1 Introduction

In classical thin solid structures, such as beams, plates and shells, studying the buckling phenomenon is of high interest. Similar to mechanical buckling, thermal stresses may result in buckling when certain conditions are met for boundary conditions and types of thermal stresses. With the introduction of novel types of materials, such as functionally graded materials (FGM) or carbon nanotubes (CNTs), recent researches of investigators are focused on these classes of materials.

Thermal stability of functionally graded material plates is well-documented in the open literature in the past decade. Some authors investigated the linear thermal buckling response of functionally graded material plates [1–5] and some others used a completely nonlinear analysis to obtained the critical buckling temperature as well as post-buckling equilibrium paths of the plate [6–12].

Carbon nanotubes (CNTs) have exceptional thermomechanical properties which made them as a candidate for reinforcement of the composites [13]. It is reported that nonuniform distribution of CNTs may be achieved through a powder metallurgy process [14]. Therefore the concept of FGMs and CNTs may be achieved together through a nonuniform distribution of CNTs through a specific direction [15]. This class of materials are known as functionally graded carbon nanotube reinforced composites (FG-CNTRC).

Uniaxial and biaxial buckling of composite plates reinforced with single walled carbon nanotubes is reported by Jafari et al. [16]. Ehselby-Mori-Tanaka and modified rule of mixtures approaches are used to estimate the properties of the nanocomposite media. Solution of this research is suitable only for plates with all edge simply supported. Exact closed form expressions are provided through the Navier solution method. Mechanical buckling response of FG-CNTRC composite plates with various boundary conditions and subject to various combinations of compression/tension loads is investigated by Lei et al. [17]. Solution of this method is based on an element-free kp-Ritz method. Since nonsymmetric distribution of CNTs across the thickness of the plate results in nonlinear bending rather than the buckling/postbuckling problem, only symmetric distributions of CNTs across the plate thickness are considered. Zhang et al. [18] studied the buckling response of FG-CNTRC plates resting on the Winkler elastic foundation. Elastic foundation is of conventional type which acts in compression as well as in tension. First order plate theory is used to obtain the governing stability equations. Similar to the investigation of Lei et al. [17], uniaxial compression, biaxial compression and combined compression/tension types of buckling are considered in this research. Material properties of the composite media are obtained according to a modified rule of mixtures approach. In two other studies, Lei et al. [19] and Zhang et al. [20] investigated the buckling behaviour of moderately thick skew plates with or without contact with elastic foundation using mesh-free methods based on a first order shear deformation theory. For the case of an FG-CNTRC plate integrated with two identical piezoelectric layers, Wu and Chang [21] studied the bifurcation buckling based on Reissner’s mixed variational theorem. In this study, three dimensional elasticity formulation is used rather than a two dimensional lumped plate theory formulation. Plate is subjected to in-plane biaxial compression and finite element method is used to solve the stability equations.

Shen and Zhu [22] investigated the buckling and post-buckling responses of sandwich plates with CNTRC face sheets in thermal environment. It is shown that characteristics of CNTs in face sheets are influential on buckling and post-buckling features of the sandwich plate. Shen and Zhang [23] studied the thermal buckling and postbuckling of FG-CNTRC rectangular plates with symmetric distribution of CNTs across the plate thickness. It is reported that, plates with intermediate volume fraction of CNTs have not, necessarily, intermediate critical buckling temperatures. Zhang et al. [24], also, studies the post-buckling of FG-CNTRC rectangular plates with edges elastically restrained against translation and rotation under the von-Kármán plate assumptions. Rafiee et al. [25] proposed a closed form solution for thermal postbuckling of rectangular plates made of FG-CNTRC plates integrated with smart piezoelectric layers with all edges simply supported.

Similar to stability analysis of FG-CNTRC plates, many studies are available on geometrically nonlinear static response [26–29], vibrational behaviour [30–36], dynamic response [37] and dynamic stability [38] of plates made from FG-CNTRC.

As the above literature survey reveals, while the literature is wealth enough on the subject of mechanical stability of FG-CNTRC plates, the only available works on the subject of thermal stability of FG-CNTRC plates belong to Shen and Zhang [23] and Rafiee et al. [25]. The above-mentioned researches, however, are confined to the case of plates with all edges simply supported. Present investigation aims to analyze the thermal buckling response of FG-NCTRC plate subjected to uniaixal or biaxial in-plane thermal loading via a Ritz formulation which is valid for arbitrary types of in-plane/out-of-plane boundary conditions. First order plate theory is used to obtain the governing equations of the plate. Material properties of the matrix and reinforcements are assumed to be linearly thermoelastic. Chebyshev polynomials are used as the basis shape functions suitable for arbitrary boundary conditions such as simply supported, clamped, free and sliding supported. An eigenvalue problem is established to obtain the critical buckling temperatures of temperature dependent rectangular FG-CNTRC plates. After performing the comparison studies to assure the effectiveness and correctness of the proposed formulation and solution method, parametric studies are given to explore the influences of various involved parameters such as volume fraction and dispersion profile of CNTs, geometrical properties of the plate and boundary conditions. It is shown that, all of the above-mentioned parameters are influential on buckling temperature and buckled shapes of the plate. Especially the FG-X pattern results in higher critical buckling temperature and is more suitable for thermal stability resistance.

2 Basic formulation

An FG-CNTRC rectangular plate is considered is this research. Thickness, width and length of the plate are denoted by, h, b and a, respectively. The conventional Cartesian coordinate system with its origin located at the center of the plate where \(-0.5a \le x \le +0.5a\), \(-0.5b \le y \le +0.5b\), and \(-0.5h \le z \le +0.5h\) is considered.

