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Finite element modeling of mechanical behaviors of piezoelectric nanoplates with flexoelectric effects

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Nguyễn Gia Hào

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O R I G I NA L

Le Minh Thai · Doan Trac Luat · Van Binh Phung · Phung Van Minh · Do Van Thom

Finite element modeling of mechanical behaviors of piezoelectric nanoplates with flexoelectric effects

Received: 30 November 2020 / Accepted: 14 September 2021 / Published online: 17 November 2021

© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021

Abstract This paper uses the finite element method to simulate the mechanical, electric, and polarization behaviors of piezoelectric nanoplates resting on elastic foundations subjected to static loads, in which the flexoelectric effect is taken into consideration. The finite element formulations are established by employing a new type of shear deformation theory, which does not need any shear correction factors, but still accurately describes the stress field of the plate. The numerical results show clearly that the flexoelectric effect has a strong influence on the mechanical responses of the nanoplates. In particular, the normal stress distribution in the thickness direction is no longer linear when the flexoelectric coefficient is large enough, and this phenomenon differs completely from that of conventional plates. In addition, the distribution of the electric field and the polarization also depend on boundary conditions, which were not investigated in the published works.

Keywords Nanoplates·Elastic foundation·Flexoelectric·Bending·Finite element method 1 Introduction

Nowadays, along with the development of science and technology, piezoelectric nanostructures have been used widely in engineering practices such as sensors, actuators, energy harvesters, and so on. For these structures, flexoelectricity is a common phenomenon, especially the electric polarization inside structures due to strain gradients. There have also been several studies showing that the mechanical behavior of the nanoplates takes into account the flexoelectric effect. Yan [1] used the classical plate theory to study static bending and free vibration of piezoelectric nanoplates, taking into account the flexoelectricity based on the analytical solution. Yang et al.

[2] introduced an analytical solution to show the static bending and free vibration response of nanoplates, in which the theory considered the effects of piezoelectric and flexoelectricity. The authors also relied on Kirchhoff’s plate theory (classical plate theory—CPT) to provide the solutions to be found in explicit form. Li et al. [3] studied static bending and free vibration of the circular microplate. The equations were established from the CPT, the solution was solved analytically, and the results pointed out the effect of flexoelectric effect on the mechanical response of this plate. Still, based on the classical plate theory, Wang et al. [4] employed the finite difference solution to show the static bending response of flexoelectric nanoplates. The authors also calculated the case where one edge of the plate was clamped. He and his co-workers [5] used the CPT combined with the exact solution to show clearly the effect of both the flexoelectricity and the piezoelectricity L. M. Thai

Faculty of Special Equipments, Le Quy Don Technical University, Hanoi City, Vietnam D. T. Luat·P. V. Minh·D. V. Thom (

B

)

Faculty of Mechanical Engineering, Le Quy Don Technical University, Hanoi City, Vietnam e-mail: thom.dovan@lqdtu.edu.vn

V. B. Phung

Faculty of Aerospace Engineering, Le Quy Don Technical University, Hanoi City, Vietnam

(2)

on the mechanical response of nanoplates. Ebrahimi1 and Barati [6] researched the bucking response of flexoelectric nanoplates based on the exact solution and the classical plate theory, where the plate rested on a two-parameter elastic foundation, and the impact of the stress surface was taken into consideration. Amin and his colleagues [7] employed the classical plate theory and the flexoelectric theory to establish the analytical solution to analyze the nonlinear free vibration of functionally graded (FG) flexoelectric nanoplates. Amir et al. [8] studied the free vibration of sandwich flexoelectric plates, in which the first-order shear deformation theory and Navier’s method were mixed to give a series of free vibration responses of the plate. Ghobadi et al.

