Duality and Sheared Analytic Response in Mechanism-Based Metamaterials.

Mechanical metamaterials designed around a zero-energy pathway of deformation known as a mechanism, challenge the conventional picture of elasticity and generate complex spatial response that remains largely uncharted. Here, we present a unified theoretical framework to showing that the presence of a unimode in a 2D structure generates a space of anomalous zero-energy sheared analytic modes. The spatial profiles of these stress-free strain patterns is dual to equilibrium stress configurations. We show a transition at an exceptional point between bulk modes in structures with conventional Poisson ratios (anauxetic) and evanescent surface modes for negative Poisson ratios (auxetic). We suggest a first application of these unusual response properties as a switchable signal amplifier and filter for use in mechanical circuitry and computation.

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.

Discrete symmetries control geometric mechanics in parallelogram-based origami.

Geometric compatibility constraints dictate the mechanical response of soft systems that can be utilized for the design of mechanical metamaterials such as the negative Poisson's ratio Miura-ori origami crease pattern. Here, we develop a formalism for linear compatibility that enables explicit investigation of the interplay between geometric symmetries and functionality in origami crease patterns. We apply this formalism to a particular class of periodic crease patterns with unit cells composed of four arbitrary parallelogram faces and establish that their mechanical response is characterized by an anticommuting symmetry. In particular, we show that the modes are eigenstates of this symmetry operator and that these modes are simultaneously diagonalizable with the symmetric strain operator and the antisymmetric curvature operator. This feature reveals that the anticommuting symmetry defines an equivalence class of crease pattern geometries that possess equal and opposite in-plane and out-of-plane Poisson's ratios. Finally, we show that such Poisson's ratios generically change sign as the crease pattern rigidly folds between degenerate ground states and we determine subfamilies that possess strictly negative in-plane or out-of-plane Poisson's ratios throughout all configurations.

Robotic swimming in curved space via geometric phase.

Locomotion by shape changes or gas expulsion is assumed to require environmental interaction, due to conservation of momentum. However, as first noted in [J. Wisdom, Science 299, 1865-1869 (2003)] and later in [E. Guéron, Sci. Am. 301, 38-45 (2009)] and [J. Avron, O. Kenneth, New J. Phys, 8, 68 (2006)], the noncommutativity of translations permits translation without momentum exchange in either gravitationally curved spacetime or the curved surfaces encountered by locomotors in real-world environments. To realize this idea which remained unvalidated in experiments for almost 20 y, we show that a precision robophysical apparatus consisting of motors driven on curved tracks (and thereby confined to a spherical surface without a solid substrate) can self-propel without environmental momentum exchange. It produces shape changes comparable to the environment's inverse curvatures and generates movement of [Formula: see text] cm per gait. While this simple geometric effect predominates over short time, eventually the dissipative (frictional) and conservative forces, ubiquitous in real systems, couple to it to generate an emergent dynamics in which the swimming motion produces a force that is counter-balanced against residual gravitational forces. In this way, the robot both swims forward without momentum and becomes fixed in place with a finite momentum that can be released by ceasing the swimming motion. We envision that our work will be of use in a broad variety of contexts, such as active matter in curved space and robots navigating real-world environments with curved surfaces.

Topological invariant and anomalous edge modes of strongly nonlinear systems.

Despite the extensive studies of topological states, their characterization in strongly nonlinear classical systems has been lacking. In this work, we identify the proper definition of Berry phase for nonlinear bulk waves and characterize topological phases in one-dimensional (1D) generalized nonlinear Schrödinger equations in the strongly nonlinear regime, where the general nonlinearities are beyond Kerr-like interactions. Without utilizing linear analysis, we develop an analytic strategy to demonstrate the quantization of nonlinear Berry phase due to reflection symmetry. Mode amplitude itself plays a key role in nonlinear modes and controls topological phase transitions. We then show bulk-boundary correspondence by identifying the associated nonlinear topological edge modes. Interestingly, anomalous topological modes decay away from lattice boundaries to plateaus governed by fixed points of nonlinearities. Our work opens the door to the rich physics between topological phases of matter and nonlinear dynamics.

Conformal elasticity of mechanism-based metamaterials.

Deformations of conventional solids are described via elasticity, a classical field theory whose form is constrained by translational and rotational symmetries. However, flexible metamaterials often contain an additional approximate symmetry due to the presence of a designer soft strain pathway. Here we show that low energy deformations of designer dilational metamaterials will be governed by a scalar field theory, conformal elasticity, in which the nonuniform, nonlinear deformations observed under generic loads correspond with the well-studied-conformal-maps. We validate this approach using experiments and finite element simulations and further show that such systems obey a holographic bulk-boundary principle, which enables an analytic method to predict and control nonuniform, nonlinear deformations. This work both presents a unique method of precise deformation control and demonstrates a general principle in which mechanisms can generate special classes of soft deformations.

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.

Hidden symmetries generate rigid folding mechanisms in periodic origami.