Plate is made from a polymeric matrix reinforced with single walled carbon nanotube (SWCNT). Distribution of SWCNT across the plate thickness may be uniform (referred to as UD) or functionally graded (referred to as FG) [26, 30–32, 39, 40]. In this research, three types of FG distribution of CNTs and the UD case are considered. FG-V, FG-O and FG-X CNTRC are the functionally graded distribution of carbon nanotubes through the thickness direction of the rectangular composite plate.

Generally, the effective thermo-mechanical properties of the FG-CNTRC rectangular plate are obtained using the well-known homogenization schemes, such as Mori-Tanaka scheme[41] or the rule of mixtures [42]. For the sake of simplicity, in the present research, the rule of mixtures is used to obtain the properties of the composite plate. However to account for the scale dependent properties of nanocomposite media, efficiency parameters are introduced. Accordingly, the effective material properties may be written as [43–47]

$$\begin{aligned}&E_{11}=\eta _1 V_{CN} E_{11}^{CN}+V_{m}E^{m}\nonumber \\&\dfrac{\eta _2}{E_{22}}=\dfrac{V_{CN}}{E_{22}^{CN}}+\dfrac{V_m}{E^m}\nonumber \\&\dfrac{\eta _3}{G_{12}}=\dfrac{V_{CN}}{G_{12}^{CN}}+\dfrac{V_m}{G^m} \end{aligned}$$
(1)

In the above equations, \(\eta _1, \eta _2\) and \(\eta _3\) are the so called efficiency parameters and as mentioned earlier are introduced to account for the size dependent material properties of the plate. These constants are chosen to equal the obtained values of Young modulus and shear modulus from the present modified rule of mixtures with the results obtained according to the molecular dynamics simulations [43]. Besides, \(E_{11}^{CN}\), \(E_{22}^{CN}\) and \(G_{12}^{CN}\) are the Young’s modulus and shear modulus of SWCNTs, respectively. Furthermore, \(E^m\) and \(G^m\) indicate the corresponding properties of the isotropic matrix.

In Eq. (1), volume fraction of CNTs and matrix are denoted by \(V_{CN}\) and \(V_m\), respectively which satisfy the condition

$$V_{CN}+V_m=1$$
(2)

As mentioned earlier, three types of functionally graded CNTRC plates are considered. These types along with the UD type are the considered patterns of CNT dispersion through the thickness of the plate. In Table 1 distribution function of CNTs across the plate thickness is provided.

Table 1 Volume fraction of CNTs as a function of thickness coordinate for various cases of CNTs distribution
Full size table

It is easy to check from Table 1 that, all of these types have the same value of volume fraction. The total volume fraction across the plate thickness in all of these cases is equal to \(V_{CN}^*\). In FG-X type distribution of CNT is maximum near the top and bottom surfaces whereas the mid-plane is free of CNT. For FG-O, however, top and bottom surfaces are free of CNTs and the mid-surface of the plate is enriched with CNTs. In FG-V, the top surface is enriched with CNT and the bottom one is free of CNT. In UD type, each surface of the plate though the thickness has the same volume fraction of CNTs.

The effective Poisson ratio depends weakly on position [43, 44] and is expressed as

$$\nu _{12}=V_{CN}^* \nu _{12}^{CN}+V_{m}\nu ^{m}$$
(3)

Following the Shapery model, longitudinal and transverse thermal expansion coefficients are expressed as [43]

$$\begin{aligned}&\alpha _{11}=\dfrac{ V_{CN} E_{11}^{CN}\alpha _{11}^{CN}+V_{m}E^{m}\alpha ^m}{ V_{CN} E_{11}^{CN}+V_{m}E^{m}}\nonumber \\&\alpha _{22}=\left( 1+\nu _{12}^{CN}\right) V_{CN}\alpha _{22}^{CN}+\left( 1+\nu ^{m}\right) V_{m}\alpha ^{m}-\nu _{12}\alpha _{11} \end{aligned}$$
(4)

where in the above equation, \(\alpha _{11}^{CN}, \alpha _{22}^{CN}\) and \(\alpha ^{m}\) are the thermal expansion coefficients of the constituents.

First order shear deformation theory (FSDT) of plates suitable for moderately thick and thick plates is used in this study to estimate the kinematics of the plate [48]. According to the FSDT, displacement components of the plate may be written in terms of characteristics of the mid-surface of the plate and cross section rotations as

$$\begin{aligned}&u\left( x,y,z\right) =u_0\left( x,y \right) +z\varphi _x\left( x,y\right) \nonumber \\&v\left( x,y,z\right) =v_0\left( x,y \right) +z\varphi _y\left( x,y\right) \nonumber \\&w\left( x,y,z\right) =w_0\left( x,y \right) \end{aligned}$$
(5)

In the above equation \(u,\;v\), and w are the through-the-length, through-the-width and through-the-thickness displacements, respectively. Mid-plane characteristics of the plate are designated with a subscript 0. Besides, transverse normal rotations about the x and y axes are denoted by \(\varphi _{y}\) and \(\varphi _{x}\), respectively.