[9] explored the nonlinear static bending of flexoelectric nanoplates. The analytic solution was incorporated into classical Kirchhoff’s plate theory to find the displacement and electric field of these plates subjected to the thermo-electromagnetic force. Based on published papers, Ghobadi and his team [10] investigated the static bending response of FG nanoplates, taking into account the effects of temperature, electric, and flexoelectric fields. Giannakopoulos et al. [11] introduced an antiplane dynamic flexoelectric issue, which was defined as a dielectric solid with electric polarization and flexoelectricity gradients owing to strain gradients. Qu and colleagues [12] investigated the torsion of a rectangular cross-sectional flexoelectric semiconductor rod. It was based on the macroscopic theory of flexoelectric semiconductors. Using double power series expansion of the coordinates inside the cross section, a one-dimensional model was established from the three-dimensional theory. Yang et al. [13] researched multilayer phononic crystal band structures with flexoelectricity. In this work, the classical plate theory of Kirchhoff was used to derive the finite element formulations.

From the summary of previously published works, it can be seen that these studies mainly relied on the classical plate theory to determine the mechanical response of nanoplates. It can be understood that the CPT is the simple theory to establish the equilibrium equation of structures. However, it cannot show all the behavior of nanoplates when taking into account the effects of the flexoelectric effect. Currently, there is no study showing the distribution of stress, electric field, and polarization of nanoplates based on shear deformation theories that correctly show the mechanical response of the plate, including the satisfaction of stress boundary conditions, has to be zero at the surfaces of the plate. As a result, this paper aims to use the finite element method to simulate the mechanical, electric, and polarization behaviors of piezoelectric nanoplates resting on an elastic foundation subjected to static load, taking into account the flexoelectric effect by employing the finite element method and the new type of shear deformation theory.

The remainder of this work is structured as follows: Sect.2develops finite element formulas to address the bending issue of nanoplates while accounting for the flexoelectric effect. Section3continues to provide comparisons to demonstrate the validity of the suggested theory and the calculation program coded in the MATLAB environment. Section4presents a set of findings from the static bending response of nanoplates, in which the impact of flexoelectric on the plate’s stress distribution, electric field, and polarization is clearly shown. Section5highlights the study’s major points.

2 Finite element model of piezoelectric nanoplate in static bending problem

Consider a mechanical structure of the piezoelectric nanoplate and the elastic foundation placed in the Cartesian coordinate system with the lengtha, widthb, and thicknesshas shown in Fig.1.

In the analysis of plate structures, there are many different theories, such as classical deformation theory, first-order shear deformation theory, and higher-order shear deformation theory. The classical shear deforma- tion (CDT) theory is only suitable for thin structures because the effect of shear deformation is not taken into account, and the first-order shear deformation theory (FSDT) gives the acceptable results. However, it needs to add a shear correction factor into calculations. Higher-order shear deformation theories do not require a shear correction coefficient, but their finite element formulations are more complicated than those of the CDT and FSDT. As a result, Shimpi [14] developed a simple plate theory by isolating the transverse displacement component into bending and shear parts. The most amazing feature of Shimpi’s approach is that it has fewer unknown variables and governing equations than the CDT and FSDT, and does not need a shear correction factor. According to Shimpi’s work, some four unknown shear deformation theories were modified using some shape functions, such as polynomial functions [15], sinusoidal functions [16], and hyperbolic functions [17].

This work uses the simple form theory proposed by Shimpi with different shape functions such as sinusoidal functions as in works [18,19]. The main advantage of this theory is simplicity, no need for any shear correc- tions since it is satisfied that the shear stress boundary at the top and bottom surface of the plate is zero, and it accurately describes the mechanical response of the plate.

(3)

Fig. 1 The model of a piezoelectric nanoplate resting on a two-parameter elastic foundation

According to the shear deformation theory based on hyperbolic sine functions [18,19], the displacements u,v, andwin thex-,y-, andz-directions, respectively, depending on the coordinates of one point, have the following forms:

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

ux(x,y,z) −z∂wb(x,y,0)

∂xf(z)∂ws(x,y,0)

∂x vy(x,y,z) −z∂wb(x,y,0)

∂yf(z)∂ws(x,y,0)

∂y wz(x,y,z)wb(x,y,0) +ws(x,y,0)

(1)

with f(z)zζ(z),ζ(z)h.sinhzz.cosh12, whereux, vy, andwzare the displacements in thex-,y-, andz-directions at one point within the plate;wbandwsare the bending displacement and shear displacement in thez-direction. Therefore, to determine the longitudinal displacements and tangential displacements at any point with the coordinates (x,y,z), two main components,wbandwsneed to be defined.

