Kaijun Yi
email: kaijun.yi@bit.edu.cn

Manuel Collet

Broadening low-frequency bandgaps in locally resonant piezoelectric metamaterials by negative capacitance

Keywords: metamaterial, bandgap, piezoelectric material, negative capacitance, vibration and noise control

This paper combines negative capacitance (NC) with inductance (L) to enlarge lowfrequency bandgap width in locally resonant piezoelectric metamaterials. The studied metamaterials are obtained by directly bonding patches on the surfaces of host structures, then connecting patches to shunts. Shunts with NC and L in series and in parallel are both studied. Analytical expressions of the bandgap ranges are derived, which reveal that the bandgap size is increased not simply because the NC enhancing the material's electro-mechanical coupling factor, but in a more complicated way. Parametric studies are performed to analytically investigate the tuning properties of the LR bandgap by NC. Results demonstrate that by modifying NC value, the LR bandgap size can be significantly increased. Numerical simulations are done to verify the effects of the broadened bandgap on vibration transmission and reveal the limitations of the used analytical model. Practical implementation of the shunts are also discussed, recommendations on choosing the shunt configurations and NC values are given. This paper gives a theoretical guideline on designing piezoelectric metamaterials with bandgap effects at desired frequency ranges for practical applications like low-frequency vibration and noise reduction or isolation.

Introduction

In recent years, the emerging metamaterials based on local resonators provide new ways to deal with low-frequency vibration and noise issues [START_REF] Liu | Locally resonant sonic materials[END_REF][START_REF] Zhu | A chiral elastic metamaterial beam for broadband vibration suppression[END_REF][START_REF] Jung | Realisation of a locally resonant metamaterial on the automobile panel structure to reduce noise radiation[END_REF][START_REF] Droz | Improving sound transmission loss at ring frequency of a curved panel using tunable 3D-printed small-scale resonators[END_REF][START_REF] Timorian | Numerical investigations and experimental measurements on the structural dynamic behaviour of quasi-periodic meta-materials[END_REF][START_REF] Cai | Design and numerical validation of quasi-zero-stiffness metamaterials for very lowfrequency band gaps[END_REF]. Metamaterials present sub-wavelength bandgaps (at frequencies much lower than the first Bragg bandgap), within which wave propagation is prohibited. Therefore, using the bandgaps, one can totally remove the resonant modes or isolate vibration transmission. Metamaterials based on passive resonators are first studied. However, they have narrow bandgaps and are not adaptable to working condition or environment changes. These drawbacks limit they applications in practice. Recently, piezoelectric materials with external shunting circuits have been integrated into structures, results in so-called piezoelectric (smart) metamaterials [START_REF] Collet | Wave Motion Optimization in Periodically Distributed Shunted Piezocomposite Beam Structures[END_REF][START_REF] Collet | Adaptive metacomposites for vibroacoustic control applications[END_REF][START_REF] Ouisse | A piezo-shunted Kirigami auxetic lattice for adaptive elastic wave filtering[END_REF][START_REF] Bao | Vibration reduction for smart periodic structures via periodic piezoelectric arrays with nonlinear interleaved-switched electronic networks[END_REF][START_REF] Li | Creating the Coupled Band Gaps in Piezoelectric Composite Plates by Interconnected Electric Impedance[END_REF][START_REF] Wang | Tunable and active phononic crystals and metamaterials[END_REF]. In such metamaterials, the locations and widths of bandgaps can be easily tuned by just varying the circuits, no modification is needed on the mechanical structures. Different shunting circuits, such as resonant shunts and negative capacitance (NC) circuits, have been proposed to create and tune bandgaps in piezoelectric metamaterials. However, there are still challenges in creating wide and controllable bandgaps at low frequencies in these metamaterials, because of the necessity of using large inductance or lack of robustness. In this paper, we are going to demonstrate that combining resonant shunts with NC could be an effective way to realize wide and tunable bandgaps at low frequencies.

Piezoelectric shunt techniques are first proposed for vibration reduction of flexible structures. Hagood and von Flotow [START_REF] Hagood | Damping of structural vibrations with piezoelectric materials and passive electrical networks[END_REF] demonstrated that a piezoelectric patch connected with an inductance (L) behaves like a mechanical resonator. This is because the L and the intrinsic capacitance of the patch form a resonant shunt. Based on this feature, patches shunted with L are periodically bonded on a bar to create locally resonant (LR) bandgaps in Thorp et al's work [START_REF] Thorp | Attenuation and localization of wave propagation in rods with periodic shunted piezoelectric patches[END_REF]. It was shown that bandgaps are created in the bar near the resonance frequency of the circuit at sub-wavelength frequencies. The location of the bandgap can be tuned by varying the L value. Airoldi and Ruzzene [START_REF] Airoldi | Design of tunable acoustic metamaterials through periodic arrays of resonant shunted piezos[END_REF] studied the effective bending stiffness of a periodic beam with distributed patches shunted by L and resistance (R). They demonstrated that the beam's effective bending stiffness presents resonant feature near the resonance frequency of the circuit, just like the effective mass of a metamaterial with mechanical resonators does near the resonance frequency.

Therefore, it was concluded that the periodic beam with patches shunted by L is an electro-mechanical version of the locally resonant metamaterials. Following these works, the conception of locally resonant piezoelectric metamaterial is expanded to plates [START_REF] Casadei | Broadband vibration control through periodic arrays of resonant shunts: experimental investigation on plates[END_REF][START_REF] Casadei | Piezoelectric resonator arrays for tunable acoustic waveguides and metamaterials[END_REF][START_REF] Chen | Wave propagation and attenuation in plates with periodic arrays of shunted piezo-patches[END_REF][START_REF] Sugino | An analytical framework for locally resonant piezoelectric metamaterial plates[END_REF][START_REF] Xiao | A lightweight adaptive hybrid laminate metamaterial with higher design freedom for wave attenuation[END_REF]. However, due to the resonance nature, bandgaps in these metamaterials are narrow.

