spectra of the structures relative to jamming in flat Euclidean space, and metric jammed systems also possess new types of vibrational mode that couple particle and shape degrees of freedom. Our work provides a theoretical framework that unifies our understanding of solidification processes that take place on deformable media and lays the groundwork to exploit jamming for the control and stabilization of shape in self-assembly processes.

Direct Imaging of "Patch-Clasping" and Relaxation in Robust and Flexible Nanoparticle Assemblies

Polymer patching on inorganic nanoparticles (NPs) enables multifunctionality and directed self-assembly into nonclosely packed optical and mechanical metamaterials. However, experimental demonstration of such assemblies has been scant due to challenges in leveraging patch-induced NP-NP attractions and understanding NP self-assembly dynamics. Here we use low-dose liquid-phase transmission electron microscopy to visualize the dynamic behaviors of tip-patched triangular nanoprisms upon patch-clasping, where polymer patches interpenetrate to form cohesive bonds that connect NPs. Notably, these bonds are longitudinally robust but rotationally flexible. Patch-clasping is found to allow highly selective tip-tip assembly, interconversion between dimeric bowtie and sawtooth configurations, and collective structural relaxation of NP networks. The integration of single particle tracking, polymer physics theory, and molecular dynamics simulation reveals the macromolecular origin of patch-clasping-induced NP dynamics. Our experiment-computation integration can aid the design of stimuli-responsive nanomaterials, such as topological metamaterials for chiral sensors, waveguides, and nanoantennas.

Emergent nonlocal combinatorial design rules for multimodal metamaterials

Combinatorial mechanical metamaterials feature spatially textured soft modes that yield exotic and useful mechanical properties. While a single soft mode often can be rationally designed by following a set of tiling rules for the building blocks of the metamaterial, it is an open question what design rules are required to realize multiple soft modes. Multimodal metamaterials would allow for advanced mechanical functionalities that can be selected on the fly. Here we introduce a transfer matrix-like framework to design multiple soft modes in combinatorial metamaterials composed of aperiodic tilings of building blocks. We use this framework to derive rules for multimodal designs for a specific family of building blocks. We show that such designs require a large number of degeneracies between constraints, and find precise rules on the real space configuration that allow such degeneracies. These rules are significantly more complex than the simple tiling rules that emerge for single-mode metamaterials. For the specific example studied here, they can be expressed as local rules for tiles composed of pairs of building blocks in combination with a nonlocal rule in the form of a global constraint on the type of tiles that are allowed to appear together anywhere in the configuration. This nonlocal rule is exclusive to multimodal metamaterials and exemplifies the complexity of rational design of multimode metamaterials. Our framework is a first step towards a systematic design strategy of multimodal metamaterials with spatially textured soft modes.

Rigidity percolation in a random tensegrity via analytic graph theory

Functional structures from across the engineered and biological world combine rigid elements such as bones and columns with flexible ones such as cables, fibers, and membranes. These structures are known loosely as tensegrities, since these cable-like elements have the highly nonlinear property of supporting only extensile tension. Marginally rigid systems are of particular interest because the number of structural constraints permits both flexible deformation and the support of external loads. We present a model system in which tensegrity elements are added at random to a regular backbone. This system can be solved analytically via a directed graph theory, revealing a mechanical critical point generalizing that of Maxwell. We show that even the addition of a few cable-like elements fundamentally modifies the nature of this transition point, as well as the later transition to a fully rigid structure. Moreover, the tensegrity network displays a collective avalanche behavior, in which the addition of a single cable leads to the elimination of multiple floppy modes, a phenomenon that becomes dominant at the transition point. These phenomena have implications for systems with nonlinear mechanical constraints, from biopolymer networks to soft robots to jammed packings to origami sheets.

Nonlinear Topological Mechanics in Elliptically Geared Isostatic Metamaterials

Despite the extensive studies of topological systems, the experimental characterizations of strongly nonlinear topological phases have been lagging. To address this shortcoming, we design and build elliptically geared isostatic metamaterials. Their nonlinear topological transitions can be realized by collective soliton motions, which stem from the transition of nonlinear Berry phase. Endowed by the intrinsic nonlinear topological mechanics, surface polar elasticity and dislocation-bound zero modes can be created or annihilated as the topological polarization reverses orientation. Our approach integrates topological physics with strongly nonlinear mechanics and promises multiphase structures at the micro- and macroscales.