Longitudinal and shear strain components are defined as follows:

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

ε

⎧⎨

εx x

εyy

γx y

⎫⎬

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

z∂2wb

∂x2f(z)2ws

∂x2 εbx +sx

z∂2wb

∂y2f(z)2ws

∂y2 εby+sy

z∂2wb

∂x∂y −2f(z)2ws

∂x∂y γbx y+sx y

⎫⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎭ z

⎧⎨

εbx

εby

γbx y

⎫⎬

εb

+f(z)

⎧⎨

εsx

εsy

γsx y

⎫⎬

εs

γ γx z

γyz

∂ζ (z)

∂z

⎧⎪

⎪⎨

⎪⎪

∂ws

∂x

∂ws

∂y

⎫⎪

⎪⎬

⎪⎪

∂ζ (z)

∂z γ0

(2)

In this work, only the strain gradients in thex- andy-axes are considered, and the strain gradient in the z-axis is zero. It is obvious due to the displacement field chosen. This means that the strain gradients in the x- andy-axes are much higher than that in the thickness direction. Then, the strain gradient is expressed as follows:

η

⎧⎪

⎪⎨

⎪⎪

ηx x z2wb

∂x2∂f(z)

∂z

2ws

∂x2 ηyyz2wb

∂y2∂f(z)

∂z

2ws

∂y2

⎫⎪

⎪⎬

⎪⎪

2xw2b

2yw2b

+∂f(z)

∂z

2xw2s

2yw2s

ηb+∂f(z)

∂z ηs (3)

(4)

When the flexoelectric effect is taken into consideration, the stress components and electric displacement vector for a nanoscale dielectric material are expressed as follows:

Ti j ci j klεkleki jEk

i j mfki j mEk

Pi ci j kεj k+κi jEk+ fi j klηj kl (4)

in whichci j kl,eki j, fki j m, andκi jare the components of elastic, piezoelectric, flexoelectric, and permittivity constant tensors; they are the material parameters. Ti j is the stress tensor, which is similar to that of the traditional elastic foundation. Pi is the electric displacement vector, andi j mis the moment stress tensor or the higher-order stress tensor.

From the expressions of the strain components, we have specific expressions of stress and electric dis- placement vectors as follows:

T

c11c12 0 c12c11 0 0 0 c66

⎧⎨

εx

εy

γx y

⎫⎬

⎭−e31

⎧⎨

⎩ 1 1 0

⎫⎬

EzCbε− ˜E (5) S

sx z

syz

c66 0 0 c66

γ Csγ (6)

Ψ x x z

yyz

f14 1

1

Ez (7)

Pz0e31

εx x+εyy

+κ33Ez+ f14

ηx x z+ηyyz

(8) wheref14f3113andf14f3223[20].

Herein, some assumptions are made as follows: there is no external electric field acting on the plate, the electric displacement equals the electric polarization. Hence, the last element in Eq. (8) f14

ηx x z+ηyyz

is the polarization, which is caused by the strain gradients in this plate; and this element depends on the coordinate zof the plate due to the derivative of the functionf (z)in the expression of the strain gradient.

The electrical field is calculated from the partial derivative of electrical potential as follows:

Ez∂ϕ

∂z (9)

According to Gaussian’s law in electrostatics, the electric displacement is given by the formula:

∂Pz0

∂z 0 (10)

From Eqs. (8), (9), and (10), we have:

ϕ e31 2a33

2w

∂x2 +2w

∂y2

z2+c1+c2z (11)

wherec1andc2are the coefficients to be found. With the open-circuit condition, we have the following conditions:

Pz0

±h 2

0 (12)

(5)

Then, the following coefficient can be found:

c2f14 a33

2w

∂x2 +2w

∂y2

(13) Substituting Eqs. (11) and (13) into Eq. (9), we obtain the expression of the internal electric field as follows:

Eze31 κ33

εx x+εyy

+ f14 κ33

2wb

∂x2 +2wb

∂y2

+ f14 κ33

2ws

∂x2 +2ws

∂y2

∂f(z)

∂z e31

κ33

2wb

∂x2 +2wb

∂y2

z+e31 κ33

2ws

∂x2 +2ws

∂y2

f(z) + f14

κ33

2wb

∂x2 +2wb

∂y2

+ f14 κ33

2ws

∂x2 +2ws

∂y2

∂f(z)

∂z (14)

Note that the expression of the electric fieldEzdepends on the coordinatez, functionf(z), and the derivative of the functionf (z). Therefore,Ezmay convert roughly to linear or nonlinear form depending on the value of f14; it means the flexoelectric effect will significantly affect the distribution ofEzover the thickness direction, which will be specified in the example below. This is a very interesting problem. However, there are no published works dealing with this issue.

With the open-circuit condition, the electric Gibbs free energy has the following form:

U 1 2

V

εTT +γTS+ηTΨ

dV 1

2

V

εTCε +γTSεTe31

⎧⎨

⎩ 1 1 0

⎫⎬

EzηT f14 1

1

Ez

⎠dV

−1 2

V

εTe31.e31 k33

⎧⎨

⎩ 1 1 0

⎫⎬

2ws

∂x2 +2ws

∂y2

f(z)

⎠dV

−1 2

V

εTe31

⎧⎨

⎩ 1 1 0

⎫⎬

f14

k33 2wb

∂x2 +2wb

∂y2

⎞⎠dV −1 2

V

εTe31

⎧⎨

⎩ 1 1 0

⎫⎬

f14

k33 2ws

∂x2 +2ws

∂y2

∂f(z)

∂z

⎠dV

−1 2

V

ηT f14

1 1

e31 k33

2wb

∂x2 +2wb

∂y2

.z

dV −1 2

V

ηT f14

1 1

e31 k33

2ws

∂x2 +2ws

∂y2

f(z)

dV

−1 2

V

ηT f14

1 1

f14 k33

2wb

∂x2 +2wb

∂y2

dV −1 2

V

ηT f14

1 1

f14 k33

2ws

∂x2 +2ws

∂y2

∂f(z)

∂z

dV (15) Since the plate is rested on an elastic foundation, the potential energy expression of the plate, taking into account the effect of the elastic foundation, has the following form:

Ufound 1 2

S

"

kww2+ks

"

∂w

∂x 2

+ ∂w

∂y 2##

dS (16)

wherekwandksare the two coefficients of the elastic foundation.

In this paper, a four-node element is used. Each node has six degrees of freedom:

qe

$4 i1

wbi, wsi, ∂wb

∂x

i

, ∂ws

∂x

i

, ∂wb

∂y

i

, ∂ws

∂y

i

T

(17) where

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩ wb

$4 i1

Hiwbi+Hi+1 ∂wb

∂x

i

+Hi+2 ∂wb

∂y

i

Hbqe ws

$4 i1

Hiwsi +Hi+1

∂ws

∂x

i

+Hi+2 ∂ws

∂y

i

Hsqe

(18)

(6)

in which,Hjis the Hermit interpolation function.

Then, the displacement vector at any point in the element is interpolated through the element’s nodal displacement vector as follows:

u

wb, ws, ∂wb

∂x

, ∂ws

∂x

, ∂wb

∂y

, ∂ws

∂y T

H.qe (19) According to this definition, the strain vectors are computed through the nodal displacement vector as follows:

εb

⎢⎢

2Hb

x2

2Hb

y2

−22Hb

x y

⎥⎥

qe B1qe; εs

⎢⎢

2Hs

x2

2Hs

y2

−22Hs

x y

⎥⎥

qe B2qe (20)

γ0 'Hs

x

Hs

y

(

qe B3qe; ηb

'−2Hb

x2

2Hb

y2

(

qe B4qe; ηs

'−2Hs

x2

2Hs

y2

(

qe B5qe (21)

So Eq. (15) is rewritten as follows:

Ue 1 2qeT

Ve

)

zBT1 + f(z)BT2

Cb

zB1+ f(z)B2

+ BT3CsB3

* dVqe

−1 2qTe

Ve

zB1T + f(z)BT2

e31I3e31 k33

2Hb

∂x2 +2Hb

∂y2

.z

dVqe

−1 2qTe

Ve

zB1T + f(z)BT2e31.e31 k33 I3

2Hs

∂x2 +2Hs

∂y2

f(z)

dVqe

−1 2qTe

Ve

zB1T + f(z)BT2

e31I3f14

k33 2Hb

∂x2 +2Hb

∂y2

dVqe

−1 2qTe

Ve

zBT1 + f(z)B2T

e31I3 f14

k33 2Hs

∂x2 +2Hs

∂y2

∂f(z)

∂z

dVqe

−1 2qTe

Ve

BT4 +∂f(z)

∂z BT5

f14I2e31

k33 2Hb

∂x2 +2Hb

∂y2

.z

dVqe

−1 2qTe

Ve

BT4 +∂f(z)

∂z BT5

f14I2e31

k33 2Hs

∂x2 +2Hs

∂y2

f(z)

dVqe

−1 2qTe

Ve

BT4 +∂f(z)

∂z BT5

f14I2 f14

k33 2Hb

∂x2 +2Hb

∂y2 dVqe

−1 2qTe

Ve

BT4 +∂f(z)

∂z BT5

f14I2 f14 k33

2Hs

∂x2 +2Hs

∂y2

∂f(z)

∂z

dVqe (22)

withI3{1,1,0}TI2{1,1}T.

Equation (22) can be written in short form as follows:

Ue 1

2qTe Keqe (23)

(7)

Fig. 2 Boundary conditions of piezoelectric nanoplates

Equation (16) relates to the element nodal displacement vector as follows:

Uefound 1 2qeT

Se

⎜⎜

⎜⎝kw(Hb+Hs)T(Hb+Hs)+ks

⎜⎜

⎜⎝ Hb

∂x +Hs

∂x

THb

∂x +Hs

∂x

+ Hb

∂y +Hs

∂y

THb

∂y +Hs

∂y

⎟⎟

⎟⎠

⎟⎟

⎟⎠dSqe

1

2qeTKefqe (24)

The external force exerted on the plate has the following formula:

We

Se

(wb+ws)TFSdSqTe

Se

(Hb+Hs)TFSdS

Fe

qTeFe (25)

So the static equilibrium equation of the piezoelectric nanoplate the flexoelectricity is obtained from the minimal equationUe,Uefound, andWeis as follows:

Ke+Kef

qe Fe (25)

By solving Eq. (26), the displacement and stress fields will be obtained, then substituting them into Eqs. (8) and (14),EzandPzwill be obtained due to strain gradients of this plate.

In this paper, some boundary conditions are defined as follows [18]:

• As one edge is simply supported:

wb0, ws0, ∂wb

∂y 0, ∂ws

∂y 0 at x 0,a wb0, ws0, ∂wb

∂x 0, ∂ws

∂x 0 at y0,b

(27a)

• As one edge is clamped:

wb0, ws0, ∂wb

∂x 0, ∂wb

∂y 0, ∂ws

∂x 0, ∂ws

∂y 0 (27b)

andSrepresents for a simply supported boundary condition,Crepresents a clamped supported boundary condition. Some boundary conditions used in this paper are shown in Fig.2.