To enlarge LR bandgaps in piezoelectric metamaterials, Wang et al. [START_REF] Wang | Vibration attenuations induced by periodic arrays of piezoelectric patches connected by enhanced resonant shunting circuits[END_REF][START_REF] Wang | Large low-frequency vibration attenuation induced by arrays of piezoelectric patches shunted with amplifier-resonator feedback circuits[END_REF] proposed an amplifier-resonator feedback circuit to broad bandgaps. Numerical and experimental results in their works demonstrate that the bandgap is broadened, but is still not wide enough. The conception of varying shunts' parameters gradually in space to create "rainbow traps" is explored in [START_REF] Cardella | Manipulating waves by distilling frequencies: a tunable shunt-enabled rainbow trap[END_REF][START_REF] Yi | Programmable metamaterials with digital synthetic impedance circuits for vibration control[END_REF]. This kind of design shows good effects in increasing bandgap ranges, but the number of unit cells needed to realize a "rainbow trap" with acceptable wave isolation level could be large. Shunts with multiple resonance frequencies are also proposed. The multiple resonant effects could be realized using multi-branch shunts [START_REF] Airoldi | Wave propagation control in beams through periodic multibranch shunts[END_REF] or digital circuits with properly designed control law [START_REF] Wang | Multi-resonant piezoelectric shunting induced by digital controllers for subwavelength elastic wave attenuation in smart metamaterial[END_REF][START_REF] Sugino | Design and analysis of piezoelectric metamaterial beams with synthetic impedance shunt circuits[END_REF]. Limited by the complexity of multi-branch shunts and processing speed in digital circuits, the number of resonance frequency is within 3 at present. Except resonant shunts, NC circuits are also proposed to control bandgaps in piezoelectric metamaterials [START_REF] Collet | Semiactive optimization of 2d wave dispersion into shunted piezo-composite systems for controlling acoustic interaction[END_REF][START_REF] Tateo | Experimental characterization of a bi-dimensional array of negative capacitance piezo-patches for vibroacoustic control[END_REF][START_REF] Bao | Elastic wave manipulation in piezoelectric beam metastructure using electronic negative capacitance dual-adjacent/staggered connections[END_REF]. NC can cancel a part of the intrinsic capacitance of a patch, therefore to enhance the electro-mechanical coupling effect [START_REF] Collet | Diffusion Draft : and Plates Energy Diffusion By Mean of Distributed Shunted Patches[END_REF][START_REF] Berardengo | Improved resistive shunt by means of negative capacitance: new circuit, performances and multi-mode control[END_REF]. Recently, it is also shown that NC can increase or decrease the effective static stiffness of the structure covered by the patches [START_REF] Yi | Flexural waves focusing through shunted piezoelectric patches[END_REF]. This property has been explored to enlarge bandgaps in piezoelectric metamaterials. In these applications, patches shunted with NC can be directly bonded on the surfaces of structures [START_REF] Fan | A wave-based design of semi-active piezoelectric composites for broadband vibration control[END_REF], the NCs are used to change the local stiffness therefore to tune the bandgaps induced by Bragg scattering effects. In such a manner, the integrity of the structures is preserved. However, in order to obtain wide and low-frequency bandgaps, the NC needs to significantly reduce the stiffness of the host structures, i.e., the NC need to work at the vicinity of its unstable zone, which is impractical. In other studies [START_REF] Yy Chen | Band gap control in an active elastic metamaterial with negative capacitance piezoelectric shunting[END_REF][START_REF] Zhu | Experimental study of an adaptive elastic metamaterial controlled by electric circuits[END_REF][START_REF] Yy Chen | An adaptive metamaterial beam with hybrid shunting circuits for extremely broadband control of flexural waves[END_REF][START_REF] Zhou | Actively controllable flexural wave band gaps in beam-type acoustic metamaterials with shunted piezoelectric patches[END_REF], patches shunted with NC are used to control the stiffness of local resonators in passive metamaterials, therefore to broaden the bandgaps caused by passive resonators. In this way, it is more easy to obtain wide bandgaps at low frequencies, however extra local resonators need to be added into the system, increasing the total mass and volume or break integrity of the host structure, which are strictly limited in many practical applications.

In this paper, L and NC are combined to realize low and wide bandgaps in piezoelectric metamaterials. It should be noted that combining L and NC are previously studied for vibration reduction of structural modes [START_REF] Marneffe | Active and passive vibration isolation and damping via shunted transducers[END_REF][START_REF] Berardengo | Piezoelectric resonant shunt enhancement by negative capacitances: Optimisation, performance and resonance cancellation[END_REF], but few works are done to increase bandgap size. The studied metamaterials are obtained by directly bonding patches on the surfaces of a host structure, therefore to avoid adding too much extra mass and keep the integrity of the original structure. A beam is used as the host structure in this study.

Note that the proposed conception in this paper can be naturally extended to other types of structures, like plates and shells. Combining NC with L in the shunts remains the resonant nature of the shunts. Therefore, LR bandgaps at low frequencies can still be obtained in the proposed piezoelectric metamaterials. Effects of NC on the LR bandgap are analytically and numerically studied in the following sections.

Theoretical analysis

Shunting configurations

NC could be connected with L in series or in parallel, as shown in Fig. 1. In these figures, C 0 indicates NC, the piezoelectric patch made of PZT is represented by a current source I eq (t) in parallel with a capacitance C s p . All these two shunting configurations and the case with only L are studied in this paper. The impedance (Z su ) of these different shunts are summarized in Table 1. In the table, j is the unit of complex number and ω is the angular frequency. We consider the flexural waves in the metamaterial beam. The beam is slender, Euler-Bernoulli beam theory could be used. Under harmonic excitation, the equations of the transverse motion of the beam w(x) can be written as:

Z su = jωL -ω 2 LC 0 +1 jωC 0 jωL (1-ω 2 LC 0 )

Governing equations

∂ 2 ∂x 2 [D(x) ∂ 2 w(x) ∂x 2 ] -ω 2 m(x)w(x) = 0 (1)
where, D(x) is the bending stiffness of the beam, m(x) is the mass per unit area. The unit cell has a piece-wise configuration, the length of the period is l b , as shown in Fig.

2(b).

The expressions for the piece-wise bending stiffness in a unit cell are:

D(x) =      D 1 = D b + E p (ω)b[(h b + 2h p ) 3 -h 3 b ] 12 , 0 ≤ x < l p D 2 = D b , l p ≤ x < l b in which, D b = E b bh 3 b /12
is the bending stiffness of the host beam, E b is the Young's modulus of the host beam, E p is the effective Young's modulus of the shunted piezoelectric patch [START_REF] Hagood | Damping of structural vibrations with piezoelectric materials and passive electrical networks[END_REF], expression of it is:

E p (ω) = E oc p (1 - k 2 31 1 + jωC s p Z su (ω) ), (2)
in which, k 31 = d 31 E sc p /ε σ 3 is the extensional coupling factor of the patch [41],

E oc p = E sc p /(1 -k 2 31 ) is the patch's Youngs' modulus under open-circuited (OC) condition, C s p = C T p (1 -k 2 31
) is the intrinsic capacitance of the patch under constant strain and C T p = A p ε σ 3 /h p , with A p the area of the electrode.