Unexpected bending behavior of architected 2D lattice materials

Architected two-dimensional (2D) lattice materials consisting of elastic beams are appealing because of their tunable sign of Poisson's ratio. A common belief is that such materials with positive and negative Poisson's ratios display anticlastic and synclastic curvatures, respectively, when bent in one direction. Here, we show theoretically and demonstrate experimentally that this is not true. For 2D lattices with star-shaped unit cells, we find a transition between anticlastic and synclastic bending curvatures controlled by the beam's cross-sectional aspect ratio even at a fixed Poisson's ratio. The mechanisms lay in the competitive interaction between axial torsion and out-of-plane bending of the beams and can be well captured by a Cosserat continuum model. Our result may provide unprecedented insights to the design of 2D lattice systems for shape-shifting applications.

Topological Metamaterials

The topological properties of an object, associated with an integer called the topological invariant, are global features that cannot change continuously but only through abrupt variations, hence granting them intrinsic robustness. Engineered metamaterials (MMs) can be tailored to support highly nontrivial topological properties of their band structure, relative to their electronic, electromagnetic, acoustic and mechanical response, representing one of the major breakthroughs in physics over the past decade. Here, we review the foundations and the latest advances of topological photonic and phononic MMs, whose nontrivial wave interactions have become of great interest to a broad range of science disciplines, such as classical and quantum chemistry. We first introduce the basic concepts, including the notion of topological charge and geometric phase. We then discuss the topology of natural electronic materials, before reviewing their photonic/phononic topological MM analogues, including 2D topological MMs with and without time-reversal symmetry, Floquet topological insulators, 3D, higher-order, non-Hermitian and nonlinear topological MMs. We also discuss the topological aspects of scattering anomalies, chemical reactions and polaritons. This work aims at connecting the recent advances of topological concepts throughout a broad range of scientific areas and it highlights opportunities offered by topological MMs for the chemistry community and beyond.

Extreme Spatial Dispersion in Nonlocally Resonant Elastic Metamaterials

To date, the vast majority of architected materials have leveraged two physical principles to control wave behavior, namely, Bragg interference and local resonances. Here, we describe a third path: structures that accommodate a finite number of delocalized zero-energy modes, leading to anomalous dispersion cones that nucleate from extreme spatial dispersion at 0 Hz. We explain how to design such zero-energy modes in the context of elasticity and show that many of the landmark wave properties of metamaterials can also be induced at an extremely subwavelength scale by the associated anomalous cones, without suffering from the same bandwidth limitations. We then validate our theory through a combination of simulations and experiments. Finally, we present an inverse design method to produce anomalous cones at desired locations in k space.

Synthetically non-Hermitian nonlinear wave-like behavior in a topological mechanical metamaterial

Topological mechanical metamaterials have enabled new ways to control stress and deformation propagation. Exemplified by Maxwell lattices, they have been studied extensively using a linearized formalism. Herein, we study a two-dimensional topological Maxwell lattice by exploring its large deformation quasi-static response using geometric numerical simulations and experiments. We observe spatial nonlinear wave-like phenomena such as harmonic generation, localized domain switching, amplification-enhanced frequency conversion, and solitary waves. We further map our linearized, homogenized system to a non-Hermitian, nonreciprocal, one-dimensional wave equation, revealing an equivalence between the deformation fields of two-dimensional topological Maxwell lattices and nonlinear dynamical phenomena in one-dimensional active systems. Our study opens a regime for topological mechanical metamaterials and expands their application potential in areas including adaptive and smart materials and mechanical logic, wherein concepts from nonlinear dynamics may be used to create intricate, tailored spatial deformation and stress fields greatly transcending conventional elasticity.

Dynamics of Self-Dual Kagome Metamaterials and the Emergence of Fragile Topology

Recent years have seen the discovery of systems featuring fragile topological states. These states of matter lack certain protection attributes typically associated with topology and are therefore characterized by weaker signatures that make them elusive to observe. Moreover, they are typically confined to special symmetry classes and, in general, rarely studied in the context of phononic media. In this Letter, we theoretically predict the emergence of fragile topological bands in the spectrum of a twisted kagome elastic lattice with threefold rotational symmetry, in the so-called self-dual configuration. A necessary requirement is that the lattice is a structural metamaterial, in which the role of the hinges is played by elastic finite-thickness ligaments. The interplay between the edge modes appearing in the band gaps bounding the fragile topological states is also responsible for the emergence of corner modes at selected corners of a finite hexagonal domain, which qualifies the lattice as a second-order topological insulator. We demonstrate our findings through a series of experiments via 3D scanning laser doppler vibrometry conducted on a physical prototype. The selected configuration stands out for its remarkable geometric simplicity and ease of physical implementation in the panorama of dynamical systems exhibiting fragile topology.

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.

