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*Appl. Sci.*
**2017**,
*7*(12),
1256;
https://doi.org/10.3390/app7121256

Article

Closed Form Solutions for Thermal Buckling of Functionally Graded Rectangular Thin Plates

Institute of Solid Mechanics, Beihang University, Beijing 100191, China

^{*}

Author to whom correspondence should be addressed.

Received: 31 October 2017 / Accepted: 30 November 2017 / Published: 3 December 2017

## Abstract

**:**

This work concerns the critical buckling temperature of functionally graded rectangular thin plates; the properties of functionally graded material vary continuously in accordance with the power law of thickness z. Closed form solutions for the critical thermal parameter are obtained for the plate with the following boundary condition combinations: simply supported, clamped and guided edges, under uniform, linear and nonlinear temperature fields via the separation-of-variable method. Furthermore, a new method is proposed to determine the critical buckling temperature from the critical thermal parameter. The present results coincide well with those in the literature, verifying the correctness of the present method. The influences of the length–thickness ratio, length–width ratio, power law index and initial temperature on critical buckling temperature are investigated.

Keywords:

functionally graded materials; thermal buckling; rectangular thin plates; critical temperature; closed form solution## 1. Introduction

Designable Functionally Graded Materials (FGMs) are increasingly widely used in the aerospace, nuclear, and ship industries and other various industry departments, especially in thermal environments with ultra-high temperatures and a large temperature gradient. FGMs were proposed as thermal barrier materials by Japanese scientist Koizumi [1,2] to solve many problems in the new generation of aerospace airplane thermal protection systems. The volume content of each material in FGMs is continuously distributed in the spatial position on the macroscopic scale, and the physical properties change smoothly, so that the stress concentration of FGMs can be better avoided or reduced.

Since FGMs usually work in extremely high temperature environments, in order to ensure structural stability, the critical buckling temperature of FGM structures is very important in theory and application. Many researchers have studied the thermal buckling of FGM plates based on different theories in the past two decades.

Javaheri and Eslami [3] developed closed form solutions for the thermal buckling of simply supported FGM thin plates based on the Classical thin Plate Theory (CPT). Shariat and Eslami [4] obtained closed form solutions for the thermal buckling of simply supported imperfect FGM thin plates. Kiani et al. [5] investigated the critical buckling temperature of fully clamped FGM thin plates resting on elastic foundations; they used the Bubnov–Galerkin Solution (BGS), Power Series Solution (PSS) and Semi-Levy Solution (SLS) respectively to represent the lateral displacement which cannot be obtained by the Navier solution, and compared the solutions using these different approaches. Chu et al. [6] studied the critical buckling load and buckling model of FGM thin plate with in-plane material inhomogeneity by using radial basis function, but the thermal environment was not taken into account in their work.

Shear deformation plays a more prominent role for thick plates, so shear deformation theories are more accurate than CPT in this case. Wu [7] and Bouazza et al. [8] obtained a closed form solution for thermal buckling of simply supported FGM thick plates based on the first-order shear deformation theory (FSDT), and compared the results with those of CPT. Shariat and Eslami [9] developed closed form solutions for thermal buckling of imperfect simply supported FGM thick plates based on FSDT. Lee et al. [10] obtained a finite element solution for the critical buckling temperature based on neutral surface and FSDT. Park and Kim [11] presented thermal postbuckling and vibration analyses based on FSDT by using the nonlinear finite element method. Cong and Duc [12] studied the thermal stability of eccentrically stiffened sigmoid FGM thick plate with metal–ceramic–metal layers by the Galerkin method based on FSDT.

There are also some researches based on high-order shear deformation theories. Javaheri and Eslami [13] developed closed form solutions for thermal buckling of simply supported FGM thick plates based on the third-order shear deformation theory (TSDT). Najafizadeh and Heydari [14] investigated the critical buckling temperature of clamped FGM circular thick plates based on TSDT, and compared it with results based on CPT and FSDT. Matsunaga [15] investigated the critical buckling temperature of simply supported FGM thick plates based on the 2D higher-order deformation theory that he developed, and compared the results with CPT and other lower-order shear deformation theories such as FSDT and TSDT. Zenkour and Mashat [16] studied thermal buckling of simply supported FGM thick plates based on sinusoidal shear deformation plate theory (SPT) and obtained closed form solutions.

Some researchers have used three-dimensional linear elasticity theory to analyze the thermal buckling of FGM plates. Na and Kim [17] studied the critical buckling temperature of FGM plates with various boundary conditions under uniform, linear and sinusoidal temperature fields by establishing the finite element model via three-dimensional linear elasticity theory. Subsequently, Na and Kim [18] improved their previous work and studied the thermal buckling of FGM composite structures. Malekzadeh [19] investigated the thermal buckling of FGM arbitrary straight-sided quadrilateral plates via the differential quadrature method (DQM) based on three-dimensional linear elasticity theory. Asemi et al. [20] investigated buckling of functionally graded plates under biaxial compression, shear, tension-compression, and shear-compression load conditions by establishing a full compatible three-dimensional elasticity element, but they didn’t consider the thermal environment in their work. A review paper of various modeling techniques and solution methods in thermal analysis of FGM plates was also published [21].

From the above reviews, it can be seen that closed form solutions for thermal buckling of FGM plates are only available for a few boundary conditions such as fully simply supported parallel edges. Furthermore, there is currently no literature on the closed form solution for the critical buckling temperature of FGM thin plates for other boundary conditions. Furthermore, closed form solutions can play a reference role to study the convergence and precision of various numerical methods and serve the purpose of parametric designs of structures.

In the context of closed form solutions for the critical thermal parameter of simply supported FGM rectangular thin plates, guide and clamp boundary conditions are obtained via the separation-of-variable method. A novel method is presented to determine the critical buckling temperature from the critical thermal parameter. It is assumed that the FGM properties follow the power law distribution in the thickness direction. Two types of FGM thin plates and uniform, linear and nonlinear temperature fields are taken into account. The influences of the length–thickness ratio, length–width ratio, power law index and initial temperature on the critical buckling temperature are investigated in detail.

## 2. Governing Equation of Thermal Buckling

In this work, the FGM rectangular plate of length a, width b and thickness h is investigated as shown in Figure 1.

CPT is used with the following displacement fields
where u

$$\begin{array}{ll}u=& {u}_{0}-z\frac{\partial w}{\partial x}\\ \nu =& {\nu}_{0}-z\frac{\partial w}{\partial y}\\ w=& {w}_{0}\end{array}$$

_{0}and v_{0}are the in-plane displacements on the midplane of the plate along the x and y direction, respectively; w is the midplane deflection. The strains associated with the displacement fields in Equation (1) are
$$\left[\begin{array}{c}{\epsilon}_{xx}\\ {\epsilon}_{yy}\\ {\epsilon}_{xy}\end{array}\right]=\left[\begin{array}{c}\frac{\partial {u}_{0}}{\partial x}-z\frac{{\partial}^{2}w}{\partial {x}^{2}}\\ \frac{\partial {\nu}_{0}}{\partial y}-z\frac{{\partial}^{2}w}{\partial {y}^{2}}\\ \frac{\partial {u}_{0}}{\partial y}+\frac{\partial {\nu}_{0}}{\partial x}-2z\frac{{\partial}^{2}w}{\partial x\partial y}\end{array}\right]$$

The constitutive relation for the FGM plate subjected to thermo-mechanical loadings is

$$\left[\begin{array}{c}{\sigma}_{xx}\\ {\sigma}_{yy}\\ {\sigma}_{xy}\end{array}\right]=\frac{E}{1-{\nu}^{2}}\left[\begin{array}{ccc}1& \nu & 0\\ \nu & 1& 0\\ 0& 0& \frac{1-\nu}{2}\end{array}\right]\left(\left[\begin{array}{c}{\epsilon}_{xx}\\ {\epsilon}_{yy}\\ {\epsilon}_{xy}\end{array}\right]-\Delta T\left[\begin{array}{c}\alpha \\ \alpha \\ 0\end{array}\right]\right)$$

The stress resultants can be written as

$$\begin{array}{l}\left({N}_{xx},{N}_{yy},{N}_{xy}\right)={\displaystyle {\int}_{-h/2}^{h/2}\left({\sigma}_{xx},{\sigma}_{yy},{\sigma}_{xy}\right)}\mathrm{d}z\\ \left({M}_{xx},{M}_{yy},{M}_{xy}\right)={\displaystyle {\int}_{-h/2}^{h/2}z\left({\sigma}_{xx},{\sigma}_{yy},{\sigma}_{xy}\right)}\mathrm{d}z\end{array}$$

