Voltage-controlled topological interface states for bending waves in soft dielectric phononic crystal plates

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arXiv:2210.13424v1 [cond-mat.soft] 24 Oct 2022

Voltage-controlled topological interface states for bending waves

in soft dielectric phononic crystal plates

Yingjie Chena, Bin Wub,∗, Michel Destradea,b, Weiqiu Chena,c

aKey Laboratory of Soft Machines and Smart Devices of Zhejiang Province

and Department of Engineering Mechanics,

Zhejiang University, Hangzhou 310027, P.R. China;

bSchool of Mathematical and Statistical Sciences, University of Galway, University Road, Galway, Ireland;

cSoft Matter Research Center, Zhejiang University, Hangzhou 310027, China;

Abstract

The operating frequency range of passive topological phononic crystals is generally ﬁxed

and narrow, limiting their practical applications. To overcome this diﬃculty, here we design

and investigate a one-dimensional soft dielectric phononic crystal (PC) plate system with

actively tunable topological interface states via the mechanical and electric loads. We use

nonlinear electroelasticity theory and linearized incremental theory to derive the governing

equations. First we determine the nonlinear static response of the soft dielectric PC plate

subjected to a combination of axial force and electric voltage. Then we study the motion of

superimposed incremental bending waves. By adopting the Spectral Element Method, we

obtain the dispersion relation for the inﬁnite PC plate and the transmission coeﬃcient for the

ﬁnite PC plate waveguide. Numerical results show that the low-frequency topological inter-

face state exists at the interface of the ﬁnite phononic plate waveguide with two topologically

diﬀerent elements. By simply adjusting the axial force or the electric voltage, an increase

or decrease in the frequency of the topological interface state can be realized. Furthermore,

applying the electric voltage separately on diﬀerent elements of the PC plate waveguide is

a ﬂexible and smart method to tune the topological interface state in a wide range. These

results provide guidance for designing soft smart wave devices with low-frequency tunable

topological interface states.

Keywords: Topological phononic crystal, active tunability, low-frequency interface state,

dielectric elastomer, material nonlinearity

∗Corresponding author at: School of Mathematical and Statistical Sciences, University of Galway, Univer-

sity Road, Galway, Ireland.

Email address: bin.wu@nuigalway.ie (Bin Wu)

Preprint submitted to International Journal of Solids and Structures March 22, 2024

1. Introduction

Dielectric elastomers (DEs) are a type of smart materials that respond to rapidly to

electric stimulus and develop large deformations. DEs have attracted enormous attention

from academia and industry alike, due to excellent characteristics such as high energy den-

sity, high fracture toughness and light weight, and promising potential in artiﬁcial muscles,

soft robotics, actuators, and energy harvesters (Carpi et al.,2011;Anderson et al.,2012;

Zhao and Wang,2014).

Recently, it has also been shown that applying an electric ﬁeld oﬀers an eﬀective approach

to manipulating acoustic/elastic waves in DEs via the induced ﬁnite deformations. Based on

nonlinear electroelasticity theory and associated incremental theory (Dorfmann and Ogden,

2006,2010), many investigations have been conducted to study superimposed inﬁnitesimal

waves in DEs subjected to the external mechanical and electric loads. For an inﬁnite soft

electroactive hollow tube with axial pre-stretch and axial electric ﬁeld, Su et al. (2016) ana-

lyzed the non-axisymmetric wave propagation characteristics. Shmuel and Pernas-Salom´on

(2016) employed the stable Hybrid Matrix Method to investigate the manipulation of ﬂex-

ural waves in DE ﬁlms controlled by an axial force and voltage. Galich and Rudykh (2016)

studied both the propagation of pressure and shear waves in DEs under the action of electric

stimuli. Based on the State Space Method, Wu et al. (2017) presented a theoretical analysis

of guided circumferential wave propagation in soft electroactive tubes under inhomogeneous

electromechanical biasing ﬁelds, and found it could be used for ultrasonic non-destructive

testing. Wu et al. (2020) studied nonlinear ﬁnite deformations and superimposed axisym-

metric wave in a functionally graded soft electroactive tube, when it is subjected to me-

chanical and electric biasing ﬁelds. Ziser and Shmuel (2017) showed experimentally that

the ﬂexural wave mode in a DE ﬁlm can be tuned by voltage, and that the wave velocity

can be slowed down. This setup was also modelled theoretically by Broderick et al. (2020).

