On vibration of functionally graded sandwich nanoplates in the thermal environment

Abstract

This paper deals with the free vibration response of rectangular functionally graded material sandwich nanoplates with simply supported boundary conditions. The material properties of the FGM layers are temperature-dependent and supposed to be graded continuously along the thickness direction. A simple power-law distribution in terms of the volume fractions of the material constituents is employed to obtained the effective material properties. Eringen’s nonlocal elasticity model is incorporated in order to take into account the small size effects. Two types of functionally graded material sandwich nanoplates are considered: a sandwich with functionally graded material face layers and homogeneous core, and a sandwich with homogeneous face layers and functionally graded material core. The equations of motion of the functionally graded material sandwich nanoplates are derived by using the higher shear deformation theory and the Hamilton’s variational principle, and solved using the Navier’s solutions. Several numerical results indicate the influence of the power–law index, the nonlocal parameter, the geometrical parameters of the nanoplate, and the temperature variation on the free vibration response are presented.

Keywords

Functionally graded material sandwich nanoplate free vibration nonlocal higher order shear deformation theory heat conduction

Introduction

With the rapid development of nanotechnology, increasing attention has been devoted to the nanostructures due to their outstanding electronic, physical, chemical, and mechanical properties.

In nanostructures applications, small scale effects are often observed. These effects can be captured using size-dependent continuum mechanics such as nonlocal elasticity theory [1], strain gradient theory [2], the surface elasticity theory [3], modified couple stress theories [4]. Among these theories, the nonlocal elasticity theory of Eringen is the most commonly used theory.

In recent few years, functionally graded materials (FGMs) are taken into account in the nanostructures devices and systems. A number of investigations dealing with dynamic response of functionally graded nanostructures had been studied in the scientific literature extensively. Using the finite element method (FEM), Natarajan et al. [5] employed Eringen’s differential theory to analyze vibration behavior of functionally graded nanoplates based on first-order shear deformation theory (FSDT). Hosseini-Hashemi et al. [6] proposed an analytical approach in combination with Mindlin and nonlocal elasticity plate theories to study free vibration response of thick circular/annular FGM nanoplates under various boundary conditions. Resonance of FGM micro/nanoplate by considering the nonlocal elasticity theory, strain gradient theory, and the small-scale effects is investigated by Nami et al. [7]. The same authors [8] developed an analytical solution for vibration analysis of FGM rectangular nanoplates. Nguyena et al. [9] proposed quasi-3D theory for free vibration, bending, and buckling analysis of FGM nanoplates taking into account the thickness stretching effect and by using an efficient computational approach. Hosseini and Jamalpoor [10] analyzed dynamic behavior of double-FGM viscoelastic nanoplates in thermal environment and resting on Pasternak elastic foundation by considering surface effects. Based on three-dimensional (3D) nonlocal elasticity theory, free vibration of exponential FGM simply supported micro/nanoplates is investigated by Salehipour et al. [11]. They have also [12] used modified couple stress to investigate free vibration response of FGM micro/nanoplates. Salehipour et al. [13] proposed a modified nonlocal theory for vibration analysis of FGM simply supported rectangular micro/nanoplates using FSDT and 3D nonlocal elasticity theory. Zare et al. [14] developed an analytical method to study vibration response of FGM rectangular nanoplates with various boundary conditions. Belkorissat et al. [15] proposed a new nonlocal refined four variable theory for temperature-dependent free vibration of FGM nanoplates. A sinusoidal deformation plate theory in combination with nonlocal elasticity theory is used by Mechab et al. [16] to investigate the small-scale effects on vibration response of FGM nanoplate. Daneshmeh et al. [17] analyzed vibration behaviors of the FGM nanoplate using higher order shear deformation plate theory (HSDT). Ansari et al. [18] presented new solution method (variational differential quadrature) based on 3D nonlocal elasticity theory for bending and free vibration of two types of functionally graded nanoplates. Barati and Shahverdi [19] reported an analytical solution for thermomechanical vibration of FGM nanoplates under different thermal loadings using four-variable plate theory and by considering neutral surface position. The effect of surface stress and thermal loading on the buckling and vibration of FGM nanoplates is carried out by Ansari et al. [20] using a developed nonclassical plate model. Bessaim et al. [21] proposed a nonlocal quasi-3D trigonometric plate theory to investigated micro/nanoscale FGM plates with considering thickness stretching effects. A zeroth-order shear deformation theory is presented by Bounouara et al. [22] to studied free vibration behavior of FGM nanoscale plates resting on elastic foundation. Differential cubature and quadrature–Bolotin methods are used by Kolahchi et al. [23] to study the dynamic stability response of embedded piezoelectric nanoplates subjected to an applied voltage using visco-nonlocal-piezoelasticity theories. Panyatong et al. [24] developed a second-order shear deformation plate theory (SSDT) for vibration analysis of power law, sigmoid, and exponential FGM nanoplates with temperature-dependent material properties. Mechab et al. [25] employed two-variable refined plate theory to investigate free vibration of porous FGM nanoplate resting on Winkler–Pasternak elastic foundations. Nonlinear vibration analysis of piezoelectric nanoplates under thermoelectric loads and various boundary conditions using the differential quadrature method (DQM) is adopted by Liu et al. [26]. Hosseini et al. [27] studied the effect of thermomechanical loads and small-scale effects on double viscoelastic FGM nanoplate system surrounded by an elastic foundation. The same authors [28] investigated thermomechanical vibration response of multi nanoplate system. Jandaghian and Rahmani [29] proposed an analytical solution for free vibration of FGM simply supported piezoelectric nanoscale plates using Eringen’s nonlocal Kirchhoff plate theory. Sobhy and Radwan [30] used a new quasi-3D nonlocal hyperbolic plate theory to analyze buckling and free vibration of FGM nanoplates considering the thickness-stretching effects. Vibration analysis of three-layered exponentially FGM nanoplate with piezomagnetic face layers is discussed by Arefi et al. [31] using the FSDT. Shahverdi and Barati [32] developed a nonlocal strain-gradient elasticity model to examine the vibration characteristics of porous FGM nanoplates under hygrothermal loading. Barati [33] carried out dynamic of nanoplates constructed from porous FGM and metal foam materials considering the effects of both stiffness softening and stiffness hardening using Galerkin’s method. Natural frequencies analysis of rotating circular two directional FGM nanoplate is reported by Mahinzare et al. [34] by employing the modified couple stress theory based on the FSDT. Arefi et al. [35] employed two-variable sinusoidal shear deformation theory to study vibration response of a sandwich FGM nanoplate with FGM core and piezoelectric face layers. Norouzzadeh and Ansari [36] analyzed free vibration response of FGM rectangular and circular nanoplates using a new matrix–vector form of the governing equations of motion. Using isogeometric finite element approach, nonlinear transient response of FGM nanoplate with porosity-dependent material properties under effect of transverse dynamic loads is carried out by Phung-Van et al. [37].

The purpose of the present work is to investigate the free vibration behavior of simply supported functionally graded sandwich nanoplates due to heat conduction. The variations of material properties along the thickness direction are temperature-dependent and vary according to a power function. The temperature distribution solution takes into account the thermal conductivity, the volume fraction index, and the sandwich scheme. Governing equations are derived from the Hamilton’s variational principle combined with the Navier’s solutions for simply supported FGM sandwich plates. Closed form solutions for free vibration of sandwich nanoplates are obtained in this paper taking into account the strain energies due to the applied thermal loads and the generalized higher order shear deformation theories.

FGM sandwich plates

Consider a functionally graded sandwich nanoplate as shown in Figure 1. The nanoplate is composed of three layers. The top face of the nanoplate is at $z = \pm h / 2$ while the bottom face is at $z = \pm h / 2$ . The vertical positions of the bottom and the top are denoted by $h_{0} = - h / 2$ , $h_{3} = h / 2$ , respectively. The two interfaces are at the positions $h_{1}$ , $h_{2}$ . Two kinds of sandwich nanoplates are used: sandwich nanoplates with functionally graded face sheets and ceramic core (FGM-A), and sandwich nanoplates with functionally graded core and homogeneous face sheets (FGM-B).

Figure 1.

Configuration of functionally graded sandwich nanoplates.