Substituting Equations (1)–(3) into Equation (4) yields
where

$$\begin{array}{l}\left[\begin{array}{c}{N}_{xx}\\ {M}_{xx}\end{array}\right]=\left[\begin{array}{c}{\displaystyle {\int}_{-h/2}^{h/2}\left\{\frac{E}{1-{\nu}^{2}}\left[\frac{\partial {u}_{0}}{\partial x}-z\frac{{\partial}^{2}w}{\partial {x}^{2}}+\nu \left(\frac{\partial {\nu}_{0}}{\partial y}-z\frac{{\partial}^{2}w}{\partial {y}^{2}}\right)\right]\right\}}\mathrm{d}z-{N}_{T}\\ {\displaystyle {\int}_{-h/2}^{h/2}\left\{\frac{Ez}{1-{\nu}^{2}}\left[\frac{\partial {u}_{0}}{\partial x}-z\frac{{\partial}^{2}w}{\partial {x}^{2}}+\nu \left(\frac{\partial {\nu}_{0}}{\partial y}-z\frac{{\partial}^{2}w}{\partial {y}^{2}}\right)\right]\right\}}\mathrm{d}z-{M}_{T}\end{array}\right]\\ \left[\begin{array}{c}{N}_{yy}\\ {M}_{yy}\end{array}\right]=\left[\begin{array}{c}{\displaystyle {\int}_{-h/2}^{h/2}\left\{\frac{E}{1-{\nu}^{2}}\left[\frac{\partial {\nu}_{0}}{\partial y}-z\frac{{\partial}^{2}w}{\partial {y}^{2}}+\nu \left(\frac{\partial {u}_{0}}{\partial x}-z\frac{{\partial}^{2}w}{\partial {x}^{2}}\right)\right]\right\}}\mathrm{d}z-{N}_{T}\\ {\displaystyle {\int}_{-h/2}^{h/2}\left\{\frac{Ez}{1-{\nu}^{2}}\left[\frac{\partial {\nu}_{0}}{\partial y}-z\frac{{\partial}^{2}w}{\partial {y}^{2}}+\nu \left(\frac{\partial {u}_{0}}{\partial x}-z\frac{{\partial}^{2}w}{\partial {x}^{2}}\right)\right]\right\}}\mathrm{d}z-{M}_{T}\end{array}\right]\\ \left[\begin{array}{c}{N}_{xy}\\ {M}_{xy}\end{array}\right]=\left[\begin{array}{c}{\displaystyle {\int}_{-h/2}^{h/2}\left[\frac{E}{2\left(1+\nu \right)}\left(\frac{\partial {u}_{0}}{\partial y}+\frac{\partial {\nu}_{0}}{\partial x}-2z\frac{{\partial}^{2}w}{\partial x\partial y}\right)\right]}\mathrm{d}z\\ {\displaystyle {\int}_{-h/2}^{h/2}\left[\frac{Ez}{2\left(1+\nu \right)}\left(\frac{\partial {u}_{0}}{\partial y}+\frac{\partial {\nu}_{0}}{\partial x}-2z\frac{{\partial}^{2}w}{\partial x\partial y}\right)\right]}\mathrm{d}z\end{array}\right]\end{array}$$

$$\begin{array}{l}{N}_{T}={\displaystyle {\int}_{-h/2}^{h/2}\frac{E\alpha \Delta T}{1-\nu}\mathrm{d}z}\\ {M}_{T}={\displaystyle {\int}_{-h/2}^{h/2}\frac{E\alpha \Delta Tz}{1-\nu}\mathrm{d}z}\end{array}$$

From Equation (5), it can be seen that the in-plane loading and bending of the plate couple together because there is in-plane stretching in the geometrical midplane due to the materials’ asymmetry with respect to the midplane. However, the coupling can be eliminated by choosing the neutral surface instead of the geometrical midplane as a reference surface in which there are no in-plane displacements [22] (p. 1091).

The in-plane displacements of the midplane can be expressed as
where z

$${u}_{0}={z}_{0}\frac{\partial w}{\partial x},\phantom{\rule{24pt}{0ex}}{\nu}_{0}={z}_{0}\frac{\partial w}{\partial y}$$

_{0}is the distance between the geometrical midplane and the neutral surface. Substituting Equation (7) into Equation (5) yields
$$\begin{array}{l}\left[\begin{array}{c}{N}_{xx}\\ {M}_{xx}\end{array}\right]=\left[\begin{array}{c}{\displaystyle {\int}_{-h/2}^{h/2}\left[\frac{E\left({z}_{0}-z\right)}{1-{\nu}^{2}}\left(\frac{{\partial}^{2}w}{\partial {x}^{2}}+\nu \frac{{\partial}^{2}w}{\partial {y}^{2}}\right)\right]}\mathrm{d}z-{N}_{T}\\ {\displaystyle {\int}_{-h/2}^{h/2}\left[\frac{Ez\left({z}_{0}-z\right)}{1-{\nu}^{2}}\left(\frac{{\partial}^{2}w}{\partial {x}^{2}}+\nu \frac{{\partial}^{2}w}{\partial {y}^{2}}\right)\right]}\mathrm{d}z-{M}_{T}\end{array}\right]\\ \left[\begin{array}{c}{N}_{yy}\\ {M}_{yy}\end{array}\right]=\left[\begin{array}{c}{\displaystyle {\int}_{-h/2}^{h/2}\left[\frac{E\left({z}_{0}-z\right)}{1-{\nu}^{2}}\left(\frac{{\partial}^{2}w}{\partial {y}^{2}}+\nu \frac{{\partial}^{2}w}{\partial {x}^{2}}\right)\right]}\mathrm{d}z-{N}_{T}\\ {\displaystyle {\int}_{-h/2}^{h/2}\left[\frac{Ez\left({z}_{0}-z\right)}{1-{\nu}^{2}}\left(\frac{{\partial}^{2}w}{\partial {y}^{2}}+\nu \frac{{\partial}^{2}w}{\partial {x}^{2}}\right)\right]}\mathrm{d}z-{M}_{T}\end{array}\right]\\ \left[\begin{array}{c}{N}_{xy}\\ {M}_{xy}\end{array}\right]=\left[\begin{array}{c}{\displaystyle {\int}_{-h/2}^{h/2}\left(\frac{E\left({z}_{0}-z\right)}{1+\nu}\frac{{\partial}^{2}w}{\partial x\partial y}\right)}\mathrm{d}z\\ {\displaystyle {\int}_{-h/2}^{h/2}\left(\frac{Ez\left({z}_{0}-z\right)}{1+\nu}\frac{{\partial}^{2}w}{\partial x\partial y}\right)}\mathrm{d}z\end{array}\right]\end{array}$$

From Equation (8), it can be seen that the in-plane loading and bending of the plate have been decoupled.

The total potential energy of a plate in a thermal environment is [23,24]

$$U=\frac{1}{2}{\displaystyle {\int}_{V}^{}\left\{{\sigma}_{xx}\left({\epsilon}_{xx}-\alpha \Delta T\right)+{\sigma}_{yy}\left({\epsilon}_{yy}-\alpha \Delta T\right)+{\sigma}_{xy}{\epsilon}_{xy}\right\}}\mathrm{d}V$$

Using Equation (8) and employing the Euler equations to minimize the total potential energy function yields the equilibrium equations of an FGM plate

$$\begin{array}{l}\frac{\partial {N}_{xx}}{\partial x}+\frac{\partial {N}_{xy}}{\partial y}=0\\ \frac{\partial {N}_{xy}}{\partial x}+\frac{\partial {N}_{yy}}{\partial y}=0\\ \frac{{\partial}^{2}{M}_{xx}}{\partial {x}^{2}}+\frac{{\partial}^{2}{M}_{yy}}{\partial {y}^{2}}+2\frac{{\partial}^{2}{M}_{xy}}{\partial x\partial y}+{N}_{xx}\frac{{\partial}^{2}w}{\partial {x}^{2}}+2{N}_{xy}\frac{{\partial}^{2}w}{\partial x\partial y}+{N}_{yy}\frac{{\partial}^{2}w}{\partial {y}^{2}}=0\end{array}$$

