Buckling analysis of nanocomposite sandwich plates with piezoelectric face sheets based on flexoelectricity and first-order shear deformation theory

Abstract

In piezoelectric materials and at the nano-scale, there is a coupling between electrical polarization and strain gradients fields, which is called flexoelectricity. The effects of this phenomenon seem to be negligible in micro/macro scales. The current study has attempted to have a cohesive concentration on the buckling behaviors of sandwich plates. To achieve the abovementioned aim with a higher accuracy, the flexoelectric effect assumes to be existing on the top and bottom face sheets and the core is a composite plate. Also, based on statistics, the first-order shear deformation theory seems to lead to more accurate results. Therefore, in the present research we follow this method to obtain results. The analytical method is applied to solve higher order governing equations. In addition, the critical buckling voltage is calculated considering the flexoelectricity, and it is found that the effects of flexoelectricity play significant roles in determining the critical buckling voltage. Moreover, it is revealed that the thickness of the flexoelectric face sheets and the aspect ratio of the sandwich plate play the same role in critical buckling load variations. It means that the critical buckling load decreases when the thickness of the flexoelectric face sheets or the aspect ratio of the sandwich plate increases and vice versa. The results of the present work can be used for the optimum design and control of similar systems such as micro-electro-mechanical and nano-electromechanical devices.

Keywords

Flexoelectricity buckling sandwich plates first-order shear deformation theory nanocomposite core

Introduction

Nowadays, researchers are trying to move the science borders faster than before. Among the most popular contents, nanocomposites [1–3], functionally graded materials (FGMs) [4–9] and piezoelectric materials [10–13] have attracted the attention of scientists because of their individual features. Using applications of piezoelectric nanostructures like generators and sensors, provided the demand of thorough knowledge about nano-scale material’s behaviors. Most of the previous studies used the fundamental form of piezoelectricity which denoted on non-centrosymetric crystals. Haghshenas and Ghorbanpour Arani [14] studied the nonlocal vibration of a piezoelectric polymeric nanoplate carrying nanoparticle via Mindlin plate theory. They considered a nano-electro-mechanical sensor made of polyvinylidene fluoride (PVDF) carrying a nanoparticle with different masses at any position. Also, the nonlinear dynamic analysis and vibration of imperfect FG sandwich plates with piezoelectric actuators on elastic foundations were investigated by Duc and Cong [15]. They used Reddy's higher-order shear deformation plate theory to model the displacement field and applied a combination of electrical, damping and thermal loads on sandwich plate.

In comparison with piezoelectricity theory, flexoelectricity is valid for different types of dielectric materials even if their crystals are centrosymetric in core. As another expression, piezoelectricity represents the linear coupling between electrical and mechanical variables, while flexoelectricity denotes the linear coupling between strain gradient and polarization [16] and strain and polarization gradient [17]. Despite the piezoelectric effect, flexoelectric effect is negligible in micro-macro scale and its effect becomes visible when the scales incline to nano-scales. Duc et al. [18], Yudin and Tagantsev [16], and Zubko et al. [19] were among the first scholars who investigated the effects of flexoelectricity in solids. In recent years, researchers endeavor to obtain a comprehensive understanding on this frail effect by using experimental and theoretical results. For example, Ray [20] provided the exact solutions for the flexoelectric responses of flexoelectric nano-beams. Based on a circular cylindrical shell, Hu and Li [21] provided a hybrid flexoelectric-piezoelectric design to demonstrate its mechanical behaviors like bending and buckling. As said before, flexoelectric effects play a remarkable role in sensors. Due to this great importance, Abdollahi and Arias [22] tried to examine constructive and destructive interaction between piezoelectricity and flexoelectricity in flexural sensors and actuators. In order to have an analytical understanding of buckling and vibrational behaviors of a beam Liang et al. [23] developed modified beam models with the consideration of the surface, flexoelectric, and non-local electric effects. Li et al. [24] reformulated a size-dependent flexoelectric theory for isotropic dielectrics to identify the contribution of each strain gradient component. Wang and Wang [25] provided an analytical model for nano-scale unimorph piezoelectric energy harvesters with flexoelectric effect. They indicated the effect of flexoelectricity, thickness, position of piezoelectric layer and thickness ratio of substructure layer to piezoelectric layer and the proof mass. In another work, Liang et al. [26] studied the buckling and vibrational behaviors of a nano-film with flexoelectric effect. The results of this study successfully showed that the critical buckling loads and natural frequency are enhanced by the flexoelectricity consideration. They used piezoelectric lead magnesium niobate (PMN) as an example material. Rong Zhang and Yiang Jiang [27] utilized a modified Kirchhoff plate model to examine the influences of flexoelectricity on electromechanical coupling behaviors of nano-plates. A lot of experiments have been done by Ma and Cross [28] to examine the flexoelectricity quantitatively [29,30] by measuring the flexoelectric coefficients of ferroelectric ceramics. Moreover, some of initial theories have been suggested to account the flexoelectricity effect [31,32]. According to this, the flexoelectric effect on the electro-mechanical coupling behaviors of piezoelectric nanostructures has been studied [33 –37]. Li et al. [38] applied flexoelectric sensors for natural modal signal analysis of conical shells. Their results showed that the distributions of the total signal and both components are sinusoidal in the circumferential direction.