FGM-A: sandwich nanoplates with FGM faces and ceramic core

The sandwich nanoplate FGM-A is composed of three sheets, a ceramic core and two functionally graded sheets graded from metal to ceramic. The volume fraction of the nanoplate face sheets varies according to a power–law function along the thickness direction as

\begin{array}{l} V^{(1)} (z) = {(\frac{z - h_{0}}{h_{1} - h_{0}})}^{k}, h_{0} \leq z \leq h_{1} \\ V^{(2)} (z) = 1, h_{1} \leq z \leq h_{2} \\ V^{(3)} (z) = {(\frac{z - h_{3}}{h_{2} - h_{3}})}^{k}, h_{2} \leq z \leq h_{3} \end{array}

(1)

where k is the volume fraction index

(k \geq 0)

FGM-B: sandwich nanoplates with FGM core and homogeneous faces

The bottom layer in this sandwich is fully metal and the top layer is fully ceramic, while the core is graded from metal to ceramic

\begin{array}{l} V^{(1)} (z) = 0, h_{0} \leq z \leq h_{1} \\ V^{(2)} (z) = {(\frac{z - h_{1}}{h_{2} - h_{1}})}^{k}, h_{1} \leq z \leq h_{2} \\ V^{(3)} (z) = 1, h_{2} \leq z \leq h_{3} \end{array}

(2)

The material properties of sandwich nanoplate $P^{(n)}$ of layer $n (n = 1, 2, 3)$ , such as the modulus of elasticity E, the thermal conductivity coefficient $K$ , and the thermal expansion coefficient α, can be written as

P^{(n)} (z) = P_{m} + (P_{c} - P_{m}) V^{(n)} (z)

(3)

where

P_{c}

and

P_{m}

are the temperature-dependent properties of the ceramic and metal, respectively.

Temperature field

Consider an FGM sandwich nanoplate in high temperature environment. The temperature field assumed to be uniform over the sandwich surface. The temperature of the bottom and the top sandwich surfaces is $T_{m}$ and $T_{p}$ , respectively. The material properties are temperature-dependent and are changed along the thickness direction as the following expression [38]

P = P_{0} (P_{- 1} T^{- 1} + 1 + P_{1} T + P_{2} T^{2} + P_{3} T^{3})

(4)

where

T = T_{0} + Δ T

T_{0,}

and

Δ T

are the ambient temperature

(T_{0} = 300 K)

and the temperature change, respectively.

P_{0}

P_{1}

P_{2}

, and

P_{3}

are coefficients of temperature.

The variation of temperature through the thickness is obtained by solving the steady-state heat transfer equation as [39]

- \frac{d}{d z} (K (z) \frac{d T}{d z}) = 0

(5)

To solve this problem, the boundary conditions $T = T_{b}$ at the bottom surface $(z = - h / 2)$ and $T = T_{t}$ at the top surface $(z = h / 2)$ are used. The temperature distribution is obtained as [40]

T (z) = T_{b} + (T_{t} - T_{b}) Θ^{(n)}

(6)

where

\begin{array}{l} Θ^{(1)} = \frac{\int_{h_{0}}^{z} (1 / K^{(1)} (z)) d z}{\sum_{n = 1}^{3} \int_{h_{n - 1}}^{h_{n}} (1 / K^{(n)} (z)) d z}, h_{0} \leq z \leq h_{1} \\ Θ^{(2)} = \frac{\int_{h_{0}}^{h_{1}} (1 / K^{(1)} (z)) d z + \int_{h_{1}}^{z} (1 / K^{(2)} (z)) d z}{\sum_{n = 1}^{3} \int_{h_{n - 1}}^{h_{n}} (1 / K^{(n)} (z)) d z}, h_{1} \leq z \leq h_{2} \\ Θ^{(3)} = \frac{\int_{h_{0}}^{h_{1}} (1 / K^{(1)} (z)) d z + \int_{h_{1}}^{h_{2}} (1 / K^{(2)} (z)) d z + \int_{h_{2}}^{z} (1 / K^{(3)} (z)) d z}{\sum_{n = 1}^{3} \int_{h_{n - 1}}^{h_{n}} (1 / K^{(n)} (z)) d z}, h_{2} \leq z \leq h_{3} \end{array}

(7)

For the sandwich nanoplate FGM-A, the solution of the previous equation can be expressed by means of polynomial series as

\begin{array}{l} T^{1} (z) = T_{b} + \frac{(T_{1} - T_{b})}{C} \sum_{i = 0}^{N} \frac{{(- 1)}^{i}}{i k + 1} {(\frac{k_{c m}}{k_{m}})}^{i} {(\frac{z - h_{0}}{h_{1} - h_{0}})}^{i k + 1} \\ T^{2} (z) = T_{1} + (T_{2} - T_{1}) (\frac{z - h_{1}}{h_{2} - h_{1}}) \\ T^{3} (z) = T_{t} + \frac{(T_{2} - T_{t})}{C} \sum_{i = 0}^{N} \frac{{(- 1)}^{i}}{i k + 1} {(\frac{k_{c m}}{k_{m}})}^{i} {(\frac{z - h_{3}}{h_{2} - h_{3}})}^{i k + 1} \end{array}

(8)

where

C = \sum_{i = 0}^{N} \frac{{(- 1)}^{i}}{i k + 1} {(\frac{k_{c m}}{k_{m}})}^{i}

(9)

and for the sandwich nanoplate FGM-B

\begin{array}{l} T^{1} (z) = T_{b} + (T_{1} - T_{m}) (\frac{z - h_{0}}{h_{1} - h_{0}}) \\ T^{2} (z) = T_{1} + \frac{(T_{2} - T_{1})}{C} \sum_{i = 0}^{N} \frac{{(- 1)}^{i}}{i k + 1} {(\frac{k_{c m}}{k_{m}})}^{i} {(\frac{z - h_{1}}{h_{2} - h_{1}})}^{i k + 1} \\ T^{3} (z) = T_{t} + (T_{2} - T_{t}) (\frac{z - h_{3}}{h_{2} - h_{3}}) \end{array}

(10)

where

\begin{array}{l} T_{1} = T_{m} + (T_{p} - T_{m}) \frac{\int_{h_{0}}^{h_{1}} (1 / K^{(1)} (z)) d z}{\sum_{n = 1}^{3} \int_{h_{n - 1}}^{h_{n}} (1 / K^{(n)} (z)) d z} \\ T_{2} = T_{m} + (T_{p} - T_{m}) \frac{\int_{h_{0}}^{h_{1}} (1 / K^{(1)} (z)) d z + \int_{h_{1}}^{h_{2}} (1 / K^{(2)} (z)) d z}{\sum_{n = 1}^{3} \int_{h_{n - 1}}^{h_{n}} (1 / K^{(n)} (z)) d z} \end{array}

(11)

T_{1}

and

T_{2}

are the temperature value at the positions

h_{1}

and

h_{2}

, respectively.

Theory of nonlocal elasticity

The nonlocal elasticity theory of Eringen [1], it is assumed that the stress tensor at a point is function of the strain tensor at all the points of the continuum. Hence the nonlocal constitutive equation can be defined by the following relation as [41]

(1 - μ \nabla^{2}) σ_{i j}^{N L} = σ_{i j}^{L} = C : ε for i, j = x, y, z

(12)

σ_{i j}^{N L}

and

σ_{i j}^{L}

are nonlocal and local stress tensor.

C

and

ε

denote the elastic constants and the strain, respectively.

\nabla = \partial / \partial x + \partial / \partial y

represents the differential operator and

μ = {(e_{0} a)}^{2}

represents the small-scale effect where

a

denotes the internal characteristic length and

e_{0}

is the material nonlocal scaling component.

Displacement field

Based on the higher order shear deformation plate theory, the generalized displacements field of a material point located at $(x, y, z)$ in the sandwich nanoplate may be expressed as the following form

\begin{array}{l} u (x, y, z) = u_{0} - z \frac{\partial w_{0}}{\partial x} + f (z) φ_{1} \\ v (x, y, z) = v_{0} - z \frac{\partial w_{0}}{\partial y} + f (z) φ_{2} \\ w (x, y, z) = w_{0} \end{array}

(13)

where

(u_{0}, v_{0}, w_{0})

denote the displacements and

(φ_{1}, φ_{2})

denote the rotations of transverse normals on the plane

z = 0

. The displacement field of the present higher order shear deformation plate theory is obtained by setting [42]

f (z) = z (1 - \frac{3 z^{2}}{2 h^{2}} + \frac{2 z^{4}}{5 h^{4}})

(14)

The displacement field of the classical thin plate theory (CPT) is obtained by setting $f (z) = 0$ , whereas the displacement of the first-order shear deformation plate theory FSDT is obtained by setting $f (z) = z$ .