Substituting Equation (8) into Equation (10) determines the distance z
or
where
where the Young’s modulus E, mass density ρ, Poisson’s ratio ν, thermal conductivity k and thermal expansion coefficient α of FGM vary continuously with thickness z as
where P stands for E, ν, α, k and ρ, respectively, and the subscripts t and b stand for the top and bottom surfaces of the plate, respectively. n is the power law index indicating the material variation profile through the thickness, and n = 0 represents a homogeneous plate. The material properties are also related to temperature by the following relation:
where P

_{0}and yields the thermal buckling that governs the equation of an FGM thin rectangular plate as
$${z}_{0}=\frac{{B}_{11}+{G}_{1}}{{A}_{11}+{G}_{0}}$$

$$\left({D}_{11}+{G}_{2}-\frac{{\left({B}_{11}+{G}_{1}\right)}^{2}}{{A}_{11}+{G}_{0}}\right)\left(\frac{{\partial}^{4}w}{\partial {x}^{4}}+2\frac{{\partial}^{4}w}{\partial {x}^{2}\partial {y}^{2}}+\frac{{\partial}^{4}w}{\partial {y}^{4}}\right)-{G}_{0}\left(\frac{{\partial}^{2}w}{\partial {x}^{2}}+\frac{{\partial}^{2}w}{\partial {y}^{2}}\right)=0$$

$$\left(\frac{{\partial}^{4}w}{\partial {x}^{4}}+2\frac{{\partial}^{4}w}{\partial {x}^{2}\partial {y}^{2}}+\frac{{\partial}^{4}w}{\partial {y}^{4}}\right)+{\widehat{G}}_{0}\left(\frac{{\partial}^{2}w}{\partial {x}^{2}}+\frac{{\partial}^{2}w}{\partial {y}^{2}}\right)=0$$

$$\begin{array}{l}{\widehat{G}}_{0}=\frac{-{G}_{0}}{{D}_{11}+{G}_{2}-\frac{{\left({B}_{11}+{G}_{1}\right)}^{2}}{{A}_{11}+{G}_{0}}}\\ {G}_{i}=-{\displaystyle {\int}_{-\frac{h}{2}}^{\frac{h}{2}}{z}^{i}\frac{\alpha E}{1-\nu}\left(T-{T}_{0}\right)}\mathrm{d}z\phantom{\rule{24pt}{0ex}}\left(i=0,1,2\right)\\ \left({A}_{ij},{B}_{ij},{D}_{ij}\right)={\displaystyle {\int}_{-\frac{h}{2}}^{\frac{h}{2}}{Q}_{ij}\left(1,z,{z}^{2}\right)}\mathrm{d}z\phantom{\rule{24pt}{0ex}}\left(i,j=1,2,6\right)\end{array}$$

$$\begin{array}{l}{Q}_{11}={Q}_{22}=\frac{E}{1-{\nu}^{2}}\\ {Q}_{12}={Q}_{21}=\frac{\nu E}{1-{\nu}^{2}}\\ {Q}_{66}=\frac{E}{2\left(1+\nu \right)}\end{array}$$

$$P=\left({P}_{\mathrm{t}}-{P}_{\mathrm{b}}\right){\left(\frac{1}{2}+\frac{z}{h}\right)}^{n}+{P}_{\mathrm{b}}$$

$$\frac{P}{{P}_{0}}={P}_{-1}T{(z)}^{-1}+1+{P}_{1}T(z)+{P}_{2}T{(z)}^{2}+{P}_{3}T{(z)}^{3}$$

_{−1}, P_{0}, P_{1}, P_{2}and P_{3}are the constants only related to the material, and T is the temperature function.The temperature distribution of the uniform temperature field is
which is independent of thickness z.

$$T\left(z\right)={T}_{0}+\Delta T$$

The temperature distribution of the linear temperature field is

$$T\left(z\right)={T}_{0}+\Delta T\left(\frac{1}{2}+\frac{z}{h}\right)$$

The temperature changes from T

_{0}to T_{0}+ ΔT linearly as the thickness varies from −h/2 to h/2.The nonlinear temperature field is from the one-dimensional thermal conduction equation:

$$-\frac{\mathrm{d}}{\mathrm{d}z}\left[k\left(z\right)\frac{\mathrm{d}T}{\mathrm{d}z}\right]=0$$

Note that the thermal conductivity of materials k(z) is independent of temperature T, so the analytical solution of Equation (20) can be obtained as

$$T\left(z\right)={T}_{0}+\Delta T\frac{{\displaystyle {\int}_{-\frac{h}{2}}^{z}\frac{1}{k\left(z\right)}}\mathrm{d}z}{{\displaystyle {\int}_{-\frac{h}{2}}^{\frac{h}{2}}\frac{1}{k\left(z\right)}}\mathrm{d}z}$$

Substituting Equation (16) into Equation (21) yields the nonlinear temperature field as
where
k

$$T\left(z\right)={T}_{0}+\Delta T\cdot f\left(z\right)$$

$$\begin{array}{c}f\left(z\right)=\frac{{\displaystyle \sum _{j=0}^{5}\frac{{(-1)}^{j}\Delta {k}_{}^{j}}{\left(j\times n+1\right){k}_{\mathrm{b}}^{j}}{\left(\frac{z}{h}+\frac{1}{2}\right)}^{j\times n+1}}}{{\displaystyle \sum _{j=0}^{5}\frac{{(-1)}^{j}\Delta {k}_{}^{j}}{\left(j\times n+1\right){k}_{\mathrm{b}}^{j}}}}\hfill \\ \Delta k={k}_{\mathrm{t}}-{k}_{\mathrm{b}}\hfill \end{array}$$

_{t}and k_{b}are thermal conductivities at the top and bottom surfaces of the plate, respectively.## 3. Closed Form Solutions of Thermal Buckling

After establishing the thermal buckling governing equation, the material model and temperature fields, the buckling solutions including thermal buckling mode function and critical buckling temperature are obtained below in separation-of-variable form.

Assuming that Equation (12) has the following separation-of-variable solution:
where w is the buckling mode function, and
where ϑ and η are unknown spatial eigenvalues. Substituting Equations (24) and (25) into Equation (12) yields the relation of ϑ, η and ${\widehat{G}}_{0}$ as

$$w\left(x,y\right)=\varphi \left(x\right)\psi \left(y\right)$$

$$\varphi \left(x\right)=A{\mathrm{e}}^{\vartheta x},\psi \left(y\right)=B{\mathrm{e}}^{\eta y}$$

$$\left({\widehat{G}}_{0}+{\vartheta}^{2}+{\eta}^{2}\right)\left({\vartheta}^{2}+{\eta}^{2}\right)=0$$

From Equation (26), we have the following relations

$$\begin{array}{l}{\vartheta}_{1,2}^{}=\pm {\alpha}_{2}^{}=\pm \sqrt{-{\eta}^{2}}\\ {\vartheta}_{3,4}^{}=\pm \mathrm{i}{\alpha}_{1}^{}=\pm \mathrm{i}\sqrt{{\eta}^{2}+{\widehat{G}}_{0}}\\ {\eta}_{1,2}^{}=\pm {\beta}_{2}^{}=\pm \sqrt{-{\vartheta}^{2}}\\ {\eta}_{3,4}^{}=\pm \mathrm{i}{\beta}_{1}^{}=\pm \mathrm{i}\sqrt{{\vartheta}^{2}+{\widehat{G}}_{0}}\end{array}$$

Therefore, the functions ϕ(x) and ψ(y) have the following closed forms:
where the five unknown variables α

$$\begin{array}{c}\varphi (x)={A}_{2}\mathrm{cos}{\alpha}_{1}x+{B}_{2}\mathrm{sin}{\alpha}_{1}x+{C}_{2}\mathrm{cosh}{\alpha}_{2}x+{D}_{2}\mathrm{sinh}{\alpha}_{2}x\\ \psi (y)={A}_{1}\mathrm{cos}{\beta}_{1}y+{B}_{1}\mathrm{sin}{\beta}_{1}y+{C}_{1}\mathrm{cosh}{\beta}_{2}y+{D}_{1}\mathrm{sinh}{\beta}_{2}y\end{array}$$

_{1}, α_{2}, β_{1}, β_{2}and ${\widehat{G}}_{0}$ are involved; then, four equations are needed for a unique solution. From Equation (27), one can obtain three relations of these five variables:
$$\begin{array}{l}{\alpha}_{1}^{2}+{\alpha}_{2}^{2}={\widehat{G}}_{0}\\ {\beta}_{1}^{2}+{\beta}_{2}^{2}={\widehat{G}}_{0}\end{array}$$

$${\alpha}_{1}{}^{2}+{\beta}_{1}{}^{2}={\vartheta}^{2}+{\eta}^{2}+2{\widehat{G}}_{0}$$

Substituting ϑ = ±iα

_{1}, η = ±iβ_{1}into Equation (30) yields
$${\alpha}_{1}{}^{2}+{\beta}_{1}{}^{2}={\widehat{G}}_{0}$$

Another two transcendental eigenvalue equations can be derived by using the two pairs of boundary conditions of four edges. The boundary conditions used in this study are presented in Table 1 where n denotes the outer normal direction of the edge.