Electromechanical responses of the nano-plate with consideration of the flexoelectricity effect were presented by Yang et al. [39]. They utilized Kirchhoff plate theory and Hamilton’s principle to explore the influence of flexoelectricity on the electromechanical coupling behavior of a simply supported (SSSS) piezoelectric nano-plate. Simulation results on the electroelastic fields indicate that the flexoelectric effect is size-dependent, which is more prominent for thinner plates. Although many efforts have been devoted to the study of the properties of macroscopic piezoelectric materials, there are much fewer studies investigating the properties of nano-scale piezoelectric materials with electromechanical coupling and also using first-order shear deformation theory (FSDT).

The aforementioned ideas intrigued us to investigate the mechanical behaviors of sandwich plates using FSDT. Sandwich plates include a nanocomposite core between two flexoelectric face sheets where electric filed applied on face sheets. Composite core is reinforced by carbon nanotubes (CNTs) fiber where CNTs are assumed to be uniformly distributed in the thickness direction. Also, effective material properties of composite core is estimated through the rule of mixture. The simply supported sandwich plate resting on the Pasternak foundation includes spring foundation and shear layer. Utilizing Hamilton’s principle higher order governing equations of motion are derived and solved. The effects of applied voltage, CNT volume fraction, aspect ratio, and coefficient of elastic medium on the critical buckling load of sandwich flexoelectric plates are discussed. The numerical results are partially validated by using different methods (kp-Ritz method and Analytical method) where good agreement was found between results.

Sandwich plate modeling

In this section, attention is concentrated on the bending and buckling of a sandwich plate with length a, width b, and thickness h, as shown in Figure 1. Also, $h_{c}$ is the thickness of central composite core and $h_{t}$ and $h_{b}$ are the thickness of top and bottom flexoelectric face sheets, respectively, which $h = h_{t} + h_{c} + h_{b}$ . Composite core is made of polymer matrix and uniformly distributed CNTs fibers. A Cartesian coordinate system (x, y, z) is utilized to describe the plate with the z-axis being along the thickness direction and the x − y plane sitting on the mid-plane of plate. The top and bottom flexoelectric face sheets are subjected to external voltage which is applied along the thickness of each layer. Sandwich plate is rested on Pasternak elastic foundation.

This research makes use of FSDT to model displacement fields. This theory is more accurate than classical plate theory (CPT) which has used in pervious flexoelectric works. Based on FSDT, the displacements of an arbitrary point in the sandwich plate for both flexoelectric face sheets and nanocomposite core can be denoted as [40]

\tilde{u} (x, y, z, t) = u (x, y, t) - z [\frac{\partial}{\partial x} w (x, y, t) + ϕ_{x} (x, y, t)]

(1)

\tilde{v} (x, y, z, t) = v (x, y, t) - z [\frac{\partial}{\partial y} w (x, y, t) + ϕ_{y} (x, y, t)]

(2)

\tilde{w} (x, y, z, t) = w (x, y, t)

(3)

where u, v, and w are the displacement components along the x, y, and z directions, respectively. Also,

ϕ_{x}

and

ϕ_{y}

are rotation of middle surface about x and y-axes, respectively.

Nanocomposite core

Nanocomposite core is made of polymer matrix and uniformly distributed CNTs fibers where fibers are at x direction. The strain components for nanocomposite core can be defined as [41]

ε_{x x} c^{=} \frac{\partial \tilde{u}}{\partial x}

(4)

ε_{y y} c^{=} \frac{\partial \tilde{v}}{\partial y}

(5)

γ_{x y} c^{=} \frac{\partial \tilde{u}}{\partial y} + \frac{\partial \tilde{v}}{\partial x}

(6)

γ_{x z} c^{=} \frac{\partial \tilde{u}}{\partial z} + \frac{\partial \tilde{w}}{\partial x}

(7)

γ_{y z} c^{=} \frac{\partial \tilde{v}}{\partial z} + \frac{\partial \tilde{w}}{\partial y}

(8)

where

ε

and

γ

are the normal and shear strains, respectively, and superscript c denotes composite core. The constitutive equations for nanocomposite core can be presented as [42]

\begin{array}{l} σ_{x x}^{c} = Q_{11} ε_{x x}^{c} + Q_{12} ε_{y y}^{c} \\ σ_{y y}^{c} = Q_{12} ε_{x x}^{c} + Q_{22} ε_{y y}^{c} \\ τ_{x y}^{c} = Q_{66} γ_{x y}^{c} \\ τ_{y z}^{c} = Q_{44} γ_{y z}^{c} \\ τ_{x z}^{c} = Q_{55} γ_{x z}^{c} \end{array}

(9)

in which

σ

and

τ

introduce normal and shear stresses, respectively. Q is elastic constants of composite core can be determined as

[\begin{matrix} Q_{11} \\ Q_{12} \\ Q_{21} \\ Q_{22} \end{matrix}] = [\begin{matrix} \frac{E_{11}}{1 - v_{12} v_{21}} \\ \frac{v_{12} E_{11}}{1 - v_{12} v_{21}} \\ \frac{v_{21} E_{11}}{1 - v_{12} v_{21}} \\ \frac{E_{22}}{1 - v_{12} v_{21}} \end{matrix}], [\begin{matrix} Q_{44} \\ Q_{55} \\ Q_{66} \end{matrix}] = [\begin{matrix} G_{23} \\ G_{13} \\ G_{12} \end{matrix}]

(10)

where

E_{11}

E_{22}

_, and

G

indicate longitudinal, transversely elastic moduli, and shear modulus, respectively, and v is Poisson’s ratio.

