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Physics Letters A ••• (••••) •••–•••

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Physics Letters A www.elsevier.com/locate/pla

Nonlinear bandgap properties in a nonlocal piezoelectric phononic crystal nanobeam Denghui Qian a,∗ , Zhiyu Shi b , Chengwei Ning c , Jianchun Wang d a

Jiangsu Province Key Laboratory of Structure Engineering, College of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011, Jiangsu, China b State Key Laboratory of Mechanics and Control of Mechanical Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, Jiangsu, China c China National Aeronautical Radio Electronics Research Institute, Shanghai 200241, China d China Ship Scientiﬁc Research Center, Wuxi 214082, Jiangsu, China

a r t i c l e

i n f o

Article history: Received 11 June 2019 Accepted 4 July 2019 Available online xxxx Communicated by B. Malomed Keywords: Piezoelectric phononic crystal nanobeam Nonlocal effect Nonlinearity Plane wave expansion method Electro-mechanical coupling ﬁelds

a b s t r a c t Applying nonlocal elasticity theory, von Kármán type nonlinear strain-displacement relation and plane wave expansion (PWE) method to Euler-Bernoulli beam, the calculation method of band structure of a nonlinear nonlocal piezoelectric phononic crystal (PC) nanobeam is proposed and formulized. In order to investigate the properties of wave propagating in the nanobeam in detail, band gaps of ﬁrst four orders are picked, and the corresponding inﬂuence rules of electro-mechanical coupling ﬁelds, nonlocal effect and geometric parameters on band gaps are studied. During the researches, external electrical voltage and axial force are chosen as the inﬂuencing parameters related to electro-mechanical coupling ﬁelds. Scale coeﬃcient is chosen as the inﬂuencing parameter corresponding to nonlocal effect. Length ratio between materials PZT-4 and epoxy and height-width ratio are chosen as the inﬂuencing parameters of geometric parameters. Moreover, all the inﬂuence rules are compared to those in linear nanobeam. The results are expected to be of help for the design of micro and nano devices based on piezoelectric periodic nanobeam. © 2019 Elsevier B.V. All rights reserved.

1. Introduction In the last thirty years, different kinds of PCs, which are artiﬁcial periodic composite elastic structures, have attracted the attention of many experts and scholars on account of the existence of unique elastic/acoustic wave bandgap characteristics [1]. In order to meet the engineering requirements, the design ideas of traditional PCs have been applied to the frequently-used rods, beams, plates and some other basic elastic structures [2,3]. Moreover, PCs based on the multi-physics coupling have also been researched, such as: piezomagnetic PCs, piezoelectric PCs, magneto-electroelastic PCs and so on [4–6]. Based on the conversion between mechanical and electric ﬁelds, regulation of band gaps can be realized effectively to piezoelectric PCs, which can be further applied to realize the active control of band gaps. All the PCs mentioned above are in macroscopic size, the orders of magnitude of corresponding frequency regions of band gaps are from hertz (Hz) to megahertz (MHz) generally. But recently, with the rapid devel-

*

Corresponding author. E-mail address: [email protected] (D. Qian).

https://doi.org/10.1016/j.physleta.2019.07.006 0375-9601/© 2019 Elsevier B.V. All rights reserved.

opment of nanotechnology in all ﬁelds, a batch of researches on new-style PCs at nanoscale have been emerged [7,8], which increase the orders of magnitude of corresponding frequency regions of band gaps to gigahertz (GHz) even terahertz (THz). Therefore, if piezoelectric and nano materials are combined and then introduced to PCs, the piezoelectric PC nanostructures will be formed, which will gather the excellent properties of piezoelectric material, nano material and PC, and display the new physical characteristics by the coupling of components. Intensive investigations of piezoelectric PC nanostructures will promote the produce of new-style piezoelectric nano-devices, and provide new possibilities of the application of nano electro mechanical system (NEMS). For the moment, researches on piezoelectric PC nanostructures were few [9,10], but piezoelectric nanostructures have obtained a mass of studies. Previous work has demonstrated that classical continuum theory cannot be applied to nanostructures on account of the existence of size dependence. However, some higher-order continuum theories were proposed, and one of the most commonly used is nonlocal elastic theory. Nonlocal theory was proposed by Eringen [11,12], it assumes that the stress at a reference point is associated with the strain ﬁeld of every point in the nanostructure, which has been veriﬁed by the simulation of