Engineering zero modes in transformable mechanical metamaterials

In the field of flexible metamaterial design, harnessing zero modes plays a key part in enabling reconfigurable elastic properties of the metamaterial with unconventional characteristics. However, only quantitative enhancement of certain properties succeeds in most cases rather than qualitative transformation of the metamaterials' states or/and functionalities, due to the lack of systematic designs on the corresponding zero modes. Here, we propose a 3D metamaterial with engineered zero modes, and experimentally demonstrate its transformable static and dynamic properties. All seven types of extremal metamaterials ranging from null-mode (solid state) to hexa-mode (near-gaseous state) are reported to be reversibly transformed from one state to another, which is verified by the 3D-printed Thermoplastic Polyurethanes prototypes. Tunable wave manipulations are further investigated in 1D-, 2D- and 3D-systems. Our work sheds lights on the design of flexible mechanical metamaterials, which can be potentially extended from the mechanical to the electro-magnetite, the thermal or other types.

A hydrogel-based mechanical metamaterial for the interferometric profiling of extracellular vesicles in patient samples

The utility of mechanical metamaterials for biomedical applications has seldom been explored. Here we show that a metamaterial that is mechanically responsive to antibody-mediated biorecognition can serve as an optical interferometric mask to molecularly profile extracellular vesicles in ascites fluid from patients with cancer. The metamaterial consists of a hydrogel responsive to temperature and redox activity functionalized with antibodies to surface biomarkers on extracellular vesicles, and is patterned into micrometric squares on a gold-coated glass substrate. Through plasmonic heating, the metamaterial is maintained in a transition state between a relaxed form and a buckled state. Binding of extracellular vesicles from the patient samples to the antibodies on the hydrogel causes it to undergo crosslinking, induced by free radicals generated via the activity of horseradish peroxidase conjugated to the antibodies. Hydrogel crosslinking causes the metamaterial to undergo fast chiral re-organization, inducing amplified changes in its mechanical deformation and diffraction patterns, which are detectable by a smartphone camera. The mechanical metamaterial may find broad utility in the sensitive optical immunodetection of biomolecules.

Tensor electromagnetism and emergent elasticity in jammed solids

The theory of mechanical response and stress transmission in disordered, jammed solids poses several open questions of how nonperiodic networks-apparently indistinguishable from a snapshot of a fluid-sustain shear. We present a stress-only theory of emergent elasticity for a nonthermal amorphous assembly of grains in a jammed solid, where each grain is subjected to mechanical constraints of force and torque balance. These grain-level constraints lead to the Gauss's law of an emergent U(1) tensor electromagnetism, which then accounts for the mechanical response of such solids. This formulation of amorphous elasticity has several immediate consequences. The mechanical response maps exactly to the static, dielectric response of this tensorial electromagnetism with the polarizability of the medium mapping to emergent elastic moduli. External forces act as vector electric charges, whereas the tensorial magnetic fields are sourced by momentum density. The dynamics in the electric and magnetic sectors naturally translate into the dynamics of the rigid jammed network and ballistic particle motion, respectively. The theoretical predictions for both stress-stress correlations and responses are borne out by the results of numerical simulations of frictionless granular packings in the static limit of the theory in both 2D and 3D.

Finite-Frequency Topological Maxwell Modes in Mechanical Self-Dual Kagome Lattices

In this Letter, an elastic twisted kagome lattice at a critical twist angle, called self-dual kagome lattice, is shown to exhibit peculiar finite-frequency topological modes which emerge when certain conditions are satisfied. These states are topologically reminiscent of the zero energy (floppy) modes of Maxwell lattices, but they occur at a finite frequency in the band gap of the self-dual kagome lattice. Thus, we present a completely new class of topological modes that share similarities with both the zero frequency floppy modes in Maxwell lattices and the finite energy in-gap modes in topological insulators. We envision the presented mathematical and numerical framework to be invaluable for many technological advances pertaining to wave phenomena, such as reconfigurable waveguide designs.

Rigidity transitions in zero-temperature polygons

We study geometrical clues of a rigidity transition due to the emergence of a system-spanning state of self-stress in underconstrained systems of individual polygons and spring networks constructed from such polygons. When a polygon with harmonic bond edges and an area spring constraint is subject to an expansive strain, we observe that convexity of the polygon is a necessary condition for such a self-stress. We prove that the cyclic configuration of the polygon is a sufficient condition for the self-stress. This correspondence of geometry and rigidity is akin to the straightening of a one dimensional chain of springs to rigidify it. We predict the onset of the rigidity transition and estimate the transition strain using purely geometrical methods. These findings help determine the rigidity of an area-preserving polygon just by knowing its geometry. Since two-dimensional spring networks can be considered as a network of polygons, we look for similar geometric features in underconstrained spring networks under isotropic expansive strain. We observe that all polygons attain convexity at the rigidity transition such that the fraction of convex, but not cyclic, polygons predicts the onset of the rigidity transition. Acyclic polygons in the network correlate with larger tensions, forming effective force chains.