The linear mass is:

m(x) =      m 1 = ρ b bh b + 2ρ p bh p , 0 ≤ x < l p m 2 = ρ b bh b , l p ≤ x < l b
in which, ρ b and ρ p are the densities of the host beam and the patch, respectively.

Transfer matrix method

The transfer matrix (TM) method is used to calculate the dispersion curves. The sate vector y(x) is used in the TM method. Therefore, Eq. ( 1) is rewritten into the form:

∂y(x) ∂x = C(x, ω)y(x) (3)
where, C(x, ω) is a periodic and piecewise function:

C(x, ω) =      C 1 (x, ω), 0 ≤ x < l p C 2 (x, ω), l p ≤ x < l b
For the transverse motion, the state vector is y

(x) = [w(x) θ(x) M (x) Q(x)] T , in which, the slope is θ(x) = ∂w(x)/∂x, the bending moment is M (x) = -D(x)∂ 2 w(x)/∂x 2
and the shear force is Q(x) = -D(x)∂ 3 w(x)/∂x 3 , and:

C i (x, ω) =          0 1 0 0 0 0 -1 D i (x) 0 0 0 0 1 -ω 2 m i (x) 0 0 0          , i = 1, 2 (4)
Equation ( 3) describes a 1D medium with periodic coefficients. Using the Floquet theorem, the state vectors of the two ends of a unit cell are related through

y(x + l b ) = λy(x) (5)
in which, λ = e jkl b is the Floquet multiplier, with k the wavenumber.

The state vectors of the two ends can also be related through a TM T:

y(x + l b ) = Ty(x) (6)
Eqs. ( 5) and ( 6) together form an eigenvalue problem for the Floquet multiplier and the state vector:

(T -λI)y(x) = 0 [START_REF] Collet | Wave Motion Optimization in Periodically Distributed Shunted Piezocomposite Beam Structures[END_REF] in which, I is a diagonal matrix with all terms equal to one.

For the TM between the two ends of the studied unite cell (Fig. 2(b)), considering the continuities of displacement and force at the interface between the two segments, T is given by:

T = T 2 T 1 (8)
where, T 1 relates the two state vectors at x = 0 and x = l p , while T 2 relates the two state vectors at x = l p and x = l b . T 1 and T 2 can be expressed as [START_REF] Airoldi | Design of tunable acoustic metamaterials through periodic arrays of resonant shunted piezos[END_REF]:

T 1 = e l b C 1 , T 2 = e (l b -lp)C 2 (9)
After obtaining the TM, the Floquet multiplier can be obtained by solving the eigenvalue problem in Eq. ( 7) when the frequency is given, the wavenumber is then deduced according to λ = e jkl b .

Analytical estimation of LR bandgap range

The bandgaps in passive locally resonant metamaterials are caused by negative parameters like negative density or bulk modulus [START_REF] Zhou | Analytic model of elastic metamaterials with local resonances[END_REF]. In locally resonant piezoelectric metamaterials, it has been mentioned that bandgaps for flexural waves are associated with negative bending stiffness [START_REF] Zhang | Sound transmission loss of metamaterial thin plates with periodic subwavelength arrays of shunted piezoelectric patches[END_REF]. However, no analytical expressions have yet been given to directly estimate the ranges of bandgaps. Such expressions could be very useful to analyze the influences of NC and other parameters on the bandgap sizes and locations.

Using the effective medium theory, the effective bending stiffness of the whole unit cell can be approximately expressed as [START_REF] Airoldi | Design of tunable acoustic metamaterials through periodic arrays of resonant shunted piezos[END_REF]:

D ef f (ω) = D A D b (1 -χ)D A + χD b ( 10
)
in which, χ = l p /l b , it is the covering ratio of the patch, D A is the bending stiffness of the part covered by patches, the expression of it is:

D A (ω) = D b + D sc p 1 -k 2 31 - k 2 31 1 -k 2 31 D sc p 1 1 + jωZ su C s p (11) in which, D sc p = E sc p b[(h b + 2h p ) 3 -h 3 b ]/12
is the bending moment of the patches under SC condition.

When the configuration of the shunt is given, namely, the expression of Z su is determined, according to Eqs. ( 10) and [START_REF] Li | Creating the Coupled Band Gaps in Piezoelectric Composite Plates by Interconnected Electric Impedance[END_REF], one can easily deduce the expressions of the effective bending stiffness and the range where the stiffness is negative. To make the analysis more general, the following non-dimensional parameters are used hereinafter along with the previously defined χ = l p /l b :

γ = D b /D sc p , α = C 0 /C s p
α is a real number, it could be regarded as the non-dimensional NC value.

For the case with only L in the shunt, the frequency range where the effective bending stiffness becomes negative is

ω LC 1 - k 2 31 γ(1 -k 2 31 ) + 1 < ω < ω LC (γ + 1 -χ)(1 -k 2 31 ) γ(1 -k 2 31 ) + 1 -χ , (12)
in which, ω LC = 1/ LC s p is the resonance frequency of the circuit. Eq. ( 12) also gives the bandgap range (which will be verified later in this section), it can be seen that the lower boundary of the bandgap depends on the resonance frequency of the circuit (ω LC ), the ratio of the bending stiffness of the host beam to that of the short-circuited patch (γ) and the extensional coupling factor (k 31 ). The upper boundary not only depends on the three factors listed above, but also depends on the covering ratio of the patch (χ).

If χ = 1, which means the patches cover all the surfaces of the host beam, the upper boundary is exactly equal to ω LC , this is the case studied in [START_REF] Sugino | An investigation of electroelastic bandgap formation in locally resonant piezoelectric metastructures[END_REF]. If χ < 1, namely, the patches are shorter than the host beam in a unit cell, the upper boundary is lower than

ω LC .
Due to the active nature of NCs [START_REF] Tateo | Experimental characterization of a bi-dimensional array of negative capacitance piezo-patches for vibroacoustic control[END_REF], they may cause stability issues in some circumstances, which must be avoided in applications. Therefore, before presenting the expressions of bandgap ranges for the cases with NCs in the shunts, the stable zone for the NC values are shown first. In this paper, the L and R values are not limited to be positive, since one can obtain negative L and R using synthetic or digital circuits. Therefore, the studied piezoelectric metamaterial beam is stable as long as a positive bending stiffness of it is guaranteed when ω = 0. According to this, we can deduce the stable zone for NC. It is found that the stable zone of NC is the same no matter it is in series or in parallel with L. The stable zone is

α < - γ(1 -k 2 31 ) + 1 (γ + 1)(1 -k 2 31 )
, or α > -1.

The stable zone includes two sets, as shown graphically in Fig. 3. The limit of the left set depends on γ, k 31 and C s p (recall that α = C 0 /C s p ), the limit of the right set only depends on C s p .