The effective material properties of nanocomposite core can be given through extended mixture rule approach as follows

E_{11} = η_{1} V^{C N T} E_{11}^{C N T} + V^{P} E^{P}

(11a)

\frac{η_{2}}{E_{22}} = \frac{V^{C N T}}{E_{22}^{C N T}} + \frac{V^{P}}{E^{P}}

(11b)

\frac{η_{3}}{G_{12}} = \frac{V^{C N T}}{G_{12}^{C N T}} + \frac{V^{P}}{G^{P}}

(11c)

ν_{12} = V^{C N T} ν_{12}^{C N T} + V^{P} ν^{P}

(11d)

ρ_{c} = V^{C N T} ρ^{C N T} + V^{P} ρ^{P}

(11e)

where superscript P denotes polymer matrix;

η_{i}

(j = 1,2, 3) is the CNT efficiency parameter.

V^{C N T}

and

V^{P}

are volume fractions for CNT and polymer matrix, respectively, and

V^{C N T} + V^{P} = 1

Flexoelectric face sheets

In order to account the flexoelectric effect, the extended form of linear piezoelectricity theory is applied, in which the coupling between strain gradient and polarization is considered.

For simplicity, the effects of higher order terms are ignored [19,43,44]. Therefore, the general expression of the internal energy density for flexoelectric face sheets can be expressed as [44]

\cup = \frac{1}{2} a_{k l} P_{k} P_{l} + \frac{1}{2} c_{i j k l} ε_{i j} ε_{k l} + d_{i j k} ε_{i j} P_{k} + \frac{1}{2} b_{i j k l} P_{i, j} P_{k, l} + f_{i j k l} u_{i, j k} P_{l} + e_{i j k l} ε_{i j} P_{k, l}

(12)

where

P_{i}

ε_{i j}

a_{k l}

c_{i j k l}

_, and

d_{i j k}

are, respectively, polarization, strain, dielectric, elastic, and piezoelectric constant tensors. The polarization gradient and polarization gradient coupling tensor represents by

b_{i j k l}

f_{i j k l}

is the strain gradient and polarization coupling tensor.

e_{i j k l}

denotes the strain and polarization gradient coupling and

f_{i j k l} = - e_{i j k l}

[45,46]. Consequently, the constitutive equations can be derived for flexoelectric face sheets as [31]

σ_{i j} = \frac{\partial \cup}{\partial ε_{i j}} = c_{i j k l} ε_{k l} + d_{i j k} P_{k} + e_{i j k l} P_{k, l}

(13)

τ_{i j m} = \frac{\partial \cup}{\partial u_{i, j m}} = f_{i j m k} P_{k}

(14)

E_{i} = \frac{\partial \cup}{\partial P_{i}} = a_{i j} P_{j} + d_{j k i} ε_{j k} + f_{j k l i} u_{j, k l}

(15)

E_{i j} = \frac{\partial \cup}{\partial P_{i, j}} = b_{i j k l} P_{k, l} + e_{k l i j} ε_{k l}

(16)

where

σ_{i j}

E_{i}

τ_{i j m}

_, and

E_{i j}

represent stress, electrical field, higher order stress, and higher order electrical field tensors, respectively. Also the strain relations can be represented as follows [47]

{\begin{cases} ε_{x x} = \frac{\partial \tilde{u}}{\partial x}, \\ ε_{y y} = \frac{\partial \tilde{v}}{\partial y}, \\ ε_{x y} = \frac{1}{2} (\frac{\partial \tilde{v}}{\partial x} + \frac{\partial \tilde{u}}{\partial y}), \end{cases} and {\begin{cases} γ_{x x z} = \frac{\partial ε_{x x}}{\partial z}, \\ γ_{y y z} = \frac{\partial ε_{y y}}{\partial z}, \\ γ_{x y z} = \frac{\partial (2 ε_{x y})}{\partial z}, \end{cases} and γ_{x z} = γ_{y z} = 0

(17)

In order to present constitutive equations in a simple and expanded form, some assumptions should be considered as: $c_{11} = c_{1111}$ , $c_{66} = c_{1212}$ , $d_{31} = d_{311}$ , $a_{33} = a_{3333}$ , and $b_{33} = b_{3333}$ , $f_{1133} = f_{2233}$ = $f_{19}$ [48]. According to Figure 1, the electric field $E_{i}$ exists in the z direction only [49] and using equations (15) to (17), the higher order electric field $E_{i j}$ for flexoelectric face sheets based on FDST can be derived as

\begin{array}{l} E_{z} = a_{33} P_{z} + d_{31} (- z (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{2} w}{\partial y^{2}}) + \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + z (\frac{\partial ϕ_{x}}{\partial x} + \frac{\partial ϕ_{y}}{\partial y})) \\ + f_{19} (- (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{2} w}{\partial y^{2}}) + (\frac{\partial ϕ_{x}}{\partial x} + \frac{\partial ϕ_{y}}{\partial y})) \end{array}