Jandron and Henann (2018) demonstrated the tunable eﬀect of electric load on the linearized

wave propagation in inﬁnite periodic composite DEs, based on Finite Element Method sim-

ulations.

Phononic crystals (PCs), which are essentially artiﬁcial periodic composites, have at-

tracted intensive interests because of their outstanding characteristics in steering acous-

tic/elastic waves. Ascribed to the Bragg scattering (Kushwaha et al.,1993), local resonance

(Liu et al.,2000) or inertial ampliﬁcation (Yilmaz et al.,2007) mechanisms, the existence

of a band gap (BG) is the most important feature possessed by PCs, wherein the propa-

gation of acoustic/elastic wave is forbidden. Due to the exotic BG and the dispersive pass

band properties, PCs can be applied to realize peculiar wave behaviors, such as negative re-

fraction (Feng et al.,2006;Zhang and Liu,2004), cloaking (Zhang et al.,2011;Chen et al.,

2017) and one-way propagation (Fleury et al.,2014;Chen et al.,2019b).

To achieve active control of wave propagation, many studies have been carried out to de-

sign PCs with wide tunable BGs, especially by using electric stimuli to tune the BGs in soft

DE PCs. For the ﬁrst time, Shmuel and deBotton (2012) investigated the electrostatically

tunable BGs of incremental shear waves propagating perpendicular to the neo-Hookean ideal

DE periodic laminates by the transfer matrix method. Galich and Rudykh (2017) re-checked

the problem studied by Shmuel and deBotton (2012) and found that the BGs are not aﬀected

directly by the electric load for the shear waves propagating perpendicular to the layers in the

neo-Hookean ideal DE laminates, which corrected the conclusion of Shmuel and deBotton

(2012). Shmuel (2013) analyzed the propagation characteristics of incremental anti-plane

shear waves in ﬁnitely extensible Gent DE ﬁber composites, and provided the ﬁrst accurate

demonstration of electrostatically tunable BGs in the DE composites. In addition, Getz et al.

(2017) showed that the complete BGs of a soft dielectric ﬁber composite can be tuned by

electric voltage, due to the resulting changes in geometry and physical properties of the struc-

ture. Getz and Shmuel (2017) designed PC plates composed of two DE phases, and achieved

voltage-controlled BGs, which can be used for active waveguides and isolators. By adjusting

the axial force and electric voltage applied to one-dimensional (1D) PC cylinders made of

DE materials, Wu et al. (2018) studied active tunability of superimposed longitudinal wave

propagation. For a periodic compressible DE laminate, Chen et al. (2020) shed light on the

inﬂuence of pre-stress and electric stimuli on the nonlinear response and small-amplitude

longitudinal and shear wave propagation behaviors. For more details on tunable and active

PCs, the interested readers are referred to a recent review paper by Wang et al. (2020b).

Inspired by the concept of topological interface state in electronic systems, attention

has been devoted in recent years to topological PC systems with particular topologically

protected interface or edge states, which are unidirectional and immune to backscatter-

ing (Xiao et al.,2015;Ma et al.,2019). The topological invariants, named Berry phase

(Zak,1989) for two-dimensional (2D) systems and Zak phase (Atala et al.,2013) for 1D

systems, play an important role in characterizing the topological properties of band struc-

tures for PCs. Mimicking the quantum Hall eﬀect, a ﬁrst type of 2D topological PCs

breaks the time-reversal symmetry with gyroscopes (Nash et al.,2015), time-modulated

materials (Chen et al.,2019a), or external ﬂow ﬁelds (Khanikaev et al.,2015). The uni-

directional topologically protected interface state in this type of topological PCs was ob-

served experimentally by Fleury et al. (2014). A second type of 2D topological PCs breaks

the spatial-inversion symmetry while conserving time-reversal symmetry, which support the

pseudospin-dependent edge states and are named as quantum spin Hall topological insulators

(Brendel et al.,2018;Zhang et al.,2017). Because of the spin-orbit mechanism, quantum

spin Hall topological PCs may feature forward and backward edge states by relying on ap-

propriate polarization excitation (Yu et al.,2018). A third avenue is to break the inversion

or mirror symmetry of 2D topological PCs, where the quantum valley Hall eﬀect provides

topologically protected interface states between two parts with opposite valley vortex states

(Pal and Ruzzene,2017;Wang et al.,2020a). For 1D topological systems, the topological

transition process can be achieved by breaking spatial symmetry. Hence, topological inter-

face states were observed in waveguides composed of base elements with diﬀerent topological

properties (Xiao et al.,2015;Yin et al.,2018).