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D. Qian et al. / Physics Letters A ••• (••••) •••–•••

2

2 ∂ 2 w (x, t ) 1 ∂ w (x, t ) εx = − z + 2 ∂x ∂ x2

Fig. 1. Schematic conﬁgure of a piezoelectric PC nanobeam.

where w (x, t ) is the ﬂexural displacement along z-direction. Assuming the electric ﬁeld E z only exists in z-direction, the relation between the electric potential φ and E z is given by:

Ez = − atomic lattice dynamics and phonon dispersion experiment [12]. Originally, the theory was mainly applied to study the inﬂuences of nonlocal effect on the mechanical properties of elastic nanostructures [13–15]. But in recent years, many experts and scholars have begun to expand such a theory to piezoelectric nanostructures. Based on the nonlocal theory, the free vibration of piezoelectric nanobeams with the coupling of thermo-electric-mechanical ﬁelds were researched by Ke et al. [16]. It was found that nonlocal effect has obvious inﬂuences on the free frequencies and vibration modes. Further, Asemi et al. [17] studied the vibration performances of piezoelectric nanoplates in the Pasternak type elastic medium in detail by combining nonlocal theory and Kirchhoff plate model. In some cases, the vibration amplitude and thickness of piezoelectric nanobeam and nanoplate have similar size, so nonlinear strain-displacement relation should be considered to represent the mechanical properties of structures more precisely. By introducing von Kármán nonlinear strain-displacement relation to nonlocal Timoshenko piezoelectric nanobeam model, Ansari et al. [18] researched the vibration characteristics of nonlinear piezoelectric nanobeams in a postbuckling state. Besides, by introducing von Kármán nonlinear strain-displacement relation to nonlocal Kirchhoff piezoelectric nano plate model, the nonlinear dynamic characteristics of graphene/piezoelectric laminated ﬁlms in sensing moving loads were studied by Li et al. [19]. Recently, based on the nonlocal theory, Euler-Bernoulli beam model and PWE method, the bandgap properties of a linear piezoelectric PC nanobeam were investigated by Qian [20]. Results demonstrated that the band gaps can be regulated and controlled effectively by changing the corresponding parameters of thermo-electro coupling ﬁelds, nonlocal effect and geometric size. In this paper, nonlocal theory, von Kármán type nonlinear strain-displacement relation and PWE method are combined and applied to calculate the band structures of a proposed nonlinear nonlocal piezoelectric PC nanobeam, which is interesting since other studies with respect to nano-PCs applying PWE method [20–23] not considering the nonlinearity or size-dependence. For the sake of understanding wave characteristics propagating in this nanobeam, the inﬂuence rules of electro-mechanical coupling ﬁelds, nonlocal effect and geometric parameters on band gaps of ﬁrst four orders are investigated in detail. Moreover, the effects of nonlinearity on inﬂuence rules above are also analyzed.

(1)

∂φ ∂z

(2)

Considering the nanobeam is composed of piezoelectric and elastic materials, the constitutive equations can be divided into two parts: PZT-4 and epoxy. For PZT-4, the electro-mechanical coupling constitutive equations can be written as [16]:

∂ 2 σx = c 11 εx − e 31 E z (3) ∂ x2 ∂2 Dz D z − μ2 = e 31 εx + κ33 E z (4) ∂ x2 where σx and D z are the axial stress along x-direction and the

σx − μ2

electrical displacement along z-direction, respectively. c 11 , e 31 and κ33 denote the elastic, piezoelectric and dielectric constants, separately. In addition, μ represents the scale coeﬃcient introduced by the nonlocal effect. Ignoring the free electric charge, Gauss’s law requires that:

∂ Dz =0 ∂z

(5)

Substituting Eqs. (1), (2), (4) to (5), and applying the electrical boundary conditions φ( h2 ) = V , φ(− h2 ) = 0, electric ﬁeld E z can be determined as:

Ez =

e 31 ∂ 2 w (x, t )

∂ x2

κ33

z−

V

(6)

h

By substituting Eqs. (1), (6) to (3), the axial stress can be expressed by w (x, t ) as:

σx of PZT-4

2 ∂ 2 σx V ∂ 2 w (x, t ) 1 ∂ w (x, t ) = e − cz + c 31 11 h 2 ∂x ∂ x2 ∂ x2

σx − μ2

(7)

where c = c 11 + e 231 /κ33 . The strain energy ΠS of PZT-4 in a unit cell is given by:

ΠS =

b

a1 h/2 (σx εx − D z E z )dzdx

2

(8)

0 −h/2

Substituting Eqs. (1), (6) to (8), it gives:

ΠS =

1

a1 −M x

2

2 ∂ 2 w (x, t ) 1 ∂ w (x, t ) + dx N x 2 ∂x ∂ x2

0

2. Model and method By periodically repeating two different materials: PZT-4 and epoxy, the piezoelectric PC nanobeam is constructed. As shown in Fig. 1, a cartesian coordinate system is established and the cross section of beam is rectangular. The lengths of PZT-4 and epoxy in a unit cell are deﬁned by a1 and a2 , respectively. Hence, the lattice constant a = a1 + a2 . The width and thickness of beam are represented as b and h, separately. Besides, an external electrical voltage V is applied to PZT-4 and an external axial force P 0 is applied to the nanobeam. Based on the Euler-Bernoulli theory and von Kármán type nonlinear strain-displacement relation [24], the axial strain along x-direction εx can be written as:

h

−

b

a1 2 Dzz

2 0 −h 2

e 31 ∂ 2 w (x, t )

κ33

∂ x2

h /2

− Dz

V

h

dzdx

(9)

h /2

where M x = b −h/2 σx zdz and N x = b −h/2 σx dz represent the bending moment and axial normal force respectively. The kinetic energy ΠK of PZT-4 in a unit cell can be calculated as:

ΠK =

1

a1

ρ1 bh

2

∂ w (x, t ) ∂t

2 dx

0

where

ρ1 represents the mass density of PZT-4.

(10)

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Moreover, the work ΠF of PZT-4 in a unit cell done by the external forces can be written as:

ΠF =

1

a1

2

2 ∂ w (x, t ) (NP + NE ) dx ∂x

(11)

0

where N P and N E represent the normal forces induced by the axial force P 0 and external electrical voltage V , which can be written as: N P = P 0 , N E = bV e 31 . The Hamilton’s principle can be expressed as:

t (δΠS + δΠF − δΠK )dt = 0

(12)

∂ ∂ ∂ w (x, t ) ∂ 2 Mx ∂ w (x, t ) + ( + N N + N ) x P E ∂x ∂x ∂x ∂x ∂ x2 ∂ 2 w (x, t ) = ρ1 bh ∂t2

(13)

Moreover, from Eq. (7), it can be obtained:

Nx − μ

Nx

∂ x2

M x − μ2

1

= bV e31 + bhc 11 2

∂ w (x, t ) ∂x

2 (14)

∂ 2 Mx bh3 ∂ 2 w (x, t ) = − c 12 ∂ x2 ∂ x2

(15)

Assuming that the inﬂuence of nonlinear strain-displacement relation on axial normal force N x can be ignored, then Eq. (14) can be reduced to:

N x − μ2

∂ 2 Nx = bV e31 ∂ x2

(16)

The solution of Eq. (16) can be expressed as: x

N x (x) = Xe μ + Y e

x

−μ

+ bV e31

(17)

where X and Y are constants. From Eqs. (13), (15) and (17), the governing equation of PZT-4 with the vibration regarded as a simple harmonic can be obtained as:

∂ ∂ 2 w (x) ∂ w (x) ∂2 − m ( x ) n ( x ) 1 1 ∂x ∂x ∂ x2 ∂ x2 3

x x ∂ w (x) ∂ − + μ2 3 Xe μ + Y e μ ∂x ∂x 3

x ∂ ∂ ∂ w ( x ) − x ∂ w (x) + μ2 3 n1 (x) − Xe μ + Y e μ ∂x ∂x ∂x ∂x 2

∂ = ω2 p 1 (x) w (x) − μ2 2 p 1 (x) w (x) (18) ∂x 3

where m1 (x) = bh c, n1 (x) = P 0 + 2bV e 31 , p 1 (x) = ρ1 bh. 12 For epoxy, the elastic constitutive equation can be written as:

σx − μ2

∂ 2 σx = E εx ∂ x2

where E is the Young’s modulus. By substituting Eqs. (1) to (19), the axial stress be expressed as:

σx − μ2

∂ ∂ 2 w (x) ∂ w (x) ∂2 − m ( x ) n ( x ) 2 2 ∂x ∂x ∂ x2 ∂ x2 3

x x ∂ w (x) ∂ w (x) ∂ ∂3 − + μ2 3 n2 (x) + μ2 3 Xe μ + Y e μ ∂x ∂x ∂x ∂x x ∂ w (x) ∂ μx − Xe + Y e μ − ∂x ∂x 2

2 2 ∂ p 2 (x) w (x) (21) = ω p 2 (x) w (x) − μ ∂ x2 3

By substituting Eqs. (9)–(11) into (12), it gives:

2

Similar to the formula derivation of the governing equation of PZT-4, the ﬁnial governing equation of epoxy is:

where m2 (x) = bh E, n2 (x) = P 0 , p 2 (x) = ρ2 bh. ρ2 is the mass den12 sity of epoxy. Applying m(x), n(x) and p (x) to express (m1 (x), m2 (x)), (n1 (x), n2 (x)) and ( p 1 (x), p 2 (x)), Eqs. (18) and (21) can be uniformly rewritten as:

0

2∂

3

(19)

∂ ∂ 2 w (x) ∂ w (x) ∂2 − m ( x ) n ( x ) ∂x ∂x ∂ x2 ∂ x2 3

x x ∂ w (x) ∂ ∂3 ∂ w (x) − + μ2 3 n(x) + μ2 3 Xe μ + Y e μ ∂x ∂x ∂x ∂x x ∂ w (x) ∂ μx − Xe + Y e μ − ∂x ∂x 2

∂ (22) = ω2 p (x) w (x) − μ2 2 p (x) w (x) ∂x

According to the periodicity of nanobeam and Floquet-Bloch periodic boundary condition [25], the following relations should be satisﬁed:

w (x + a) = w (x)e ika

(23)

m(x + a) = m(x)

(24)

n(x + a) = n(x)

(25)

p (x + a) = p (x)

(26)

Substituting Eqs. (23)–(26) to (22), it gives:

a ∂ w (x) ∂ 3 μx μa − μx − μ + Xe e − 1 Y e e − 1 ∂x ∂ x3 a ∂ w (x) ∂ μx μa − μx − μ =0 Xe e − 1 + Y e e −1 − ∂x ∂x

μ2

As known, Eq. (27) is satisﬁed no matter x takes any value. That is to say, constants X and Y must be equal to zero. Then, Eq. (22) can be reduced to:

3 ∂ ∂ 2 w (x) ∂ w (x) ∂ w (x) ∂2 2 ∂ − + m ( x ) n ( x ) μ n ( x ) ∂x ∂x ∂x ∂ x2 ∂ x2 ∂ x3 2

∂ (28) = ω2 p (x) w (x) − μ2 2 p (x) w (x) ∂x

Based on the theory of one dimensional PWE method [20], it can be obtained from Eq. (28):

2

2 ∂ 2 σx ∂ 2 w (x, t ) 1 ∂ w (x, t ) = − E z + E 2 ∂x ∂ x2 ∂ x2

(20)

k + G m G − G k + G

G

σx of epoxy can

(27)

2

+ k + G n G − G k + G

3

+ μ2 k + G n G − G k + G w k G

2

= ω2 p G − G + μ2 k + G p G − G w k G (29) G

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Fig. 2. (a) Band structures of nonlocal piezoelectric PC nanobeams with and without nonlinear considered, (b) inﬂuences of nonlinearity on the fourth band gap with different external electrical voltages. (For interpretation of the colors in the ﬁgure(s), the reader is referred to the web version of this article.) Table 1 Material parameters of PZT-4 [16] and epoxy. Material

ρ /kg m−3

PZT-4 Epoxy

7500 1180

E /GPa

c 11 /GPa

e 31 /C m−2

κ33 /C V−1 m−1

λ1 /N m−2 K−1

132

−4.1

7.124 × 10−9

4.738 × 105

76

If the number of reciprocal-lattice vectors is chosen as N, Eq. (29) can be expressed by a matrix formulation as:

[ MG ] + [ N G ] − ω2 [ P G ] × w (G ) = [0]

(30)

where

[ MG ]i j = (k + G i )2m(G i − G j )(k + G j )2

(31)

[ N G ]i j = (k + G i )n(G i − G j )(k + G j ) + μ2 (k + G i )3n(G i − G j )(k + G j ) 2

2

[ P G ]i j = p (G i − G j ) + μ (k + G i ) p (G i − G j )

(32) (33)

Eq. (30) represents a generalized eigenvalue problem for ω2 . Finally, the band structure of a nonlinear piezoelectric PC nanobeam with nonlocal effect, can be obtained by solving the equation for each Bloch wave vector limited in the irreducible ﬁrst Brillouin zone (1BZ). All the material parameters of PZT-4 and epoxy used in the calculations are shown in Table 1. Moreover, η = μ/a is introduced to denote the ratio of nonlocal size in a unit cell. If the nonlinear strain-displacement relation is not considered, the linear piezoelectric PC nanobeam with nonlocal effect will be formed and the characteristic equation still remains as Eq. (30). But the item n1 (x) will be changed to P 0 + bV e 31 . 3. Numerical results and analyses 3.1. Band structures of linear and nonlinear nonlocal piezoelectric PC nanobeams The band structure of nonlocal piezoelectric PC nanobeam with the consideration of nonlinearity is shown in Fig. 2(a). As a comparison, band structure of the same nanobeam but with linear strain-displacement relation is also displayed in the ﬁgure. During the calculation, all the material parameters are shown in Table 1. Besides, a1 = a2 = 50 nm, b = h = 10 nm, V = 1 V, P 0 = 1 × 10−8 N and η = 0.1. As shown in the ﬁgure, band gaps of ﬁrst four orders are labeled by the gray region, which will be investigated in detail in the following sections. By comparing the two band structures, considering nonlinearity makes all the bands down if the external electrical voltage V is considered to be 1 V. In order to further analyze the effects of nonlinearity on band

gaps, Fig. 2(b) gives the differences of starting frequencies and bandgap widths between the fourth order of nonlinear and linear nanobeams with different external electrical voltages. As shown in the ﬁgure, with the change of V , the effects of nonlinearity on the band gap are different. Nonlinearity can not only lower the bands as shown in Fig. 2(a) but also raise them. But if V is larger than a certain value, the band structures of nonlocal piezoelectric PC nanobeams with and without nonlinearity considered will coincide as shown in Fig. 2(b). Moreover, why the external electrical voltage is chosen as the independent variable to study the differences of nonlinear and linear nanobeams can be attributed to the change of item n1 (x) in the characteristic equations of the two kinds of nanobeams. 3.2. Inﬂuence rules of electro-mechanical coupling ﬁelds Fig. 3(a) displays the inﬂuences of external electrical voltage V on starting frequencies f s and bandgap widths f w of band gaps of ﬁrst four orders. During the calculation, all the parameters except for V are same as those in Fig. 2(a), and V is from −50 V to 50 V. As displayed in the ﬁgure, all the starting frequencies and bandgap widths of four orders decrease to be zero with the increase of V . The value of V corresponding to that the starting frequency or bandgap width exactly equals to zero is called as the critical voltage. By increasing the order, critical voltage increases. For the starting frequencies, all of them keep decreasing with V increases until it reaches the critical voltage. But for the widths of band gaps, several ups and downs are needed before they decrease to zero. As an application example, the external electrical voltage corresponding to the highest peak of the ups and downs can be used to achieve the widest band gap of a certain order. In addition, by comparing Fig. 3(a) and 2(b), the critical voltage of the fourth band gap is almost 10 V if nonlinearity is considered, but the differences of the fourth band gap between nonlinear and linear nanobeams are continued until V equals to almost 20 V. That means the critical voltage of the fourth band gap of linear nanobeam is almost 20 V, which illustrates that the consideration of nonlinearity reduces the critical voltage. The inﬂuences of external axial force P 0 on starting frequencies f s and bandgap widths f w of band gaps of ﬁrst four orders are shown in Fig. 3(b). Here, all the parameters except for P 0 are same as those in Fig. 2(a), and P 0 is from −5 × 10−7 N to 5 × 10−7 N.