(18)

E_{z x} = b_{3133} P_{z, z}

(19)

E_{z y} = b_{3233} P_{z, z}

(20)

E_{z z} = b_{33} P_{z, z} - f_{19} (- z (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{2} w}{\partial y^{2}}) + \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + z (\frac{\partial ϕ_{x}}{\partial x} + \frac{\partial ϕ_{y}}{\partial y}))

(21)

When the flexoelectric face sheet is under an electric potential $φ$ across z direction (across its thickness), the equilibrium equation should be satisfied as [31,46]

E_{z} + \frac{\partial φ}{\partial z} - E_{z x, x} - E_{z y, y} - E_{z z, z} = 0

(22)

where

φ

is the electric potential in z direction. When there is not any free electric charge on the flexoelectric face sheet, the Gauss’s law can be written as [31]

- k φ_{, z z} + P_{z, z} = 0

(23)

where

k = k_{0} k_{b}

k_{0} = 8.85 \times 10^{- 12} C V^{- 1} m^{- 1}

is the permittivity of the air and

k_{b} = 6.62

is the background permittivity of BaTiO₃ when its electric field is across the polarization direction [50]. Using equations (18) to (23) and the electric boundary conditions

E_{i j} n_{j} = 0

φ_{(\frac{h}{2})} = V_{0}

_, and

φ_{(- \frac{h}{2})} = 0

for flexoelectric face sheets, the electric potential can be derived in terms of transverse displacement, rotations of middle surface and applied voltage

V_{0}

\begin{array}{l} φ (x, y, z, t) = \frac{d_{31} (z^{2} - \frac{h^{2}}{4}) (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{2} w}{\partial y^{2}} - \frac{\partial ϕ_{x}}{\partial x} - \frac{\partial ϕ_{y}}{\partial y})}{2 (a_{33} k + 1)} \\ + \frac{V_{0}}{h} z + \frac{V_{0}}{2} + \frac{f_{19} z (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{2} w}{\partial y^{2}} - \frac{\partial ϕ_{x}}{\partial x} - \frac{\partial ϕ_{y}}{\partial y})}{(a_{33} k + 1)} \\ - \frac{f_{19} h (e^{λ_{2} z} - {(e^{λ_{2} z})}^{- 1}) (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{2} w}{\partial y^{2}} - \frac{\partial ϕ_{x}}{\partial x} - \frac{\partial ϕ_{y}}{\partial y}) {(e^{\frac{λ_{2} h}{2}} - {(e^{\frac{λ_{2} h}{2}})}^{- 1})}^{- 1}}{2 (a_{33} k + 1)} \\ + \frac{b_{33} k d_{31}}{{(a_{33} k + 1)}^{2}} (1 - \frac{e^{λ_{2} z} + {(e^{λ_{2} z})}^{- 1}}{e^{\frac{λ_{2} h}{2}} + {(e^{\frac{λ_{2} h}{2}})}^{- 1}}) (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{2} w}{\partial y^{2}} - \frac{\partial ϕ_{x}}{\partial x} - \frac{\partial ϕ_{y}}{\partial y}) \\ - \frac{f_{19}}{(a_{33} k + 1)} (1 - \frac{e^{λ_{2} z} + {(e^{λ_{2} z})}^{- 1}}{e^{\frac{λ_{2} h}{2}} + {(e^{\frac{λ_{2} h}{2}})}^{- 1}}) (\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}) \end{array}

(24)

and using equations (23) and (24), the polarization is

\begin{array}{l} P_{z} = 2 \frac{k d_{31} z (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{2} w}{\partial y^{2}} - \frac{\partial ϕ_{x}}{\partial x} - \frac{\partial ϕ_{y}}{\partial y})}{2 (a_{33} k + 1)} + \frac{k V_{0}}{h} + \frac{k f_{19} (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{2} w}{\partial y^{2}} - \frac{\partial ϕ_{x}}{\partial x} - \frac{\partial ϕ_{y}}{\partial y})}{(a_{33} k + 1)} \\ - \frac{b_{33} k^{2} d_{31}}{{(a_{33} k + 1)}^{2}} (e^{λ_{2} z} λ_{2} - \frac{λ_{2}}{e^{λ_{2} z}}) (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{2} w}{\partial y^{2}} - \frac{\partial ϕ_{x}}{\partial x} - \frac{\partial ϕ_{y}}{\partial y}) {(e^{\frac{λ_{2} h}{2}} + {(e^{\frac{λ_{2} h}{2}})}^{- 1})}^{- 1} \\ - \frac{k f_{19} h}{2 (a_{33} k + 1)} (e^{λ_{2} z} λ_{2} + \frac{λ_{2}}{e^{λ_{2} z}}) (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{2} w}{\partial y^{2}} - \frac{\partial ϕ_{x}}{\partial x} - \frac{\partial ϕ_{y}}{\partial y}) {(e^{\frac{λ_{2} h}{2}} - {(e^{\frac{λ_{2} h}{2}})}^{- 1})}^{- 1} \\ + \frac{k f_{19}}{(a_{33} k + 1)} (e^{λ_{2} z} λ_{2} - \frac{λ_{2}}{e^{λ_{2} z}}) (\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}) {(e^{\frac{λ_{2} h}{2}} + {(e^{\frac{λ_{2} h}{2}})}^{- 1})}^{- 1} \end{array}