According to FSDT, in-plane strain components are written in terms of mid-plane strains and change in curvatures. Besides, through-the-thickness shear strain components are assumed to be constant. Therefore, one may write

$$\begin{aligned} \left\{ \begin{array}{c} \varepsilon _{xx} \\ \varepsilon _{yy}\\ \gamma _{xy}\\ \gamma _{xz}\\ \gamma _{yz} \end{array}\right\} =\left\{ \begin{array}{c} \varepsilon _{xx0} \\ \varepsilon _{yy0}\\ \gamma _{xy0}\\ \gamma _{xz0}\\ \gamma _{yz0} \end{array}\right\} + z\left\{ \begin{array}{c} \kappa _{xx} \\ \kappa _{yy}\\ \kappa _{xy}\\ \kappa _{xz}\\ \kappa _{y z} \end{array}\right\} \end{aligned}$$
(6)

In this study, a rectangular plate under the action of uniform heating is under investigation. At least on two parallel edges of the plate normal to edge displacements are restrained. Under such conditions, the plate cannot go thermal expansion at least along one direction and compressive thermal stresses are induced. Therefore, buckling phenomenon may happen. It is known that buckling is a nonlinear phenomenon which should be obtained under geometrically nonlinear analysis. Meanwhile, when only bifurcation buckling is of interest, stability equations are linearised. Therefore, to obtain the stability equations associated with the onset of buckling, strain components may be assumed to be infinitesimal and therefore linear strain-displacement relations suffice. In such solution method, the potential energy due to the prebuckling forces should be included as is shown in the next.

Considering the above mentioned discussions, the components of the strain associated to the mid-surface of the plate are linearised and are equal to

$$\begin{aligned} \left\{ \begin{array}{c} \varepsilon _{xx0} \\ \varepsilon _{yy0}\\ \gamma _{xy0}\\ \gamma _{xz0}\\ \gamma _{yz0}\end{array} \right\} = \left\{ \begin{array}{c} u_{0,x} \\ v_{0,y}\\ u_{0,y}+v_{0,x}\\ \varphi _{x}+w_{0,x} \\ \varphi _{y}+w_{0,y} \end{array}\right\} \end{aligned}$$
(7)

and the components of change in curvature compatible with the FSDT are

$$\begin{aligned} \left\{ \begin{array}{c} \kappa _{xx}\\ \kappa _{yy}\\ \kappa _{xy}\\ \kappa _{xz}\\ \kappa _{yz} \end{array} \right\} =\left\{ \begin{array}{c} \varphi _{x,x}\\ \varphi _{y,y}\\ \varphi _{x,y}+\varphi _{y,x}\\ 0\\ 0 \end{array}\right\} \end{aligned}$$
(8)

where in the above equations \(()_{,x}\) and \(()_{,y}\) denote the derivatives with respect to the x and y directions, respectively.

For linear thermoelastic materials, stress field may be written as a linear function of strain field and temperature change as

$$\begin{aligned} \left\{ \begin{array}{c} \sigma _{xx}\\ \sigma _{yy}\\ \tau _{yz}\\ \tau _{xz}\\ \tau _{xy}\end{array}\, \right\} = \begin{bmatrix} Q_{11}&Q_{12}&0&0&0\\ Q_{12}&Q_{22}&0&0&0\\ 0&0&Q_{44}&0&0\\ 0&0&0&Q_{55}&0\\ 0&0&0&0&Q_{66} \end{bmatrix} \left\{ \begin{array}{c} {\varepsilon }_{xx}-\alpha _{11}\Delta T\\ {\varepsilon }_{yy}-\alpha _{22}\Delta T\\ {\gamma }_{yz}\\ \gamma _{xz}\\ \gamma _{xy} \end{array}\right\} \end{aligned}$$
(9)

where, T and \(T_0\) are, respectively, the elevated temperature and reference temperature, respectively. Besides, \(Q_{ij}\)’s \((i,j=1,2,4,5,6)\) are the reduced material stiffness coefficients compatible with the plane-stress conditions and are obtained as follow [49]

$$\begin{aligned}&Q_{11}=\frac{E_{11}}{1-\nu _{12}\nu _{21}},\;\;\; Q_{22}=\frac{E_{22}}{1-\nu _{12}\nu _{21}},\;\;\; Q_{12}=\frac{\nu _{21}E_{11}}{1-\nu _{12}\nu _{21}},\;\;\;\nonumber \\&Q_{44}=G_{23},\;\;\;Q_{55}=G_{13},\;\;\;Q_{66}=G_{12} \end{aligned}$$
(10)

Stress resultants of the FSDT may be obtained upon integration of stress field through the thickness. Stress resultant components in this case become [48]

$$\begin{aligned}&\left\{ \begin{array}{c} N_{xx} \\ N_{yy}\\ N_{xy}\\ M_{xx}\\ M_{yy}\\ M_{xy}\\ Q_{xz} \\ Q_{yz}\end{array}\right\} = \int _{-0.5h}^{+0.5h}\left\{ \begin{array}{c} \sigma _{xx} \\ \sigma _{yy}\\ \tau _{xy}\\ \ z\sigma _{xx} \\ z\sigma _{yy}\\ z\tau _{xy}\\ \kappa \tau _{xz} \\ \kappa \tau _{yz} \end{array}\right\} dz \end{aligned}$$
(11)

In the above equation, \(\kappa\) is the shear correction factor of FSDT. As known, adoption of a shear correction factor results in more accurate buckling loads and somehow compensate the errors due to the assumption of uniform transverse strains. Evaluation of accurate shear correction factor for FG-CNTRC plates is not straightforward since this factor depends on the boundary conditions, material properties, geometry and loading type. However, approximate values of \(\kappa =1\), \(\kappa =5/6\) or \(\kappa =\pi ^2/12\) are used extensively. In this research, the shear correction factor is set equal to \(\kappa =5/6\).