The buckling mode functions have the separation-of-variable form as shown in Equation (24); then, the two transcendental eigenvalue equations can be derived separately by considering the boundary conditions with respect to x and y, respectively. Substituting the two eigenfunctions in Equation (28) into the boundary conditions given in Table 1 yields the transcendental eigenvalue equations which are shown in Table 2 [25,26,27].

By using Equation (29), Equation (31) and two transcendental eigenvalue equations corresponding to specific boundary conditions of a rectangular plate, we can solve the five variables α

_{1}, α_{2}, β_{1}, β_{2}and ${\widehat{G}}_{0}$. For clarity of discussion, the solved ${\widehat{G}}_{0}$ from eigenvalue equations is denoted by ${\widehat{G}}_{0\mathrm{cr}}$.Note that ${\widehat{G}}_{0\mathrm{cr}}$ is a critical thermal buckling parameter in which the critical temperature is involved, but in general it is hard or impossible to directly extract the explicit form of the critical buckling temperature T
where ${\widehat{G}}_{0}$(T) is given in Equation (14); it is a function of temperature, material properties and plate thickness, and T

_{cr}from ${\widehat{G}}_{0\mathrm{cr}}$. In the present study, a novel method of determining T_{cr}is proposed. To achieve this aim, an algebraic equation for the critical buckling temperature is given as
$${\widehat{G}}_{0}(T)={\widehat{G}}_{0\mathrm{cr}}$$

_{cr}can be solved numerically from Equation (31). For three simple cases, analytical solutions can be found; the details are given in the following.#### 3.1. Case 1 Temperature-Independent Homogeneous Thin Plates

For homogeneous isotropic plates, the material properties are independent of geometry and temperature. According to Equation (14), ${\widehat{G}}_{0}$ can be written as
then, the analytical critical temperature can be solved from Equation (31). For a uniform temperature field, it is
where ΔT

$${\widehat{G}}_{0}=\frac{-{G}_{0}}{{D}_{11}+{G}_{2}-\frac{{\left({B}_{11}+{G}_{1}\right)}^{2}}{{A}_{11}+{G}_{0}}}=\frac{\left(1+\nu \right)\alpha \left({\displaystyle {\int}_{-h/2}^{h/2}T\mathrm{d}z-{T}_{0}h}\right)}{\frac{{h}^{3}}{12}-\left(1+\nu \right)\alpha \left({\displaystyle {\int}_{-h/2}^{h/2}T{z}^{2}\mathrm{d}z-\frac{{T}_{0}{h}^{3}}{12}}\right)}$$

$$\Delta {T}_{\mathrm{cr},\mathrm{uni}}=\frac{{\widehat{G}}_{0\mathrm{cr}}{h}^{2}}{\alpha \left(1+\nu \right)\left({\widehat{G}}_{0\mathrm{cr}}{h}^{2}+12\right)}$$

_{cr}= T_{cr}− T_{0}; for a linear temperature field, it is
$$\Delta {T}_{\mathrm{cr},\mathrm{line}}=\frac{2{\widehat{G}}_{0\mathrm{cr}}{h}^{2}}{\alpha \left(1+\nu \right)\left({\widehat{G}}_{0\mathrm{cr}}{h}^{2}+12\right)}$$

It can be seen that, for the temperature-independent homogeneous isotropic thin plates, we have

$$\Delta {T}_{\mathrm{cr},\mathrm{line}}=2\Delta {T}_{\mathrm{cr},\mathrm{uni}}$$

The nonlinear temperature field does not exist for homogeneous isotropic plates since the thermal conductivities at the top and bottom surfaces are the same. Substituting k

_{t}= k_{b}into Equation (23), we can find that the temperature distribution of the nonlinear temperature field reduces to the linear temperature field.#### 3.2. Case 2 Temperature-Independent FGM Thin Plates

In this case, the FGM thin plates have temperature-independent material properties which vary with the thickness following the power law in Equation (16). For a uniform temperature field, ${\widehat{G}}_{0}$ follows from Equation (14) that

$${\widehat{G}}_{0}=\frac{\Delta T\cdot {\displaystyle {\int}_{-h/2}^{h/2}\frac{\alpha E}{1-\nu}}\mathrm{d}z}{{\displaystyle {\int}_{-h/2}^{h/2}\frac{E{z}^{2}}{1-{\nu}^{2}}}\mathrm{d}z-\Delta T\cdot {\displaystyle {\int}_{-h/2}^{h/2}\frac{\alpha E{z}^{2}}{1-\nu}}\mathrm{d}z-\frac{{\left({\displaystyle {\int}_{-h/2}^{h/2}\frac{Ez}{1-{\nu}^{2}}}\mathrm{d}z-\Delta T\cdot {\displaystyle {\int}_{-h/2}^{h/2}\frac{\alpha Ez}{1-\nu}}\mathrm{d}z\right)}^{2}}{{\displaystyle {\int}_{-h/2}^{h/2}\frac{E}{1-{\nu}^{2}}}\mathrm{d}z-\Delta T\cdot {\displaystyle {\int}_{-h/2}^{h/2}\frac{\alpha E}{1-\nu}}\mathrm{d}z}}$$

Then, the critical buckling temperature is
where

$$\Delta {T}_{\mathrm{cr}}=\frac{{\widehat{G}}_{0\mathrm{cr}}{J}_{0}{J}_{1}+{\widehat{G}}_{0\mathrm{cr}}{J}_{2}{J}_{5}-2{\widehat{G}}_{0\mathrm{cr}}{J}_{3}{J}_{4}+{J}_{0}{J}_{5}-\sqrt{\mathsf{\Theta}}}{2\left({\widehat{G}}_{0\mathrm{cr}}{J}_{0}{J}_{2}-{\widehat{G}}_{0\mathrm{cr}}{J}_{4}{}^{2}+{J}_{0}{}^{2}\right)}$$

$$\begin{array}{ll}\Theta =& {\widehat{G}}_{0\mathrm{cr}}^{2}{J}_{0}^{2}{J}_{0}^{1}-2{\widehat{G}}_{0\mathrm{cr}}^{2}{J}_{0}{J}_{1}{J}_{2}{J}_{5}-4{\widehat{G}}_{0\mathrm{cr}}^{2}{J}_{0}{J}_{1}{J}_{3}{J}_{4}+4{\widehat{G}}_{0\mathrm{cr}}^{2}{J}_{0}{J}_{2}{J}_{3}^{2}\\ & +4{\widehat{G}}_{0\mathrm{cr}}^{2}{J}_{1}{J}_{4}^{2}{J}_{5}+{\widehat{G}}_{0\mathrm{cr}}^{2}{J}_{2}^{2}{J}_{5}^{2}-4{\widehat{G}}_{0\mathrm{cr}}^{2}{J}_{2}{J}_{3}{J}_{4}{J}_{5}-2{\widehat{G}}_{0\mathrm{cr}}{J}_{0}^{2}{J}_{1}{J}_{5}\\ & +4{\widehat{G}}_{0\mathrm{cr}}{J}_{0}^{2}{J}_{2}^{3}+2{\widehat{G}}_{0\mathrm{cr}}{J}_{0}{J}_{2}{J}_{5}^{2}-4{\widehat{G}}_{0\mathrm{cr}}{J}_{0}{J}_{3}{J}_{4}{J}_{5}+{J}_{0}^{2}{J}_{2}^{5}\end{array}$$

The critical buckling temperature for linear and nonlinear temperature fields can be obtained similarly. Table 3 presents the integrals J

_{0}~ J_{5}which can be calculated numerically, and f(z) is given in Equation (23). It is noteworthy that the critical buckling temperature is irrelevant to the initial temperature for temperature-independent materials.#### 3.3. Case 3 Temperature-Dependent Homogeneous Plates for a Uniform Temperature Field