(25)

Substitution of equation (25) into equations (18) and (21), respectively, electrical field in the z direction $(E_{z})$ and the higher order electric field $(E_{z z})$ for top and bottom flexoelectric face sheets can be obtained as

\begin{array}{l} E_{z} = a_{33} k (2 \frac{d_{31} z (+ \frac{\partial^{2} w}{\partial x^{2}} - \frac{\partial ϕ_{x}}{\partial x} + \frac{\partial^{2} w}{\partial y^{2}} - \frac{\partial ϕ_{y}}{\partial y})}{2 (a_{33} k + 1)} + \frac{V_{0}}{h} + \frac{f_{19} (+ \frac{\partial^{2} w}{\partial x^{2}} - \frac{\partial ϕ_{x}}{\partial x} + \frac{\partial^{2} w}{\partial y^{2}} - \frac{\partial ϕ_{y}}{\partial y})}{(a_{33} k + 1)} \\ - \frac{f_{19} h}{2 (a_{33} k + 1)} (e^{λ_{2} z} λ_{2} + \frac{λ_{2}}{e^{λ_{2} z}}) (+ \frac{\partial^{2} w}{\partial x^{2}} - \frac{\partial ϕ_{x}}{\partial x} + \frac{\partial^{2} w}{\partial y^{2}} - \frac{\partial ϕ_{y}}{\partial y}) {(e^{\frac{λ_{2} h}{2}} - {(e^{\frac{λ_{2} h}{2}})}^{- 1})}^{- 1} \\ - \frac{b_{33} k d_{31}}{{(a_{33} k + 1)}^{2}} (e^{λ_{2} z} λ_{2} - \frac{λ_{2}}{e^{λ_{2} z}}) (+ \frac{\partial^{2} w}{\partial x^{2}} - \frac{\partial ϕ_{x}}{\partial x} + \frac{\partial^{2} w}{\partial y^{2}} - \frac{\partial ϕ_{y}}{\partial y}) {(e^{\frac{λ_{2} h}{2}} + {(e^{\frac{λ_{2} h}{2}})}^{- 1})}^{- 1} \\ + \frac{f_{19}}{(a_{33} k + 1)} (e^{λ_{2} z} λ_{2} - \frac{λ_{2}}{e^{λ_{2} z}}) (\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}) {(e^{\frac{λ_{2} h}{2}} + {(e^{\frac{λ_{2} h}{2}})}^{- 1})}^{- 1}) \\ + d_{31} (\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} - z \frac{\partial^{2} w}{\partial x^{2}} + z \frac{\partial ϕ_{x}}{\partial x} - z \frac{\partial^{2} w}{\partial y^{2}} + z \frac{\partial ϕ_{y}}{\partial y}) + k_{s} f_{19} (- \frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial ϕ_{x}}{\partial x} - \frac{\partial^{2} w}{\partial y^{2}} + \frac{\partial ϕ_{y}}{\partial y}) \end{array}

(26)

\begin{array}{l} E_{z z} = b_{33} k (\frac{d_{31} (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{2} w}{\partial y^{2}} - \frac{\partial ϕ_{x}}{\partial x} - \frac{\partial ϕ_{y}}{\partial y})}{(a_{33} k + 1)} \\ - \frac{f_{19} h}{(a_{33} k + 1)} (e^{λ_{2} z} λ_{2} 2^{-} \frac{λ_{2}^{2}}{e^{λ_{2} z}}) (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{2} w}{\partial y^{2}} - \frac{\partial ϕ_{x}}{\partial x} - \frac{\partial ϕ_{y}}{\partial y}) {(e^{\frac{λ_{2} h}{2}} - {(e^{\frac{λ_{2} h}{2}})}^{- 1})}^{- 1} \\ - \frac{b_{33} k d_{31}}{{(a_{33} k + 1)}^{2}} (2 e^{λ_{2} z} λ_{2} 2^{+} \frac{2 λ_{2}^{2}}{e^{λ_{2} z}}) (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{2} w}{\partial y^{2}} - \frac{\partial ϕ_{x}}{\partial x} - \frac{\partial ϕ_{y}}{\partial y}) {(e^{\frac{λ_{2} h}{2}} + {(e^{\frac{λ_{2} h}{2}})}^{- 1})}^{- 1} \\ + \frac{f_{19}}{(a_{33} k + 1)} (2 e^{λ_{2} z} λ_{2} 2^{+} \frac{2 λ_{2}^{2}}{e^{λ_{2} z}}) (\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}) {(e^{\frac{λ_{2} h}{2}} + {(e^{\frac{λ_{2} h}{2}})}^{- 1})}^{- 1}) \\ - f_{19} (\frac{\partial u}{\partial x} - z \frac{\partial^{2} w}{\partial x^{2}} + z \frac{\partial ϕ_{x}}{\partial x} + \frac{\partial v}{\partial y} - z \frac{\partial^{2} w}{\partial y^{2}} + z \frac{\partial ϕ_{y}}{\partial y}) \end{array}