Substitution of Eq. (9) into Eq. (11) with the simultaneous aid of Eqs. (5)–(8), and (10) generates the stress resultants in terms of the mid-surface characteristics of the plate as

$$\begin{aligned} \left\{ \begin{array}{c} N_{xx}\\ N_{yy}\\ N_{xy}\\ M_{xx}\\ M_{yy}\\ M_{xy}\\ Q_{yz}\\ Q_{xz} \end{array}\right\} = \begin{bmatrix} A_{11}&A_{12}&0&B_{11}&B_{12}&0&0&0\\ A_{12}&A_{22}&0&B_{12}&B_{22}&0&0&0\\ 0&0&A_{66}&0&0&B_{66}&0&0 \\ B_{11}&B_{12}&0&D_{11}&D_{12}&0&0&0\\ B_{12}&B_{22}&0&D_{12}&D_{22}&0&0&0\\ 0&0&B_{66}&0&0&D_{66}&0&0\\ 0&0&0&0&0&0&\kappa A_{44}&0\\ 0&0&0&0&0&0&0&\kappa A_{55} \end{bmatrix} \left\{ \begin{array}{c} \varepsilon _{xx0}\\ \varepsilon _{yy0}\\ \gamma _{xy0}\\ \kappa _{xx}\\ \kappa _{yy}\\ \kappa _{xy}\\ \gamma _{yz0}\\ \gamma _{xz0}\end{array}\right\} - \left\{ \begin{array}{c} N_{xx}^T\\ N_{yy}^T\\ 0 \\ M_{xx}^T\\ M_{yy}^T\\ 0\\ 0\\ 0 \end{array}\right\} \end{aligned}$$
(12)

In the above equation, the stiffness components \(A_{ij}\), \(B_{ij}\), and \(D_{ij}\) indicate the stretching, bending-stretching, and bending stiffnesses, respectively, which are calculated by

$$\left( A_{ij},B_{ij},D_{ij}\right) =\int _{-0.5h}^{+0.5h}\left( Q_{ij},zQ_{ij},z^2Q_{ij}\right) dz$$
(13)

Also, \(N_{ii}^T, M_{ii}^T, i=,x,y\) are the thermally induced force and moment resultants which are obtained upon calculation of stress resultants as

$$\begin{aligned} \begin{bmatrix} N_{xx}^T&M_{xx}^T\\ N_{yy}^T&M_{yy}^T\\ \end{bmatrix} =\int _{-0.5h}^{+0.5h} \begin{bmatrix} Q_{11}&Q_{12}\\ Q_{12}&Q_{22}\\ \end{bmatrix} \begin{bmatrix} \alpha _{11}\\ \alpha _{22}\\ \end{bmatrix}(T-T_{0})dz \end{aligned}$$
(14)

It is of worth-noting that, under uniform temperature rise and for FG-X, FG-O and UD types of CNT distribution, no thermal bending moments are produced. Besides, due to the symmetric distribution of CNTs across the thickness in these three cases, the stretching-bending coupling stiffness components, i.e. \(B_{ij}\)’s are all equal to zero.

Stability equations of the plate may be obtained with the aid of static version of the Hamilton principle [48]. At the onset of buckling, one may write

$$\delta (U+V)=0$$
(15)

where \(\delta U\) is the virtual strain energy of the thermoelastic plate which may be calculated as

$$\delta U= \int _{-0.5a}^{+0.5a}\int _{-0.5b}^{+0.5b}\int _{-0.5h}^{+0.5h}\left( \sigma _{xx}\delta \varepsilon _{xx}+\sigma _{yy}\delta \varepsilon _{yy}+\tau _{xy}\delta \gamma _{xy}+\kappa \tau _{xz}\delta \gamma _{xz}+\kappa \tau _{yz}\delta \gamma _{yz}\right) dzdydx$$
(16)

and \(\delta V\) is the virtual potential energy of the prebuckling loads due to the constant uniform heating. It is of worth-noting that, even under uniform heating, bifurcation phenomenon may not occur. Considering the distributed patterns of CNTs across the plate thickness, in this study, three symmetric patterns and one nonsymmetric pattern are taken into consideration. As mentioned earlier in FG-V type of CNT dispersion and even in uniform heating, thermal moments are generated. Obviously, the induced thermal moments enforce the plate to deflect unless the thermally induced moments are surpassed by the supports. As known, clamped and sliding supported edges are capable of applying the additional bending moment at the support when is necessary, however, simply supported and free edges are unable to reveal such feature. Therefore, thermal buckling occurs for arbitrary type of FG-CNTRC plate with combinations of clamped and sliding supported edges. However, for plates with at least one edge simply supported or free, distribution of CNTs across the thickness should be symmetric.

Considering that the above conditions are met for the occurrence of thermal bifurcation buckling, prebuckling forces may be obtained according to the in-plane boundary conditions. For the case when in-plane displacement components are equal to zero at the edges (all edges are restrained again thermal expansion), prebuckling forces are equal to thermally induced loads. Therefore, the potential energy of the prebuckling forces at the onset of buckling is equal to

$$\delta V=- \int _{-0.5a}^{+0.5a}\int _{-0.5b}^{+0.5b} \left( N_{xx}^Tw_{0,x}\delta w_{0,x}+ N_{yy}^Tw_{0,y}\delta w_{0,y}\right) dydx$$
(17)

However, when two parallel edges of the plate are not restrained to move in normal to edge direction, in that direction, only thermal expansion is generated and thermal forces are not induced. In this case, prebuckling thermal forces should be obtained using the same process developed by Jones [50]. In a plate with free boundary conditions along \(y=\pm b/2\), while the two other edges are restrained against thermal expansion, the prebuckling forces are obtained by Jones [50]. The work done by such thermally induced forces at the onset of buckling may be written as

$$\delta V=- \int _{-0.5a}^{+0.5a}\int _{-0.5b}^{+0.5b} \left( \dfrac{A_{11}}{A_{22}}N_{yy}^T -N_{xx}^T\right) w_{0,x}\delta w_{0,x} dydx$$
(18)