Referring to Equations (14) and (17), in this case ${\widehat{G}}_{0}$ can be written as
where

$${\widehat{G}}_{0}=\frac{12{\alpha}_{0}(1+{\nu}_{0}{\mathsf{\Upsilon}}_{1}){\mathsf{\Upsilon}}_{2}\Delta T}{{h}^{2}[1-{\alpha}_{0}(1+{\nu}_{0}{\mathsf{\Upsilon}}_{1}){\mathsf{\Upsilon}}_{2}\Delta T]}$$

$$\begin{array}{cc}{\mathsf{\Upsilon}}_{1}=& \frac{{\nu}_{-1}}{{T}_{0}+\Delta T}+1+{\nu}_{1}({T}_{0}+\Delta T)+{\nu}_{2}{({T}_{0}+\Delta T)}^{2}+{\nu}_{3}{({T}_{0}+\Delta T)}^{3}\\ {\mathsf{\Upsilon}}_{2}=& \frac{{\alpha}_{-1}}{{T}_{0}+\Delta T}+1+{\alpha}_{1}({T}_{0}+\Delta T)+{\alpha}_{2}{({T}_{0}+\Delta T)}^{2}+{\alpha}_{3}{({T}_{0}+\Delta T)}^{3}\end{array}$$

It can be proved that ΔT
where the coefficients are given in Appendix A. Equation (42) can be solved by the numerical methods to obtain the critical buckling temperature for a uniform temperature field. In general, the critical buckling temperature should be the smallest positive solution of the nine-order algebraic equation. It can be seen that the initial temperature T

_{cr}is one of the roots of the following ninth-order algebraic equation
$${A}_{9}{x}^{9}+{A}_{8}{x}^{8}+{A}_{7}{x}^{7}+{A}_{6}{x}^{6}+{A}_{5}{x}^{5}+{A}_{4}{x}^{4}+{A}_{3}{x}^{3}+{A}_{2}{x}^{2}+{A}_{1}x+{A}_{0}=0$$

_{0}affects the critical buckling temperature of temperature-dependent homogeneous plates under a uniform temperature field. The results provided later demonstrate the effect of T_{0}for temperature-dependent FGM plates under all kinds of thermal environments.#### 3.4. Case 4 Other Situations

Except for the above three cases, it is difficult to obtain the explicit expression of the critical buckling temperature, and Equation (32) must be solved numerically for T

_{cr}.In this study, the secant method is used to solve
because it is difficult to determine the derivative d${\widehat{G}}_{0}$/dT. Let x = T and

$${\widehat{G}}_{0}\left(T\right)-{\widehat{G}}_{0\mathrm{cr}}=0$$

$$g\left(x\right)={\widehat{G}}_{0}\left(T\right)-{\widehat{G}}_{0\mathrm{cr}}$$

Given two initial values x

_{−1}and x_{0}, the iterative scheme of the secant method is
$${x}_{k+1}={x}_{k}-\frac{g\left({x}_{k}\right)\left({x}_{k}-{x}_{k-1}\right)}{g\left({x}_{k}\right)-g\left({x}_{k-1}\right)},\phantom{\rule{24pt}{0ex}}\left(k=0,1,\cdots \right)$$

When x
the iteration stops, where the relative tolerance η > 0.

_{k}and x_{k}_{−1}satisfy
$$\frac{\left|{x}_{k}-{x}_{k-1}\right|}{\left|{x}_{k}\right|}\le \eta $$

## 4. Numerical Results and Discussion

Two types of FGM thin plates are considered in this section. Type-I FGM plate is temperature-dependent; it is made from stainless steel (SUS304) and Silicon nitride (Si

_{3}N_{4}). The material properties are shown in Table 4. Type-II FGM plate is temperature-independent; it is made from aluminum as its bottom surface and aluminum oxide as its top surface. E_{t}= 70 GPa, α_{t}= 23 × 10^{−6}/K, E_{b}= 380 GPa, α_{b}= 7.4 × 10^{−6}/K, ν = 0.3. Plates with different boundary conditions investigated below are shown in Figure 2.Firstly, the results of the present method are compared with those in the literature to validate the correctness of this method. Table 5 and Table 6 compare the critical buckling temperature ΔT

_{cr}of simply supported type-II FGM square plate based on different theories. Table 7 compares ΔT_{cr}of fully clamped type-II FGM rectangular plate. It can be seen that the present results coincide well with available results.Table 8 and Table 9 show ΔT

_{cr}of type-I FGM thin plate under a uniform temperature field with different boundary conditions. The results indicate that fully clamped (CCCC) plates have the largest ΔT_{cr}, and ΔT_{cr}decreases as the power law index n increases. Table 10 and Table 11 give ΔT_{cr}of Type-I FGM thin plate under different temperature fields. Due to the small difference between the temperature distribution of the linear temperature field and nonlinear temperature field, ΔT_{cr}under these two temperature fields is close. Furthermore, ΔT_{cr}under a uniform temperature field is obviously lower than that under the other two temperature fields, so a uniform temperature field affects ΔT_{cr}more significantly than the other two. Table 12 presents ΔT_{cr}of Type-I FGM thin plate with different initial temperatures T_{0}.In addition, the critical temperatures are also plotted in Figure 3, Figure 4, Figure 5 and Figure 6 for different thickness–length ratios h/a, power law indexes n, length–width ratios a/b and initial temperatures T

_{0}for type-I FGM plate. Figure 3 shows the relation between ΔT_{cr}and h/a for type-I FGM plate. The critical buckling temperature ΔT_{cr}increases as h/a increases since the thicker plates have larger bending rigidities than the thinner ones. One can also see that the critical buckling temperature of rectangular CCCC plate is the highest, whereas that of the CSCS plate is the lowest; this is because the CCCC plate has the highest bending rigidity compared with the other two cases. Furthermore, the uniform temperature field influences ΔT_{cr}more remarkably than the other two temperature fields.Figure 4 demonstrates that ΔT

_{cr}decreases as n decreases. It can be concluded from Equation (16) that the volume fraction of metal increases as n increases, and metallic materials have a lower critical buckling temperature than ceramic materials; this is because the thermal expansion coefficient α of ceramic materials is less than that of metallic materials.Figure 5 shows the relation between ΔT

_{cr}and a/b for different n. It is observed that, with a/b increasing, ΔT_{cr}increases gradually whatever n is; this indicates that the longer plates have a higher critical buckling temperature for different n. Homogeneous isotropic ceramic plate has the highest ΔT_{cr}and metallic plate has the lowest, and the volume fraction of the metal component increases as n increases.Figure 6 indicates that ΔT

_{cr}of type-I FGM plate decreases almost linearly as T_{0}increases. For temperature-independent material, the ΔT_{cr}stays at a constant value, thus T_{cr}has the same increment as that of T_{0}. However, for type-I FGM plate, which is made from temperature-dependent materials, ΔT_{cr}decreases almost linearly as T_{0}increases, and T_{cr}increases almost linearly, but the increment of T_{cr}is less than that of T_{0}.## 5. Conclusions

In this study, the closed form analytical solutions were obtained via the separation-of-variable method to determine the thermal buckling of FGM thin plates with simply supported, clamped and guided edges. A novel method to find the critical buckling temperature was given; the explicit expressions of the critical buckling temperature were presented for three special cases.

Two types of FGM thin plates and uniform, linear and nonlinear temperature fields were taken into account in this work. The present results coincide well with those in the literature, which illustrates the correctness of this present method. It follows that the critical buckling temperature of FGM thin plates decreases as the length–thickness ratio and power law index increase, and increases as the length–width ratio increases, and the initial temperature affects the critical buckling temperature of temperature-dependent materials. The uniform temperature field has the greatest influence on the critical buckling temperature, and the critical buckling temperatures become very close for both linear and nonlinear temperature fields.

## Acknowledgments

The work is supported by the National Natural Science Foundation of China (11372021, 11672019).

## Author Contributions

Y.X. conceived the research ideas; Z.W. derived equations, calculated and analyzed numerical results; both authors participated in the writing of this article.

## Conflicts of Interest

The authors declare no conflict of interest.