(27)

where

λ_{2} = \sqrt{\frac{1 + k a_{33}}{k b_{33}}}

After the derivation of electric field terms, the stresses of flexoelectric face sheets can be determined from the constitutive as follows

{\begin{array}{c} σ_{x x} \\ σ_{y y} \\ σ_{x y} \end{array}} = [\begin{array}{c} c_{11} & c_{12} & 0 \\ c_{21} & c_{22} & 0 \\ 0 & 0 & c_{66} \end{array}] {\begin{array}{c} ε_{x x} \\ ε_{y y} \\ 2 ε_{x y} \end{array}} + [\begin{array}{c} k d_{31} & - k_{s} k f_{19} \\ k d_{32} & - k_{s} k f_{19} \\ 0 & 0 \end{array}] {\begin{array}{c} \frac{\partial φ}{\partial z} \\ - \frac{\partial^{2} φ}{\partial z^{2}} \end{array}}

(28a)

{\begin{array}{c} τ_{x x z} \\ τ_{y y z} \\ τ_{x y z} \end{array}} = [\begin{array}{c} - k_{s} k f_{19} & 0 \\ - k_{s} k f_{19} & 0 \\ 0 & 0 \end{array}] {\begin{array}{c} \frac{\partial φ}{\partial z} \\ - \frac{\partial^{2} φ}{\partial z^{2}} \end{array}}

(28b)

in equation (28),

k_{s}

is shear correction factor which is equal to 5/6 for FSDT.

Governing equation

In order to determine the governing equation of sandwich plates, Hamilton’s principle is employed as follows [51]

δ Π = δ \int_{t_{1}}^{t_{2}} ((\cup^{c}) + (\cup^{f}) - Σ) d t = 0

(29)

in which,

\cup

and

Σ

represent strain energy and external work, respectively. Superscripts c and f are, respectively, representative of core and face sheets.

Using equations (1) to (9), the strain energy of composite core can be represented as [52]

\cup^{c} = \int_{V} \frac{1}{2} [σ_{x x}^{c} ε_{x x}^{c} + σ_{y y}^{c} ε_{y y}^{c} + τ_{x y}^{c} γ_{x y}^{c} + τ_{x z}^{c} γ_{x z}^{c} + τ_{y z}^{c} γ_{y z}^{c}] d \forall

(30)

and using equations (1) to (3), (17), (24) to (26) and (28) the strain energy of flexoelectric face sheets can be obtained as [53]

\cup^{f} = \frac{1}{2} \int_{V} (\begin{array}{l} σ_{x x}^{f} ε_{x x}^{f} + σ_{y y}^{f} ε_{y y}^{f} + 2 σ_{x y}^{f} ε_{x y}^{f} + τ_{x x z}^{f} γ_{x x z}^{f} + τ_{y y z}^{f} γ_{y y z}^{f} + τ_{x y z}^{f} γ_{x y z}^{f} \\ + E_{z} p_{z} - k {(\frac{\partial}{\partial z} φ (x, y, z, t))}^{2} + p_{z} \frac{\partial}{\partial z} φ (x, y, z, t) \end{array}) d \forall

(31)

Pasternak foundation is capable to consider normal and transverse shear loads. The force applied on sandwich plate due to Pasternak foundation can be determined as [43,54]

F^{P a s t e r n a k f o u n d a t i o n} = [\begin{array}{l} K_{w} & K_{g x} & K_{g y} \end{array}] [\begin{array}{l} w (x, y, t) \\ - \frac{\partial^{2} w (x, y, t)}{\partial x^{2}} \\ - \frac{\partial^{2} w (x, y, t)}{\partial y^{2}} \end{array}]

(32)

where

K_{w}

is Winkler spring coefficient and

K_{g x}

K_{g y}

are shear layer parameters in x and y directions, respectively. Therefore the work of elastic medium is as follows

Σ = \frac{1}{2} \int_{A} F^{P a s t e r n a k f o u n d a t i o n} w d A

(33)

Substitution of equations (30), (31), and (33) into equation (29), the equations of motion can be obtained by setting the coefficients $δ u$ , $δ v$ , $δ w$ , $δ ϕ_{x}$ _, and $δ ϕ_{y}$ equal to zero.

