arXiv subjects

D. Zeb Rocklin

Publications and source records attributed to D. Zeb Rocklin.

At least 19 recordsLinked to original sources

arXiv2026

Multi-field Return Point Memory

Non-equilibrium systems display memory, a dependence not merely on their present environment but on previously applied fields. Multistable systems such as spin glasses, martensites and granular matter have exponentially many microstates consistent with an applied field, making their rich dynamics difficult to control. Control and order can be achieved through the concept of partial ordering, which we here generalize to systems subject to multiple control fields. We demonstrate, within the model system of the zero-temperature Ising model, that this leads to return-point memory, in which an applied sequence of fields restores the hysteretic system not only to a previous magnetization, but to a previous exact microstate. The multiplicity of fields grants more precise and complex control of the system, with different classes of operations displaying commutative and noncommutative behavior. This grants new insight into how physical systems can remember, learn, and be trained.

cond-mat.stat-mech↗

arXiv2025

Geometric Mechanics of Thin Periodic Surfaces

Thin surfaces are ubiquitous in nature, from leaves to cell membranes, and in technology, from paper to corrugated containers. Structural thinness imbues them with flexibility, the ability to easily bend under light loads, even as their much higher stretching stiffness can bear substantial stresses. When surfaces have periodic patterns of either smooth hills and valleys or sharp origami-like creases this can substantially modify their mechanical response. We show that for any such surface, there is a duality between the surface rotations of an isometric deformation and the in-plane stresses of a force-balanced configuration. This duality means that of the six possible combinations of global in-plane strain and out-of-plane bending, exactly three must be isometries. We show further that stressed configurations can be expressed in terms of both the applied deformation and the isometric deformation that is dual to the pattern of stress that arises. We identify constraints rooted in symplectic geometry on the three isometries that a single surface can generate. This framework sheds new light on the fundamental limits of the mechanical response of thin periodic surfaces, while also highlighting the role that continuum differential geometry plays in even sharply creased origami surfaces.

cond-mat.soft↗

arXiv2025

Rigidity of generic random tensegrity structures

Many mechanical structures, both engineered and biological, combine heavy rigid elements such as bones and beams with lightweight flexible ones such as cables and membranes. These are referred to as tensegrities, reflecting that cables can only support extensile tension. We model such systems via simulations of depleted triangular lattices in which we minimize the energies of tensegrities subject to strained boundary conditions. When there are equal numbers of cables and struts (which support only compressive tension), a cable and a strut together each contribute as much toward rigidity as a rod, with the two contributions being equal in the case of shear strain. Due to the highly nonaffine deformations at the rigidity transitions, the contribution of a cable (strut) can be significant even under global compression (dilation) despite a cable's inability to resist local compression. Further, we find that when neighboring elements tend to point away from one another, as is common in real systems, cables interact significantly more strongly with other cables than do cables with struts in supporting stress. These phenomena shed new light on a variety of realistic, disordered systems at the threshold of mechanical stability.

cond-mat.soft↗

arXiv2024

Novel mechanical response of parallelogram-face origami governed by topological characteristics

Origami principles are used to create strong, lightweight structures with complex mechanical response. However, identifying the fundamental physical principles that determine a sheet's behavior remains a challenge. We introduce a new analytic theory in which commonly studied origami sheets fall into distinct topological classes that predict sharply varying mechanical behavior, including effective stiffness and smoothness of mechanical response under external loads. Origami sheets with negative Poisson's ratios, such as the Miura ori, have conventional, smooth mechanical response amenable to continuum-based approaches. In contrast, positive Poisson's ratio, as in the Eggbox ori, generates a topological transition to lines of doubly degenerate zero modes that lead to dramatically softer structures with uneven, complex patterns of spatial response. These patterns interact in complicated ways with origami boundary conditions and source terms, leading to rich physical phenomena in experimentally accessible systems. This approach highlights topological mechanics, with deep connections to topologically protected quantum-mechanical systems, as a design principle for controlling the mechanical response of thin, complex sheets.

cond-mat.soft↗

arXiv2023

Orisometry formalism reveals duality and exotic nonuniform response in origami sheets

