Chiral Nonreciprocal Elasticity and Mechanical Activity

M. Shaat and H.S. Park
Journal of the Mechanics and Physics of Solids 2023; 171:105163

Abstract

There has been significant recent interest in creating, modeling, and exploiting the novel functionality afforded by odd elastic solids, which are a specific class of active matter whose behavior cannot be described by a free energy function. As a result, the mechanical behavior of such solids can be described by a non-symmetric elasticity tensor which means they can be mechanically active, and thus do work on their surroundings through quasistatic deformation cycles without energetic gain or loss terms explicitly appearing in the solid's equation of state. However, previous incarnations of such solids have required the usage of active elements coupled with robotic machinery powered by independent external energy sources to operate. As such, it is unclear whether the non-symmetric elasticity of these solids can be developed using only passive elements that do not require the usage of energy sources, and furthermore how nonreciprocity in elastic media enables non-symmetric elasticity in elastic solids or mechanical activity. In this work, we propose the notion of \emph{chiral, nonreciprocal elasticity}, which represents a generic route to enabling 2D, isotropic elastic solids exhibiting non-symmetric elasticity. Chiral, nonreciprocal elasticity describes elastic behaviors that result from coupling chirality with nonreciprocity - specifically, (1) the modulation of the elastic properties depending on the mode and direction of deformation and (2) the nonreciprocal coupling of different deformation fields, both of which enable the solid to exhibit a non-symmetric elasticity tensor. To motivate this, we introduce an isotropic 2D chiral metamaterial made of passive chiral elements that, by exploiting local geometric asymmetry, behaves in a chiral, nonreciprocal elastic fashion. We derive, based on the mechanics of a discrete model of this chiral element, the resulting continuum field equations and constitutive relationships that capture the chiral, nonreciprocal elastic behavior. Then, we establish a thermodynamic framework of energy balance and conservation of chiral, nonreciprocal elastic solids, based on which we demonstrate the ability of the proposed chiral metamaterial to act as a source of mechanical work when used in specific quasistatic deformation cycles, though no energy is dissipated by its passive elements. Finally, we demonstrate through numerical finite element simulations the practical implementation of the deformation cycles, while elucidating the specific conditions needed for the chiral metamaterial to exhibit linear, chiral nonreciprocal elastic behavior throughout the deformation cycle, and thus reveal mechanical activity.

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Double Dirac Cones and Topologically Non-Trivial Phonons for Continuous, Square Symmetric (C4v and C2v) Unit Cells

Y. Lu and H.S. Park
Physical Review B 2021; 103:064308

Abstract

Because phononic topological insulators have primarily been studied in discrete, graphene-like structures with C6v or C3v hexagonal symmetry, an open question is how to systematically achieve double Dirac cones and topologically non-trivial structures using continuous, non-hexagonal unit cells. Here, we address this challenge by presenting a novel computational methodology for the inverse design of continuous two-dimensional square phononic metamaterials exhibiting C4v and C2v symmetry. This leads to the systematic design of square unit cell topologies exhibiting a double Dirac degeneracy, which enables topologically-protected interface propagation based on the quantum spin Hall effect (QSHE). Numerical simulations prove that helical edge states emerge at the interface between two topologically distinct square phononic metamaterials, which opens the possibility of QSHE-based pseudospin-dependent transport beyond hexagonal lattices.

This paper is available in PDF form .

Topologically Switchable Behavior Induced by an Elastic Instability in a Phononic Waveguide

B.H. Nguyen, X. Zhuang, H.S. Park and T. Rabczuk
Journal of Applied Physics 2020; 127:245109

Abstract

While topological insulators have been widely studied, they typically appear in configurations and properties that are set once a structure is fabricated. As such, there is significant interest in developing topologically tunable or switchable concepts. In this work, we demonstrate that geometric nonlinearity in the form of an elastic snap-through instability can be exploited to switch the topological properties of a Timoshenko arch beam unit cell. We first demonstrate that the phonon band structure can be tuned using geometric nonlinearity and large displacement to reveal the existence of a topological phase transition point. To make this concept fully stable under the removal of the applied force, we then demonstrate the emergence of a bistable unit cell by varying the parameters of the unit cell. In doing so, we show that the bistability of the arch beam unit cell can be harnessed to design a switch that controls the topological nature of an interface between two different 1D phononic crystals.

This paper is available in PDF form .

Valley-Dependent Topologically Protected Elastic Waves Using Solid Continuous Membranes on Patterned Substrates

J-H Hong, J.H. Oh, H.S. Park and S.Y. Kim
Nanoscale 2020; 12:8997-9004

Abstract

We present a novel structure for topologically protected propagation of mechanical waves in a continuous, elastic membrane using an analog of the quantum valley Hall effect. Our system involves a thin, continuous graphene monolayer lying on a pre-patterned substrate, and as such, it can be employed across multiple length scales ranging from the nano to macroscales. This enables it to support topologically-protected waves at frequencies that can be tuned from the kHz to GHz range by either selective pre-tensioning of the overlaying membrane, or by increasing the lattice parameter of the underlying substrate. We show through numerical simulations that this continuous system is robust against imperfections, is immune to backscattering losses, and supports topologically-protected wave propagation along all available paths and angles. We demonstrate the ability to support topologically-protected interface modes using monolayer graphene, which does not intrinsically support topologically non-trivial elastic waves.