Analytical solution

In the case of simply supported sandwich plate, the displacement components can be defined according to the Navier’s method. At the edges of current model, the boundary conditions will be satisfied automatically. According to Navier’s method, the components of displacement are following as [55]

\begin{array}{l} u (x, y, t) = \sum_{m = 1}^{M} \sum_{n = 1}^{N} U_{m n} \cos (α x) \sin (β y) \\ v (x, y, t) = \sum_{m = 1}^{M} \sum_{n = 1}^{N} V_{m n} \cos (β y) \sin (α x) \\ ϕ_{x} (x, y, t) = \sum_{m = 1}^{M} \sum_{n = 1}^{N} P_{m n} \cos (α x) \sin (β y) \\ ϕ_{y} (x, y, t) = \sum_{m = 1}^{M} \sum_{n = 1}^{N} Q_{m n} \cos (β y) \sin (α x) \\ w (x, y, t) = \sum_{m = 1}^{M} \sum_{n = 1}^{N} W_{m n} \sin (α x) \sin (β y) \end{array}

(34)

in which

α = \frac{m π}{a}

and

β = \frac{n π}{b}

U_{m n}, V_{m n}, P_{m n}, Q_{m n}

_, and

W_{m n}

are unknown coefficients of each mode. Eventually, the equations of motion can be defined as a matric form as

[\begin{array}{l} K_{11} & K_{12} & K_{13} & K_{14} & K_{15} \\ K_{21} & K_{22} & K_{23} & K_{24} & K_{25} \\ K_{31} & K_{32} & K_{33} & K_{34} & K_{35} \\ K_{41} & K_{42} & K_{43} & K_{44} & K_{45} \\ K_{51} & K_{52} & K_{53} & K_{54} & K_{55} \end{array}] [\begin{array}{l} U_{m n} \\ V_{m n} \\ P_{m n} \\ Q_{m n} \\ W_{m n} \end{array}] = [\begin{array}{l} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}]

(35)

The arrays of matric [K] are obtained by substituting equation (34) into the governing equation of motion.

Numerical results and discussions

In this section, numerical results and discussions are presented to investigate the flexoelectric effect on buckling and bending analysis of sandwich plate which is includes a CNT-Polymer composite as core and $B a T i O_{3}$ as flexoelectric face sheets. The properties of CNT fibers are defined in Tables 1 and 2 [56].poly{(m-phenylene-vinylene)-co-[(2,5-dioctoxy-p-phenylene) vinylene]} (PmPV) is considered as matrix [56,57], where the material properties of PmPV are expressed in Table 3. Also, The properties for $B a T i O_{3}$ as flexoelectric face sheets are indicated in Table 4 [34,44] and it should be noted that $c_{11} = c_{22}$ , $c_{12} = c_{21}$ _, and $d_{32} = d_{31}$ .

Table 1.

The mechanical properties of CNT fibers at $T = 300^{\circ} k$ [56].

$E_{11}^{C N T} (T P a)$	$E_{22}^{C N T} (T P a)$	$G_{12}^{C N T} (T P a)$	$v_{12}^{C N T}$	$ρ^{C N T} (k g / m^{3})$
$5.6466$	$7.08$	$1.9445$	$0.175$	$1400$

Table 2.

The CNT efficiency parameters for different volume fractions of CNT [56].

$V^{C N T}$	$η_{1}$	$η_{2} = η_{3}$
0.11	0.149	0.934
0.14	0.150	0.941
0.17	0.149	1.381

Table 3.

The properties of PmPV at $T = 300^{\circ} k$ [56,57].

$E^{P} (G P a)$	$G^{P}$	$v^{P}$	$ρ^{P} (k g / m^{3})$
$2.1$	$E^{P} / 2 (1 + v^{P})$	$0.34$	$1150$

Table 4.

The properties of $B a T i O_{3}$ as flexoelectric face sheets [34,44].

$ρ_{f} (k g / m^{3})$	$c_{11} (G P a)$	$c_{12} (G P a)$	$c_{66} (G P a)$	$a_{33} (V m / C)$	$b_{33} (J m^{3} / C^{2})$	$d_{31} (V / m)$	$f_{19} (V)$
$6020$	$167.55$	$78.15$	$44.7$	$0.79 \times 10^{8}$	$1 \times 10^{- 9}$	$3.5 \times 10^{8}$	$10$

In addition $G_{12}^{=} G_{13}^{=}_{G}^{23}$ . The geometrical and mechanical properties of sandwich plate are considered as follows

h_{c} = 40 (μ m), h_{f} = 20 (n m), a = 50 h, \frac{a}{b} = 1, K_{w} = 10^{9} (N / m^{3}), K_{g x} = 100 (N / m), K^{*} = \frac{K_{g y}}{K_{g x}}

To ensure validation, the effects of mode number on the non-dimensional buckling load of CNT reinforced composite square plates compared with Lei et al. [58] which has used kp-Ritz method for obtaining buckling load and Timoshenko and Gere [59] which has used an analytical method for earning buckling load in Table 5.

Table 5.

Comparison among the critical buckling load of composite square plates reinforced by CNT in present study, Lei et al. [58] and Timoshenko and Gere [59].

Mode number	Lei et al. [58] (kp-Ritz method)	Timoshenko and Gere [59] (Analytical method)	Present work (Navier’s method)	% Error respect with
Mode number	Lei et al. [58] (kp-Ritz method)	Timoshenko and Gere [59] (Analytical method)	Present work (Navier’s method)	Lei et al. [58]	Timoshenko and Gere [59]
1	39.1744	39.4761	39.3896	0.55	0.22
2	60.4874	61.6814	61.3391	1.41	0.55
3	105.391	109.656	108.439	2.89	1.11
4	152.501	157.904	156.501	2.62	0.89

Moreover, in order to show the validity of this study in a more touchable way, critical buckling load versus the CNT volume fraction in current study and Lei et al. [58] presented in Figure 2.