Origami metamaterial design enables drastic qualitative changes in the response properties of a thin sheet via the addition of a repeating pattern of folds based around a rigid folding motion. Known also as a mechanism, this folding motion will have a very small energy cost when applied uniformly; and yet uniform activation of such remains highly difficult to observe, these sheets instead generically displaying nonuniform response patterns which are not yet well understood. Here, we present a purely geometric continuum theory which captures the nonuniform, nonlinear response to generic loading as composed locally of the planar mechanism, as well as previously identified ``twist'' and ``bend'' modes which enable the patterned sheet to curve out of the plane across long distances. Our numerical analysis confirms that these three modes govern the observed nonuniform response, varying smoothly across the sheet according to three PDEs which guarantee compatibility. In analogy with the recently solved case of planar mechanism metamaterials, these ``Orisometries'' (origami + isometries, so named by us) are subextensive but infinite in number, with each mode displaying, in the linear limit, ``sheared analytic'' spatial patterns which are controlled by the Poisson's ratio of the uniform folding mechanism. Furthermore, the ``planar'' mechanism-based deformation patterns superimpose with a mathematically dual space of ``non-planar'' twist/bend deformations to span the available soft linear response. Together, our findings furnish the first quantification of the number of soft response modes available, as well as the first intuitive quantification of their spatial distribution.

cond-mat.soft↗

arXiv2023

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 novel 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 fundamentally new 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.

cond-mat.soft↗

arXiv2022

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 modes and characterize topological phases in one-dimensional (1D) generalized nonlinear Schrödinger equations in the strongly nonlinear regime. 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. We propose passive photonic and active electrical systems that can be experimentally implemented. Our work opens the door to the rich physics between topological phases of matter and nonlinear dynamics.

cond-mat.dis-nn↗

arXiv2022

Duality and Sheared Analytic Response in Mechanism-Based Metamaterials

Mechanical metamaterials designed around a zero-energy pathway of deformation, known as a mechanism, have repeatedly challenged the conventional picture of elasticity. However, the complex spatial deformations these structures are able to support beyond the uniform mechanism remain largely uncharted. Here we present a unified theoretical framework, showing that the presence of any uniform mechanism in a two-dimensional structure fundamentally changes its elastic response by admitting a family of non-uniform zero-energy deformations. Our formalism reveals a mathematical duality between these stress-free strains, which we term "sheared analytic modes" and the supported spatial profiles of stress. These modes undergo a transition from bulk periodic response to evanescent surface response as the Poisson's ratio $ν$ of the mechanism is tuned through an exceptional point at $ν=0$. We suggest a first application of these unusual response properties as a switchable signal amplifier and filter for use in mechanical circuitry and computation.

cond-mat.soft↗

arXiv2022

Discrete symmetries control mechanical response 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 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 which possess equal and opposite in-plane and out-of-plane Poisson's ratios.

cond-mat.soft↗

arXiv2022

Locomotion without force, and impulse via dissipation: Robotic swimming in curved space via geometric phase

Locomotion by shape changes (spermatozoon swimming, snake slithering, bird flapping) or gas expulsion (rocket firing) is assumed to require environmental interaction, due to conservation of momentum. As first noted in (Wisdom, 2003) and later in (Guéron, 2009) and (Avron et al, 2006), in curved space or spacetime the non-commutativity of translations permits translation without momentum exchange, just as falling cats and lizards can self-deform to reorient in flat space without environmental interaction. Translation in curved space can occur not only in gravitationally induced curved spacetime (where translation is predicted to be on the order of $10^{-23}$ m per gait cycle) but also in the curved surfaces encountered by locomotors in real-world environments. Here 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 (impulse) via shape changes that can generate gauge potentials that manifest as translations. Our system produces shape changes comparable to the environment's inverse curvatures and generates from zero momentum forward movement of $10^{-1}$ cm per gait cycle even while resisted by weak gravitational and frictional forces. Dissipation via friction eventually arrests the robot but also imbues it with momentum which can be released upon a cessation of shape changes. This work demonstrates how the interaction between environmental curvature, active driving and geometric phases yields rich, exotic phenomena.