When the shunts is composed of L in series with NC, the equivalent capacitance of the circuit becomes C eq = αC s p /(1 + α) (see Fig. 1 (a)). Consequently, the resonance frequency of the circuit is ω LC = 1/ LC eq . If L is fixed, resonance frequency of the circuit will change with the NC value. Usually, our interest is to obtain bandgaps at targeted frequency ranges. Accordingly, in our studies, we assume that resonance frequency of the circuit is fixed, which means L is varying according to the applied NC value. In this case, the bandgap range becomes

               ω LC 1 - 1 1 + α k 2 31 γ(1 -k 2 31 ) + 1 < ω < ω LC γ(1 -k 2 31 ) + (1 -χ)(1 - k 2 31 1+α ) γ(1 -k 2 31 ) + 1 -χ , α > -1 ω LC γ(1 -k 2 31 ) + (1 -χ)(1 - k 2 31 1+α ) γ(1 -k 2 31 ) + 1 -χ < ω < ω LC 1 - 1 1 + α k 2 31 γ(1 -k 2 31 ) + 1 , α < - γ(1 -k 2 31 ) + 1 (γ + 1)(1 -k 2 31 ) (14)
From the above expressions, it can be seem that NC influences simultaneously the lower and upper boundaries of the bandgap when χ < 1. For the critical situation with χ = 1, only the lower or upper boundary (depends on the α value) will be modified by NC.

Comparing Eqs. ( 12) and ( 14), it is also observed that the effects of NC on the bandgap boundaries are not simply due to the modification of the material eletro-mechanical coupling factor (which becomes k 31 α/(α + 1) when the NC is connected in series).

When L is in parallel with NC, the equivalent capacitance of the circuit becomes

C eq = (1 + α p )C s p (see Fig. 1 (b)
). In this case, the bandgap range is

               ω LC (γ + 1)(1 -k 2 31 ) γ(1 -k 2 31 ) + 1 - k 2 31 1+α < ω < ω LC (γ + 1 -χ)(1 -k 2 31 ) γ(1 -k 2 31 ) + (1 -χ)(1 - k 2 31 1+α ) , α > -1 ω LC (γ + 1 -χ)(1 -k 2 31 ) γ(1 -k 2 31 ) + (1 -χ)(1 - k 2 31 1+α ) < ω < ω LC (γ + 1)(1 -k 2 31 ) γ(1 -k 2 31 ) + 1 - k 2 31 1+α , α < - γ(1 -k 2 31 ) + 1 (γ + 1)(1 -k 2 31 ) (15)
It is also observed that NC modifies simultaneously the upper and lower boundaries if χ < 1, and only one of the boundaries if χ = 1.

To verify the accuracy of Eqs. ( 12), ( 14) and ( 15), the negative bending stiffness ranges predicted by them are compared with the bandgap ranges obtained using dispersion curves, the results are shown in Figures 4. In these simulations, the host beam is made of aluminum and the patches are made of PZT-5H, the geometric and material parameters are described in Table 2. For the NC in series with L, α = -1.5 is used as an example, and for the NC in parallel with L, α = -0.7 is used. The shadowed area in each figure indicates the range where the effective bending stiffness is negative, it is obtained using one of the expressions in Eqs. ( 12), ( 14) and [START_REF] Airoldi | Design of tunable acoustic metamaterials through periodic arrays of resonant shunted piezos[END_REF]. It can be observed that the bandgaps completely overlap the negative bending stiffness ranges, which verifies the accuracy of using Eqs. ( 12), ( 14) and [START_REF] Airoldi | Design of tunable acoustic metamaterials through periodic arrays of resonant shunted piezos[END_REF] to predict the bandgaps. It is also observed that the NC enlarges the negative bending stiffness ranges, therefore leading to wider bandgaps. More details on how NC and other parameters of the piezoelectric metamaterial influence the bandgap will be discussed later.

Tuning characteristics of LR bandgap by NC

In this section, first, the influences of NC value on bandgap ranges are studied. Then, effects of NC on the wave attenuation ability within bandgaps are analyzed. The bandgap ranges are predicted using Eqs. ( 14) and ( 15), the wave attenuation properties within bandgaps are obtained using the TM method.

In the simulations to study the tuning characteristics of bandgap ranges, γ = 0.23 12), ( 14) and [START_REF] Airoldi | Design of tunable acoustic metamaterials through periodic arrays of resonant shunted piezos[END_REF]. and k 31 = -0.39 are used, they are corresponding to the dimensional parameters listed in Table 2. For the parameter χ, three different values of it are used, they are χ = 0.5, χ = 0.7 and χ = 1, the aim of using 3 different χ values is to reveal the influences of it on the tunable features of bandgap. Recall that, the stable zone of NC includes two sets, namely, the left set α < -[γ(1 -

k 2 31 ) + 1]/(γ + 1)(1 -k 2 31
) and the right set α > -1, as shown in Fig. 3. Accordingly, influences of NC will be analyzed separately in these two sets.

The cases with L and NC in series are first analyzed. seen that the required inductance value decreases as the bandgap enlarges. This makes the circuit become more easy to be realized in practice since large analog inductance is difficult to obtain. Note that if the shunts are realized using digital techniques, large inductance value is no longer a problem [START_REF] Yi | Programmable metamaterials with digital synthetic impedance circuits for vibration control[END_REF].

Figure 6 shows the variation of bandgap boundaries and size along with α in the right set of its stable zone. It can be observed that increasing the |α| results in increase of the upper and lower boundaries if χ < 1. Basically, the upper boundary gains fast than the lower one, leading to enlargement of the bandgap size. The lower boundary becomes less sensitive to α as χ gets larger. Consequently, the bandgap size can be improved more for larger χ. For the special situation with χ = 1, the upper boundary is left unchanged by NC, the largest bandgap can be achieved. The equivalent capacitance is negative in these circumstances, therefore, negative inductance values are also required according to

L = 1/ω 2 LC C eq .
It is also shown that inductance with a larger absolute value is needed to have wider bandgaps.