As can be observed in Table 5 and Figure 2, there are good agreement among the results of present study, Lei et al. [58] and Timoshenko and Gere [59].

The critical buckling load of nano-sandwich plate versus the voltage applied on top and bottom face sheets for different volume fractions of CNT is shown in Figure 3. By increasing the volume fraction of CNT, the stiffness of structure rises and so, the stability and critical buckling load increase. It is also found that by increasing the voltage which applied on face sheets, the critical buckling load decreases. That is because the biaxial compressive and tensile forces will be generated in the nano-sandwich plate by applying positive and negative voltages on the flexoelectric face sheets, respectively. The compressive/tensile forces will in turn enhance/reduce the stiffness of the flexoelectric face sheets, and hence cause to higher/lower critical buckling load. Therefore, the buckling behavior of nano-sandwich plate with flexoelectric face sheets is controllable by varying the external voltage.

Figure 4 presents the CNT volume fraction effect on the critical buckling load of the sandwich plate versus aspect ratio. This figure approved that increasing aspect ratio of sandwich plate leads to decrease critical buckling load of current model. In fact the increasing aspect ratio causes to decrease system’s stability and therefore critical buckling load decreases. Also, increasing of CNT volume fraction has a positive effect on the critical buckling load, where this enhancement of CNT volume fraction increases the total stiffness of nanocomposite core. But after a special a/b (about 2.5), the curves converge together and it means from this special a/b onward the variations of CNT volume fractions become negligible. In addition, the effect of CNTs volume fraction is more significant at square plate.

Figure 5 indicates the variations of critical buckling load versus thickness of flexoelectric face sheets subjected to different external voltages. It is observed that when the thickness of flexoelectric face sheets increases, the flexibility of sandwich plate increases and leads to increase the energy dissipation in system.

Figure 6 shows the effects of CNT volume fraction and thickness of flexoelectric face sheets on critical buckling load of nano-sandwich plate. It can be concluded that critical buckling load rise up when the CNT volume fraction increase. In addition, in larger values of flexoelectric face sheet’s thickness the critical buckling load becomes smaller.

Figure 7 is used to demonstrate the effects of Pasternak shear constants on the buckling behavior of sandwich plate. $K_{g x}$ represents the Pasternak shear constant in the x direction and $K^{*}$ can be defined as $K^{*} = K_{g y} / K_{g x}$ . Figure 7 shows that by using Pasternak foundation the critical buckling load increases and this enlargement becomes more prominent when the shear constant in x direction to shear layer constant in y direction ratio $(K^{*})$ increases in value. Also, in a determined $K^{*}$ the critical buckling load enhances when $K_{g x}$ increases. So, increasing $K_{g x}$ leads to increase stability of sandwich plate.

In order to examination the effects of Winkler constant under different Pasternak shear constants this study provides Figure 8. It is clearly illustrated that critical buckling load increases with increasing the Pasternak shear constants and it means by $K_{g x}, K_{g y}$ enlargement, the stability of model rises. Also, Figure 8 reveals that larger value of subgrade modulus results larger value of critical buckling load and stability.

Figure 9 depicts the 3D model of critical buckling load for various aspect ratios subjected to different flexoelectric face sheet’s thickness. When flexoelectric face sheet’s thickness has higher magnitude, the critical buckling load has smaller values. Consequently, the stability of sandwich plate decreases with increasing flexoelectric face sheet’s thickness in each aspect ratio. It is worth mentioning that, in larger magnitudes of aspect ratio the critical buckling load become smaller in each flexoelectric face sheet’s thickness.

Figure 10 shows the 3D model of variation of Pasternak shear constant in the x & y directions respect to the critical buckling load. It can be observed from Figure 10 that the critical buckling load moves upward if each one of Pasternak shear constant in the x & y directions increases. It seems to be obvious that increasing $K_{g x}$ and $K_{g y}$ leads to increase stability of sandwich plate which leads to larger magnitude of critical buckling load.

Conclusions

In this paper, based on FSDT the critical buckling load of sandwich plates with consideration the effects of flexoelectricity on the top and bottom face sheets has investigated. For governing equations elicitation, the Hamilton’s principle and Navier’s methods have been hired. The results indicate that the flexoelectric effects play a significant role in critical buckling load of nano-sandwich plates and this role is more prominent when thinner plates have used as face sheets. This study clearly illustrates that in larger value of thickness of face sheets the critical buckling load has smaller value. It is also shows that the critical buckling load is sensitive to applied electrical voltage on face sheets. Also, the increasing volume fraction of CNT fibers causes to increase strength of composite core plate and consequently the critical buckling load of sandwich plate increase. The results demonstrate that there is a good agreement with other results. The current work can claim that the results seem to be so helpful for comprehensive understanding about flexoelectrical buckling behaviors of sandwich plates and their relations. The results of this research are hoped to apply in design and manufacturing of flexoelectric systems as smart materials.

Footnotes

Acknowledgements

The authors would like to thank the reviewers for their valuable comments and suggestions to improve the clarity of this study.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors are grateful to University of Kashan for supporting this work by Grant No. 682579/1. They would also like to thank the Iranian Nanotechnology Development Committee for their financial support.

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