3 Solution procedure

While the stability equations and the associated boundary conditions may be obtained through the application of Green theorem to the expression (15), energy based methods also may be used to solve the stability equation (15). In the present research, Ritz method with Chebyshev basis polynomials is used to derive the stability equations in a matrix representation. Accordingly, each of the essential variables may be expanded via Chebyshev polynomials and auxiliary functions such that

$$\begin{aligned}&u_0(x,y)=R^u(x,y)\sum _{i=1}^{N_x}\sum _{j=1}^{N_y} U_{ij}P_i(x)P_j(y)\nonumber \\&v_0(x,y)=R^v(x,y)\sum _{i=1}^{N_x}\sum _{j=1}^{N_y} V_{ij}P_i(x)P_j(y)\nonumber \\&w_0(x,y)=R^w(x,y)\sum _{i=1}^{N_x}\sum _{j=1}^{N_y} W_{ij}P_i(x)P_j(y)\nonumber \\&\varphi _x(x,y)=R^x(x,y)\sum _{i=1}^{N_x}\sum _{j=1}^{N_y} X_{ij}P_i(x)P_j(y)\nonumber \\&\varphi _y(x,y)=R^y(x,y)\sum _{i=1}^{N_x}\sum _{j=1}^{N_y} Y_{ij}P_i(x)P_j(y) \end{aligned}$$
(19)

where in the above equation \(P_i(x)\) and \(P_j(y)\) are the i-th and \(j-\)th Chebyshev polynomials of the first kind which are defined by

$$\begin{aligned}&P_i(x)=\cos ((i-1) \arccos (2x/a))\nonumber \\&P_j(y)=\cos ((j-1) \arccos (2y/b)) \end{aligned}$$
(20)

Besides, functions \(R^{\alpha }(x,y), \alpha =u,v,w,x,y\) are the boundary functions corresponding to the essential boundary conditions. It is known that in Ritz family methods, adoption of a shape function depends only on the essential boundary condition. Four types of boundary conditions are used in this study. clamped (C), simply supported (S), free (F) and sliding supported (X). For a clamped edge all of the in-plane and out-of-plane essential variables are restrained. For a simply supported edge normal and tangential displacements, lateral displacement and tangential slope are restrained and for a sliding supported edge only normal slope is restrained. At a free edge none of the boundary conditions are essential. Therefore, the essential variables associated to each of these cases may be written as

$$\begin{aligned}&\text {For\, a\, clamped\, edge{:}}\nonumber \\&x=\pm a/2{:}\,u_{0}=v_{0}=w_0=\varphi _x=\varphi _y=0\nonumber \\&y=\pm b/2{:}\,u_{0}=v_{0}=w_0=\varphi _x=\varphi _y=0\nonumber \\&\text {For\, a\, simply\, supported\, edge{:}}\nonumber \\&x=\pm a/2{:}\,u_{0}=v_{0}=w_0=\varphi _y=0\nonumber \\&y=\pm b/2{:}\,u_{0}=v_{0}=w_0=\varphi _x=0\nonumber \\&\text {For\, a\, sliding\, supported\, edge{:}}\nonumber \\&x=\pm a/2{:}\,u_{0}=v_{0}=\varphi _x=0\nonumber \\&y=\pm b/2{:}\,u_{0}=v_{0}=\varphi _y=0\nonumber \\&\text {For\, a\, free\, edge{:}}\nonumber \\&x=\pm a/2{:}\,-\nonumber \\&y=\pm b/2{:}\,- \end{aligned}$$
(21)

The shape functions of the Ritz method should be chosen according to the above essential variables. All of the Chebyshev functions are nonzero at both ends of the interval. Therefore, auxiliary functions \(R^\alpha , \alpha =u,v,w,x,y\) should satisfy the essential boundary conditions on each edge of the plate. Each of the functions \(R^\alpha , \alpha =u,v,w,x,y\) may be written as

$$R^\alpha (x,y)=\left( 1+\dfrac{2x}{a}\right) ^p\left( 1-\dfrac{2x}{a}\right) ^q\left( 1+\dfrac{2y}{b}\right) ^r\left( 1-\dfrac{2y}{b}\right) ^s$$
(22)

Each of the variables p, q, r and s depends on the essential boundary conditions and are equal to zero or one. For instance in a plate with clamped boundaries on \(x=-0.5a\) and \(x=+0.5a\), simply supported at \(y=-0.5b\) and sliding supported at \(y=+0.5b\), auxiliary functions are as

$$\begin{aligned}&R^u(x,y)=\left( 1+\dfrac{2x}{a}\right) \left( 1-\dfrac{2x}{a}\right) \left( 1+\dfrac{2y}{b}\right) \left( 1-\dfrac{2y}{b}\right) \nonumber \\&R^v(x,y)=\left( 1+\dfrac{2x}{a}\right) \left( 1-\dfrac{2x}{a}\right) \left( 1+\dfrac{2y}{b}\right) \left( 1-\dfrac{2y}{b}\right) \nonumber \\&R^w(x,y)=\left( 1+\dfrac{2x}{a}\right) \left( 1-\dfrac{2x}{a}\right) \left( 1+\dfrac{2y}{b}\right) \nonumber \\&R^x(x,y)=\left( 1+\dfrac{2x}{a}\right) \left( 1-\dfrac{2x}{a}\right) \left( 1+\dfrac{2y}{b}\right) \nonumber \\&R^y(x,y)=\left( 1+\dfrac{2x}{a}\right) \left( 1-\dfrac{2x}{a}\right) \left( 1-\dfrac{2y}{b}\right) \end{aligned}$$
(23)