(8)

Table 1 Non-dimensional deflectionwof the plate resting on the two-parameter elastic foundation Kw Ks a/h10

[21] [22] This work

8 elements

10×10 elements 12×12 elements 14×14 elements

1 5 3.3455 3.3455 3.3797 3.3643 3.3560 3.3512

10 2.7505 2.7504 2.7743 2.7635 2.7578 2.7545

15 2.3331 2.3331 2.3508 2.3428 2.3386 2.3361

20 2.0244 2.0244 2.0382 2.0320 2.0287 2.0268

81 5 2.8422 2.8421 2.8667 2.8557 2.8499 2.8464

10 2.3983 2.3983 2.4163 2.4083 2.4040 2.4015

15 2.0730 2.0730 2.0868 2.0806 2.0774 2.0755

20 1.8245 1.8244 1.8355 1.8306 1.8280 1.8265

625 5 1.3785 1.3785 1.3835 1.3816 1.3804 1.3797

10 1.2615 1.2615 1.2658 1.2642 1.2632 1.2626

15 1.1627 1.1627 1.1665 1.1650 1.1642 1.1637

20 1.0782 1.0782 1.0815 1.0802 1.0795 1.0791

3 Verification study

In this section, some comparison problems will be carried out to verify the accuracy of the proposed theory and mathematical model. The numerical results of the deflection and stress in this work are compared with those of exact published data.

Example 1 Consider a plate resting on an elastic foundation with dimensionsab0.2 m,ha/10 and a/200. The material properties areE 320.24 GPa, Poisson’s ratio of 0.26. The plate isSSSSand under a uniformly distributed loadq0. The non-dimensional elastic foundation parameters are calculated as:

⎧⎪

⎪⎨

⎪⎪

Kw Kwa4 D Ks Ksa2

D

(28)

with

D E h3 12

1−ν2 (29)

The non-dimensional deflection at the center point of the plate is defined by the following formula:

w 103D q0a4 w

a 2,b

2

(30) The comparative results obtained by this work, the differential quadrature method [21], and the analytical method [22] are presented in Table1with the increase in the mesh size, and one can see that with the mesh size of 12×12, the accuracy is acceptable. Therefore, in all the following investigations, this mesh size will be used.

Example 2 Consider a square plate witha/b1, a/h10,E 380 GPa, andν 0.3. The plate is fully simply supported, and the load is bi-sinusoidal type as follows:

qq0sinπx a

sinπy

b

(31) Non-dimensional stresses are calculated as:

⎧⎪

⎪⎨

⎪⎪

σx∗∗, σy∗∗

σx

q0

a 2,b

2,z

, σy

q0

a 2,b

2,z τx y∗∗ τx y(0,0,z)

q0

(32)

(9)

a b

c d

Fig. 3 The stress distributions in the thickness direction of the square plate under bi-sinusoidal load

For different volume fraction indexn, the stress distributions in the thickness direction are presented in Fig.3. Herein, the numerical results of this work are compared with those of Hiroyuki’s exact solution [23]. It can be seen that the stresses are distributed nonlinearly in the thickness direction, and they reach their maximum values at the top and bottom surfaces.

Example 3 Finally, non-dimensional maximum defection of theSSSSnanoplate withh20 nm,ab 50 h, and material properties c11102 GPa; c12 31 GPa;c3335.50 GPa;e31 −17.05 C/m2;k33 1.76.10–8C/(Vm);f1410–7C/m is considered. The plate is under a uniformly distributed load ofq0 0.05 MPa. The numerical results of this work compared with those of Yang et al.’s analytical solution [2] are presented in Fig.4. It can see that the data has good agreement.

4 Numerical results

In this section, a piezoelectric nanoplate withh20 nm,ab50h, and the material propertiesc11 102 GPa;c1231 GPa;c3335.50 GPa;e31 −17.05 C/m2;k331.76.10–8C/(Vm) is considered; the uniformly distributed load isq00.05 MPa. The plate is rested on a two-parameter elastic foundation with kwandks. The boundary conditions are as follows:

• Fully simply supported plate: SSSS

• Fully clamped plate: CCCC

• Two opposite edges are clamped, the other edges are simply supported: CSCS

• Two adjacent edges are clamped, the other edges are simply supported: CCSS

(10)

Fig. 4 Non-dimensional deflection atyb/2 taking into account the flexoelectric effect

Non-dimensional parameters are calculated as follows:

w∗ 103c11h3

12q0a4 w; f14 f14

f140; Kw kwa4 Df

; Ks ksa2 Df

; Df c11h3 ) 12

σx, τx y, τx z

* 1 q0

)σx, τx y, τx z

*

(33) with f140 10–7C/m.