The linear strain components associated with the displacements can be expressed as

\begin{matrix} {\begin{matrix} ε_{x x} \\ ε_{y y} \\ γ_{x y} \end{matrix}} = {\begin{matrix} ε_{x x}^{0} \\ ε_{y y}^{0} \\ γ_{x y}^{0} \end{matrix}} + z {\begin{matrix} ε_{x x}^{1} \\ ε_{y y}^{1} \\ γ_{x y}^{1} \end{matrix}} + f (z) {\begin{matrix} ε_{x x}^{2} \\ ε_{y y}^{2} \\ γ_{x y}^{2} \end{matrix}} \\ ε_{z z} = 0, {\begin{matrix} γ_{y z} \\ γ_{x z} \end{matrix}} = \frac{d f (z)}{d z} {\begin{matrix} γ_{y z}^{0} \\ γ_{x z}^{0} \end{matrix}} \end{matrix}

(15)

where

\begin{array}{l} \{\begin{matrix} ε_{x x}^{0} \\ ε_{y y}^{0} \\ γ_{x y}^{0} \end{matrix}\} = \{\begin{matrix} \frac{\partial u_{0}}{\partial x} \\ \frac{\partial v_{0}}{\partial y} \\ \frac{\partial v_{0}}{\partial x} + \frac{\partial u_{0}}{\partial y} \end{matrix}\}, \{\begin{matrix} ε_{x x}^{1} \\ ε_{y y}^{1} \\ γ_{x y}^{1} \end{matrix}\} = - \{\begin{matrix} \frac{\partial^{2} w_{0}}{\partial x^{2}} \\ \frac{\partial^{2} w_{0}}{\partial y^{2}} \\ 2 \frac{\partial^{2} w_{0}}{\partial x \partial y} \end{matrix}\} \\ \{\begin{matrix} ε_{x x}^{2} \\ ε_{y y}^{2} \\ γ_{x y}^{2} \end{matrix}\} = \{\begin{matrix} \frac{\partial φ_{1}}{\partial x} \\ \frac{\partial φ_{2}}{\partial y} \\ \frac{\partial φ_{2}}{\partial x} + \frac{\partial φ_{1}}{\partial y} \end{matrix}\}, \{\begin{matrix} γ_{y z}^{0} \\ γ_{x z}^{0} \end{matrix}\} = \{\begin{matrix} φ_{1} \\ φ_{2} \end{matrix}\} \end{array}

(16)

Stress–strain relations

In order to capture the effect of small scale, the constitutive relations based on nonlocal elasticity theory can be defined as

[1 - μ \nabla^{2}] {\{\begin{matrix} σ_{x x} \\ σ_{y y} \\ \begin{matrix} \begin{matrix} σ_{y z} \\ σ_{x z} \end{matrix} \\ σ_{x y} \end{matrix} \end{matrix}\}}^{(n)} = {[\begin{matrix} \begin{matrix} \begin{matrix} Q_{11} & Q_{12} & 0 \\ Q_{12} & Q_{22} & 0 \\ \begin{matrix} 0 \\ 0 \\ 0 \end{matrix} & \begin{matrix} 0 \\ 0 \\ 0 \end{matrix} & \begin{matrix} Q_{44} \\ 0 \\ 0 \end{matrix} \end{matrix} & \begin{matrix} 0 \\ 0 \\ \begin{matrix} 0 \\ Q_{55} \\ 0 \end{matrix} \end{matrix} \end{matrix} & \begin{matrix} 0 \\ 0 \\ \begin{matrix} 0 \\ 0 \\ Q_{66} \end{matrix} \end{matrix} \end{matrix}]}^{(n)} \{\begin{matrix} ε_{x x} - α^{(n)} (z) T^{(n)} (z) \\ ε_{y y} - α^{(n)} (z) T^{(n)} (z) \\ \begin{matrix} \begin{matrix} γ_{y z} \\ γ_{x z} \end{matrix} \\ γ_{x y} \end{matrix} \end{matrix}\}

(17)

Q_{11}^{(n)} = Q_{22}^{(n)} = \frac{E^{(n)} (z, T)}{1 - υ^{2} (z, T)}, Q_{12}^{(n)} = υ Q_{11}^{(n)}, Q_{44}^{(n)} = Q_{55}^{(n)} = Q_{66}^{(n)} = \frac{E^{(n)} (z, T)}{2 (1 + υ (z, T))}

(18)

The membrane force $N$ , the bending moment forces $M$ and $P$ , and the resultant shear force $Q$ are defined as

\begin{matrix} \{\begin{matrix} N_{x x} \\ N_{y y} \\ N_{x y} \end{matrix}\} = \sum_{n = 1}^{3} \int_{h_{n - 1}}^{h_{n}} {\{\begin{array}{l} σ_{x x} \\ σ_{y y} \\ σ_{x y} \end{array}\}}^{(n)} d z, \{\begin{array}{l} M_{x x} \\ M_{y y} \\ M_{x y} \end{array}\} = \sum_{n = 1}^{3} \int_{h_{n - 1}}^{h_{n}} {\{\begin{array}{l} σ_{x x} \\ σ_{y y} \\ σ_{x y} \end{array}\}}^{(n)} z d z \\ \{\begin{array}{l} P_{x x} \\ P_{y y} \\ P_{x y} \end{array}\} = \sum_{n = 1}^{3} \int_{h_{n - 1}}^{h_{n}} {\{\begin{array}{l} σ_{x x} \\ σ_{y y} \\ σ_{x y} \end{array}\}}^{(n)} f (z) d z, \{\begin{array}{l} Q_{y z} \\ Q_{x z} \end{array}\} = \sum_{n = 1}^{3} \bar{K} \int_{h_{n - 1}}^{h_{n}} {\{\begin{array}{l} σ_{y z} \\ σ_{x z} \end{array}\}}^{(n)} \frac{d f (z)}{d z} d z \end{matrix}

(19)

where

\bar{K}

is the shear correction factor of the FSDT. The stress resultants of the sandwich nanoplate in function of the strains are given by

\{\begin{array}{l} \{N\} \\ \{M\} \\ \{P\} \end{array}\} = [\begin{array}{l} [A] & [B] & [C] \\ [B] & [D] & [F] \\ [C] & [F] & [H] \end{array}] \{\begin{array}{l} \{ε^{0}\} \\ \{ε^{1}\} \\ \{ε^{2}\} \end{array}\}, \{\begin{array}{l} R_{y z} \\ R_{x z} \end{array}\} = [\begin{matrix} J_{44} & 0 \\ 0 & J_{55} \end{matrix}] \{\begin{matrix} γ_{y z}^{0} \\ γ_{x z}^{0} \end{matrix}\}

(20)

where

\begin{array}{l} \{N\} = {\{\begin{array}{l} N_{x x} & N_{y y} & N_{x y} \end{array}\}}^{T}, \{M\} = {\{\begin{array}{l} M_{x x} & M_{y y} & M_{x y} \end{array}\}}^{T}, \{P\} = {\{\begin{array}{l} P_{x x} & P_{y y} & P_{x y} \end{array}\}}^{T} \\ \{ε^{0}\} = {\{\begin{array}{l} ε_{x x}^{0} & ε_{y y}^{0} & γ_{x y}^{0} \end{array}\}}^{T}, \{ε^{1}\} = {\{\begin{array}{l} ε_{x x}^{1} & ε_{y y}^{1} & γ_{x y}^{1} \end{array}\}}^{T}, \{ε^{2}\} = {\{\begin{array}{l} ε_{x x}^{2} & ε_{y y}^{2} & γ_{x y}^{2} \end{array}\}}^{T} \end{array}

(21)

The coefficients $A_{i j}$ , $B_{i j}$ , $D_{i j}$ , $C_{i j}$ , $F_{i j}$ , and $H_{i j}$ are defined as

\begin{array}{l} \{A_{i j}, B_{i j}, D_{i j}, C_{i j}, F_{i j}, H_{i j}\} = \sum_{n = 1}^{3} \int_{h_{n - 1}}^{h_{n}} {Q_{i j}}^{(n)} \{1, z, z^{2}, f (z), z f (z), {f (z)}^{2}\} d z, (i, j = 1, 2, 6) \\ J_{i i} = \sum_{n = 1}^{3} \int_{h_{n - 1}}^{h_{n}} {Q_{i i}}^{(n)} {[\frac{d f (z)}{d z}]}^{2} d z, (i = 4, 5) \end{array}

(22)

Variational statements

The Hamilton’s principle is used to obtain the equations of motion of the FGM sandwich nanoplate

δ \int_{t_{2}}^{t_{1}} (U ‒ T) d t = 0

(23)

where

U

and

T

are the strain energy and the kinetic energy of the FGM sandwich nanoplate, respectively.