For the cases with L and NC in parallel, Figs 7 and8 show the results corresponding to α inside the left set and the right set of its stable zone, respectively. Comparing to the case with L and NC in series, it is observed that in this case, the bandgap boundaries are oppositely increased within the left set and decreased within the right set. The trends (see Eq. ( 13)). Therefore, in Fig. 9 (a) and (c), α

* (γ + 1)(1 -k 2 31 )/[γ(1 -k 2 31 ) + 1] is
used as the horizontal axis to eliminate the influences of the shifting of the stable zone's boundary. From Fig. 9 (a), it can be seen that the enlargement trend of the gap size caused by the varying of α is independent on γ. Considering γ itself, from Fig. 9 (b) it is observed that the gap size varies non-linearly as γ increases, the gap size enlarges in the beginning then decreases after a maximum. There is an optimal value for γ in order to have the largest bandgap size, this value is independent on the NC value. The influences of k 31 are studied in Fig. that NC broadens the bandgap size, but degenerates the wave attenuation ability. The degeneration is due to the enhancement of damping in the circuit by NC, since damping weakens the resonant behavior of the circuit. Nonetheless, the degeneration is mild, the wave attenuation effects are strong within these bandgaps, as will be seen from the forced responses of the metamaterial beam in the following section.

In summary, studies in this section reveal how NC affects the bandgap when the piezoelectric metamaterial beam has different geometrical and material parameters. These results can be used as guidelines to design bandgaps at desired frequency ranges.

Numerical simulations

The main tuning characteristics of bandgap by NC are verified using finite element 2. NC is in series with L in these simulations.

In practice, resistance of the circuit is inevitable. Therefore, a small resistance R = 50 Ω is also included in the shunt in series with the other parts.

First of all, accuracy of the FE models is checked and improved to be acceptable. The patches are kept OC to take into account the electro-mechanical coupling effects in these simulations. Natural frequencies of the first 10 non-rigid modes are used to check the convergence of the simulations. The meshes of FE models are refined until the simulated natural frequencies vary no more than 1% when the degrees of freedom are doubled.

Secondly, the intrinsic capacitance value of the patch in the FE models is estimated.

This capacitance value is necessary in order to find the required inductance value when ω LC is designated. In the analytical studies, C s p is involved in the process to determine inductance values (for example, through L = 1/ω 2 LC C s p ). However, this analytical C s p can not be directly used in the FE simulations to deign the L values, because it overestimates the capacitance of the patch in the FE model. To demonstrate this, the same L value is used in both the analytical and FE simulations, the analytically predicted bandgap range and the gap observed in the finite metamaterial beam are compared in Fig. 11 for the case with LR shunts. The analytical bandgap range (calculated using Eq. ( 12)) is indicated by the shadow in Fig. 11. The transmission function curve obtained in the FE simulation using the same parameters as those in the analytical calculation is labeled with "original", the frequency range with relatively low response level is the bandgap. It can be seen that the gap observed in the FE result is at higher frequency range than the analytical one, which verifies that the analytical C s p overestimates the capacitance value of the patch in the FE model, leading to a smaller L than the required one. Therefore, the capacitance in the FE simulation is corrected to be C s * p = 0.84C s p . Using this corrected capacitance, the corrected L value for the FE simulation is obtained. The curve with label "corrected" in Fig. 11 is the FE result corresponding to the corrected L value. One can see that the gap in the FE simulation now has good coincidence with the analytical one.

After the above procedures, it is able to verify the effects of NC on the bandgaps.

4 shunts with different NC and L parameters are studied for each of the two unit cell lengths, the parameters of shunts are listed in Table 3 (shunt 1 to 4). The case with LR shunts is used as reference, the corresponding shunting parameters are also given in Table 3. Recall that, C s * p in the table is the corrected capacitance of the patch in the FE model.

f LC is the resonance frequency of the circuit in Hz. Figure 12 shows the transmission function curves for different studied cases. The predicted bandgap locations using Eqs. In conclusion of this section, the FE results clearly verify the tuning characteristics of the bandgap observed in the analytical studies, therefore, the analytical model could be used to predict the tuning trend of the bandgap, however, to precisely calculate the size and location of a bandgap, FE simulations are more suitable.

Discussions

To implement shunts with L and NC in practice, one can use analog circuits to synthesize negative capacitance and large inductance. In this way, it is recommended to use L in series with NC and choose the NC value from the left set of stable zone. This choice could avoid using simultaneously negative capacitance and negative inductance in the shunts, it also benefits from the smaller inductance needed for wider bandgaps (see Figs. 5). The required inductance and negative capacitance can be realized using the circuits in Fig. 13 (a) and (b), respectively [START_REF] Wang | Large low-frequency vibration attenuation induced by arrays of piezoelectric patches shunted with amplifier-resonator feedback circuits[END_REF][START_REF] Tateo | Experimental characterization of a bi-dimensional array of negative capacitance piezo-patches for vibroacoustic control[END_REF]. The synthesized inductance and negative capacitance are respectively

L = R 1 R 3 R 4 R 2 C, C 0 = - R 3 R 2 C.
A more advanced way to realize shunts with L and NC is using programmable digital circuits [START_REF] Yi | Programmable metamaterials with digital synthetic impedance circuits for vibration control[END_REF]. A digital circuit usually is composed of voltage measuring block, digital signal processing (DSP) block and voltage controlled current source block. Digital circuit measures the voltage on the electrodes of PZT patches and feeds back current to the same patches, therefore desired impedance is established between the terminals of patches. 12) and ( 14). The used shunting parameters in different simulations are listed in Table 3. To simulate behaviors of shunts with L and NC, the inverse expression of the shunt's impedance is programmed in the DSP block in Laplace domain as a transfer function.

Since the impedance is realized digitally, no limitation is constrained on the inductance value and the configurations of L with NC, namely, the L and NC could either in series or in parallel, and it is free to choose NC value within the whole stable zone, as long as the implemented transfer function in the DSP block is causal and the whole system is stable.

The above discussed synthetic and digital circuits both need power supply in practice.

One possible solution is introducing energy harvesting circuits into the system, the harvested energy maybe used to power low-powered electron devices in the circuits. Future work will be done to verify this idea.

Conclusions

In this paper, piezoelectric metamaterials shunted with combined L and NC are analytically and numerically studied. The metamaterial is obtained by periodically distributing piezoelectric patches on the surfaces of a host structure. A slender beam is used as host structure in this paper as an representative. Shunts with NC and L in series and in parallel are both studied. Major conclusions of this paper are:

(1) NC enlarges the frequency regions of negative stiffness caused by resonant shunts, therefore leading to wider bandgaps;

(2) NC enlarges the LR bandgap size and also shifts the bandgap to lower or higher frequency range when the host beam is not fully covered by the patches. As the covering ratio of the patches increases, the gap size becomes even wider but the shifting effect of the bandgap location weakens;

(3) Except the covering ratio of patches, other geometric and material parameters of the metamaterial do not change the enlargement trend of the bandgap size caused by NC.

(4) The analytical model based on the Euler-Bernoulli beam theory and effective medium theory is useful to predict the tuning characteristics of bandgaps by NC. However, it can not precisely predict the sizes and locations of bandgaps, for such demand, FE models are more suitable.