Finally substitution of Eq. (19) into the Eqs. (15) results in an eigenvalue problem as

$$\left( \mathbf{K ^\mathbf{e }}-\mathbf{K ^\mathbf{g }}\right) {\mathbf{X }}=0$$
(24)

where \(\mathbf{K ^\mathbf{e }}\) is the elastic stiffness matrix which is originated from Eq. (16) and \(\mathbf{K ^\mathbf{g }}\) is originated from the geometrical stiffness matrix from Eqs. (17) or (18). Each of these matrices has \(5\times N_x\times N_y\) rows and columns. The above system should be treated as a standard eigenvalue problem to obtain the critical buckling temperature. It should be mentioned that, since the temperature dependency of the constituents is taken into consideration, a successive procedure should be carried out to obtain the critical buckling temperature. To initiate the process, at first, thermomechanical properties of the CNTs and matrix are evaluated at reference temperature. Thermomechanical properties are then computed at the obtained temperature and the eigenvalue problem is repeated again to extract a new critical buckling temperature. This procedure continues until a converged critical buckling temperature is achieved.

4 Numerical results and discussion

Present study aims to analyse the thermal buckling behaviour of FG-CNTRC moderately thick rectangular plates with temperature dependent material properties. In this section, at first, convergence and comparison studies are conducted. Afterwards, parametric studies are performed to examine the influences of involved parameters. In the rest of this manuscript the following convention is established for boundary conditions. For instance, in an SCSX plate, the first letter is associated to \(x=-0.5a\), the second letter is the boundary condition at \(y=-0.5b\), the third letter denotes the boundary conditions at \(x=+0.5a\) and finally the last letter is associated with the boundary at \(y=+0.5b\). Unless otherwise stated, Poly (methyl methacrylate), referred to as PMMA, is selected for the matrix with material properties \(E^{m}=(3.52-0.0034T)\) GPa, \(\nu ^m=0.34\) and \(\alpha ^m= 45(1+0.0005\Delta T)\)10−6/K. In calculation of elasticity modulus of matrix \(T=T_0+\Delta T\) where \(T_{0}=300\,{\rm K}\) is the reference temperature and T is measured in Kelvin. (10,10) armchair SWCNT is chosen as the reinforcement. Elasticity modulus, shear modulus, Poisson’s ratio and thermal expansion coefficient of SWCNT are highly dependent to temperature. Shen and Xiang [51] reported these properties at four certain temperature levels, i.e. \(T=300, 400, 500\) and 700 K. The magnitudes of \(E_{11}, E_{22}, G_{12}, \alpha _{11}, \alpha _{22}\) and \(\nu _{12}\) for CNTs at these four specific temperatures are given in Table 2.

Table 2 Thermo-mechanical properties of (10,10) armchair SWCNT at specific temperatures [51] (tube length = 9.26 nm, tube mean radius = 0.68 nm, tube thickness = 0.067 nm)
Full size table

Since this study aims to analyse the critical buckling temperature of FG-CNTRC plates under temperature dependent assumptions, it is necessary to obtain the thermomechanical properties of CNTs as a continuous function of temperature. For this purpose, for each of the thermomechanical properties of the CNT, a third order interpolation is done to obtain the properties of CNT as a function of temperature. As a result of interpolation, each of the properties may be written as

$$\begin{aligned}&E_{11}^{CN}(T) \mathrm {(TPa)}=6.3998-4.338417\times 10^{-3} T+7.43\times 10^{-6} T^2-4.458333\times 10^{-9} T^3 \nonumber \\&E_{22}^{CN}(T) \mathrm {(TPa)}=8.02155-5.420375\times 10^{-3} T+9.275\times 10^{-6} T^2-5.5625\times 10^{-9} T^3 \nonumber \\&G_{12}^{CN}(T) \mathrm {(TPa)}=1.40755+3.476208\times 10^{-3} T-6.965\times 10^{-6} T^2+4.479167\times 10^{-9} T^3\nonumber \\&\alpha _{11}^{CN}(T) \mathrm {(10^{-6}/K)}=-1.12515+0.02291688T-2.887\times 10^{-5}T^2+1.13625\times 10^{-8}T^3 \nonumber \\&\alpha _{22}^{CN}(T) \mathrm {(10^{-6}/K)}=5.43715-0.9.84625\times 10^{-4}T+2.9\times 10^{-7} T^2+1.25\times 10^{-11}T^3 \nonumber \\&\nu _{12}^{CN}=0.175 \end{aligned}$$
(25)

Han and Elliott [52] performed a molecular dynamics simulation to obtain the mechanics properties of nanocomposites reinforced with SWCNT. However in their analysis, the effective thickness of CNT is assumed to be at least 0.34 nm. The thickness of CNT as reported should be at least 0.142 nm [53]. Therefore molecular dynamics simulation of Han and Elliott [52] is re-examined [43]. The so-called efficiency parameters, as stated earlier, are chosen to match the data obtained by the modified rule of mixture of the present study and the molecular dynamics simulation results [43]. For three different volume fractions of CNTs, these parameters are as: \(\eta _1=0.137\) and \(\eta _2=1.022\) for \(V_{CN}^*=0.12\). \(\eta _1=0.142\) and \(\eta _2=1.626\) for \(V_{CN}^*=0.17\). \(\eta _1=0.141\) and \(\eta _2=1.585\) for \(V_{CN}^*=0.28\). For each case, the efficiency parameter \(\eta _3\) is equal to \(0.7\eta _2\). The shear modulus \(G_{13}\) is taken equal to \(G_{12}\) whereas \(G_{23}\) is taken equal to \(1.2 G_{12}\) [43].