Universal scaling for disordered viscoelastic matter near the onset of rigidity

The onset of rigidity in interacting liquids, as they undergo a transition to a disordered solid, is associated with a rearrangement of the low-frequency vibrational spectrum. In this Letter, we derive scaling forms for the singular dynamical response of disordered viscoelastic networks near both jamming and rigidity percolation. Using effective-medium theory, we extract critical exponents, invariant scaling combinations, and analytical formulas for universal scaling functions near these transitions. Our scaling forms describe the behavior in space and time near the various onsets of rigidity, for rigid and floppy phases and the crossover region, including diverging length scales and timescales at the transitions.

Omnimodal topological polarization of bilayer networks: Analysis in the Maxwell limit and experiments on a 3D-printed prototype

Periodic networks on the verge of mechanical instability, called Maxwell lattices, are known to exhibit zero-frequency modes localized to their boundaries. Topologically polarized Maxwell lattices, in particular, focus these zero modes to one of their boundaries in a manner that is protected against disorder by the reciprocal-space topology of the lattice's band structure. Here, we introduce a class of mechanical bilayers as a model system for designing topologically protected edge modes that couple in-plane dilational and shearing modes to out-of-plane flexural modes, a paradigm that we refer to as "omnimodal polarization." While these structures exhibit a high-dimensional design space that makes it difficult to predict the topological polarization of generic geometries, we are able to identify a family of mirror-symmetric bilayers that inherit the in-plane modal localization of their constitutive monolayers, whose topological polarization can be determined analytically. Importantly, the coupling between the layers results in the emergence of omnimodal polarization, whereby in-plane and out-of-plane edge modes localize on the same edge. We demonstrate these theoretical results by fabricating a mirror-symmetric, topologically polarized kagome bilayer consisting of a network of elastic beams via additive manufacturing and confirm this finite-frequency polarization via finite element analysis and laser-vibrometry experiments.

Duality, Hidden Symmetry, and Dynamic Isomerism in 2D Hinge Structures

Recently, a new type of duality was reported in some deformable mechanical networks that exhibit Kramers-like degeneracy in phononic spectrum at the self-dual point. In this work, we clarify the origin of this duality and propose a design principle of 2D self-dual structures with arbitrary complexity. We find that this duality originates from the partial central inversion (PCI) symmetry of the hinge, which belongs to a more general end-fixed scaling transformation. This symmetry gives the structure an extra degree of freedom without modifying its dynamics. This results in dynamic isomers, i.e., dissimilar 2D mechanical structures, either periodic or aperiodic, having identical dynamic modes, based on which we demonstrate a new type of wave guide without reflection or loss. Moreover, the PCI symmetry allows us to design various 2D periodic isostatic networks with hinge duality. At last, by further studying a 2D nonmechanical magnonic system, we show that the duality and the associated hidden symmetry should exist in a broad range of Hamiltonian systems.

Jammed solids with pins: Thresholds, force networks, and elasticity

The role of fixed degrees of freedom in soft or granular matter systems has broad applicability and theoretical interest. Here we address questions of the geometrical role that a scaffolding of fixed particles plays in tuning the threshold volume fraction and force network in the vicinity of jamming. Our two-dimensional simulated system consists of soft particles and fixed "pins," both of which harmonically repel overlaps. On the one hand, we find that many of the critical scalings associated with jamming in the absence of pins continue to hold in the presence of even dense pin latices. On the other hand, the presence of pins lowers the jamming threshold in a universal way at low pin densities and a geometry-dependent manner at high pin densities, producing packings with lower densities and fewer contacts between particles. The onset of strong lattice dependence coincides with the development of bond-orientational order. Furthermore, the presence of pins dramatically modifies the network of forces, with both unusually weak and unusually strong forces becoming more abundant. The spatial organization of this force network depends on pin geometry and is described in detail. Using persistent homology, we demonstrate that pins modify the topology of the network. Finally, we observe clear signatures of this developing bond-orientational order and broad force distribution in the elastic moduli which characterize the linear response of these packings to strain.

Band Theory and Boundary Modes of High-Dimensional Representations of Infinite Hyperbolic Lattices

Periodic lattices in hyperbolic space are characterized by symmetries beyond Euclidean crystallographic groups, offering a new platform for classical and quantum waves, demonstrating great potential for a new class of topological metamaterials. One important feature of hyperbolic lattices is that their translation group is nonabelian, permitting high-dimensional irreducible representations (irreps), in contrast to abelian translation groups in Euclidean lattices. Here we introduce a general framework to construct wave eigenstates of high-dimensional irreps of infinite hyperbolic lattices, thereby generalizing Bloch's theorem, and discuss its implications on unusual mode counting and degeneracy, as well as bulk-edge correspondence in hyperbolic lattices. We apply this method to a mechanical hyperbolic lattice, and characterize its band structure and zero modes of high-dimensional irreps.