In the next step, experiments will be done to study the influences of real NC and L circuits on the bandgaps and to reveal the limitations of such piezoelectric metamaterials under practical constrains.

Figure 1 :

 1 Figure 1: Different shunting configurations: (a) L in series with NC; (b) L in parallel with NC. C 0 indicates NC. The piezoelectric patch made of PZT is represented by a current source I eq (t) in parallel with a capacitance C s p .

Figure 2 (

 2 Figure 2 (a) shows the studied metamaterial beam with periodically distributed shunted piezoelectric patches, a unit cell of it is illustrated in Fig. 2 (b).

Figure 2 :

 2 Figure 2: (a) The studied piezoelectric metamaterial beam, (b) a unit cell of the metamaterial.

Figure 3 :

 3 Figure 3: Graphical illustration of the stable zone in Eq. (13).

Figure 4 :

 4 Figure 4: Dispersion curves of the flexural waves in the piezoelectric metamaterial beam for three different shunts: (a) shunt with only L, (b) shunt with L and NC in series, α = -1.5 and (c) shunt with L and NC in parallel, α = -0.7. Shadowed area in each figure corresponding to the frequency range where the negative effective bending stiffness is negative, predicted using one of the expressions in Eqs. (12), (14) and[START_REF] Airoldi | Design of tunable acoustic metamaterials through periodic arrays of resonant shunted piezos[END_REF].

Figure 5 (Figure 5 :

 55 Figs. 5 (a) and (b), it can be seen that for the situations with χ < 1, NC decreases simultaneously the upper and lower boundaries of the LR bandgap. The lower boundary drops faster than the upper one as |α| decreases, therefore, the bandgap size enlarges. As χ gets larger, the upper boundary becomes less sensitive to α, and the bandgap is broadened more in these situations. The bandgap can be enlarged the most when χ = 1 since the upper boundary becomes immune to α. From Figs. 5(a), (b) and (c), it is also

Figure 6 :

 6 Figure 6: Variation of the bandgap boundaries (a-c) and size (d) and size along with α in the right set of its stable zone for the covering ratios (a) χ = 0.5, (b) χ = 0.7 and (c) χ = 1 when the NC is in series with the L. The shadowed areas indicate the bandgap regions. The required L values for different α are also illustrated in Figs. 6(a), (b) and (c), L 0 is the required inductance when no NC is connected with L.

Figure 7 :Figure 8 :

 78 Figure 7: Variation of the bandgap boundaries (a-c) and size (d) and size along with α in the left set of its stable zone for the covering ratios (a) χ = 0.5, (b) χ = 0.7 and (c) χ = 1 when the NC is in parallel with the L. The shadowed areas indicate the bandgap regions. The required L values for different α are also illustrated in Figs. 7(a), (b) and (c), L 0 is the required inductance when no NC is connected with L.

Figure 9 :

 9 Figure 9: Influences of (a-b) γ and (c-d) k 31 on the tuning characteristics of bandgap size by NC: (a) χ = 0.8, k 31 = -0.39, (b) χ = 0.8, k 31 = -0.39, (c) χ = 0.8, γ = 0.23 and (d) χ = 0.8, γ = 0.23.

Figure 10 :

 10 Figure 10: Influences of NC on the wave attenuation ability within the bandgap.

(

  FE) simulations. The simulations are done in commercial software COMSOL. The FE beam model has 10 unit cells, a harmonic boundary force is applied on the left end of the beam, the right end of the beam is free. Transmission function between the displacement of the right end and the excitation force will be studied during the discussions. Two different unit cell lengths are used, they are l b = 52 mm and l b = 100 mm, the aim is to verify the influences of the covering ratio on the effects of NC. The other geometric and material parameters are the same as in Table

Figure 11 :

 11 Figure 11: Transmissibility at the right end of the metamaterial beam when a boundary transverse force is applied at the left end for the case with LR shunts. Shadowed area indicates the analytically predicted gap region using Eq. (12), results corresponding to the original and corrected L are both shown in the figure.

( 12 )

 12 and (14) are also indicated in these figures through the horizontal colorful slats. For the tuning trend of the bandgap size and location caused by NC, it can be seen from the transmission curves in Fig.12(a) that, when the covering ratio of the patch is close to one, NC decreases the lower boundaries of the gap, and has little influence on the upper ones, therefore, leading to a wider bandgap, just as illustrated in Figs.5(c); when the covering ratio is far from one, the results in Fig.12(b) clearly show that NC decreases simultaneously the upper and lower boundaries of the gap, the lower boundary drops faster, therefore, shifting as well as enlarging of the bandgap are both observed, which is coincident with the results in Figs.5(a). From Fig.12, it can also be seen that, with regard to the values of the bandgap sizes and locations, the FE and analytical results have obvious differences when there are NCs in the shunts, which indicates that the analytical model based on Euler-Bernoulli beam theory and sub-wavelength homogenization theory can not precisely capture the bandgap' features.

Figure 12 :

 12 Figure 12: Transmission function curves for different patch covering ratios and shunts with different parameters: (a) l b = 52 mm, l p = 50 mm and (b) l b = 100 mm, l p = 50 mm. The horizontal colorful slats in each figure indicate the predicted bandgap locations using Eqs. (12) and (14). The used shunting parameters in different simulations are listed in Table3.

Figure 13 :

 13 Figure 13: (a) Antoniou's circuit to synthesize inductance and (b) typical circuit to synthesize negative capacitance.

Table 1 :

 1 Impedance of different shunts

	only L	L and NC in series	L and NC in parallel

Table 2 :

 2 Geometric and material parameters of the unit cell

		Beam	Piezoelectric patch
	Length	l b =60 mm	l p =50 mm
	Width	b=50 mm	b=50 mm
	Thickness	h b =3 mm	h p =0.5 mm
	Young's modulus	E b =70 Gpa	E sc p =60.6 Gpa
	Density	ρ b =2700 kg/m 3 ρ p =7500 kg/m 3
	Coupling constant	\	d 31 =-2.74e-10 C/N
	Relative permittivity under constant stress	\	ε σ 3 =3400

Table 3 :

 3 Resonance frequency of the circuit and the corresponding electrical parameters used in the FE simulations

		L (H) R (Ω) C 0 /C s * p	f LC (Hz)
	reference 12.00 50	\	100
	shunt 1	4.00	50	-1.5	100
	shunt 2	2.40	50	-1.25	100
	shunt 3	2.00	50	-1.2	100
	shunt 4	1.57	50	-1.15	100


Amir Darabi

Manuel Collet

Michael J Leamy

Experimental Realization of a Reconfigurable Electroacoustic Topological Insulator

A substantial challenge in guiding elastic waves is the presence of reflection and scattering at sharp edges, defects, and disorder. Recently, mechanical topological insulators have sought to overcome this challenge by supporting back-scattering resistant wave transmission. In this Letter, we propose and experimentally demonstrate the first reconfigurable electroacoustic topological insulator exhibiting an analog to the quantum valley Hall effect (QVHE). Using programmable switches, this phononic structure allows for rapid reconfiguration of domain walls and thus the ability to control back-scattering resistant wave propagation along dynamic interfaces for phonons lying in static and finite-frequency regimes. Accordingly, a graphene-like Polyactic Acid (PLA) layer serves as the host medium, equipped with periodically arranged and bonded piezoelectric patches, resulting in two Dirac cones at the K-points. The PZT patches are then connected to negative capacitance external circuits to break inversion symmetry and create nontrivial topologically-protected bandgaps. As such, topologically protected interface waves are demonstrated numerically and validated experimentally for different predefined trajectories over a broad frequency range.