## Appendix A

A_{9} | ${\nu}_{0}{\alpha}_{3}{\nu}_{3}$ |

A_{8} | ${\nu}_{0}\left(8{T}_{0}{\gamma}_{1}+{\gamma}_{2}\right)$ |

A_{7} | ${\nu}_{0}\left(28{T}_{0}{}^{2}{\gamma}_{1}+7{T}_{0}{\gamma}_{2}+{\gamma}_{3}\right)$ |

A_{6} | ${\nu}_{0}\left(56{T}_{0}{}^{3}{\gamma}_{1}+21{T}_{0}{}^{2}{\gamma}_{2}+6{T}_{0}{\gamma}_{3}+{\gamma}_{4}\right)+{\alpha}_{3}$ |

A_{5} | ${\nu}_{0}\left(70{T}_{0}{}^{4}{\gamma}_{1}+35{T}_{0}{}^{3}{\gamma}_{2}+15{T}_{0}{}^{2}{\gamma}_{3}+5{T}_{0}{\gamma}_{4}+{\gamma}_{5}\right)+\left(5{T}_{0}{\alpha}_{3}+{\alpha}_{2}\right)$ |

A_{4} | $\begin{array}{l}{\nu}_{0}\left(56{T}_{0}{}^{5}{\gamma}_{1}+35{T}_{0}{}^{4}{\gamma}_{2}+20{T}_{0}{}^{3}{\gamma}_{3}+10{T}_{0}{}^{2}2{\gamma}_{4}+4{T}_{0}{\gamma}_{5}+{\gamma}_{6}\right)\\ +\left(10{T}_{0}{}^{2}{\alpha}_{3}+4{T}_{0}{\alpha}_{2}+{\alpha}_{1}\right)\end{array}$ |

A_{3} | $\begin{array}{l}{\nu}_{0}\left(28{T}_{0}{}^{6}{\gamma}_{1}+21{T}_{0}{}^{5}{\gamma}_{2}+15{T}_{0}{}^{4}{\gamma}_{3}+10{T}_{0}{}^{3}{\gamma}_{4}+6{T}_{0}{}^{2}{\gamma}_{5}+3{T}_{0}{\gamma}_{6}+{\gamma}_{7}\right)\\ +(10{T}_{0}{}^{3}{\alpha}_{3}+6{T}_{0}{}^{2}{\alpha}_{2}+3{T}_{0}{\alpha}_{1}+1)\end{array}$ |

A_{2} | $\begin{array}{l}{\nu}_{0}\left(8{T}_{0}{}^{7}{\gamma}_{1}+7{T}_{0}{}^{6}{\gamma}_{2}+6{T}_{0}{}^{5}{\gamma}_{3}+5{T}_{0}{}^{4}{\gamma}_{4}+4{T}_{0}{}^{3}{\gamma}_{5}+3{T}_{0}{}^{2}{\gamma}_{6}+2{T}_{0}{\gamma}_{7}+{\gamma}_{8}\right)\\ +\left(5{T}_{0}{}^{4}{\alpha}_{3}+4{T}_{0}{}^{3}{\alpha}_{2}+3{T}_{0}{}^{2}{\alpha}_{1}+2{T}_{0}+{\alpha}_{-1}\right)-\frac{{\hat{G}}_{0cr}{h}^{2}}{{\alpha}_{0}+\left({\hat{G}}_{0\mathrm{cr}}{h}^{2}+12\right)}\end{array}$ |

A_{1} | $\begin{array}{l}{\nu}_{0}\left(\begin{array}{l}{T}_{0}{}^{8}{\gamma}_{1}+{T}_{0}{}^{7}{\gamma}_{2}+{T}_{0}{}^{6}{\gamma}_{3}+{T}_{0}{}^{5}{\gamma}_{4}+{T}_{0}{}^{4}{\gamma}_{5}\\ +{T}_{0}{}^{3}{\gamma}_{6}+{T}_{0}{}^{2}{\gamma}_{7}+{T}_{0}{\gamma}_{8}+{\nu}_{-1}{\alpha}_{-1}\end{array}\right)\\ +\left({T}_{0}{}^{5}{\alpha}_{3}+{T}_{0}{}^{4}{\alpha}_{2}+{T}_{0}{}^{3}{\alpha}_{1}+{T}_{0}{}^{2}+{T}_{0}{\alpha}_{-1}\right)-\frac{2{\hat{G}}_{0\mathrm{cr}}{h}^{2}{T}_{0}}{{\alpha}_{0}\left({\hat{G}}_{0\mathrm{cr}}{h}^{2}+12\right)}\end{array}$ |

A_{0} | $-\frac{{\hat{G}}_{0cr}{h}^{2}{T}_{0}}{{\alpha}_{0}\left({\hat{G}}_{0cr}{h}^{2}+12\right)}$ |

where

$$\begin{array}{l}{\gamma}_{1}={\alpha}_{3}{\nu}_{3}\\ {\gamma}_{2}={\alpha}_{2}{\nu}_{3}+{\alpha}_{3}{\nu}_{2}\\ {\gamma}_{3}={\alpha}_{1}{\nu}_{3}+{\alpha}_{2}{\nu}_{2}+{\alpha}_{3}{\nu}_{1}\\ {\gamma}_{4}={\alpha}_{1}{\nu}_{2}+{\alpha}_{2}{\nu}_{1}+{\alpha}_{3}+{\nu}_{3}\\ {\gamma}_{5}={\alpha}_{1}{\nu}_{1}+{\nu}_{3}{\alpha}_{-1}+{\nu}_{-1}{\alpha}_{3}+{\alpha}_{2}+{\nu}_{2}\\ {\gamma}_{6}={\nu}_{2}{\alpha}_{-1}+{\nu}_{-1}{\alpha}_{2}+{\nu}_{1}+{\alpha}_{1}\\ {\gamma}_{7}={\nu}_{-1}{\alpha}_{1}+{\nu}_{1}{\alpha}_{-1}+1\\ {\gamma}_{8}={\alpha}_{-1}+{\nu}_{-1}\end{array}$$

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**Figure 3.**The relation between the critical buckling temperature ΔT

_{cr}and the length–thickness ratio h/a of type-I FGM rectangular thin plate (a/b = 2, n = 2, T

_{0}= 300 K).

**Figure 4.**The relation between the critical buckling temperature ΔT

_{cr}and power law index n of type-I FGM rectangular thin plate (a/h = 100, a/b = 2, T

_{0}= 300 K).

**Figure 5.**The relation between the critical buckling temperature ΔT

_{cr}and the length–width ratio a/b of type-I FGM, ceramic and metal rectangular thin plate (a/h = 100, n = 2, T

_{0}= 300 K).

**Figure 6.**(

**a**) The relation between ΔT

_{cr}and the initial temperature T

_{0}of type-I FGM rectangular thin plate (a/h = 100, a/b = 2, n = 2); (

**b**) The relation between T

_{cr}and the initial temperature T

_{0}of type-I FGM rectangular thin plate (a/h = 100, a/b = 2, n = 2).

Simple Support (S) | $w=0$ | ${M}_{n}=0$ |

Guide (G) | $\frac{\partial w}{\partial n}=0$ | ${Q}_{n}=0$ |

Clamp (C) | $w=0$ | $\frac{\partial w}{\partial n}=0$ |

B.C. | Eigenvalue Equations for y = 0 and b | Eigenvalue Equations for x = 0 and a |
---|---|---|

S-S | $\mathrm{sin}{\beta}_{1}b=0$ | $\mathrm{sin}{\alpha}_{1}a=0$ |

G-G | $\mathrm{sin}{\beta}_{1}b=0$ | $\mathrm{sin}{\alpha}_{1}a=0$ |

C-C | $\frac{1-\mathrm{cos}{\beta}_{1}b\mathrm{cosh}{\beta}_{2}b}{\mathrm{sin}{\beta}_{1}b\mathrm{sinh}{\beta}_{2}b}=\frac{{\beta}_{1}^{2}-{\beta}_{2}^{2}}{2{\beta}_{1}{\beta}_{2}}$ | $\frac{1-\mathrm{cos}{\alpha}_{1}a\mathrm{cosh}{\alpha}_{2}a}{\mathrm{sin}{\alpha}_{1}a\mathrm{sinh}{\alpha}_{2}a}=\frac{{\alpha}_{1}^{2}-{\alpha}_{2}^{2}}{2{\alpha}_{1}{\alpha}_{2}}$ |

S-G | $\mathrm{cos}{\beta}_{1}b=0$ | $\mathrm{cos}{\alpha}_{1}a=0$ |

S-C | ${\beta}_{2}\mathrm{tan}{\beta}_{1}b={\beta}_{1}\mathrm{tanh}{\beta}_{2}b$ | ${\alpha}_{2}\mathrm{tan}{\alpha}_{1}a={\alpha}_{1}\mathrm{tanh}{\alpha}_{2}a$ |