Stiffening of under-constrained spring networks under isotropic strain

Disordered spring networks are a useful paradigm to examine macroscopic mechanical properties of amorphous materials. Here, we study the elastic behavior of under-constrained spring networks, i.e. networks with more degrees of freedom than springs. While such networks are usually floppy, they can be rigidified by applying external strain. Recently, an analytical formalism has been developed to predict the scaling behavior of the elastic network properties close to this rigidity transition. Here we numerically show that these predictions apply to many different classes of spring networks, including phantom triangular, Delaunay, Voronoi, and honeycomb networks. The analytical predictions further imply that the shear modulus G scales linearly with isotropic stress T close to the rigidity transition. However, this seems to be at odds with recent numerical studies suggesting an exponent between G and T that is smaller than one for some network classes. Using increased numerical precision and shear stabilization, we demonstrate here that close to the transition a linear scaling, G ∼ T, holds independent of the network class. Finally, we show that our results are not or only weakly affected by finite-size effects, depending on the network class.

The Stability of a Hydrodynamic Bravais Lattice

We present the results of a theoretical investigation of the stability and collective vibrations of a two-dimensional hydrodynamic lattice comprised of millimetric droplets bouncing on the surface of a vibrating liquid bath. We derive the linearized equations of motion describing the dynamics of a generic Bravais lattice, as encompasses all possible tilings of parallelograms in an infinite plane-filling array. Focusing on square and triangular lattice geometries, we demonstrate that for relatively low driving accelerations of the bath, only a subset of inter-drop spacings exist for which stable lattices may be achieved. The range of stable spacings is prescribed by the structure of the underlying wavefield. As the driving acceleration is increased progressively, the initially stationary lattices destabilize into coherent oscillatory motion. Our analysis yields both the instability threshold and the wavevector and polarization of the most unstable vibrational mode. The non-Markovian nature of the droplet dynamics renders the stability analysis of the hydrodynamic lattice more rich and subtle than that of its solid state counterpart.

Anomalous elasticity of a cellular tissue vertex model

Vertex models, such as those used to describe cellular tissue, have an energy controlled by deviations of each cell area and perimeter from target values. The constrained nonlinear relation between area and perimeter leads to new mechanical response. Here we provide a mean-field treatment of a highly simplified model: a uniform network of regular polygons with no topological rearrangements. Since all polygons deform in the same way, we only need to analyze the ground states and the response to deformations of a single polygon (cell). The model exhibits the known transition between a fluid/compatible state, where the cell can accommodate both target area and perimeter, and a rigid/incompatible state. We calculate and measure the mechanical resistance to various deformation protocols and discover that at the onset of rigidity, where a single zero-energy ground state exists, linear elasticity fails to describe the mechanical response to even infinitesimal deformations. In particular, we identify a breakdown of reciprocity expressed via different moduli for compressive and tensile loads, implying nonanalyticity of the energy functional. We give a pictorial representation in configuration space that reveals that the complex elastic response of the vertex model arises from the presence of two distinct sets of reference states (associated with target area and target perimeter). Our results on the critically compatible tissue provide a new route for the design of mechanical metamaterials that violate or extend classical elasticity.

Thermal Fluctuations of Singular Bar-Joint Mechanisms

A bar-joint mechanism is a deformable assembly of freely rotating joints connected by stiff bars. Here we develop a formalism to study the equilibration of common bar-joint mechanisms with a thermal bath. When the constraints in a mechanism cease to be linearly independent, singularities can appear in its shape space, which is the part of its configuration space after discarding rigid motions. We show that the free-energy landscape of a mechanism at low temperatures is dominated by the neighborhoods of points that correspond to these singularities. We consider two example mechanisms with shape-space singularities and find that they are more likely to be found in configurations near the singularities than others. These findings are expected to help improve the design of nanomechanisms for various applications.

Continuum Field Theory for the Deformations of Planar Kirigami

Mechanical metamaterials exhibit exotic properties that emerge from the interactions of many nearly rigid building blocks. Determining these properties theoretically has remained an open challenge outside a few select examples. Here, for a large class of periodic and planar kirigami, we provide a coarse-graining rule linking the design of the panels and slits to the kirigami's macroscale deformations. The procedure gives a system of nonlinear partial differential equations expressing geometric compatibility of angle functions related to the motion of individual slits. Leveraging known solutions of the partial differential equations, we present an illuminating agreement between theory and experiment across kirigami designs. The results reveal a dichotomy of designs that deform with persistent versus decaying slit actuation, which we explain using the Poisson's ratio of the unit cell.