The total strain energy of FGM sandwich nanoplate can be written as

U = U_{p} + U_{T}

(24)

The variation of the strain energy of the nanoplate is written as

U_{p} = \frac{1}{2} \int_{V}^{} [σ_{x x}^{(n)} ε_{x x} + σ_{y y}^{(n)} ε_{y y} + σ_{x y}^{(n)} γ_{x y} + σ_{y z}^{(n)} γ_{y z} + σ_{x z}^{(n)} γ_{x z}] d V

(25)

The strain energies due to the thermal load can be expressed as

U_{T} = \frac{1}{2} \int_{V}^{} [σ_{x x}^{T (n)} d_{x x} + 2 σ_{x y}^{T (n)} d_{x y} + σ_{y y}^{T (n)} d_{y y}] d V

(26)

then

U_{T} = \frac{1}{2} \int_{V}^{} σ_{x x}^{T (n)} ({(\frac{\partial u}{\partial x})}^{2} + {(\frac{\partial v}{\partial x})}^{2} + {(\frac{\partial w}{\partial x})}^{2}) + σ_{y y}^{T (n)} ({(\frac{\partial u}{\partial y})}^{2} + {(\frac{\partial v}{\partial y})}^{2} + {(\frac{\partial w}{\partial y})}^{2}) d V

(27)

The initial stresses can be expressed as

{\{\begin{matrix} σ_{x x}^{T} \\ σ_{y y}^{T} \\ σ_{x y}^{T} \end{matrix}\}}^{(n)} = {[\begin{matrix} Q_{11} & Q_{12} & 0 \\ Q_{12} & Q_{22} & 0 \\ 0 & 0 & Q_{66} \end{matrix}]}^{(n)} [\begin{matrix} α^{(n)} (z) Δ T^{(n)} (z) \\ α^{(n)} (z) Δ T^{(n)} (z) \\ 0 \end{matrix}]

(28)

where

Δ T^{(n)} (z) = T^{(n)} (z) - T_{0}

(29)

The kinetic energy of the sandwich nanoplate at any moment can be written as

T = \frac{1}{2} \int_{0}^{L} \int_{A}^{} ρ (z, T) ({(\frac{\partial u}{\partial t})}^{2} + {(\frac{\partial v}{\partial t})}^{2} + {(\frac{\partial w}{\partial t})}^{2}) d A d x

(30)

By inserting equation (24) to (30) into equation (23) and applying equations (25) and (27), equations of motion for FGM sandwich nanoplate can be obtained as follows

(A_{11} + A_{11}^{T}) \frac{\partial^{2} u_{0}}{\partial x^{2}} + (A_{66} + A_{22}^{T}) \frac{\partial^{2} u_{0}}{\partial y^{2}} + (A_{12} + A_{66}) \frac{\partial^{2} v_{0}}{\partial x \partial y} - (B_{11} + B_{11}^{T}) \frac{\partial^{3} w_{0}}{\partial x^{3}} - (B_{12} + 2 B_{66} + B_{22}^{T}) \frac{\partial^{3} w_{0}}{\partial x \partial y^{2}} + (C_{11} + C_{11}^{T}) \frac{\partial^{2} φ_{x}}{\partial x^{2}} + (C_{66} + C_{22}^{T}) \frac{\partial^{2} φ_{x}}{\partial y^{2}} + (C_{12} + C_{66}) \frac{\partial^{2} φ_{y}}{\partial x \partial y} = (1 - μ \nabla^{2}) [I_{0} \frac{\partial^{2} u_{0}}{{\partial t}^{2}} - I_{1} \frac{\partial^{3} w_{0}}{{\partial x \partial t}^{2}} + I_{3} \frac{\partial^{2} φ_{x}}{{\partial t}^{2}}]

(31)

(A_{12} + A_{66}) \frac{\partial^{2} u_{0}}{\partial x \partial y} + (A_{22} + A_{22}^{T}) \frac{\partial^{2} v_{0}}{\partial y^{2}} + (A_{66} + A_{11}^{T}) \frac{\partial^{2} v_{0}}{\partial x^{2}} - (B_{22} + B_{22}^{T}) \frac{\partial^{3} w_{0}}{\partial y^{3}} - (B_{12} + 2 B_{66} + B_{11}^{T}) \frac{\partial^{3} w_{0}}{\partial x^{2} \partial y} + (C_{12} + C_{66}) \frac{\partial^{2} φ_{x}}{\partial x \partial y} + (C_{22} + C_{22}^{T}) \frac{\partial^{2} φ_{y}}{\partial y^{2}} + (C_{66} + C_{11}^{T}) \frac{\partial^{2} φ_{y}}{\partial x^{2}} = (1 - μ \nabla^{2}) [I_{0} \frac{\partial^{2} v_{0}}{{\partial t}^{2}} - I_{1} \frac{\partial^{3} w_{0}}{{\partial y \partial t}^{2}} + I_{3} \frac{\partial^{2} φ_{y}}{{\partial t}^{2}}]

(32)

- (B_{11} + B_{11}^{T}) \frac{\partial^{3} u_{0}}{\partial x^{3}} - (B_{12} + 2 B_{66} + B_{22}^{T}) \frac{\partial^{3} u_{0}}{\partial x \partial y^{2}} - (B_{12} + 2 B_{66} + B_{11}^{T}) \frac{\partial^{3} v_{0}}{\partial x^{2} \partial y} - (B_{22} + B_{22}^{T}) \frac{\partial^{3} v_{0}}{\partial y^{3}} + (D_{11} + D_{11}^{T}) \frac{\partial^{4} w_{0}}{\partial x^{4}} + (2 D_{12} + 4 D_{66} + D_{11}^{T} + D_{22}^{T}) \frac{\partial^{4} w_{0}}{\partial x^{2} \partial y^{2}} + (D_{22} + D_{22}^{T}) \frac{\partial^{4} w_{0}}{\partial y^{4}} + (1 - μ \nabla^{2}) [A_{11}^{T} \frac{\partial^{2} w_{0}}{\partial x^{2}} + A_{22}^{T} \frac{\partial^{2} w_{0}}{\partial y^{2}}] - (F_{11} + F_{11}^{T}) \frac{\partial^{3} φ_{x}}{\partial x^{3}} - (F_{12} + 2 F_{66} + F_{22}^{T}) \frac{\partial^{3} φ_{x}}{\partial x \partial y^{2}} - (F_{12} + 2 F_{66} + F_{11}^{T}) \frac{\partial^{3} φ_{y}}{\partial x^{2} \partial y} - (F_{22} + F_{22}^{T}) \frac{\partial^{3} φ_{y}}{\partial y^{3}} = (1 - μ \nabla^{2}) [I_{0} \frac{\partial^{2} w_{0}}{{\partial t}^{2}} + I_{1} (\frac{\partial^{3} u_{0}}{\partial x {\partial t}^{2}} + \frac{\partial^{3} v_{0}}{\partial y {\partial t}^{2}}) - I_{2} (\frac{\partial^{4} w_{0}}{{\partial x}^{2} {\partial t}^{2}} + \frac{\partial^{4} w_{0}}{{\partial y}^{2} {\partial t}^{2}}) + I_{4} (\frac{\partial^{3} φ_{x}}{\partial x {\partial t}^{2}} + \frac{\partial^{3} φ_{y}}{\partial y {\partial t}^{2}})]

(33)

(C_{11} + C_{11}^{T}) \frac{\partial^{2} u_{0}}{\partial x^{2}} + (C_{66} + C_{22}^{T}) \frac{\partial^{2} u_{0}}{\partial y^{2}} + (C_{12} + C_{66}) \frac{\partial^{2} v_{0}}{\partial x \partial y} - (F_{11} + F_{11}^{T}) \frac{\partial^{3} w_{0}}{\partial x^{3}} - (F_{12} + 2 F_{66} + F_{22}^{T}) \frac{\partial^{3} w_{0}}{\partial x \partial y^{2}} + (H_{11} + H_{11}^{T}) \frac{\partial^{2} φ_{x}}{\partial x^{2}} + (H_{66} + H_{22}^{T}) \frac{\partial^{2} φ_{x}}{\partial y^{2}} - J_{44} φ_{x} + (H_{12} + H_{66}) \frac{\partial^{2} φ_{y}}{\partial x \partial y} = (1 - μ \nabla^{2}) [I_{3} \frac{\partial^{2} u_{0}}{{\partial t}^{2}} - I_{4} \frac{\partial^{3} w_{0}}{{\partial x \partial t}^{2}} + I_{5} \frac{\partial^{2} φ_{x}}{{\partial t}^{2}}]

(34)

(C_{12} + C_{66}) \frac{\partial^{2} u_{0}}{\partial x \partial y} + (C_{22} + C_{22}^{T}) \frac{\partial^{2} v_{0}}{\partial y^{2}} + (C_{66} + C_{11}^{T}) \frac{\partial^{2} v_{0}}{\partial x^{2}} - (F_{12} + 2 F_{66} + F_{11}^{T}) \frac{\partial^{3} w_{0}}{\partial x^{2} \partial y} - (F_{22} + F_{22}^{T}) \frac{\partial^{3} w_{0}}{\partial y^{3}} + (H_{12} + H_{66}) \frac{\partial^{2} φ_{x}}{\partial x \partial y} + (H_{66} + H_{11}^{T}) \frac{\partial^{2} φ_{y}}{\partial x^{2}} + (H_{22} + H_{22}^{T}) \frac{\partial^{2} φ_{y}}{\partial y^{2}} - J_{55} φ_{y} = (1 - μ \nabla^{2}) [I_{3} \frac{\partial^{2} v_{0}}{{\partial t}^{2}} - I_{4} \frac{\partial^{3} w_{0}}{{\partial y \partial t}^{2}} + I_{5} \frac{\partial^{2} φ_{y}}{{\partial t}^{2}}]

(35)

where

\{A_{i j}^{T}, B_{i j}^{T}, D_{i j}^{T}, C_{i j}^{T}, F_{i j}^{T}, H_{i j}^{T}\} = - \sum_{n = 1}^{3} \int_{h_{n - 1}}^{h_{n}} \frac{α^{(n)} (z, T) E^{(n)} (z, T) Δ T (z)}{1 - υ} \{1, z, z^{2}, f (z), z f (z), {f (z)}^{2}\} d z, (i, j = 1, 2)