We consider the zero-energy deformations of periodic origami sheets with generic crease patterns. Using a mapping from the linear folding motions of such sheets to force-bearing modes in conjunction with the Maxwell-Calladine index theorem we derive a relation between the number of linear folding motions and the number of rigid body modes that depends only on the average coordination number of the origami's vertices. This supports the recent result by Tachi [T. Tachi, Origami 6, 97-108 (2015)] which shows periodic origami sheets with triangular faces exhibit two-dimensional spaces of rigidly foldable cylindrical configurations. We also find, through analytical calculation and numerical simulation, branching of this configuration space from the flat state due to geometric compatibility constraints that prohibit finite Gaussian curvature. The same counting argument leads to pairing of spatially varying modes at opposite wavenumber in triangulated origami, preventing topological polarization but permitting a family of zero-energy deformations in the bulk that may be used to reconfigure the origami sheet.

Correlated Rigidity Percolation and Colloidal Gels.

Rigidity percolation (RP) occurs when mechanical stability emerges in disordered networks as constraints or components are added. Here we discuss RP with structural correlations, an effect ignored in classical theories albeit relevant to many liquid-to-amorphous-solid transitions, such as colloidal gelation, which are due to attractive interactions and aggregation. Using a lattice model, we show that structural correlations shift RP to lower volume fractions. Through molecular dynamics simulations, we show that increasing attraction in colloidal gelation increases structural correlation and thus lowers the RP transition, agreeing with experiments. Hence, the emergence of rigidity at colloidal gelation can be understood as a RP transition, but occurs at volume fractions far below values predicted by the classical RP, due to attractive interactions which induce structural correlation.

Mechanical diffraction reveals the role of passive dynamics in a slithering snake.

Limbless animals like snakes inhabit most terrestrial environments, generating thrust to overcome drag on the elongate body via contacts with heterogeneities. The complex body postures of some snakes and the unknown physics of most terrestrial materials frustrates understanding of strategies for effective locomotion. As a result, little is known about how limbless animals contend with unplanned obstacle contacts. We studied a desert snake, Chionactis occipitalis, which uses a stereotyped head-to-tail traveling wave to move quickly on homogeneous sand. In laboratory experiments, we challenged snakes to move across a uniform substrate and through a regular array of force-sensitive posts. The snakes were reoriented by the array in a manner reminiscent of the matter-wave diffraction of subatomic particles. Force patterns indicated the animals did not change their self-deformation pattern to avoid or grab the posts. A model using open-loop control incorporating previously described snake muscle activation patterns and body-buckling dynamics reproduced the observed patterns, suggesting a similar control strategy may be used by the animals. Our results reveal how passive dynamics can benefit limbless locomotors by allowing robust transit in heterogeneous environments with minimal sensing.

Curved boundaries and chiral instabilities - two sources of twist in homeotropic nematic tori.

Many liquid crystalline systems display spontaneous breaking of achiral symmetry, as achiral molecules aggregate into large chiral domains. In confined cylinders with homeotropic boundary conditions, chromonic liquid crystals - which have a twist elastic modulus which is at least an order of magnitude less than their splay and bend counter parts - adopt a twisted escaped radial texture (TER) to minimize their free energy, whilst 5CB - which has all three elastic constants roughly comparable - does not. In a recent series of experiments, we have shown that 5CB confined to tori and bent cylindrical capillaries with homeotropic boundary conditions also adopts a TER structure resulting from the curved nature of the confining boundaries [P. W. Ellis et al., Phys. Rev. Lett., 2018, 247803]. We shall call this microscopic twist, as the twisted director organization not only depends on the confinement geometry but also on the values of elastic moduli. Additionally, we demonstrate theoretically that the curved geometry of the boundary induces a twist in the escaped radial (ER) texture. Moving the escaped core of the structure towards the center of the torus not only lowers the splay and bend energies, but lowers the energetic cost of this distinct source for twist that we shall call geometric twist. As the torus becomes more curved, the ideal location for the escaped core approaches the inner radius of the torus.

Topology and Geometry of Spin Origami.

Kagome antiferromagnets are known to be highly frustrated and degenerate when they possess simple, isotropic interactions. We consider the entire class of these magnets when their interactions are spatially anisotropic. We do so by identifying a certain class of systems whose degenerate ground states can be mapped onto the folding motions of a generalized "spin origami" two-dimensional mechanical sheet. Some such anisotropic spin systems, including Cs_{2}ZrCu_{3}F_{12}, map onto flat origami sheets, possessing extensive degeneracy similar to isotropic systems. Others, such as Cs_{2}CeCu_{3}F_{12}, can be mapped onto sheets with nonzero Gaussian curvature, leading to more mechanically stable corrugated surfaces. Remarkably, even such distortions do not always lift the entire degeneracy, instead permitting a large but subextensive space of zero-energy modes. We show that for Cs_{2}CeCu_{3}F_{12}, due to an additional point group symmetry associated with the structure, these modes are "Dirac" line nodes with a double degeneracy protected by a topological invariant. The existence of mechanical analogs thus serves to identify and explicate the robust degeneracy of the spin systems.