Linear viscoelastic properties of the vertex model for epithelial tissues

Epithelial tissues act as barriers and, therefore, must repair themselves, respond to environmental changes and grow without compromising their integrity. Consequently, they exhibit complex viscoelastic rheological behavior where constituent cells actively tune their mechanical properties to change the overall response of the tissue, e.g., from solid-like to fluid-like. Mesoscopic mechanical properties of epithelia are commonly modeled with the vertex model. While previous studies have predominantly focused on the rheological properties of the vertex model at long time scales, we systematically studied the full dynamic range by applying small oscillatory shear and bulk deformations in both solid-like and fluid-like phases for regular hexagonal and disordered cell configurations. We found that the shear and bulk responses in the fluid and solid phases can be described by standard spring-dashpot viscoelastic models. Furthermore, the solid-fluid transition can be tuned by applying pre-deformation to the system. Our study provides insights into the mechanisms by which epithelia can regulate their rich rheological behavior.

Progress in Topological Mechanics

Topological mechanics is rapidly emerging as an attractive field of research where mechanical waveguides can be designed and controlled via topological methods. With the development of topological phases of matter, recent advances have shown that topological states have been realized in the elastic media exploiting analogue quantum Hall effect, analogue quantum spin Hall effect, analogue quantum valley Hall effect, higher-order topological physics, topological pump, topological lattice defects and so on. This review aims to introduce the experimental and theoretical achievements with defect-immune protected elastic waves in mechanical systems based on the abovementioned methods, respectively. From these discussions, we predict the possible perspective of topological mechanics.

Energetic rigidity. II. Applications in examples of biological and underconstrained materials

This is the second paper devoted to energetic rigidity, in which we apply our formalism to examples in two dimensions: Underconstrained random regular spring networks, vertex models, and jammed packings of soft particles. Spring networks and vertex models are both highly underconstrained, and first-order constraint counting does not predict their rigidity, but second-order rigidity does. In contrast, spherical jammed packings are overconstrained and thus first-order rigid, meaning that constraint counting is equivalent to energetic rigidity as long as prestresses in the system are sufficiently small. Aspherical jammed packings on the other hand have been shown to be jammed at hypostaticity, which we use to argue for a modified constraint counting for systems that are energetically rigid at quartic order.

Energetic rigidity. I. A unifying theory of mechanical stability

Rigidity regulates the integrity and function of many physical and biological systems. This is the first of two papers on the origin of rigidity, wherein we propose that "energetic rigidity," in which all nontrivial deformations raise the energy of a structure, is a more useful notion of rigidity in practice than two more commonly used rigidity tests: Maxwell-Calladine constraint counting (first-order rigidity) and second-order rigidity. We find that constraint counting robustly predicts energetic rigidity only when the system has no states of self-stress. When the system has states of self-stress, we show that second-order rigidity can imply energetic rigidity in systems that are not considered rigid based on constraint counting, and is even more reliable than shear modulus. We also show that there may be systems for which neither first- nor second-order rigidity imply energetic rigidity. The formalism of energetic rigidity unifies our understanding of mechanical stability and also suggests new avenues for material design.

Dionysian Hard Sphere Packings Are Mechanically Stable at Vanishingly Low Densities

High strength-to-weight ratio materials can be constructed by either maximizing strength or minimizing weight. Tensegrity structures and aerogels take very different paths to achieving high strength-to-weight ratios but both rely on internal tensile forces. In the absence of tensile forces, removing material eventually destabilizes a structure. Attempts to maximize the strength-to-weight ratio with purely repulsive spheres have proceeded by removing spheres from already stable crystalline structures. This results in a modestly low density and a strength-to-weight ratio much worse than can be achieved with tensile materials. Here, we demonstrate the existence of a packing of hard spheres that has asymptotically zero density and yet maintains finite strength, thus achieving an unbounded strength-to-weight ratio. This construction, which we term Dionysian, is the diametric opposite to the Apollonian sphere packing which completely and stably fills space. We create tools to evaluate the stability and strength of compressive sphere packings. Using these we find that our structures have asymptotically finite bulk and shear moduli and are linearly resistant to every applied deformation, both internal and external. By demonstrating that there is no lower bound on the density of stable structures, this work allows for the construction of arbitrarily lightweight high-strength materials.