(36)

Solution procedure

The functions of the displacements field that satisfy simply supported boundary conditions are developed as Fourier series as follows

\begin{array}{l} \{u_{0} {, φ}_{x}\} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \{U_{m n}, X_{m n}\} \cos (λ x) \sin (β y) e^{i ω t} \\ \{v_{0} {, φ}_{y}\} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \{V_{m n}, Y_{m n}\} \sin (λ x) \cos (β y) e^{i ω t} \\ w_{0} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} W_{m n} \sin (λ x) \sin (β y) e^{i ω t} \end{array}

(37)

where

U_{m n}, V_{m n}, W_{m n}, X_{m n},

and

Y_{m n}

are arbitrary parameters and

λ = m π / L

β = n π / L

. Substituting equation (37) into equations (31 –35) give

([K] - ω^{2} [M]) \{\begin{matrix} U_{m n} \\ V_{m n} \\ \begin{matrix} W_{m n} \\ X_{m n} \\ Y_{m n} \end{matrix} \end{matrix}\} = \{\begin{matrix} 0 \\ 0 \\ \begin{matrix} 0 \\ 0 \\ 0 \end{matrix} \end{matrix}\}

(38)

where [K] and [M] are the rigidity matrix and mass matrix, respectively.

[L] = [\begin{matrix} \begin{matrix} K_{11} & K_{12} & \begin{matrix} K_{13} & K_{14} & K_{15} \end{matrix} \end{matrix} \\ \begin{matrix} K_{12} & K_{22} & \begin{matrix} K_{23} & K_{24} & K_{25} \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} K_{13} & K_{23} & \begin{matrix} K_{33} & K_{34} & K_{35} \end{matrix} \end{matrix} \\ \begin{matrix} K_{14} & K_{24} & \begin{matrix} K_{34} & K_{44} & K_{45} \end{matrix} \end{matrix} \\ \begin{matrix} K_{15} & K_{25} & \begin{array}{l} K_{35} & K_{45} & K_{55} \end{array} \end{matrix} \end{matrix} \end{matrix}], [M] = [\begin{matrix} \begin{matrix} M_{11} & M_{12} & \begin{matrix} M_{13} & M_{14} & M_{15} \end{matrix} \end{matrix} \\ \begin{matrix} M_{21} & M_{22} & \begin{matrix} M_{23} & M_{24} & M_{25} \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} M_{13} & M_{23} & \begin{matrix} M_{33} & M_{34} & M_{35} \end{matrix} \end{matrix} \\ \begin{matrix} M_{14} & M_{24} & \begin{matrix} M_{34} & M_{44} & M_{45} \end{matrix} \end{matrix} \\ \begin{matrix} M_{15} & M_{25} & \begin{matrix} M_{35} & M_{45} & M_{55} \end{matrix} \end{matrix} \end{matrix} \end{matrix}]

(39)

The elements of the matrix [K] and [M] are given as

\begin{array}{l} K_{11} = (A_{11} + A_{11}^{T}) λ^{2} + (A_{66} + A_{22}^{T}) β^{2} \\ K_{12} = λ β (A_{12} + A_{66}) \\ \begin{array}{l} K_{13} = - (B_{11} + B_{11}^{T}) λ^{3} - λ β^{2} (B_{12} + 2 B_{66} + B_{22}^{T}) \\ K_{14} = (C_{11} + C_{11}^{T}) λ^{2} + (C_{66} + C_{22}^{T}) β^{2} \\ \begin{array}{l} \begin{array}{l} K_{15} = λ β (C_{12} + C_{66}) \\ K_{22} = (A_{66} + A_{11}^{T}) λ^{2} + (A_{22} + A_{22}^{T}) β^{2} \end{array} \\ K_{25} = (C_{66} + C_{11}^{T}) λ^{2} + (C_{22} + C_{22}^{T}) β^{2} \\ \begin{array}{l} K_{23} = - (B_{22} + B_{22}^{T}) β^{3} - β λ^{2} (B_{12} + 2 B_{66} + B_{11}^{T}) \\ K_{24} = K_{15} \\ \begin{array}{l} K_{25} = (C_{66} + C_{11}^{T}) λ^{2} + (C_{22} + C_{22}^{T}) β^{2} \\ K_{33} = (D_{11} + D_{11}^{T}) λ^{4} + (2 D_{12} + 4 D_{66} + D_{11}^{T} + D_{22}^{T}) {λ^{2} β}^{2} + (D_{22} + D_{22}^{T}) β^{4} + L (A_{11}^{T} λ^{2} + A_{22}^{T} β^{2}) \\ \begin{array}{l} K_{34} = - (F_{11} + F_{11}^{T}) λ^{3} - λ β^{2} (F_{12} + {2 F}_{66} + F_{22}^{T}) \\ K_{35} = - (F_{22} + F_{22}^{T}) β^{3} - β λ^{2} (F_{12} + {2 F}_{66} + F_{11}^{T}) \\ \begin{array}{l} K_{44} = J_{44} + (H_{11} + H_{11}^{T}) λ^{2} + (H_{66} + H_{22}^{T}) β^{2} \\ K_{45} = λ β (H_{12} + H_{66}) \\ K_{55} = J_{55} + (H_{66} + H_{11}^{T}) λ^{2} + (H_{22} + H_{22}^{T}) β^{2} \end{array} \end{array} \end{array} \end{array} \end{array} \end{array} \end{array}

(40)

and

\begin{array}{l} M_{11} = {L I}_{0} \\ M_{13} = - λ L I_{1} \\ \begin{array}{l} M_{14} = L I_{3} \\ M_{22} = L I_{0} \\ \begin{array}{l} M_{23} = - β L I_{1} \\ M_{15} = L I_{3} \\ \begin{array}{l} M_{33} = {L I}_{0} - L I_{2} (λ^{2} + β^{2}) \\ M_{34} = - λ L I_{4} \\ \begin{array}{l} M_{35} = - β {L I}_{4} \\ M_{44} = {L I}_{5} \\ M_{55} = {L I}_{5} \end{array} \end{array} \end{array} \end{array} \end{array}

(41)

where

M_{12} = M_{14} = M_{15} = M_{45} = 0

, and

L = 1 + μ (λ^{2} + β^{2})

. For the CPT

\begin{array}{l} K_{14} = K_{15} = K_{24} = K_{25} = K_{34} = K_{35} = K_{44} = K_{45} = K_{55} = 0 \\ M_{14} = M_{15} = M_{24} = M_{25} = M_{34} = M_{35} = M_{44} = M_{45} = M_{55} = 0 \end{array}

(42)

Results

In the present article, numerical results are obtained to investigate the influence of high temperature on vibration of functionally graded sandwich nanoplates. The equations of motion are performed based on generalized higher order shear deformation plate theory. The considered sandwich nanoplate is simply supported. The material chosen is a mixture of Titanium alloy Ti-6Al-4V as metal and Zirconia ZrO₂ as ceramic (Table 1). Two types of FGM sandwich nanoplates with different schemes are used. The (1–1–1) FGM sandwich nanoplate $(h_{1} = - h / 6 and h_{2} = h / 6)$ , the (1–2–1) FGM sandwich plate $(h_{1} = - h / 4 and h_{2} = h / 4)$ and the (2–2–1) FGM sandwich plate $(h_{1} = - h / 10 and h_{2} = 3 h / 10)$ .

Table 1.

Temperature-dependent coefficients of elastic modulus $E (P a)$ , coefficient of thermal expansion $α (K^{- 1})$ , mass density $ρ (K g / m^{3})$ , thermal conductivity $k (W / m K)$ , and Poisson’s ratio $υ$ .

	P₀	P_-1	P₁	P₂	P₃
ZrO₂
E	244.27⁺⁹	0	–1371^–03	1214^–06	–3681^–10
α	12.766⁻⁰⁶	0	–1491^–03	1006^–05	–6778^–11
ρ	3000	0	1133^–04	0	0
k	1.8	0	0	0	0
v	0.3	0	0	0	0
Ti-4V-6Al
E	122.56⁺⁹	0	4.586^–4	0	0
α	7.5788^–6	0	6.638^–4	–3.147^–6	0
ρ	4429	0	0	0	0
k	7.82	0	0	0	0
v	0.3	0	0	0	0

The dimensionless natural frequency can be expressed as

\bar{ω} = \frac{ω a^{2}}{h} \sqrt{\frac{(1 - v^{2}) ρ_{0}}{E_{0}}}

(43)

To validate our formulations, a comparison study is performed between the dimensionless natural frequencies $(\hat{ω} = ω h \sqrt{ρ / G})$ of a homogeneous square nanoplate obtained using the proposed theory and those obtained by Aghababaei and Reddy [43] based on CPT, FSDT and third-order shear deformation theory (TSDT), Belkorissat et al. [15] based on hyperbolic refined plate theory and Mechab et al. [16] based on sinusoidal deformation plate theory (see Table 2). The used material properties are $E = 30 \times 10^{6}$ and $υ = 0.3$ with geometric parameters $a = 10, a / b = 1$ and $a / h = 10$ . We can see from Table 2 that the results of dimensionless frequencies obtained based on the CPT are noticeably greater than the results obtained based on other theories.