Curvature-Induced Twist in Homeotropic Nematic Tori.

We confine a nematic liquid crystal with homeotropic anchoring to stable toroidal droplets and study how geometry affects the equilibrium director configuration. In contrast to the case of cylindrical confinement, we find that the equilibrium state is chiral-a twisted and escaped radial director configuration. Furthermore, we find that the magnitude of the twist distortion increases as the ratio of the ring radius to the tube radius decreases; we confirm this with computer simulations of optically polarized microscopy textures. In addition, numerical calculations also indicate that the local geometry indeed affects the magnitude of the twist distortion. We further confirm this curvature-induced twisting using bent cylindrical capillaries.

Transformable topological mechanical metamaterials.

Mechanical metamaterials are engineered materials whose structures give them novel mechanical properties, including negative Poisson's ratios, negative compressibilities and phononic bandgaps. Of particular interest are systems near the point of mechanical instability, which recently have been shown to distribute force and motion in robust ways determined by a nontrivial topological state. Here we discuss the classification of and propose a design principle for mechanical metamaterials that can be easily and reversibly transformed between states with dramatically different mechanical and acoustic properties via a soft strain. Remarkably, despite the low energetic cost of this transition, quantities such as the edge stiffness and speed of sound can change by orders of magnitude. We show that the existence and form of a soft deformation directly determines floppy edge modes and phonon dispersion. Finally, we generalize the soft strain to generate domain structures that allow further tuning of the material.

Mechanical Weyl Modes in Topological Maxwell Lattices.

We show that two-dimensional mechanical lattices can generically display topologically protected bulk zero-energy phonon modes at isolated points in the Brillouin zone, analogs of massless fermion modes of Weyl semimetals. We focus on deformed square lattices as the simplest Maxwell lattices, characterized by equal numbers of constraints and degrees of freedom, with this property. The Weyl points appear at the origin of the Brillouin zone along directions with vanishing sound speed and move away to the zone edge (or return to the origin) where they annihilate. Our results suggest a design strategy for topological metamaterials with bulk low-frequency acoustic modes and elastic instabilities at a particular, tunable finite wave vector.

Rigidity percolation by next-nearest-neighbor bonds on generic and regular isostatic lattices.

We study rigidity percolation transitions in two-dimensional central-force isostatic lattices, including the square and the kagome lattices, as next-nearest-neighbor bonds ("braces") are randomly added to the system. In particular, we focus on the differences between regular lattices, which are perfectly periodic, and generic lattices with the same topology of bonds but whose sites are at random positions in space. We find that the regular square and kagome lattices exhibit a rigidity percolation transition when the number of braces is ∼LlnL, where L is the linear size of the lattice. This transition exhibits features of both first-order and second-order transitions: The whole lattice becomes rigid at the transition, and a diverging length scale also exists. In contrast, we find that the rigidity percolation transition in the generic lattices occur when the number of braces is very close to the number obtained from Maxwell's law for floppy modes, which is ∼L. The transition in generic lattices is a very sharp first-order-like transition, at which the addition of one brace connects all small rigid regions in the bulk of the lattice, leaving only floppy modes on the edge. We characterize these transitions using numerical simulations and develop analytic theories capturing each transition. Our results relate to other interesting problems, including jamming and bootstrap percolation.

Self-assembly of three-dimensional open structures using patchy colloidal particles.

Open structures can display a number of unusual properties, including a negative Poisson's ratio, negative thermal expansion, and holographic elasticity, and have many interesting applications in engineering. However, it is a grand challenge to self-assemble open structures at the colloidal scale, where short-range interactions and low coordination number can leave them mechanically unstable. In this paper we discuss the self-assembly of three-dimensional open structures using triblock Janus particles, which have two large attractive patches that can form multiple bonds, separated by a band with purely hard-sphere repulsion. Such surface patterning leads to open structures that are stabilized by orientational entropy (in an order-by-disorder effect) and selected over close-packed structures by vibrational entropy. For different patch sizes the particles can form into either tetrahedral or octahedral structural motifs which then compose open lattices, including the pyrochlore, the hexagonal tetrastack and the perovskite lattices. Using an analytic theory, we examine the phase diagrams of these possible open and close-packed structures for triblock Janus particles and characterize the mechanical properties of these structures. Our theory leads to rational designs of particles for the self-assembly of three-dimensional colloidal structures that are possible using current experimental techniques.

Organization of strongly interacting directed polymer liquids in the presence of stringent constraints.

The impact of impenetrable obstacles on the energetics and equilibrium structure of strongly repulsive directed polymers is investigated. As a result of the strong interactions, regions of severe polymer depletion and excess are found in the vicinity of the obstacle, and the associated free-energy cost is found to scale quadratically with the average polymer density. The polymer-polymer interactions are accounted for via a sequence of transformations: from the 3D line liquid to a 2D fluid of Bose particles to a 2D composite fermion fluid and, finally, to a 2D one-component plasma. The results presented here are applicable to a range of systems consisting of noncrossing directed lines.