arXiv subjects

Ian Frankel

Publications and source records attributed to Ian Frankel.

11 recordsLinked to original sources

arXiv2026

Single-particle limit of a topological edge state in a locally resonant band gap

Topological metamaterials promise unprecedented wave control. Here, we theoretically and numerically investigate a one-dimensional Su-Schrieffer-Heeger (SSH)-inspired stiffness dimer modified with a local resonator, which imparts a frequency-dependent effective stiffness to the unit cell. The resonator introduces an attenuation singularity: at a frequency at which the effective stiffness vanishes, the spatial attenuation of waves diverges. By tuning a dimerization parameter, we migrate this singularity from one band gap to the other via an intermediate flat-band state, transferring the dominant local-resonance character between the gaps without closing either gap and while preserving the underlying band topology. Crucially, when the resulting topological edge state intersects the attenuation singularity, the edge state collapses onto a single boundary particle, forming a single-particle mode (SPM). This yields an inverse participation ratio of exactly unity, the theoretical limit for localization in a discrete system. Moreover, this extreme localization can be realized at low frequencies, below the first Bragg-type band gap. Further, we demonstrate that while random disorder detunes this mode, merely tuning the boundaries stabilizes the single-particle mode over a broad parameter range. Our findings provide a clear pathway to designing ultra-localized edge states in low-frequency regimes, where band topology guarantees the edge mode and local resonance drives its single-particle confinement.

physics.app-ph↗

arXiv2025

Electric-Field-Controlled Chemical Reaction via Piezo-Chemistry Creates Programmable Material Stiffness

The spatial and temporal control of material properties at a distance has yielded many unique innovations including photo-patterning, 3D-printing, and architected material design. To date, most of these innovations have relied on light, heat, sound, or electric current as stimuli for controlling the material properties. Here, we demonstrate that an electric field can induce chemical reactions and subsequent polymerization in composites via piezoelectrically-mediated transduction. The response to an electric field rather than through direct contact with an electrode is mediated by a nanoparticle transducer, i.e., piezoelectric ZnO, which mediates reactions between thiol and alkene monomers, resulting in tunable moduli as a function of voltage, time, and the frequency of the applied AC power. The reactivity of the mixture and the modulus of a na\"ive material containing these elements can be programmed based on the distribution of the electric field strength. This programmability results in multi-stiffness gels. Additionally, the system can be adjusted for the formation of an electro-adhesive. This simple and generalizable design opens new avenues for facile application in adaptive damping and variable-rigidity materials, adhesive, soft robotics, and potentially tissue engineering.

physics.chem-ph↗

arXiv2024

Minimizing finite viscosity enhances relative kinetic energy absorption in bistable mechanical metamaterials but only with sufficiently fine discretization: a nonlinear dynamical size effect

Bistable mechanical metamaterials have shown promise for mitigating the harmful consequences of impact by converting kinetic energy into stored strain energy, offering an alternative and potentially synergistic approach to conventional methods of attenuating energy transmission. In this work, we numerically study the dynamic response of a one-dimensional bistable metamaterial struck by a high speed impactor (where the impactor velocity is commensurate with the sound speed), using the peak kinetic energy experienced at midpoint of the metamaterial compared to that in an otherwise identical linear system as our performance metric. We make five key findings: 1) The bistable material can counter-intuitively perform better (to nearly 1000x better than the linear system) as the viscosity decreases (but remains finite), but only when sufficiently fine discretization has been reached (i.e. the system approaches sufficiently close to the continuum limit); 2) This discretization threshold is sharp, and depends on the viscosity present; 3) The bistable materials can also perform significantly worse than linear systems (for low discretization and viscosity or zero viscosity); 4) The dependence on discretization stems from the partition of energy into trains of solitary waves that have pulse lengths proportional to the unit cell size, where, with intersite viscosity, the solitary wave trains induce high velocity gradients and thus enhanced damping compared to linear, and low-unit-cell-number bistable, materials; and 5) When sufficiently fine discretization has been reached at low viscosities, the bistable system outperforms the linear one for a wide range of impactor conditions. The first point is particularly important, as it shows the existence of a nonlinear dynamical size effect, where, given a protective layer of some thickness...

nlin.PS↗

arXiv2024

Customizable wave tailoring materials enabled by nonlinear bilevel inverse design

Passive transformation of waves via nonlinear systems is ubiquitous in settings ranging from acoustics to optics and electromagnetics. Passivity is of particular importance for responding rapidly to stimuli and nonlinearity enormously expands signal transformability compared to linear systems due to the breaking of superposition. It is well known that different types of nonlinearity yield vastly different effects on propagating signals, which raises the question of ``what precise nonlinearity is the best for a given wave tailoring application?'' Considering a one-dimensional spring-mass chain as a testbed, we couple the shape optimization of structures for tailored nonlinear constitutive responses with reduced-order nonlinear dynamical inverse design. Using minimization of peak kinetic energy transmission from impact as a case study, we identify ideal nonlinear constitutive responses and the geometries needed to achieve them. As part of this, we show the large sensitivity of this metric to small changes in nonlinearity, and thus the need for high precision, free-form nonlinearity tailoring. We validate our predictions using impact experiments in a chain of nonlinear springs and masses. This work sets the foundation for broader passive nonlinear mechanical wave tailoring material design, with applications to computing, signal processing, shock mitigation, and autonomous materials.