The need for lossless information carriers has generated increased interest in topologically-protected structures in the past few years. The study of topological insulators originated with electronic states in condensed matter physics [1][2][3][4], and were later studied in electromagnetic materials [5][6][7][8]. Recently, topological insulators have been investigated for phononic systems (elastic waves in solids) to control statical floppy modes [9][10][11][12][13][14][15] and dynamical edge waves [16,17]. These mechanical structures are induced by analogs to the quantum Hall [18], quantum spin Hall [19], or quantum valley Hall effects [20], and inherently support topological edge states resistant to back-scattering at sharp interfaces, discrete defects, or continuous disorder [21,22].

Mechanical TIs mimicking the quantum Hall effect (QHE) require active means to break time-reversal symmetry [18,23,24] and have been achieved using a weak magnetic field [25]; gyroscopes or rotating frames [26][27][28]; or by varying material properties in time and space [29][30][31]. In contrast, mechanical TIs mimicking the quantum spin Hall effect (QSHE) [19,21] may be achieved passively through breaking of spatial inversion symmetry. These topological insulators have been investigated numerically, and tested experimentally for elastic waves in thin plates [22,32,33], and in discrete systems composed of masses and linear springs [15,16,19,29]. Compared to active TIs, they i) do not require external energy input, ii) feature both forward-and backward-propagating edge modes, and iii) typically retain time-reversal symmetry [START_REF] Landau | Electrodynamics of continuous media[END_REF].

A third approach for achieving a mechanical TI results from mimicking the quantum valley Hall effect (QVHE), which breaks inversion symmetry in a simpler system. The quantum valley Hall effect was first predicted theoretically in graphene [START_REF] Rycerz | [END_REF][36][37], and later observed in solid-state devices [38][39][40], photonic crystals [41][42][43], and graphene bi-layers [44][45][46]. Unlike QSHE, only one set of degenerate Dirac cones is required, which reduces the geometrical complexity of designing TIs for elastic media. Recently, QVHE has been extended to phononic systems to exhibit valley edge states [20] by utilizing: i) anisotropic scatterers in sonic crystals [47], and ii) arrays of resonators or different inclusion types in thin plates [24,48,49]. Mechanical topological insulators proposed to date lack an easy means of reconfigurabilty, which is essential for enabling important TI-based applications. One potential means to overcome this issue, as explored in this Letter, is to employ shunted piezoelectric disks in which the system's mechanical impedance can be altered dramatically using negative capacitance circuits [50][51][52]. Dynamic reconfigurability of such structures can then be obtained through simple ON/OFF switching of these external circuits. By doing so, we propose and experimentally verify the first electroacoustic topological insulator which exhibits topologically-protected edge states. This reconfigurable structure is composed of an elastic hexagonal lattice (made from polylactic acid, PLA, plastic) whose unit cell contains two shunted piezoelectric (PZT) disks, each connected to a negative capacitance circuit by an ON/OFF switch. Closing one or the other circuit results in the breaking of mirror symmetry and yields mechanical behavior analogous to the QVHE.

I. RESULTS

A. Graphene-like unit cell

Figure 1(a) displays the schematic of the unit cell composed of an h h = 0.5 mm thick PLA layer, with two bonded piezoelectric disks, used to mimick the band structure of graphene. This unit cell is then periodically repeated in the lattice directions to form the entire material system (see Supplementary Note 1 for more details). Each of the PZT disks employed has a diameter of 7 mm, and a thickness of h p = 0.5 mm, and is connected to a negative capacitance circuit through a digital ON/OFF switch. Red dashed lines in Fig. 1(b) plot the band structure of the unit cell when both of the switches are off (i.e., both of PZT disks experience open circuit conditions), documenting two Dirac cones at the edge of the unit cell where two distinct Lamb modes [START_REF] Lamb | Proc. R. Soc. Lond. A[END_REF] meet (approximately f 1 = 45 kHz, f 2 = 73 kHz). Note that these curves are computed only for the frequency range of interest; hence, other Dirac cones are not visible. Supplementary Note 1 provides full discussion on the computation details for the entire frequency spectrum (0 -90 kHz).

B. Breaking inversion symmetry

The next step in configuring the TI requires breaking mirror symmetry and separating the folded Dirac cones. As such, one of the switches in Fig. 1(b) is set to ON, thus connecting the PZT (green disk) to the external negative capacitance circuit (Fig. 1(a)), which provides a significant change in the elastic modulus of the disk [51,52,[START_REF] Hagood | [END_REF]; the other PZT (blue disk) remains disconnected. This then enables breaking C 6 symmetry and creates a topological bandgap at the location of the Dirac cone. The circuit includes two resistors (R 1 , R 2 ), one capacitor (C 0 ), and an operational amplifier, which yields an effective negative capacitance of C = -(R 2 C 0 )/R 1 . Note that, the resistor R 0 prevents saturation of the paralleled capacitance C 0 , which can cause instability of the PZT disk. Blue solid lines in Fig. 1(b) plot the band structure of the described unit cell when the green PZT disk is connected in series with a negative capacitance of C = -1.7 nF , reporting two complete frequency bandgaps at the location of the Dirac modes. The stated negative capacitance is achieved by placing R 1 = 10,R 2 = 17 Ω, and C 0 = 1nF , resulting in an optimal bandgap width. The band structure for other values of negative capacitance are provided in Supplementary Note 1.

For each of the bands bounding the topological gaps in Fig. 1(e), the valley Chern numbers are computed numerically to be C v = ±1/2 (as labeled on the graph). The computation of these numbers and the corresponding Berry curvatures are detailed in Supplementary Note 2. According to the bulk-edge correspondence principle [41,55], for each of these bandgaps, the total Chern number is equal to the summation of the Chern numbers (∆C v ) for all the modes below the gap. If two structures with opposite total Chern numbers for bands share an interface (i.e., one structure with green disk shunted and the other with blue disk shunted), one U-shape helical edge mode will be present at the interface (|∆C v | = 1).