G-C | ${\beta}_{1}\mathrm{tan}{\beta}_{1}b=-{\beta}_{2}\mathrm{tanh}{\beta}_{2}b$ | ${\alpha}_{1}\mathrm{tan}{\alpha}_{1}a=-{\alpha}_{2}\mathrm{tanh}{\alpha}_{2}a$ |

Integral | Uniform Temperature Field | Linear Temperature Field | Nonlinear Temperature Field |
---|---|---|---|

J_{0} | ${\int}_{-h/2}^{h/2}\frac{\alpha E}{1-\nu}}\mathrm{d}z$ | ${\int}_{-h/2}^{h/2}\frac{\alpha E}{1-\nu}}\left(\frac{1}{2}+\frac{z}{h}\right)\mathrm{d}z$ | ${\int}_{-h/2}^{h/2}\frac{\alpha E}{1-\nu}}f\left(z\right)\mathrm{d}z$ |

J_{1} | ${\int}_{-h/2}^{h/2}\frac{E{z}^{2}}{1-{\nu}^{2}}}\mathrm{d}z$ | ${\int}_{-h/2}^{h/2}\frac{E{z}^{2}}{1-{\nu}^{2}}\left(\frac{1}{2}+\frac{z}{h}\right)}\mathrm{d}z$ | ${\int}_{-h/2}^{h/2}\frac{E{z}^{2}}{1-{\nu}^{2}}}f\left(z\right)\mathrm{d}z$ |

J_{2} | ${\int}_{-h/2}^{h/2}\frac{\alpha E{z}^{2}}{1-\nu}}\mathrm{d}z$ | ${\int}_{-h/2}^{h/2}\frac{\alpha E{z}^{2}}{1-\nu}\left(\frac{1}{2}+\frac{z}{h}\right)}\mathrm{d}z$ | ${\int}_{-h/2}^{h/2}\frac{\alpha E{z}^{2}}{1-\nu}f\left(z\right)}\mathrm{d}z$ |

J_{3} | ${\int}_{-h/2}^{h/2}\frac{Ez}{1-{\nu}^{2}}}\mathrm{d}z$ | ${\int}_{-h/2}^{h/2}\frac{Ez}{1-{\nu}^{2}}\left(\frac{1}{2}+\frac{z}{h}\right)}\mathrm{d}z$ | ${\int}_{-h/2}^{h/2}\frac{Ez}{1-{\nu}^{2}}f\left(z\right)}\mathrm{d}z$ |

J_{4} | ${\int}_{-h/2}^{h/2}\frac{\alpha Ez}{1-\nu}}\mathrm{d}z$ | ${\int}_{-h/2}^{h/2}\frac{\alpha Ez}{1-\nu}\left(\frac{1}{2}+\frac{z}{h}\right)}\mathrm{d}z$ | ${\int}_{-h/2}^{h/2}\frac{\alpha Ez}{1-\nu}}f\left(z\right)\mathrm{d}z$ |

J_{5} | ${\int}_{-h/2}^{h/2}\frac{E}{1-{\nu}^{2}}}\mathrm{d}z$ | ${\int}_{-h/2}^{h/2}\frac{E}{1-{\nu}^{2}}\left(\frac{1}{2}+\frac{z}{h}\right)}\mathrm{d}z$ | ${\int}_{-h/2}^{h/2}\frac{E}{1-{\nu}^{2}}}f\left(z\right)\mathrm{d}z$ |

Property | Material Type | P_{−1} | P_{0} | P_{1} | P_{2} | P_{3} |
---|---|---|---|---|---|---|

E | Si_{3}N_{4} | 0 | 348.43 × 10^{9} | −3.070 × 10^{-4} | 2.160 × 10^{−7} | −8.946 × 10^{−11} |

SUS304 | 0 | 201.04 × 10^{9} | 3.079 × 10^{-4} | −6.534 × 10^{−7} | 0 | |

ν | Si_{3}N_{4} | 0 | 0.2400 | 0 | 0 | 0 |

SUS304 | 0 | 0.3262 | −2.002 × 10^{-4} | 3.797 × 10^{−7} | 0 | |

α | Si_{3}N_{4} | 0 | 5.8723 × 10^{-6} | 9.095 × 10^{-4} | 0 | 0 |

SUS304 | 0 | 12.330 × 10^{-6} | 8.086 × 10^{-4} | 0 | 0 | |

ρ | Si_{3}N_{4} | 0 | 2370 | 0 | 0 | 0 |

SUS304 | 0 | 8166 | 0 | 0 | 0 | |

k | Si_{3}N_{4} | 0 | 9.19 | 0 | 0 | 0 |

SUS304 | 0 | 12.04 | 0 | 0 | 0 |

**Table 5.**Critical buckling temperature ΔT

_{cr}of SSSS type-II FGM square plate with different a/h under a uniform temperature field (a/b = 1).

n | Method | a/h = 10 | a/h = 20 | a/h = 40 | a/h = 60 | a/h = 80 | a/h = 100 |
---|---|---|---|---|---|---|---|

1 | Present | 782.016 | 197.813 | 49.600 | 22.056 | 12.409 | 7.943 |

CPT [15] | 794.377 | 198.594 | 49.648 | 22.066 | 12.412 | 7.943 | |

TSDT [15] | 757.891 | 196.257 | 49.500 | 22.037 | 12.402 | 7.939 | |

HDT5 [15] | 749.260 | 195.623 | 49.404 | 22.029 | 12.400 | 7.939 | |

5 | Present | 713.894 | 180.841 | 45.361 | 20.173 | 11.350 | 7.265 |

CPT [15] | 726.571 | 181.642 | 45.410 | 20.182 | 11.352 | 7.265 | |

TSDT [15] | 678.926 | 178.528 | 45.213 | 20.144 | 11.340 | 7.260 | |

HDT5 [15] | 669.402 | 177.805 | 45.166 | 20.134 | 11.337 | 7.259 | |

10 | Present | 733.217 | 185.863 | 46.629 | 20.737 | 11.667 | 7.468 |

CPT [15] | 746.927 | 186.731 | 46.682 | 20.747 | 11.670 | 7.469 | |

TSDT [15] | 692.519 | 183.141 | 46.455 | 20.703 | 11.657 | 7.462 | |

HDT5 [15] | 683.211 | 182.432 | 46.409 | 20.694 | 11.654 | 7.462 |

**Table 6.**Critical buckling temperature ΔT

_{cr}of SSSS type-II FGM plate with different a/b under a uniform temperature field (a/h = 100).

n | Method | a/b = 1 | a/b = 2 | a/b = 3 | a/b = 4 | a/b = 5 |
---|---|---|---|---|---|---|

1 | Present | 7.9426 | 19.8516 | 39.6876 | 67.4319 | 103.0568 |

CPT [16] | 7.9437 | 19.8594 | 39.7188 | 67.5220 | 103.2690 | |

FPT [16] | 7.9400 | 19.8358 | 39.6248 | 67.2506 | 102.6355 | |

TSDT [16] | 7.9400 | 19.8358 | 39.6248 | 67.2506 | 102.6356 | |

SSDT [16] | 7.9400 | 19.8359 | 39.6249 | 67.2510 | 102.6365 | |

2 | Present | 7.0415 | 17.5995 | 35.1851 | 59.7816 | 91.3644 |

CPT [16] | 7.0426 | 17.6065 | 35.2130 | 59.8621 | 91.5538 | |

FPT [16] | 7.0392 | 17.5853 | 35.1285 | 59.6184 | 91.9850 | |

TSDT [16] | 7.0390 | 17.5840 | 35.1234 | 59.6037 | 90.9508 | |

SSDT [16] | 7.0390 | 17.5840 | 35.1233 | 59.6034 | 90.9501 | |

5 | Present | 7.2645 | 18.1563 | 36.2964 | 61.6659 | 94.2363 |

CPT [16] | 7.2657 | 18.1642 | 36.3285 | 61.7585 | 94.4542 | |

FPT [16] | 7.2615 | 18.1380 | 36.2236 | 61.4559 | 93.7481 | |

TSDT [16] | 7.2606 | 18.1327 | 36.2025 | 61.3951 | 93.6069 | |

SSDT [16] | 7.2606 | 18.1324 | 36.2014 | 61.3921 | 93.5999 | |

10 | Present | 7.4679 | 18.6645 | 37.3115 | 63.3884 | 96.8646 |

CPT [16] | 7.4692 | 18.6731 | 37.3463 | 63.4888 | 97.1005 | |

FPT [16] | 7.4644 | 18.6427 | 37.2246 | 63.1378 | 96.2820 | |

TSDT [16] | 7.4634 | 18.6366 | 37.2006 | 63.0687 | 96.1213 | |

SSDT [16] | 7.4634 | 18.6365 | 37.2001 | 63.0673 | 96.1183 |

**Table 7.**Critical buckling temperature ΔT

_{cr}of CCCC type-II FGM rectangular plate under a uniform temperature field (a/h = 100, a/b = 2).