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Fig. 3. Inﬂuences of (a) external electrical voltage V and (b) external axial force P 0 on starting frequencies f s and bandgap widths f w of ﬁrst four band gaps.

Fig. 4. Inﬂuences of coeﬃcient

η on starting frequencies f s and bandgap widths f w of ﬁrst four band gaps of (a) nonlinear and (b) linear nanobeam.

Oppositely, all the starting frequencies and bandgap widths of four orders decrease to be zero by decreasing P 0 . Critical force is applied to describe the value of P 0 corresponding to that the starting frequency or bandgap width exactly equals to zero. With the increase of order, critical force increases. For the starting frequencies of four orders, all of them increase with the increase of P 0 since P 0 is bigger than the critical force. For the bandgap widths, all of them also increase with the increase of P 0 but after a rise and fall. Therefore, the external axial force corresponding to the peak of the rise and fall can be applied to make the band gap of a certain order widest. The region of external electrical voltage or external axial force corresponding to that the starting frequency or bandgap width equals to zero can be named as uncontrollable region. The uncontrollable regions should be avoided as far as possible when the band gaps are regulated and controlled artiﬁcially. Moreover, the inﬂuencing investigations of external electrical voltage and external axial force on band gaps play a fundamental role in realizing active control of piezoelectric PC nanobeam. 3.3. Inﬂuence rules of nonlocal effect The inﬂuences of coeﬃcient η on starting frequencies f s and bandgap widths f w of ﬁrst four band gaps with and without nonlinearity considered are displayed in Fig. 4(a) and (b), respectively. During the calculation, all the parameters except for η are same as those in Fig. 2(a), and η is from 0 to 0.5. By comparing the two ﬁgures, the impact trends of η are basically the same no matter the nonlinearity considered or not. But nonlinearity can effectively advance the critical point, that is to say the band gaps are harder to be opened with the increase of nonlocal effect if nonlinearity is considered. For the starting frequencies of four orders, all of them decrease to be zero with the increase of η . But for the bandgap

widths, several ups and downs should be experienced before they return to stable. In general, all the band gaps are harder to be opened with stronger nonlocal effect. 3.4. Inﬂuence rules of geometric parameters The inﬂuences of ratio between lengths of PZT-4 and epoxy in a unit cell a1 /a2 on starting frequencies f s and bandgap widths f w of ﬁrst four band gaps are displayed in Fig. 5. As a comparison, the inﬂuence curves of linear nanobeam are also displayed. During the calculation, all the parameters except for a1 and a2 are same as those in Fig. 2(a), and a2 is invariable as 50 nm. Besides, a1 is determined by the ratio a1 /a2 , and a1 /a2 is from 0 to 10. As shown in the ﬁgure, the inﬂuence rules of a1 /a2 on each band gap of nonlinear and linear nanobeams are similar. For the starting frequencies of four orders, all of them keep decreasing to zero by increasing a1 /a2 . But for the bandgap widths, several ups and downs are needed before they decrease to zero. Two, three, four and ﬁve ups and downs are needed for the ﬁrst, second, third and fourth order, respectively. Besides, with the increase of a1 /a2 , the amplitude corresponding to each peak decreases for the bandgap width of each order. By comparing the inﬂuence curves of nonlinear and linear nanobeams, considering nonlinearity makes the controllable region narrower. Moreover, the effect of nonlinearity on the last peak is more obvious than others for the bandgap widths. That is, nonlinearity increases obviously the amplitude corresponding to the last peak no matter the order. But for other peaks, the amplitudes in nonlinear nanobeam are almost consistent with those in linear nanobeam. Fig. 6(a) and (b) display the inﬂuences of height-width ratio h/b on starting frequencies f s and bandgap widths f w of ﬁrst four band gaps of nonlinear and linear nanobeams respectively. As shown, if height is much smaller than width, all the band gaps

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Fig. 5. Inﬂuences of ratio between lengths of PZT-4 and epoxy in a unit cell a1 /a2 on starting frequencies f s and bandgap widths f w of (a) ﬁrst, (b) second, (c) third and (d) fourth band gap of nonlinear and linear nanobeams.