Since this study calculates solely in the case of disregarding external voltage, only an evenly distributed mechanical loadq0is applied to the plate. Therefore, in order to clearly see the response of the electric field and the polarization (due to strain gradientPz f14

ηx x z+ηyyz

) corresponding to this mechanical load, this work gives the normalized parameters of the electric field and polarization as follows:

Ez Ez

q0 V m

N

; Pz Pz

q0 C

N

; (34)

4.1 Influence of the flexoelectric effect

Consider anSSSS plate, to investigate the influence of the flexoelectric effect with the values off14are 0 and 10–7C/m (f14 1,Kw 100,Ks10). This means the plate with and without the flexoelectric effect is considered. The non-dimensional deflectionw*atyb/2 is presented in Fig.5. The distributions of the electric fieldsEz,Pz, and the stress components in the thickness direction at the center point of the plate are shown in Figs.6,7,8,9, and10. Some discussions are as follows:

• Fig. 5points out that when taking into account the flexoelectric effect,the non-dimensional maximum deflectionwof the plate decreases. However, the normal stressσx and shear stressτx zincrease (Figs.8and 10). This represents a complete difference from conventional plates and is also something to keep in mind when testing nanoplates in the presence of the flexoelectric effect.

• Figs.6,7,8,9and10show that when not taking the flexoelectric effect into the calculation (andPz0),the electric fieldEzand the normal stressσxsymmetrically across the middle plane of the plate. However, when f14is not equal to zero,Ez, and the stressσxare no longer symmetrical across the middle plane of the plate, and maximum values ofPzandσxincrease. In particular, this is completely different from traditional plates.

That is the maximum displacement ofwdecreases, the stressσx increases. On the other hand, maximum values ofσx andτx zincrease when the value off14is not equal to zero. However, the value ofτx ydecreases when taking into account the flexoelectric effect.

(11)

Fig. 5 The deflection of the plate atyb/2 in the cases of with and without the flexoelectric effect

Fig. 6 The distribution of the electric fieldEzalong the thickness direction in the cases of with and without the flexoelectric effect

Fig. 7 The distribution ofPzacross the thickness in the cases of with and without the flexoelectric effect

(12)

Fig. 8 The distribution of the stressσxacross the thickness in the cases with and without the flexoelectric effect

Fig. 9 The distribution of the stressτx y across the thickness in the cases with and without the flexoelectric effect

Fig. 10 The distribution of the stressτx z across the thickness in the cases with and without the flexoelectric effect

(13)

Fig. 11 The deflection of the plate atyb/2 with different values off14

4.2Influence of the value of f14

The appearance of the influence of the flexoelectricity effect is shown in the coefficientf14, and this will make the mechanical response of the nanoplate show a difference compared to conventional structures. SinceEz

depends on the coefficientf14as in Eq. (14) and makes the polarization-dependent onf14as in expression (8), the stress depends onf14as in expression (4), so the energy of the plate changes. Thus, the response of the nanoplate also changes. The coefficientf14will vary depending on the material, and this work investigates howf14varies in a certain range to see clearly how the flexoelectricity effect affects the deflection response, stress, electric field, and polarization of nanoplates. The computed data can be used as a reference in design, manufacturing, and practical applications.

To investigate the influence of the value off14on the behavior of the plate (SSSS,Kw 100,Ks10), the value of f14 increases gradually from 1 to 10. The deflection of the plate atyb/2 is shown in Fig.11;

the distributions along the thickness direction at the center point of the plate ofEz,Pz, and stress components are presented in Figs.12,13,14, and15. Some discussions are as follows:

• When increasing the value off14, maximum deflection of the platewdecreases (Fig.11).