cond-mat.soft↗

arXiv2021

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 novel 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 unprecedented analytic method to predict and control nonuniform, nonlinear deformations. This work both presents a novel method of precise deformation control and demonstrates a general principle in which mechanisms can generate special classes of soft deformations.

cond-mat.soft↗

arXiv2021

Soft topological modes protected by symmetry in rigid mechanical metamaterials

Topological mechanics can realize soft modes in mechanical metamaterials in which the number of degrees of freedom for particle motion is finely balanced by the constraints provided by interparticle interactions. However, solid objects are generally hyperstatic (or overconstrained). Here, we show how symmetries may be applied to generate topological soft modes even in overconstrained, rigid systems. To do so, we consider non-Hermitian topology based on non-square matrices, and design a hyperstatic material in which low-energy modes protected by topology and symmetry appear at interfaces. Our approach presents a novel way of generating softness in robust scale-free architectures suitable for miniaturization to the nanoscale.

cond-mat.soft↗

arXiv2019

Normal form for renormalization groups

The results of the renormalization group are commonly advertised as the existence of power law singularities near critical points. The classic predictions are often violated and logarithmic and exponential corrections are treated on a case-by-case basis. We use the mathematics of normal form theory to systematically group these into universality families of seemingly unrelated systems united by common scaling variables. We recover and explain the existing literature and predict the nonlinear generalization for the universal homogeneous scaling functions. We show that this procedure leads to a better handling of the singularity even in classic cases and elaborate our framework using several examples.

cond-mat.stat-mech↗

arXiv2018

Elasticity of colloidal gels: structural heterogeneity, foppy 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 the gels. 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 understand 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.

cond-mat.soft↗

arXiv2018

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 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.

cond-mat.soft↗

arXiv2018

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 Cs2ZrCu3F12, map onto flat origami sheets, possessing extensive degeneracy similar to isotropic systems. Others, such as Cs2CeCu3F12, can be mapped onto sheets with non-zero 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 sub-extensive space of zero-energy modes. We show that for Cs2CeCu3F12, due to an additional point group symmetry associated with 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.

cond-mat.str-el↗

arXiv2018

Controlling the Deformation of Metamaterials: Corner Modes via Topology

Topological metamaterials have invaded the mechanical world, demonstrating acoustic cloaking and waveguiding at finite frequencies and variable, tunable elastic response at zero frequency. Zero frequency topological states have previously relied on the Maxwell condition, namely that the system has equal numbers of degrees of freedom and constraints. Here, we show that otherwise rigid periodic mechanical structures are described by a map with a nontrivial topological degree (a generalization of the winding number introduced by Kane and Lubensky) that creates, directs and protects modes on their boundaries. We introduce a model system consisting of rigid quadrilaterals connected via free hinges at their corners in a checkerboard pattern. This bulk structure generates a topological linear deformation mode exponentially localized in one corner, as investigated numerically and via experimental prototype. Unlike the Maxwell lattices, these structures select a single desired mode, which controls variable stiffness and mechanical amplification that can be incorporated into devices at any scale.

cond-mat.soft↗

arXiv2018

Folding mechanisms at finite temperature

Folding mechanisms are zero elastic energy motions essential to the deployment of origami, linkages, reconfigurable metamaterials and robotic structures. In this paper, we determine the fate of folding mechanisms when such structures are miniaturized so that thermal fluctuations cannot be neglected. First, we identify geometric and topological design strategies aimed at minimizing undesired thermal energy barriers that generically obstruct kinematic mechanisms at the microscale. Our findings are illustrated in the context of a quasi one-dimensional linkage structure that harbors a topologically protected mechanism. However, thermal fluctuations can also be exploited to deliberately lock a reconfigurable metamaterial into a fully expanded configuration, a process reminiscent of order by disorder transitions in magnetic systems. We demonstrate that this effect leads certain topological mechanical structures to exhibit an abrupt change in the pressure -- a bulk signature of the underlying topological invariant at finite temperature. We conclude with a discussion of anharmonic corrections and potential applications of our work to the the engineering of DNA origami devices and molecular robots.

cond-mat.soft↗