This paper is available in PDF form .

Tunable Topological Bandgaps and Frequencies in a Pre-Stressed Soft Phononic Crystal

B.H. Nguyen, X. Zhuang, H.S. Park and T. Rabczuk
Journal of Applied Physics 2019; 125:095106

Abstract

Topological insulators have recently received significant attention due to the promise of lossless transport of various types of energy. Despite this interest, one outstanding issue is that the topological bandgap and the frequencies that are topologically permitted are typically fixed once the topological structure has been designed and fabricated. Therefore, an open and unresolved question concerns the ability to actively tune both the bandgap magnitude, as well as the frequencies, for which the energy is topologically protected. In this work, we report a mechanically tunable phononic topological insulator (TI) using an acoustic analog of the Quantum valley Hall effect (QVHE). We propose a phononic crystal (PC) comprised of a soft, hyperelastic material where the phononic band structure is modulated through large deformation of the structure. In doing so, space-inversion symmetry (SIS) can be broken, which leads to a phase transition between two topologically-contrasted states and the emergence of topologically-protected interface modes according to bulk-edge correspondence. We further demonstrate the robustness of this topological protection of the edge state along the interface, which demonstrates that mechanical deformation can be used to effectively tailor and tune the topological properties of elastic structures.

This paper is available in PDF form .

Inverse Design of Quantum Spin Hall-Based Phononic Topological Insulators

S.S. Nanthakumar, X. Zhuang, H.S. Park, C. Nguyen, Y. Chen and T. Rabczuk
Journal of the Mechanics and Physics of Solids 2019; 125:550-571

Abstract

We propose a computational methodology to perform inverse design of quantum spin hall effect (QSHE)-based phononic topological insulators. We first obtain two-fold degeneracy, or a Dirac cone, in the band structure using a level set-based topology optimization approach. Subsequently, four-fold degeneracy, or a double Dirac cone, is obtained by using zone folding, after which breaking of translational symmetry, which mimics the effect of strong spin-orbit coupling and which breaks the four-fold degeneracy resulting in a bandgap, is applied. We use the approach to perform inverse design of hexagonal unit cells of C6 and C3 symmetry. The numerical examples show that a topological domain wall with two variations of the designed metamaterials exhibit topologically protected interfacial wave propagation, and also demonstrate that larger topologically-protected bandgaps may be obtained with unit cells based on C3 symmetry.

This paper is available in PDF form .

Strain Tunable Phononic Topological Bandgaps in Two-Dimensional Hexagonal Boron Nitride

J-W Jiang and H.S. Park
Journal of Applied Physics 2019; 125:082511 (Invited paper: Special Issue on Strain Engineering in Functional Materials)

Abstract

The field of topological mechanics has recently emerged due to the interest in robustly transporting various types of energy in a flaw and defect-insensitive fashion. While there have been a significant number of studies based on discovering and proposing topological materials and structures, very few have focused on tuning the resulting topological bandgaps, which is critical because the bandgap frequency is fixed once the structure has been fabricated. Here, we perform both lattice dynamical calculations and molecular dynamical simulations to investigate strain effects on the phononic topological bandgaps in two-dimensional monolayer hexagonal boron nitride. Our studies demonstrate that while the topologically protected phononic bandgaps are not closed even for severely deformed hexagonal boron nitride, and are relatively insensitive to uniaxial tension and shear strains, the position of the frequency gap can be efficiently tuned in a wide range through the application of biaxial strains. Overall, this work thus demonstrates that topological phonons are robust against the effects of mechanical strain engineering, and sheds light on the tunability of the topological bandgaps in nanomaterials.

This paper is available in PDF form .

Topologically Protected Interface Phonons in Two-Dimensional Nanomaterials: Hexagonal Boron Nitride and Silicon Carbide

J-W Jiang, B-S Wang and H.S. Park
Nanoscale 2018; 10:13913-13923

Abstract

We perform both lattice dynamics analysis and molecular dynamics simulations to demonstrate the existence of topologically protected phonon modes in two-dimensional, monolayer hexagonal boron nitride and silicon carbide sheets. The topological phonon modes are found to be localized at an in-plane interface that divides these systems into two regions of distinct valley Chern numbers. The dispersion of this topological phonon mode crosses over the frequency gap, which is opened through analogy with the quantum valley Hall effect by breaking inversion symmetry of the primitive unit cells. Consequently, vibrational energy with frequency within this gap is topologically protected, resulting in wave propagation that exhibits minimal backscattering, is robust with regards to structural defects such as sharp corners, and exhibits excellent temporal stability. Our findings open up the possibility of actuating and detecting topological phonons in two-dimensional nanomaterials.

This paper is available in PDF form .