• By looking at Figs.11,12,13, and14, when the value off14is small,Ez,Pz,andσxdistribute linearly in the thickness of the plate. However, when the value off14is high, these components change nonlinearly along the plate thickness. This is also a complete difference from plates made of a homogeneous material whose mechanical properties do not vary in the thickness direction. It can be explained that in the expressions of Ez,Pz,andσx including the multiplication off14and the functiong(z) vary nonlinearly in the thickness direction. Therefore, when the value off14is small, it has a slight influence on the functiong(z). However, when the value off14is high, its effect on the functiong(z) becomes stronger. Hence, the values ofEz,Pz, andσx change nonlinearly along the thickness direction. In addition, one can see that the increase inf14 will make the law of transformation of the maximum values ofEz,Pz, andσxcompletely different from the transformation of the maximum deflectionwmax. And the surface position of the plate is always the position whereEz,Pz,andσx reach their maximum value.

• The stress componentτx y decreases gradually with increasing the value of f14. It is proportional to the decrease in deflectionw(see Fig.15), as is the case of conventional plates.

4.3 Influence of boundary condition

In this subsection, to see more clearly the influence of the boundary condition on the distribution of the electric fieldEzand the polarizationPzalong thexandyedges of the plate, four boundary conditions, SSSS, CCCC, CSCS, and CCSS (f14 3, Kw 100,Ks10) are considered. The distribution surfaces ofEz

zh2 andPz

zh2

along thexandyedges are presented in Fig.16. The value lines atyb/2 are shown in Figs.17and18. The minimum and maximum values ofw,Ez, andPzare listed in Table2. One can see that:

• Boundary conditions affect strongly not only the shapes of the distribution surfaces ofEzandPzalong the x-andy-axes, but also to the maximum values of these components.

(14)

Fig. 12 The distribution of the electric fieldEzalong the thickness direction with different values off14

Fig. 13 The distribution ofPzalong the thickness direction with different values off14

Fig. 14 The distribution ofσxalong the thickness direction with different values off14

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Fig. 15 The distribution ofτx y along the thickness direction with different values off14

Table 2 The minimum and maximum values ofw, Ez,andPzwith different boundary conditions,zh2

Value Boundary condition

SSSS CCCC SCSC CCSS

wmax 1.292 0.5403 0.7455 0.8213

Ez(max)[Vm/N] 7.845 5.595 7.576 7.446

Ez(min)[Vm/N] 0.146 4.344 5.271 5.873

Pz(max)[C/N] 8.146×10–8 4.171×10–8 6.184×10–8 6.098×10–8

Pz(min)[C/N] 2.260×10–10 4.107×10–8 5.170×10–8 5.857×10–8

• For the SSSS plate, the maximum values ofEzandPzappear in the center of the plate, and they are much greater than the minimum values at the edge of the plate.

• For the CCCC plate, the maximum values ofEzandPzobtain the maximum values in the center of the plate, and the minimum values at the edge of the plate; their magnitudes are not much different than in the case of theSSSSplate.

• For the CSCS and CCSS plates, the maximum values ofEzandPzdo not appear in the center of the plate and tend to move to the simply supported edges; their magnitudes are not much different than in the case of the SSSS plate.

• Figures17 and 18 also show clearly that when more than one edge of the plate is fixed, there will be positions, where the values ofEzandPzequal to zero. In the other worlds, there is a position, in which there is no electric field and polarization. In contrast, for theSSSSplate, this phenomenon will not occur. This is suggested for experimental measurements to determine the electric field or polarization. It is necessary to determine the measurement location to obtain the desired results.

4.4 Effect of elastic foundation

Consider a piezoelectric nanoplate under theSSSSboundary condition (f14 3). Changing the parameters of the elastic foundation between 0 and 100, the maximum displacement, stress, electric field, and polarization in the center of the plate are listed in Table3. It can be seen that when increasing the parameters of the elastic foundation, the deflection, stress, electric field, and polarization of the plate reduce significantly.

4.5 Effect of the plate thickness

Finally, to see more clearly the maximum deflection of the plate with and without the flexoelectric effect in the case of changing the plate thicknessh, this subsection uses the maximum deflection ratio as follows:

Rw wmax(without flexoelectric effect) wmax (with flexoelectric effect)

(16)

SSSS CCCC

CSCS CCSS

Fig. 16 The distributions ofPzandEzalong the edges of the plate with different boundary conditions,zh2

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