Fig. 6. Inﬂuences of height-width ratio h/b on starting frequencies f s and bandgap widths f w of ﬁrst four band gaps of (a) nonlinear and (b) linear nanobeam.

cannot be opened. If h/b is bigger than the critical value, all the starting frequencies increase with the increase of height-width ratio. For the widths of four band gaps, all of them except for the third one keep increasing with the increase of h/b after a rise and fall. But for the third order, it decreases ﬁrstly and then keeps increasing after a rise and fall. Thus, if the size of nanobeam is limited, the height-width ratio h/b corresponding to the peak of the rise and fall can be applied to make the band gap of a certain order as wide as possible. By comparing Fig. 6(a) and (b), nonlinearity increases the critical height-width ratios of all the band gaps. In addition, nonlinearity also increases the magnitudes of peaks of all the bandgap widths, which play an active role in widening the band gaps. 4. Conclusions Based on the nonlocal elasticity theory, von Kármán type nonlinear strain-displacement relation and PWE method, the band

structure of a nonlinear nonlocal piezoelectric PC nanobeam is calculated. The corresponding wave propagation properties are investigated from the aspects: inﬂuence rules of electro-mechanical coupling ﬁelds, nonlocal effect and geometric parameters. Details are as follows: 1. With the increase of voltage, all the starting frequencies of four orders keep decreasing to zero until the voltage gets to the critical voltage, but several ups and downs are undergone for bandgap widths. Oppositely, with the decrease of axial force, all of the starting frequencies and bandgap widths decrease to be zero. As an application example, the voltage or axial force corresponding to the peak of the rise and fall can be applied to make the band gap of a certain order widest. Besides, the uncontrollable regions corresponding to the zero starting frequencies and bandgap widths should be avoided as far as possible to make the band gaps be regulated and controlled artiﬁcially.

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2. With the increase of nonlocal effect, all of the starting frequencies keep decreasing to be zero, but the bandgap widths should experience several ups and downs before they return to be stable, which illustrated that stronger nonlocal effect make the band gaps harder to be opened. 3. With the increase of length of PZT-4, all the starting frequencies keep decreasing to zero, but several ups and downs are needed before the bandgap widths decrease to zero. In detail, two, three, four and ﬁve ups and downs are needed for the ﬁrst, second, third and fourth order respectively. In addition, for the bandgap width of each order, the amplitude corresponding to each peak decreases with the increase of PZT-4 length. All the band gaps cannot be opened if height is much smaller than width. With the increase of height, all the starting frequencies keep increasing, but all the bandgap widths except for the third one keep increasing after a rise and fall. The bandgap width of the third order decreases ﬁrstly and then keeps increasing after a rise and fall. 4. By comparing the nonlinear and linear nanobeams, if voltage is large enough, nonlinear has no effect on the band gaps. Beyond that, nonlinearity can not only lower but raise the bands. Besides, the consideration of nonlinearity reduces the critical voltage. By reinforcing the nonlocal effect, band gaps are harder to be opened if nonlinearity is considered. Moreover, considering nonlinearity makes the controllable regions of length ratio narrower, but which is opposite to height-width ratio. Acknowledgements This research was supported by the National Natural Science Foundation of China (No. 11847009) and the Natural Science Foundation of Suzhou University of Science and Technology (No. XKQ2018007). References [1] M. Thota, K.W. Wang, Tunable waveguiding in origami phononic structures, J. Sound Vib. 430 (2018) 93–100. [2] K.C. Chuang, Z.W. Yuan, Y.Q. Guo, et al., A self-demodulated ﬁber Bragg grating for investigating impact-induced transient responses of phononic crystal beams, J. Sound Vib. 431 (2018) 40–53. [3] D. Qian, Z. Shi, Bandgap properties in locally resonant phononic crystal double panel structures with periodically attached spring-mass resonators, Phys. Lett. A 380 (41) (2016) 3319–3325. [4] X. Guo, P. Wei, M. Lan, et al., Dispersion relations of elastic waves in onedimensional piezoelectric/piezomagnetic phononic crystal with functionally graded interlayers, Ultrasonics 70 (2016) 158–171.

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