Method | n = 0 | n = 0.5 | n = 1 | n = 2 | n = 5 | n = 10 |
---|---|---|---|---|---|---|

Present | 32.355 | 18.332 | 15.032 | 13.326 | 13.748 | 14.133 |

Kiani [5] | 33.628 | 19.053 | 15.623 | 13.850 | 14.289 | 14.690 |

**Table 8.**Critical buckling temperature ΔT

_{cr}of type-I FGM rectangular thin plate with different a/h and various B.C. under a uniform temperature field (a/b = 2, T

_{0}= 300 K).

B.C. | n | a/h = 20 | a/h = 30 | a/h = 40 | a/h = 50 | a/h = 60 | a/h = 70 | a/h = 80 |
---|---|---|---|---|---|---|---|---|

CSCS | 1 | 583.7802 | 306.6174 | 186.0370 | 123.9484 | 88.1339 | 65.7278 | 50.8301 |

2 | 547.3748 | 283.8704 | 171.2913 | 113.7826 | 80.7593 | 60.1576 | 46.4854 | |

5 | 517.0574 | 264.5071 | 158.8189 | 105.2290 | 74.5765 | 55.4993 | 42.8584 | |

CSCC | 1 | 783.2272 | 431.1364 | 267.9431 | 181.1536 | 130.0341 | 97.5959 | 75.8125 |

2 | 742.2713 | 401.3746 | 247.6355 | 166.7565 | 119.4051 | 89.4729 | 69.4245 | |

5 | 713.3132 | 376.0498 | 230.3741 | 154.5835 | 110.4567 | 82.6559 | 64.0761 | |

CCCC | 1 | 1057.5407 | 640.8390 | 413.1423 | 285.7974 | 208.3541 | 158.0883 | 123.7707 |

2 | 1013.9656 | 602.6029 | 384.3119 | 264.3487 | 192.0395 | 145.3645 | 113.6185 | |

5 | 1030.1122 | 571.4283 | 359.7632 | 246.1036 | 178.2185 | 134.6260 | 105.0764 |

Note: ‘B.C.’ stands for Boundary Condition.

**Table 9.**Critical buckling temperature ΔT

_{cr}of type-I FGM rectangular thin plate with different a/b and various B.C. under a uniform temperature field (a/h = 100, T

_{0}= 300 K).

B.C. | n | a/b = 2 | a/b = 3 | a/b = 4 | a/b = 5 | a/b = 6 | a/b = 7 |
---|---|---|---|---|---|---|---|

CSCS | 1 | 32.9286 | 57.7537 | 92.8929 | 136.4766 | 187.0319 | 243.2765 |

2 | 30.0846 | 52.8369 | 85.1409 | 125.3608 | 172.2154 | 224.5874 | |

5 | 27.7157 | 48.7288 | 78.6384 | 115.9968 | 159.6822 | 208.7213 | |

CSCC | 1 | 49.3892 | 100.7232 | 168.6840 | 248.9947 | 338.3012 | 433.8633 |

2 | 45.1642 | 92.3546 | 155.1861 | 229.9260 | 313.6459 | 403.9628 | |

5 | 41.6377 | 85.3292 | 143.7844 | 213.7327 | 292.6429 | 378.5233 | |

CCCC | 1 | 81.4595 | 176.7382 | 295.2208 | 428.0121 | 568.5417 | 711.2512 |

2 | 74.6176 | 162.6580 | 273.1802 | 398.4099 | 532.6899 | 671.3248 | |

5 | 68.8856 | 150.7570 | 254.4250 | 373.2176 | 502.7141 | 640.2733 |

**Table 10.**Critical buckling temperature ΔT

_{cr}of type-I FGM rectangular thin plate with different a/h and various B.C. (a/b = 2, n = 2, T

_{0}= 300 K).

B.C. | T.F. | a/h = 20 | a/h = 30 | a/h = 40 | a/h = 50 | a/h = 60 | a/h = 70 |
---|---|---|---|---|---|---|---|

CSCS | uniform | 547.3748 | 283.8704 | 171.2913 | 113.7826 | 80.7593 | 60.1576 |

linear | 1133.6559 | 581.2390 | 352.4874 | 235.3624 | 167.7132 | 125.2884 | |

nonlinear | 1182.5836 | 607.9852 | 369.1144 | 246.6213 | 175.8071 | 131.3707 | |

CSCC | uniform | 742.2713 | 401.3746 | 247.6355 | 166.7565 | 119.4051 | 89.4729 |

linear | 1594.4982 | 822.6864 | 507.5211 | 343.2718 | 246.8452 | 185.6015 | |

nonlinear | 1655.7434 | 859.5887 | 531.0572 | 359.4812 | 258.6364 | 194.5373 | |

CCCC | uniform | 1013.9656 | 602.6029 | 384.3119 | 264.3487 | 192.0395 | 145.3645 |

linear | 2230.3048 | 1256.9658 | 787.2933 | 541.4994 | 394.6294 | 299.7612 | |

nonlinear | 2275.4643 | 1310.1371 | 822.7439 | 566.5211 | 413.1558 | 313.9882 |

Note: ‘T.F.’ represents Temperature Field.

**Table 11.**Critical buckling temperature ΔT

_{cr}of type-I FGM rectangular thin plate with different a/b and various B.C. (a/h = 100, n = 2, T

_{0}= 300 K).

B.C. | T.F. | a/b = 2 | a/b = 3 | a/b = 4 | a/b = 5 | a/b = 6 | a/b = 7 |
---|---|---|---|---|---|---|---|

CSCS | uniform | 30.0846 | 52.8369 | 85.1409 | 125.3608 | 172.2154 | 224.5874 |

linear | 62.9589 | 110.1631 | 176.7119 | 258.9990 | 354.3651 | 460.7078 | |

nonlinear | 66.0438 | 115.5227 | 185.2298 | 271.3522 | 371.0771 | 482.1803 | |

CSCC | uniform | 45.1642 | 92.3546 | 155.1861 | 229.9260 | 313.6459 | 403.9628 |

linear | 94.2791 | 191.5108 | 319.7465 | 471.5480 | 642.0001 | 828.0680 | |

nonlinear | 98.8767 | 200.7238 | 334.8863 | 493.5001 | 671.3553 | 865.1898 | |

CCCC | uniform | 74.6176 | 162.6580 | 273.1802 | 398.4099 | 532.6899 | 671.3248 |

linear | 155.0859 | 334.9406 | 559.4693 | 816.5263 | 1101.5178 | 1417.3660 | |

nonlinear | 162.5833 | 350.7718 | 585.2726 | 853.1769 | 1149.2782 | 1475.2611 |

**Table 12.**Critical buckling temperature ΔT

_{cr}of type-I FGM rectangular thin plate with different initial temperature T

_{0}(a/h = 100, a/b = 2, n = 2).

B.C. | T.F. | T_{0} = 300 | T_{0} = 350 | T_{0} = 400 | T_{0} = 450 | T_{0} = 500 | T_{0} = 550 |
---|---|---|---|---|---|---|---|

CSCS | uniform | 30.0846 | 29.1293 | 28.2271 | 27.3736 | 26.5646 | 25.7966 |

linear | 62.9589 | 61.0372 | 59.1907 | 57.4131 | 55.6986 | 54.0411 | |

nonlinear | 66.0438 | 64.0307 | 62.0950 | 60.2304 | 58.4306 | 56.6896 | |

CSCC | uniform | 45.1642 | 43.7545 | 42.4210 | 41.1575 | 39.9584 | 38.8184 |

linear | 94.2791 | 91.4769 | 88.7779 | 86.1741 | 83.6574 | 81.2201 | |

nonlinear | 98.8767 | 95.9427 | 93.1148 | 90.3846 | 87.7440 | 85.1846 | |

CCCC | uniform | 74.6176 | 72.3631 | 70.2244 | 68.1925 | 66.259 | 64.4162 |

linear | 155.0859 | 150.7014 | 146.4609 | 142.3542 | 138.3707 | 134.4997 | |

nonlinear | 162.5833 | 157.9968 | 153.5573 | 149.2545 | 145.0775 | 141.0154 |

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