4.1 Convergence and comparison studies

In this section, convergence and comparison studies are provided. At first a convergence study is given to obtain the required number of shape functions in series expansion of the Ritz method. Convergence study is provided in Table 3. For this purpose, critical buckling temperature parameter of isotropic homogeneous plates with all edges simply supported are evaluated for various number of the assumed shape functions. Results are compared with those obtained by Shen [54] based on an exact analytical solution. It is seen than Ritz method always serves an upper bound for the critical buckling temperatures of the plate. Furthermore, After adoption of 10 shape functions in both x and y directions highly accurate numerical results are achieved. Consequently, in the rest of this work, number of shape functions in Ritz approximation is set equal to \(N_{x}=N_{y}=10\).

Table 3 Convergence study for critical buckling temperature parameter \(1000\Delta T_{cr} \alpha\) of isotropic homogeneous SSSS square plates with \(\nu =0.21\) and various side to thickness ratios
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A comparison is presented in Table 4 on critical buckling temperature parameter of cross-ply symmetric laminated plates. For the sake of comparison, characteristics of the plate are set equal to those from the references. A square plate made of an orthotropic material with material properties \(E_{11}=15E_{22}\), \(G_{12}=G_{13}=0.5E_{22}\), \(G_{23}=0.3356E_{22}\), \(\nu _{12}=0.3\) and \(\alpha _{11}=0.015\alpha _{22}\) is considered. Critical buckling temperature difference parameter in this comparison is defined as \(\Delta T \alpha _{22}\). Results of this study are presented for two layering schemes, namely [0] and [0/90/0]. In each case two width to thickness ratios are assumed. Results of the present research are compared with the results of Matsunaga [55] based on a global higher-order theory, three-dimensional elasticity-based results of Noor and Burton [56], and results of Singh et al. [57] based on meshless finite element method using third order theory. It is seen that, numerical results of this study match well with the aforementioned works which accepts the accuracy and effectiveness of the proposed solution and formulation in the present study.

Table 4 Comparison of critical buckling temperature difference parameter, \(\Delta T_{cr} \alpha _{22}\) for square SSSS cross-ply laminates
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One of the available researches on thermal buckling of CNTRC rectangular plates belongs to Shen and Zhang [23]. In the study of Shen and Zhang exact solution is presented to obtain the critical buckling temperature of FG-CNTRC plates with all edges simply-supported and symmetric pattern of CNT dispersion across the plate thickness. Unlike the present case, Shen and Zhang [23] used the classical rule of mixtures to obtain the longitudinal thermal expansion coefficient of the CNTRC. Therefore only in this example and for the sake of comparison, longitudinal thermal expansion coefficient is evaluated as

$$\alpha _{11}=V_{CN} \alpha _{11}^{CN}+V_{m}\alpha ^m$$
(26)

Critical buckling temperature of FG-CNTRC plates with two different side to thickness ratios and three different length to side ratios are evaluated and compared with those of Shen and Zhang [23]. Comparison is carried out in Table 5. It is seen that results of our study match well with those of Shen and Zhang [23].

Table 5 Critical buckling temperature, \(T_{cr}(K)\) of SSSS FG-CNTRC rectangular plates with \(V_{CN}^*=0.17\) and temperature dependent material properties
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4.2 Parametric studies

After validating the numerical results of this study with the available data in the open literature, parametric studies are conducted in this section. Numerical results are provided in Tables 6, 7, 8 and 9 and Figs. 1 and 2. In all of this section, critical buckling temperature is obtained under the assumption of temperature dependent material properties.

Table 6 presents the critical buckling temperature of FG-CNTRC plates. In this table, various combinations of simply supported and clamped conditions are considered for the edges of the plate. As stated earlier, thermal bifurcation buckling in plates with at least one edge simply supported takes place when distribution of CNTs across the thickness is symmetric with respect to the mid-plane. Therefore, only FG-X, FG-O and UD CNTRC plates are considered. Three different patterns of the CNT dispersion and three different magnitudes of volume fraction of CNTs are considered. A square plate with length to thickness ratio \(a/h=30\) is considered. From the numerical results of this table it is concluded that, for all cases, FG-X plates have higher buckling temperature than UD case and the latter dispersion profile results in higher buckling temperature in comparison to plates with FG-O type of CNT dispersion. For all of the studied cases in this table, increasing the volume fraction of CNT results in higher critical buckling temperature. However the influence of volume fraction of CNTs on critical buckling temperature is not comparable to the influence of CNT dispersion pattern. It is observed that, maximum buckling temperature belongs to a plate with all edges clamped and the minimum one belongs to the one with all edges simply supported. This is expected since clamping results in higher local flexural rigidity in comparison to simply supported edge. Considering the fact that FG-CNTRC plates are orthotropic, as seen, buckling temperature of CCCS and CCSC plates are not equal. Considering various combinations of clamped and simply supported boundary conditions in square plates, numerical results reveal that critical buckling temperature may be sorted from high to low in plates with the following boundary conditions: CCCC, CCCS, CSCS, CCSC, CCSS, CSSS, SCSC, SCSS, SSSS.