However, for topological PCs made of passive materials, the working frequency range of

topological interface/edge modes is narrow and ﬁxed. In particular, the topological interface

state in 1D systems usually emerges at a single frequency transmission peak in the overlapped

BG. Therefore, actively tunable topological PCs are designed to possess a wider operating

frequency range in practical applications. Zhou et al. (2020b) realized tunable topological

interface states in a piezoelectric rod system with periodic electric boundary conditions.

By adjusting the strain ﬁeld, Liu and Semperlotti (2018) actively tuned the topological edge

states in a 2D topological PC waveguides based on the quantum valley Hall eﬀect. Feng et al.

(2019) designed 1D magnetoelastic topological PC slabs and used a magnetic ﬁeld to tune

the topological interface states for Lamb waves contactlessly and nondestructively.

Among the many studies on tunable topological PCs, soft topological PCs are receiving

some attention because of their low operating frequency and the possibility of tunability

by external loads. Li et al. (2018) presented topological interface states in a designed soft

circular-hole PC plate that were dynamically tuned by altering the ﬁlling ratio and adjusting

the external mechanical strain. Huang et al. (2020) showed that mechanical deformations

can be used to actively tune the topological interface states in a 1D soft PC plate consist-

ing of base elements with diﬀerent topological properties. Nguyen et al. (2019) presented a

2D quantum valley Hall topological insulator, composed of soft cylinder inclusions and an

elastic matrix. They found that the mechanical deformations can be exploited to modu-

late the topological properties of the structure and tune the topologically protected states.

Chen et al. (2021) proposed a 1D soft waveguide composed of two topologically diﬀerent PC

elements, in which the low-frequency topological interface states for longitudinal waves were

tuned by the axial force in a wide frequency range. Based on the quantum valley Hall eﬀect,

Zhou et al. (2020a) designed a soft membrane-type PC consisting of a DE membrane and

metallic particles, and broadened the frequency range of the topological interface mode in

this voltage-controlled system.

However, a high density of metallic particles can result in excessive deformations of

the DE membrane, whose planar conﬁguration may collapse. Motivated by the excellent

electromechanical behaviors of DEs, here we design a 1D soft dielectric topological PC plate

with step-wise cross-sections for the incremental bending waves, where the Bragg BGs are

generated due to the geometric periodicity and the topological transition process can be

realized by changing the geometric parameter. The low-frequency topological interface states

in the soft dielectric PC plate can be actively tuned by applying electromechanical loads,

which is a smarter and more convenient method to adjust the topological properties of PC

waveguides compared with only applying a mechanical stimulus.

This paper is organized as follows. The basic formulations of nonlinear electroelasticity

theory and its linearized incremental theory are summarized in Section 2. In Section 3, we

analyze the nonlinear deformations of the designed soft dielectric PC plate with periodically

varying cross-sections. By employing the Spectral Element Method, in Section 4we derive

the dispersion relation and transmission coeﬃcient for the incremental bending waves in the

soft dielectric PC plate. The numerical results in Section 5show how the applied electric

voltage and axial force aﬀect the frequency of topological interface states in the soft dielectric

plate waveguide. Some conclusions are made in Section 6.

2. Preliminary Formulations

This section brieﬂy reviews the theoretical background of nonlinear electroelasticity and

the related linearized incremental theory. For more detailed descriptions, interested readers

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arXiv:2210.13424v1[cond-mat.soft]24Oct2022Voltage-controlledtopologicalinterfacestatesforbendingwavesinsoftdielectricphononiccrystalplatesYingjieChena,BinWub,∗,MichelDestradea,b,WeiqiuChena,caKeyLaboratoryofSoftMachinesandSmartDevicesofZhejiangProvinceandDepartmentofEngineeringMechanics,ZhejiangUniv...

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Voltage-controlled topological interface states for bending waves in soft dielectric phononic crystal plates

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