Achieving adjustable elasticity with non-affine to affine transition

For various engineering and industrial applications it is desirable to realize mechanical systems with broadly adjustable elasticity to respond flexibly to the external environment. Here we discover a topology-correlated transition between affine and non-affine regimes in elasticity in both two- and three-dimensional packing-derived networks. Based on this transition, we numerically design and experimentally realize multifunctional systems with adjustable elasticity. Within one system, we achieve solid-like affine response, liquid-like non-affine response and a continuous tunability in between. Moreover, the system also exhibits a broadly tunable Poisson's ratio from positive to negative values, which is of practical interest for energy absorption and for fracture-resistant materials. Our study reveals a fundamental connection between elasticity and network topology, and demonstrates its practical potential for designing mechanical systems and metamaterials.

Bond-space operator disentangles quasilocalized and phononic modes in structural glasses

The origin of several emergent mechanical and dynamical properties of structural glasses is often attributed to populations of localized structural instabilities, coined quasilocalized modes (QLMs). Under a restricted set of circumstances, glassy QLMs can be revealed by analyzing computer glasses' vibrational spectra in the harmonic approximation. However, this analysis has limitations due to system-size effects and hybridization processes with low-energy phononic excitations (plane waves) that are omnipresent in elastic solids. Here we overcome these limitations by exploring the spectrum of a linear operator defined on the space of particle interactions (bonds) in a disordered material. We find that this bond-force-response operator offers a different interpretation of QLMs in glasses and cleanly recovers some of their important statistical and structural features. The analysis presented here reveals the dependence of the number density (per frequency) and spatial extent of QLMs on material preparation protocol (annealing). Finally, we discuss future research directions and possible extensions of this work.

Topological floppy modes in models of epithelial tissues

Recent advances in topological mechanics have revealed unusual phenomena such as topologically protected floppy modes and states of self-stress that are exponentially localized at boundaries and interfaces of mechanical networks. In this paper, we explore the topological mechanics of epithelial tissues, where the appearance of these boundary and interface modes could lead to localized soft or stressed spots and play a role in morphogenesis. We consider both a simple vertex model (VM) governed by an effective elastic energy and its generalization to an active tension network (ATN) which incorporates active adaptation of the cytoskeleton. By analyzing spatially periodic lattices at the Maxwell point of mechanical instability, we find topologically polarized phases with exponential localization of floppy modes and states of self-stress in the ATN when cells are allowed to become concave, but not in the VM.

Fractional Excitations in Non-Euclidean Elastic Plates

We show that minimal-surface non-Euclidean elastic plates share the same low-energy effective theory as Haldane's dimerized quantum spin chain. As a result, such elastic plates support fractional excitations, which take the form of charge-1/2 solitons between degenerate states of the plate, in strong analogy to their quantum counterpart. These fractional excitations exhibit properties similar to fractional excitations in quantum fractional topological states and in Haldane's dimerized quantum spin chain, including deconfinement and braiding, as well as unique new features such as holographic properties and diodelike nonlinear response, demonstrating great potentials for applications as mechanical metamaterials.

Topology in Nonlinear Mechanical Systems

Many advancements have been made in the field of topological mechanics. The majority of the work, however, concerns the topological invariant in a linear theory. In this Letter, we present a generic prescription to define topological indices that accommodates nonlinear effects in mechanical systems without taking any approximation. Invoking the tools of differential geometry, a Z-valued quantity in terms of a topological index in differential geometry known as the Poincaré-Hopf index, which features the topological invariant of nonlinear zero modes (ZMs), is predicted. We further identify one type of topologically protected solitons that are robust to disorders. Our prescription constitutes a new direction of searching for novel topologically protected nonlinear ZMs in the future.

Elasticity of colloidal gels: structural heterogeneity, floppy modes, and rigidity

Rheological measurements of model colloidal gels reveal that large variations in the shear moduli as colloidal volume-fraction changes are not reflected by simple structural parameters such as the coordination number, which remains almost a constant. We resolve this apparent contradiction by conducting a normal-mode analysis of experimentally measured bond networks of gels of colloidal particles with short-ranged attraction. We find that structural heterogeneity of the gels, which leads to floppy modes and a nonaffine-affine crossover as frequency increases, evolves as a function of the volume fraction and is key to understanding the frequency-dependent elasticity. Without any free parameters, we achieve good qualitative agreement with the measured mechanical response. Furthermore, we achieve universal collapse of the shear moduli through a phenomenological spring-dashpot model that accounts for the interplay between fluid viscosity, particle dissipation, and contributions from the affine and non-affine network deformation.