Table 2.

Comparison of dimensionless natural frequency $\hat{ω} = ω h \sqrt{ρ / G}$ of a homogeneous nanoplate.

a/h	μ²	CPT [43]	FSDT [43]	TSDT [43]	HSDT [15]	SSDT [16]	Present
10	0	0.0963	0.0930	0.0935	0.0930	0.0934	0.0930
	1	0.0880	0.0850	0.0854	0.0850	0.0839	0.0850
	2	0.0816	0.0788	0.0791	0.0787	0.0786	0.0788
	3	0.0763	0.0737	0.0741	0.0737	0.0740	0.0737
	4	0.0720	0.0696	0.0699	0.0695	0.0691	0.0695
	5	0.0683	0.0660	0.0663	0.0659	0.0666	0.0660
20	0	0.0241	0.0239	0.0239	0.0238	−	0.0239
	1	0.0220	0.0218	0.0218	0.0218	−	0.0218
	2	0.0204	0.0202	0.0202	0.0202	−	0.0202
	3	0.0191	0.0189	0.0189	0.0189	−	0.0189
	4	0.0180	0.0178	0.0179	0.0178	−	0.0178
	5	0.0171	0.0169	0.0170	0.0169	−	0.0169

CPT: classical plate theory; FSDT: first-order shear deformation theory; TSDT: third-order shear deformation theory; SSDT: second-order shear deformation plate theory.

In Table 3, dimensionless natural frequencies $\tilde{ω} = 10 ω h \sqrt{ρ_{m} / E_{m}}$ of FGM nanoplate (Si₃N₄/SUS304) obtained in present study are compared with the previously published results of Sobhy and Radwan [30] based on quasi 3D plate theory for different values of nonlocal parameter $μ$ and power law index. The used material properties of SUS304 are: $E_{m} = 201.04 GPa$ , $ρ_{m} = 8166 kg / m^{3}$ , and $υ_{m} = 0.24$ . Whereas, Si₃N₄ are given as: $E_{c} = 348.43 GPa$ , $ρ_{c} = 2370 kg / m^{3}$ , and $υ_{c} = 0.24$ . We can see a good agreement between the obtained results and published ones is obvious, especially for the homogenous nanoplates. Also, it can be observed that the increase of side-to-thickness ratio, decrease the difference between results.

Table 3.

Comparison of dimensionless natural frequency $\tilde{ω} = 10 ω h \sqrt{ρ_{m} / E_{m}}$ of a functionally graded material (FGM) square nanoplate.

a/h	μ	Ceramic		k = 1		k = 5		Metal
a/h	μ	Sobhy [30]	Present	Sobhy [30]	Present	Sobhy [30]	Present	Sobhy [30]	Present
5	0	5.10702	5.10710	3.01860	3.01933	2.42443	2.42450	2.11261	2.11266
	0.5	4.98549	4.98557	2.94677	2.94752	2.36674	2.36682	2.06234	2.06239
	1	4.66713	4.66720	2.75859	2.75940	2.21560	2.21574	1.93064	1.93068
	1.5	4.24976	4.24982	2.51190	2.51274	2.01747	2.01765	1.75799	1.75803
	2	3.81763	3.81769	2.25648	2.25732	1.81232	1.81253	1.57923	1.57927
10	0	1.38829	1.38828	0.82250	0.82296	0.66485	0.66510	0.57695	0.57695
	0.5	1.35525	1.35524	0.80292	0.80338	0.64903	0.64927	0.56322	0.56322
	1	1.26871	1.26869	0.75165	0.75208	0.60758	0.60781	0.52725	0.52725
	1.5	1.15525	1.15524	0.68443	0.68483	0.55325	0.55346	0.48010	0.48010
	2	1.03778	1.03777	0.61484	0.61520	0.49699	0.49719	0.43128	0.43128
20	0	0.35558	0.35558	0.21083	0.21098	0.17077	0.17086	0.14799	0.14800
	0.5	0.34712	0.34712	0.20581	0.20596	0.16671	0.16679	0.14447	0.14447
	1	0.32495	0.32495	0.19267	0.19280	0.15606	0.15614	0.13524	0.13525
	1.5	0.29589	0.29589	0.17544	0.17556	0.14210	0.14218	0.12315	0.12315
	2	0.26581	0.26581	0.15760	0.15771	0.12765	0.12772	0.11063	0.11063
50	0	0.05730	0.05730	0.03398	0.03400	0.02754	0.02756	0.02385	0.02386
	0.5	0.05593	0.05593	0.03317	0.03320	0.02688	0.02690	0.02329	0.02329
	1	0.05236	0.05236	0.03105	0.03108	0.02517	0.02518	0.02180	0.02180
	1.5	0.04768	0.04768	0.02827	0.02830	0.02291	0.02293	0.01985	0.01985
	2	0.04283	0.04283	0.02540	0.02542	0.02058	0.02060	0.01783	0.01783

In Table 4, dimensionless natural frequencies of FGM plate with temperature-dependent material properties (Table 1), obtained using present theory are compared with the previously published results of Huang and Shen [44] based on TSDT and Shahrjerdi et al. [45] based on SSDT for different values of volume fraction index. The results have been found by considering the geometric values $h = 0.025 m$ and side $a = 0.2 m$ . The result for FGM plates is in between those for pure material plates, because Young’s modulus increases from pure metal to pure ceramic. It is clear that our results are in good agreement with those generated by TSDT and SSDT, which highlight the efficiency and correctness of the present model.

Table 4.

Comparison of dimensionless natural frequency $\bar{ω}$ of a functionally graded material (FGM) plate.

Temperature	k	Theory	Mode
Temperature	k	Theory	(1, 1)	(1, 2)	(2, 2)	(1, 3)	(2, 3)
T_b = 300 K T_t = 300 K	ZrO₂	Hung et al. [44]	8.273	19.261	28.962	34.873	43.070
		Shahrjerdi et al. [45]	8.333	19.613	29.785	36.084	44.942
		Present	8.280	19.345	29.217	35.288	43.785
	0.5	Hung et al. [44]	7.139	16.643	25.048	30.174	37.288
		Shahrjerdi et al. [45]	7.156	16.852	25.604	31.036	38.670
		Present	7.113	16.633	25.137	30.373	37.706
	1	Hung et al. [44]	6.657	15.514	23.345	28.120	34.747
		Shahrjerdi et al. [45]	6.700	15.774	23.959	29.040	36.176
		Present	6.658	15.559	23.504	28.392	35.236
	2	Hung et al. [44]	6.286	14.625	21.978	26.454	32.659
		Shahrjerdi et al. [45]	6.333	14.896	22.608	27.392	34.106
		Present	6.288	14.669	22.128	26.711	33.117
	Ti-6AL-4V	Hung et al. [44]	5.400	12.571	18.903	22.762	28.111
		Shahrjerdi et al. [45]	5.439	12.801	19.440	23.551	29.333
		Present	5.404	12.626	19.069	23.032	28.578
T_b = 300 K T_t = 400 K	ZrO₂	Hung et al. [44]	7.868	18.659	28.203	34.015	42.045
		Shahrjerdi et al. [45]	7.614	18.585	28.471	34.588	43.192
		Present	7.8070	18.566	28.157	34.055	42.305
	0.5	Hung et al. [44]	6.876	16.264	24.578	29.651	36.664
		Shahrjerdi et al. [45]	6.651	16.126	24.677	29.981	37.439
		Present	6.781	16.086	24.397	29.514	36.679
	1	Hung et al. [44]	6.437	15.202	22.956	27.696	34.236
		Shahrjerdi et al. [45]	6.281	15.161	23.172	28.143	35.128
		Present	6.375	15.090	22.867	27.653	34.353
	2	Hung et al. [44]	6.101	14.372	21.653	26.113	32.239
		Shahrjerdi et al. [45]	5.992	14.383	21.942	26.630	33.211
		Present	6.048	14.263	21.577	26.070	32.353
	Ti-6Al-4V	Hung et al. [44]	5.322	12.455	18.766	22.603	27.921
		Shahrjerdi et al. [45]	5.103	12.130	18.467	22.390	27.907
		Present	5.287	12.404	18.755	22.661	28.127
T_b = 300K T_t = 600K	ZrO₂	Hung et al. [44]	6.685	16.986	26.073	31.567	39.212
		Shahrjerdi et al. [45]	5.469	16.092	25.518	31.331	39.506
		Present	6.522	16.785	25.879	31.460	39.256
	0.5	Hung et al. [44]	6.123	15.169	23.166	28.041	34.789
		Shahrjerdi et al. [45]	5.255	14.431	22.647	27.734	34.887
		Present	5.925	14.868	22.833	27.732	34.588
	1	Hung et al. [44]	5.819	14.287	21.768	26.342	32.660
		Shahrjerdi et al. [45]	5.167	13.765	21.482	26.264	32.985
		Present	5.660	14.055	21.533	26.132	32.566
	2	Hung et al. [44]	5.612	13.611	20.652	24.961	30.904
		Shahrjerdi et al. [45]	5.139	13.260	20.557	25.077	31.425
		Present	5.463	13.392	20.442	24.771	30.823
	Ti-6Al-4V	Hung et al. [44]	5.118	12.059	18.175	21.898	27.045
		Shahrjerdi et al. [45]	4.836	11.655	17.802	21.604	26.954
		Present	5.091	12.003	18.168	21.960	27.266

Dimensionless natural frequency $\bar{ω}$ for different types and schemes of FGM sandwich nanoplates in function of the nonlocal parameter $μ$ is shown in Table 5 using the classical plate theory, the first order and the present higher order shear deformation theory. Temperature at the bottom surface is taken as constant $(T_{b} = 300 K)$ in all the following results (except Table 6). It is clear that the results obtained by the classical plate theory are higher than that for the other plate theories.