cond-mat.mtrl-sci↗

arXiv2023

Tailoring high precision polynomial architected material constitutive responses via inverse design

The design of specified nonlinear mechanical responses into a structure or material is a highly sought after capability, which would have a significant impact in areas such as wave tailoring in metamaterials, impact mitigation, soft robotics, and biomedicine. Here, we present a topology optimization approach to design structures for desired polynomial nonlinear behavior, wherein we formulate the problem in such a way as to decouple the nonlinear response from the stiffness. We show results across qualitatively different polynomial behaviors while achieving a high degree of precision, creating a path toward analytically tractable nonlinear dynamical systems. The approach enables access to previously difficult to design for, or hitherto unachieved, nonlinear behavior via optimized structures, which can furthermore be incorporated as unit cells of designer materials with tailored nonlinear properties.

cond-mat.mtrl-sci↗

arXiv2023

Dependence of the kinetic energy absorption capacity of bistable mechanical metamaterials on impactor mass and velocity

Using an alternative mechanism to dissipation or scattering, bistable structures and mechanical metamaterials have shown promise for mitigating the detrimental effects of impact by reversibly locking energy into strained material. Herein, we extend prior works on impact absorption via bistable metamaterials to computationally explore the dependence of kinetic energy transmission on the velocity and mass of the impactor, with strain rates exceeding $10^2$ s$^{-1}$. We observe a large dependence on both impactor parameters, ranging from significantly better to worse performance than a comparative linear material. We then correlate the variability in performance to solitary wave formation in the system and give analytical estimates of idealized energy absorption capacity under dynamic loading. In addition, we find a significant dependence on damping accompanied by a qualitative difference in solitary wave propagation within the system. The complex dynamics revealed in this study offer potential future guidance for the application of bistable metamaterials to applications including human and engineered system shock and impact protection devices.

physics.app-ph↗

arXiv2022

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, non-reciprocal, 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 new 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 exceeding conventional elasticity.

cond-mat.soft↗

arXiv2022

Duality of Topological Edge States in a Mechanical Kitaev Chain

We theoretically investigate and experimentally demonstrate the existence of topological edge states in a mechanical analog of the Kitaev chain with a non-zero chemical potential. Our system is a one-dimensional monomer system involving two coupled degrees of freedom, i.e., transverse displacement and rotation of elastic elements. Due to the particle-hole symmetry, a topologically nontrivial bulk leads to the emergence of edge states in a finite chain with fixed boundaries. In contrast, a topologically trivial bulk also leads to the emergence of edge states in a finite chain, but with free boundaries. We unravel a duality in our system that predicts the existence of the latter edge states. This duality involves the iso-spectrality of a subspace for finite chains, and as a consequence, a free chain with topologically trivial bulk maps to a fixed chain with a nontrivial bulk. Lastly, we provide the conditions under which the system can exhibit perfectly degenerate in-gap modes, akin to Majorana zero modes. These findings suggest that mechanical systems with fine-tuned degrees of freedom can be fertile testbeds for exploring the intricacies of Majorana physics.

cond-mat.mes-hall↗

arXiv2020

Meromorphic $L^2$ functions on flat surfaces

We prove a quantitative version of the non-uniform hyperbolicity of the Teichmüller geodesic flow. Namely, at each point of any Teichmüller flow line, we bound the infinitesimal spectral gap for variations of the Hodge norm along the flow line in terms of an easily estimated geometric quantity on the flat surface, which is greater than or equal to the flat systole. As applications, we strengthen results of Treviño and Smith regarding unique ergodicity of measured foliations, and give an estimate for the spectral gaps of pseudo-Anosov homeomorphisms based on the location of their axes in the moduli space of quadratic differentials.

math.GT↗

arXiv2018

CAT(-1)-Type Properties for Teichmüller Space

We show that if Teichmüller geodesics spend enough time in the thick part of moduli space, they display CAT(-1)-type properties. In particular, they exponentially contract along strongly stable leaves. As an application we prove two closing lemmas.

math.GT↗

arXiv2017

A Comparison of Period Coordinates and Teichm\"uller Distance

Let $QD^1(\mathcal{M}_{g,n})$ be the unit cotangent bundle of the moduli space of Riemann surfaces $\mathcal{M}_{g,n}$. There is a metric $d_E$ on $QD^1(\mathcal{M}_{g,n})$ that is locally bi-Lipschitz to the Euclidean metrics defined by systems of period coordinates coming from of short and moderate-length saddle connections. We show the following: if $\mathcal{M}_{g,n}$ is equipped with the Teichm\"uller metric $d_T$, then the projection $(QD^1(\mathcal{M}_{g,n}),d_E) \to (\mathcal{M}_{g,n},d_T)$ is locally a H\"older map. We give a lower bound on the exponent in terms of $g$ and $n$.

math.GT↗