Self-Assembly of Isostatic Self-Dual Colloidal Crystals

Self-dual structures whose dual counterparts are themselves possess unique hidden symmetry, beyond the description of classical spatial symmetry groups. Here we propose a strategy based on a nematic monolayer of attractive half-cylindrical colloids to self-assemble these exotic structures. This system can be seen as a 2D system of semidisks. By using Monte Carlo simulations, we discover two isostatic self-dual crystals, i.e., an unreported crystal with pmg space-group symmetry and the twisted kagome crystal. For the pmg crystal approaching the critical point, we find the double degeneracy of the full phononic spectrum at the self-dual point and the merging of two tilted Weyl nodes into one critically tilted Dirac node. The latter is "accidentally" located on the high-symmetry line. The formation of this unconventional Dirac node is due to the emergence of the critical flatbands at the self-dual point, which are linear combinations of "finite-frequency" floppy modes. These modes can be understood as mechanically coupled self-dual rhombus chains vibrating in some unique uncoupled ways. Our work paves the way for designing and fabricating self-dual materials with exotic mechanical or phononic properties.

Basement membrane stiffness determines metastases formation

The basement membrane (BM) is a special type of extracellular matrix and presents the major barrier cancer cells have to overcome multiple times to form metastases. Here we show that BM stiffness is a major determinant of metastases formation in several tissues and identify netrin-4 (Net4) as a key regulator of BM stiffness. Mechanistically, our biophysical and functional analyses in combination with mathematical simulations show that Net4 softens the mechanical properties of native BMs by opening laminin node complexes, decreasing cancer cell potential to transmigrate this barrier despite creating bigger pores. Our results therefore reveal that BM stiffness is dominant over pore size, and that the mechanical properties of 'normal' BMs determine metastases formation and patient survival independent of cancer-mediated alterations. Thus, identifying individual Net4 protein levels within native BMs in major metastatic organs may have the potential to define patient survival even before tumour formation. The ratio of Net4 to laminin molecules determines BM stiffness, such that the more Net4, the softer the BM, thereby decreasing cancer cell invasion activity.

Oligomodal metamaterials with multifunctional mechanics

Mechanical metamaterials are man-made materials with extraordinary properties that come from their geometrical structure rather than their chemical composition. For instance, they can be engineered to be extremely light and stiff; to shrink sideways when compressed, instead of expanding; or to exhibit programmable shape changes. Such properties often rely on zero-energy modes. In this work, we created a class of mechanical metamaterials with zero-energy modes that can exhibit multiple properties at the same time within a single structure. In particular, we created a metamaterial that can either shrink or expand on the side when compressed, depending on how fast it is compressed. These metamaterials could lead to novel adaptable devices for, for example, robotics and energy absorption applications.

Mechanical metamaterials are artificial composites that exhibit a wide range of advanced functionalities such as negative Poisson’s ratio, shape shifting, topological protection, multistability, extreme strength-to-density ratio, and enhanced energy dissipation. In particular, flexible metamaterials often harness zero-energy deformation modes. To date, such flexible metamaterials have a single property, for example, a single shape change, or are pluripotent, that is, they can have many different responses, but typically require complex actuation protocols. Here, we introduce a class of oligomodal metamaterials that encode a few distinct properties that can be selectively controlled under uniaxial compression. To demonstrate this concept, we introduce a combinatorial design space containing various families of metamaterials. These families include monomodal (i.e., with a single zero-energy deformation mode); oligomodal (i.e., with a constant number of zero-energy deformation modes); and plurimodal (i.e., with many zero-energy deformation modes), whose number increases with system size. We then confirm the multifunctional nature of oligomodal metamaterials using both boundary textures and viscoelasticity. In particular, we realize a metamaterial that has a negative (positive) Poisson’s ratio for low (high) compression rate over a finite range of strains. The ability of our oligomodal metamaterials to host multiple mechanical responses within a single structure paves the way toward multifunctional materials and devices.

Internally Stressed and Positionally Disordered Minimal Complexes Yield Glasslike Nonphononic Excitations

Glasses, unlike their crystalline counterparts, exhibit low-frequency nonphononic excitations whose frequencies ω follow a universal D(ω)∼ω^{4} density of states. The process of glass formation generates positional disorder intertwined with mechanical frustration, posing fundamental challenges in understanding the origins of glassy nonphononic excitations. Here we suggest that minimal complexes-mechanically frustrated and positionally disordered local structures-embody the minimal physical ingredients needed to generate glasslike excitations. We investigate the individual effects of mechanical frustration and positional disorder on the vibrational spectrum of isolated minimal complexes, and demonstrate that ensembles of marginally stable minimal complexes yield D(ω)∼ω^{4}. Furthermore, glasslike excitations emerge by embedding a single minimal complex within a perfect lattice. Consequently, minimal complexes offer a conceptual framework to understand glasslike excitations from first principles, as well as a practical computational method for introducing them into solids.