Table 5.

Dimensionless natural frequency $\bar{ω}$ of a functionally graded material (FGM) sandwich nanoplate for different theories $(Δ T = 100 K, k = 2) .$

μ	Theory	FGM-A			FGM-B
μ	Theory	1–1–1	1–2–1	2–2–1	1–1–1	1–2–1	2–2–1
0	CPT	6.2516	6.5358	6.3955	6.4700	6.4459	6.3490
	FSDT	6.0986	6.3733	6.2370	6.2911	6.2664	6.1676
	HSDT	6.1093	6.3821	6.2465	6.2891	6.2632	6.1604
0.5	CPT	5.9404	6.2075	6.0763	6.1526	6.1308	6.0405
	FSDT	5.7938	6.0517	5.9244	5.9813	5.9589	5.8670
	HSDT	5.8040	6.0602	5.9334	5.9793	5.9559	5.8601
1	CPT	5.6673	5.9194	5.7962	5.8743	5.8547	5.7703
	FSDT	5.5263	5.7695	5.6500	5.7097	5.6894	5.6035
	HSDT	5.5362	5.7776	5.6588	5.7078	5.6865	5.5969
1.5	CPT	5.4251	5.6638	5.5477	5.6277	5.6099	5.5309
	FSDT	5.2890	5.5189	5.4066	5.4689	5.4506	5.3701
	HSDT	5.2985	5.5268	5.4150	5.4670	5.4478	5.3637
2	CPT	5.2082	5.4347	5.3252	5.4070	5.3910	5.3168
	FSDT	5.0764	5.2944	5.1885	5.2533	5.2368	5.1613
	HSDT	5.0856	5.3020	5.1966	5.2516	5.2341	5.1552

CPT: classical plate theory; FSDT: first-order shear deformation theory; HSDT: higher order shear deformation plate theory.

Table 6.

Dimensionless natural frequency $\bar{ω}$ of functionally graded material (FGM) sandwich nanoplate under constant temperature rise.

Temperature	k	μ	FGM-A			FGM-B
Temperature	k	μ	1–1–1	1–2–1	2–2–1	1–1–1	1–2–1	2–2–1
T_b = 300 K T_t = 300 K	0	0	8.4323	8.4323	8.4323	7.0146	7.2437	6.8728
		0.5	8.0447	8.0447	8.0447	6.6921	6.9107	6.5569
		1	7.7060	7.7060	7.7060	6.4104	6.6197	6.2808
		1.5	7.4068	7.4068	7.4068	6.1615	6.3627	6.0370
		2	7.1399	7.1399	7.1399	5.9395	6.1334	5.8194
	0,5	0	7.3490	7.5575	7.4459	6.8094	6.8832	6.6596
		0.5	7.3490	7.2101	7.1036	6.4964	6.5667	6.3534
		1	6.7160	6.9065	6.8046	6.2229	6.2903	6.0859
		1.5	6.4553	6.6383	6.5403	5.9813	6.0460	5.8496
		2	6.2227	6.3992	6.3047	5.7658	5.8282	5.6389
	1	0	6.8823	7.1642	7.0154	6.7163	6.7278	6.5586
		0.5	6.5659	6.8349	6.6929	6.4076	6.4186	6.2571
		1	6.2895	6.5472	6.4111	6.1378	6.1483	5.9937
		1.5	6.0453	6.2929	6.1622	5.8995	5.9096	5.7610
		2	5.8275	6.0662	5.9402	5.6869	5.6967	5.5534
	2	0	6.4785	6.8092	6.6388	6.6303	6.5901	6.4591
		0.5	6.1806	6.4962	6.3336	6.3255	6.2871	6.1622
		1	5.9204	6.2227	6.0670	6.0592	6.0224	5.9027
		1.5	5.6905	5.9811	5.8314	5.8239	5.7886	5.6736
		2	5.4855	5.7656	5.6213	5.6141	5.5800	5.4691
	5	0	6.1529	6.5035	6.3301	6.5505	6.4656	6.3552
		0.5	5.8700	6.2045	6.0391	6.2494	6.1684	6.0631
		1	5.6229	5.9433	5.7848	5.9863	5.9087	5.8078
		1.5	5.4045	5.7125	5.5602	5.7539	5.6793	5.5823
		2	5.2098	5.5067	5.3599	5.5466	5.4746	5.3812
T_b = 400 K T_t = 400 K	0	0	8.0194	8.0194	8.0194	6.7438	6.9514	6.6138
		0.5	7.6507	7.6507	7.6507	6.4338	6.6318	6.3097
		1	7.3286	7.3286	7.3286	6.1629	6.3526	6.0441
		1.5	7.0441	7.0441	7.0441	5.9236	6.1060	5.8094
		2	6.7903	6.7903	6.7903	5.7102	5.8860	5.6001
	0,5	0	7.0416	7.2308	7.1292	6.5532	6.6192	6.4128
		0.5	6.7179	6.8984	6.8015	6.2519	6.3149	6.1180
		1	6.4351	6.6079	6.5151	5.9887	6.0490	5.8605
		1.5	6.1852	6.3514	6.2621	5.7562	5.8142	5.6329
		2	5.9624	6.1225	6.0365	5.5488	5.6047	5.4300
	1	0	6.6215	6.8773	6.7418	6.4663	6.4753	6.3174
		0.5	6.3171	6.5611	6.4319	6.1690	6.1776	6.0270
		1	6.0512	6.2849	6.1611	5.9093	5.9176	5.7733
		1.5	5.8162	6.0409	5.9219	5.6799	5.6878	5.5491
		2	5.6067	5.8232	5.7085	5.4752	5.4829	5.3492
	2	0	6.2580	6.5584	6.4030	6.3855	6.3467	6.2231
		0.5	5.9703	6.2569	6.1086	6.0919	6.0549	5.9370
		1	5.7189	5.9934	5.8515	5.8355	5.8000	5.6871
		1.5	5.4969	5.7607	5.6243	5.6089	5.5748	5.4663
		2	5.2988	5.5532	5.4216	5.4068	5.3739	5.2693
	5	0	5.9635	6.2833	6.1241	6.3100	6.2291	6.1248
		0.5	5.6893	5.9944	5.8426	6.0199	5.9427	5.8432
		1	5.4498	5.7421	5.5966	5.7665	5.6926	5.5972
		1.5	5.2382	5.5191	5.3793	5.5426	5.4715	5.3799
		2	5.0495	5.3202	5.1855	5.3429	5.2744	5.1861

For more details, the effect of temperature, the power law index $k$ and the nonlocal parameter $μ$ on the dimensionless natural frequency $\bar{ω}$ of various functionally graded sandwich nanoplates is illustrated in Tables 6 and 7 by using the present HSDT. Table 6 shows the effect of the constant temperature distribution on the nanoplate frequencies. The temperature through-the-thickness is uniform and take value of $300 K$ and $400 K$ . Whereas, Table 7 shows nonlinear temperature distribution. It can be seen that the fully ceramic nanoplates $(k = 0)$ have the highest values of dimensionless frequency $\bar{ω}$ because of their low thermal conductivity.

Table 7.

Dimensionless natural frequency $\bar{ω}$ of functionally graded material (FGM) sandwich nanoplate under nonlinear temperature rise.