Table 6 Thermal buckling temperature \(T_{cr}(K)\) for square FG-CNTRC plates with \(a/h=30\), various volume fraction of CNTs, different graded pattern and combinations of S and C edges
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Table 7 Thermal buckling temperature \(T_{cr}(K)\) for square FG-CNTRC plates with \(a/h=30\), various volume fraction of CNTs, different graded pattern and combinations of X and C edges
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Table 8 Critical buckling temperatures \(T_{cr}(K)\) for SSSS FG-CNTRC plates with FG-X type of CNT dispersion, various aspect and side to thickness ratios
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Table 9 Critical buckling temperatures \(T_{cr}(K)\) for FG-CNTRC plates with two parallel edges free, various types of CNT dispersion, \(V_{CN}^*=0.12\), \(b/h=50\) and various aspect ratios
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Fig. 1
figure 1

Lateral displacement mode shape of FG-CNTRC square plates with FG-X type of CNT dispersion, \(a/h=30\), volume fraction \(V_{CN}^*=0.28\) and various boundary conditions

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Fig. 2
figure 2

Lateral displacement mode shape of FG-CNTRC rectangular plates with \(a/b=2\), FG-X type of CNT dispersion, \(a/h=50\), volume fraction \(V_{CN}^*=0.12\) and various boundary conditions

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Table 7 presents the critical buckling temperature of FG-CNTRC plates with combinations of clamped and sliding supported edge. Both clamped and sliding support boundary conditions, are capable of applying the additional bending moment in the prebuckling state and therefore plate remains flat even when the distribution of material properties is not symmetric with respect to the mid-plane. Consequently, plates with combinations of C or X edges, and subjected to arbitrary type of CNT dispersion reveal the thermal bifurcation buckling. In this table, four cases of CNT dispersion and three types of volume fraction of CNTs are considered. Square plates with width to thickness ratio \(a/h=30\) are under investigation. It is seen that, critical buckling temperature of FG-V plate is also lower than those with FG-X type of CNT dispersion. Therefore from this table and Table 6 it is concluded that, FG-X type of CNT dispersion is the most influential type in thermal stability. It is observed that critical buckling temperature in FG-CNTRC plates decreases permanently when profile of the CNT through the thickness changes in order from FG-X to UD, then FG-V and finally FG-O. The key issue in higher buckling temperature of FG-X plates in comparison to the three other cases is the fact that, bending stiffnesses of the plate are much more in FG-X case since in the latter case the surfaces that are far from the mid-plane are much more enriched with CNT.

Tables 8 present the critical buckling temperatures of SSSS FG-CNTRC plates with various aspect ratios and also various side to thickness ratios. Three different magnitudes are considered for the volume fraction of CNTs whereas dispersion of CNTs is of FG-X type. Numerical results of this table reveal that, similar to square plate, increasing the volume fraction of CNTs enhances the buckling temperature of the plate. Besides, when width and thickness of the plate are constant, increasing the length of the plate results in lower critical buckling temperature.

An example of thermal buckled shapes of FG-CNTRC plates with FG-X type of CNT dispersion is presented in Fig. 1. Square plates with various boundary conditions comprising of C and X type of edges, side to thickness ratio \(a/h=30\) and volume fraction \(V_{CN}^*=0.28\) are considered. It is seen that essential boundary conditions are satisfied at the supports since at the sliding supported edge lateral displacement is not restrained. Unlike the case of fully clamped square isotropic plate which thermally buckles in a double symmetric shape, FG-CNTRC CCCC plate buckles in a shape which is symmetric along the x direction and antisymmetric along the y direction.

Table 9 is devoted to the case of plates where two parallel edges are free and the two others are restrained against thermal expansion. In such case as mentioned in Eq. (18), a uniaxial type of buckling exists since the edges \(y=\pm b/2\) are free to expand. As seen from Eq. (18), the compressive thermally induced force along the x direction is much lower than the one observed in biaxial thermal buckling. Consequently, it is expected that critical buckling temperatures for such plates are higher than those subjected to biaxial thermal loading. For this reason, a thin plate with \(b/h=50\) is considered and four different types of boundary conditions are examined. Five different aspect ratios are considered and as expected with increasing the plate length, critical buckling temperature decreases. A comparison among the results of Tables 8 and 9 reveals that, critical buckling temperature of thermally loaded SFSF plates are even higher than SSSS plates subjected to biaxial thermal compression. As an interesting result, it is observed that, critical buckling temperature of SXSX and CFXF are close to each other and approximately are equal. However, as expected the thermally buckled pattern for these two cases of boundary conditions are completely different.

Figure 2 demonstrates the thermally buckled pattern of FG-CNTRC plates with various boundary conditions, CNT volume fraction \(V_{CN}^*=0.12\), FG-X pattern of CNTs, \(a/h=50\) and \(a/b=2\). In this figure both uniaxially and biaxially loaded plates are depicted. Again, it is observed that, essential conditions are satisfied at the edges. Unlike the CCCC plate examined in Fig. 1, it is seen that buckling pattern of the plate is double symmetric. Such feature is expected since aspect ratio is of the main factors in buckled pattern of the rectangular plates.

5 Conclusion

Based on the first order shear deformation theory of plates, thermal buckling response of carbon nanotube reinforced rectangular plates is investigated in this research. Properties of the CNTs and the polymeric matrix are assumed to be temperature dependent. Only the case of uniform temperature rise parameter is considered. Distribution of CNTs across the plate thickness may be uniform or functionally graded. To obtain the thermomechanical properties of the composite plate, a refined rule of mixtures approach is developed to capture the size dependent features of CNTs. A Ritz solution with the Chebyshev basis shape functions suitable for arbitrary boundary conditions is implemented to construct the eigenvalue problem associated with the critical buckling temperature and buckled shapes of the plate. Numerical examples cover the influences of volume fraction of CNTs, dispersion pattern of CNTs, boundary conditions, plate aspect ratio and width to thickness ratio. It is shown that, in all of the studied cases, FG-X distribution of CNTs is the most efficient type for thermal buckling analysis since under such distribution, critical buckling temperature is higher in comparison to the other types of distribution of CNTs. Furthermore it is shown that, in-plane and out-of-plane boundary conditions, aspect ratio, side to thickness ratio and CNT volume fraction are all influential on critical buckling temperature as well as buckled pattern of the plate.