C. Topologically protected edge waves

The most intriguing property of topological edge states is their ability to convey waves along sharp and curved interfaces without back-scattering. For QVHE, edge states are not present at the edge of the structure with a trivial mirror (e.g., air) since the difference in Chern numbers is less than one (|∆C edge | = ±1/2); however, according to the bulk-edge correspondence principle [41,55], if two materials with opposite Chern numbers share an interface (|∆C interf ace | = ±1), topologically protected waves travel along this interface without losing their intensity. To verify these states in our system, a super cell com-posed of 10 unit cells in the y-direction is considered (see Fig. 2). This depicted strip is then repeated in the x-direction by applying Bloch boundary conditions. As depicted in Fig. 2(b), all the green disks are electrically shunted, while the blue ones experience an open circuit condition. For the upper half of the super cell, locations of the green disks are reversed compared to the lower half, resulting in an interface with a Chern number difference of |∆C interf ace | = +1. Figure 2(a) documents the band structure of the super cell, revealing the existence of a topologically-protected edge state starting from near the bulk modes on top to the bulk modes on the bottom at the location of the upper bandgap in Fig. 1(b). Furthermore, the right schematic in Fig. 2(b) provides the displacement field of the strip at 73 kHz (marked with a red star). As observed, interface wave is localized at the interface.

D. Numerical demonstration of edge states

Next we report wave propagation immune to backscattering along desired trajectories. For an ideal topological insulator, data should travel along sharp and curved trajectories without loss of intensity. For mechanical topological insulators, this advantageous behavior has significant implications for communications systems, multiplexers, and on-board mechanical logic. To illustrate such a capability, a reconfigurable interface is created between two sub-domains with opposite Chern numbers.

As such, we consider a plate composed of 8 × 8 hexagonal unit cells, each equipped with a pair of piezoelectric disks. On one side of the interface the upper disks are shunted; while for the other side the lower ones are electrically shunted (or vice-versa). Figure 3 (a) depicts the displacement field of the structure with a horizontal interface under a harmonic excitation at 73 kHz. As documented, waves clearly travel along the desired interface from the input (marked with the blue star) to the output (marked with the green star). As a second example, Fig. 3(b) displays an interface in which waves are guided along a triangular path from the source on the left side to the receiver on the bottom edge, without back-scattering or reflection at the sharp edges (again at 73 kHz). Finally, Fig. 3(c) depicts the displacement field for the case with a z-shaped interface, documenting propagation with minimal intensity loss for 73 kHz. As desired, for all three interfaces, robustness of the system is guaranteed at the location of the interface.

E. Experimental realization of edge states

We next verify the performance of the proposed reconfigurable TI by carrying out a set of experiments. The experimental setup is composed of a 5×5 hexagonal array of unit cells incorporating fifty bonded piezoelectric disks (see Supplementary Note 3 for full details on the experimental setup). Half of the PZT disks are connected to external circuits (by simply closing the attached switch) with an effective negative capacitance of C = -1.7 nF to form an interface between two domains (marked with green line), one characterized by a valley Chern number C v = 1/2, and the other by C v = -1/2, respectively. Figure 4(a) exhibits the experimentally measured RMS wavefield of a system with a horizontal interface in response to excitation at 77 kHz. This figure clearly confirms the propagation of an interface wave from the source (marked with a black star) to the receiver on the other side of the structure. Due to the imperfectness of the experimental setup, and the mass and stiffness of the connected wires and soldering material, this frequency is slightly above that predicted using numerical simulations. As shown in Fig. 4(b), the system is easily reconfigured to alter the interface location (at the same excitation frequency) simply by operation of the programmable ON/OFF switches, in this case introducing a sharp-angled trajectory. An interface wave travels from the input on the left side to the output on the top edge with minimal reflections from the sharp geometry. For both trajectories appearing in Fig. 4, time snapshots of the interface wave traveling along the trajectories are provided in Supplementary Note 3.

II. DISCUSSIONS

In summary, this Letter presents the first reconfigurable electroacoustic topological insulator to realize topological edge states analogous to the quantum valley Hall effect. The proposed phononic crystal is formed by periodically repeating a graphene-like unit cell, which is composed of two bonded piezoelectric circular disks attached to a hexagonal PLA layer. In order to break inversion symmetry, one of the two disks is connected to an external circuit with negative capacitance, while the other one is left in an open circuit state. Dispersion relationships are computed numerically, documenting a topological bandgap at the location of the Dirac one. Numerically computed and experimentally measured results illustrate immune to back-scattering wave propagation along sharp interfaces. Furthermore, by simply operating ON/OFF the desired circuits, reconfigurable interfaces are obtained and verified experimentally. The reconfigurable TI proposed in this study may be a stepping-stone material platform for implementing next generation acoustic-based wave filtering, multiplexing/demultiplexing, and logic in communication-based devices.

Figure 1 .

 1 Figure 1. Phononic crystal band structure. (a) Schematic of the phononic crystal formed by hexagonal unit cells with PLA (h0 = 0.5 mm thick) as the host layer and attached circular piezoelectric patches with thickness of h1 = 0.5 mm (d1 = 7 mm in diameter) connected to external circuits. These circuits provide negative capacitance of C = -(R2C0)/R1. (b) Comparison between the band structures in the absence (red dashed-curves with two Dirac Dirac points at 45, and 73 kHz), and in the presence (blue solidcurves with optimal bandgap at C = -1.7 nF ) of completed external circuits attached to one PZT disk.

FrequencyFigure 2 .Figure 3 .Figure 4 .

 234 Figure 2. Topologically protected interface state. (a) Band structure of a super-cell composed of a 1-by-10 array of unit cells, periodically repeated in the x-direction computed using Floquet boundary conditions. Gray curves depict bulk bands, while the blue curve depicts the protected interface mode. (b) Left: schematic of the super-cell in which green disks are electrically shunted and blue ones are left open circuited, resulting in the upper-half and lower-half subdomains experiencing opposite Chern numbers; Right: the corresponding wave distribution at a frequency of f = 73 kHz (marked with a red star), documenting wave localization at the interface. All displacements are normalized by the maximum deformation of the cell.(a) (b) (c) M ax

III. ACKNOWLEDGMENTS

The authors would like to thank the National Science Foundation for supporting this research under Grant No. 1929849.

IV. COMPETING FINANCIAL INTERESTS

The authors declare no competing financial interests.