Temperature	k	μ	FGM-A			FGM-B
Temperature	k	μ	1–1–1	1–2–1	2–2–1	1–1–1	1–2–1	2–2–1
T_b = 300 K T_t = 400 K	0	0	7.8131	7.8131	7.8131	6.5600	6.7474	6.4505
		0.5	7.4153	7.4153	7.4153	6.2290	6.4050	6.1268
		1	7.0660	7.0660	7.0660	5.9385	6.1044	5.8428
		1.5	6.7557	6.7557	6.7557	5.6805	5.8374	5.5908
		2	6.4775	6.4775	6.4775	5.4494	5.5981	5.3650
	0.5	0	6.8640	7.0404	6.9553	6.4262	6.4925	6.3129
		0.5	6.5173	6.6834	6.6042	6.1069	6.1699	6.0017
		1	6.2129	6.3700	6.2961	5.8269	5.8869	5.7288
		1.5	5.9427	6.0918	6.0225	5.5785	5.6359	5.4869
		2	5.7006	5.8423	5.7773	5.3561	5.4111	5.2704
	1	0	6.4572	6.6934	6.5758	6.3557	6.3715	6.2375
		0.5	6.1326	6.3548	6.2450	6.0414	6.0570	5.9318
		1	5.8477	6.0576	5.9547	5.7658	5.7812	5.6639
		1.5	5.5948	5.7937	5.6969	5.5214	5.5367	5.4264
		2	5.3683	5.5572	5.4660	5.3026	5.3178	5.2139
	2	0	6.1093	6.3821	6.2465	6.2891	6.2632	6.1604
		0.5	5.8040	6.0602	5.9334	5.9793	5.9559	5.8601
		1	5.5362	5.7776	5.6588	5.7078	5.6865	5.5969
		1.5	5.2985	5.5268	5.4150	5.4670	5.4478	5.3637
		2	5.0856	5.3020	5.1966	5.2516	5.2341	5.1552
	5	0	5.8351	6.1174	5.9820	6.2265	6.1647	6.0769
		0.5	5.5458	5.8101	5.6838	5.9210	5.8640	5.7822
		1	5.2921	5.5403	5.4223	5.6532	5.6005	5.5240
		1.5	5.0670	5.3010	5.1902	5.4159	5.3670	5.2952
		2	4.8655	5.0865	4.9823	5.2035	5.1582	5.0907
T_b = 300 K T_t = 600 K	0	0	5.9306	5.9306	5.9306	5.1834	5.2300	5.1860
		0.5	5.4443	5.4443	5.4443	4.7891	4.8153	4.8066
		1	5.0020	5.0020	5.0020	4.4329	4.4393	4.4650
		1.5	4.5939	4.5939	4.5939	4.1069	4.0937	4.1535
		2	4.2121	4.2121	4.2121	3.8048	3.7717	3.8662
	0.5	0	5.4450	5.4986	5.5064	5.3181	5.3711	5.3372
		0.5	5.0364	5.0717	5.0917	4.9545	5.0041	4.9901
		1	4.6678	4.6854	4.7174	4.6290	4.6755	4.6805
		1.5	4.3309	4.3311	4.3752	4.3340	4.3777	4.4010
		2	4.0191	4.0019	4.0584	4.0638	4.1049	4.1461
	1	0	5.2504	5.3123	5.2980	5.3253	5.3661	5.3517
		0.5	4.8759	4.9126	4.9126	4.9716	5.0142	5.0157
		1	4.5396	4.5520	4.5659	4.6556	4.7001	4.7167
		1.5	4.2335	4.2223	4.2499	4.3699	4.4164	4.4476
		2	3.9520	3.9171	3.9585	4.1088	4.1575	4.2028
	2	0	5.1030	5.1547	5.1287	5.3238	5.3559	5.3526
		0.5	4.7591	4.7803	4.7697	4.9786	5.0170	5.0267
		1	4.4516	4.4435	4.4477	4.6708	4.7153	4.7373
		1.5	4.1732	4.1366	4.1553	4.3930	4.4436	4.4774
		2	3.9185	3.8537	3.8867	4.1397	4.1963	4.2415
	5	0	5.0168	5.0388	5.0173	5.3187	5.3449	5.3412
		0.5	4.6994	4.6874	4.6815	4.9813	5.0178	5.0250
		1	4.4169	4.3724	4.3814	4.6809	4.7273	4.7449
		1.5	4.1625	4.0864	4.1098	4.4103	4.4663	4.4938
		2	3.9309	3.8238	3.8615	4.1639	4.2294	4.2664

Figure 2 describes the effect of temperature on the variation of Young’s modulus along the sandwich nanoplate thickness. The temperature of the bottom surface is constant $T_{b} = 300 K$ while the temperature of the top surface is varied $(Δ T = 0, 200, 400 K)$ . It is clear from this figure that the high temperature has an important influence on Young’s Modulus, where the increase of temperature decreases the values Young’s Modulus, because the temperature change influences the materials properties of the FG sandwich nanoplates and make them more flexible.

Figure 2.

Young’s modulus variation through the FGM sandwich nanoplate thickness.

The influence of nonlocal parameter $μ$ on free vibration response of FGM sandwich nanoplates (1–1–1) under different values of temperature difference $(Δ T = 0 \sim 400 K)$ is illustrated in Figure 3. The dimensionless frequency $\bar{ω}$ decreases by increasing of the parameter $μ$ . It is important to observe that the nonlocality has a softening effect on the FGM sandwich nanoplate stiffness. Accordingly, the nonlocal frequencies $(μ \neq 0)$ are lower than the local ones $(μ = 0)$ . Also, the temperature has a significant impact on the sandwich nanoplate FGM-A, where the increase of temperature decreases the frequencies.

Figure 3.

Effect of nonlocal parameter $μ$ on dimensionless frequency $\bar{ω} (k = 2, a / h = 10)$ .

Figure 4 demonstrates the effects of power law index $k$ and temperature difference on the vibration frequencies of FGM sandwich nanoplates FGM-A and FGM-B. It can be noticed that increasing of the parameter $k$ lead to a decrement in the variation of the frequencies $\bar{ω}$ . Note that the frequencies $\bar{ω}$ decrease rapidly for the values of $k \leq 2$ , then, $\bar{ω}$ decreases gradually for $k \geq 2$ wherever the sandwich type is. Because the ceramic plate is stiffer than the FGM ones, the dimensionless frequencies of the ZrO₂ nanoplates (FGM-A with $k = 0$ ) are greater than those of FGM plates. Also, power law index has more significant impact on the FGM-A plates than the FGM-B plates.

Figure 4.

Effect of power law index $k$ on dimensionless frequency $\bar{ω} (μ = 0, a / h = 10)$ .

The dimensionless natural frequencies $\bar{ω}$ in function of the geometrical parameter $b / a$ of various schemes of sandwich nanoplates due to a heat conduction are plotted in Figure 5. It can be seen that dimensionless frequencies decreases rapidly for $b / a \leq 3$ , then, $\bar{ω}$ decrease gradually for the highest values of $b / a$ wherever the sandwich type is.

Figure 5.

Effect of geometric parameter $b / a$ on dimensionless frequency $\bar{ω} (μ = 0, a / h = 10)$ .

Figure 6 presents the dimensionless frequency $\bar{ω}$ against temperature difference $Δ T$ for different nonlocal parameter $μ$ values. The frequencies $\bar{ω}$ decreases with the increasing of the parameter $μ$ and the temperature difference $Δ T$ . We can say that the inclusion of size effects leads to the reduction of the stiffness of the FGM sandwich nanoplates. Also, dimensionless frequencies decrease as the temperature difference increase. The reason is that the increasing the temperature leads to decrement of Young’s modulus and therefore decrement of FGM sandwich plate stiffness.

Figure 6.

Effect of temperature difference $Δ T$ on dimensionless frequency $\bar{ω} (k = 2, a / h = 10)$ .

Conclusion

This paper presents an analytical solution for free vibration of functionally graded sandwich nanoplates. Material properties of FGM layers are assumed to vary continuously along the thickness direction according to a simple power–law distribution. Several types of FGM sandwich nanoplates are used. Hamilton’s principle in conjunction with the Navier’s solution was employed to calculate the free vibration frequencies of FGM sandwich nanoplates based on generalized higher order shear deformation plate theory and by considering the strain energies due to the thermal loading. An exact solution for the nonlinear temperature variation across the thickness direction of the sandwich nanoplate is proposed in this research taking into account the thermal conductivity, the power law index, and the sandwich scheme.

As a result, the dimensionless frequency of FGM sandwich nanoplates is significantly influenced by temperature field, power low index distributions, the nonlocal parameter, and geometric of the plate. The numerical examples show that:

The ceramic nanoplate (FGM-A with $k = 0$ ) has the highest frequencies because it has the greater stiffness than the FGM sandwich nanoplates. Further, increasing the volume fraction index $k$ decreases the dimensionless frequencies wherever the sandwich type is.

The inclusion of size effects leads to the reduction of the stiffness of the FGM sandwich nanoplates. Thus, the dimensionless frequencies decrease with increasing of nonlocal parameter μ.

The existence of high temperature demotes the stiffness of the FGM sandwich nanoplate. Therefore, increasing the temperature leads to decrement of dimensionless frequencies.

Temperature difference has more significant impact on the FGM-A plates than the FGM-B plates.

The dimensionless frequencies decrease by increasing of the geometric parameter $b / a$ .

Footnotes

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) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Ahmed Amine Daikh

Ismail Bensaid

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