Thermo-electrical postbuckling behavior of carbon nanotubes-reinforced composite beams with piezoelectric layers and tangentially restrained ends

Abstract

An analytical investigation on the buckling and postbuckling behavior of carbon nanotube-reinforced composite beams integrated with surface-bonded piezoelectric layers under uniform temperature rise is presented in this paper. Carbon nanotubes (CNTs) are reinforced into isotropic matrix through uniform distribution and functionally graded distributions. The properties of material constituents are assumed to be temperature-dependent and effective properties of CNT-reinforced composite are estimated using an extended rule of mixture. Equilibrium equations of the beams are established based on Euler-Bernoulli theory including von Kármán nonlinearity and solved using analytical solutions and Galerkin method. Critical temperatures and postbuckling load-deflection paths are determined using an iteration algorithm. Parametric studies are performed to examine the influences of CNT distribution and volume fraction, applied voltage, in-plane and out-of-plane conditions of the ends, slenderness, and thickness ratio of layers on the critical loads and postbuckling load carrying capacity of beams. Results reveal that CNT volume fraction and degree of in-plane ends constraint have slight and significant influences on the critical temperatures and thermal postbuckling paths, respectively. The study also finds that negative and positive voltages increase and decrease the thermal buckling temperatures of piezoelectric CNT-reinforced composite beams.

Introduction

Since structural components are frequently exposed to thermal environments and gradients, thermally induced instability of these components is a problem of considerable importance. Li and Batra ¹ used analytical and shooting methods to investigate the thermal buckling and postbuckling of isotropic and homogeneous beams resting on nonlinear elastic foundations. Linear buckling problem of functionally graded material (FGM) beams under thermal loads was dealt with by Kiani and Eslami ² employing adjacent equilibrium criterion and analytical solutions. This approach was extended by Kiani and collaborators to study thermal linear buckling of FGM beams with piezoelectric layers basing on Euler-Bernoulli beam theory ³ and Timoshenko beam theory. ⁴ Thermal buckling and vibration analyses of FGM beams have been executed by Wattanasakulpong et al. ⁵ utilizing an improved third order shear deformation theory and Ritz method. Fu et al. ⁶ presented the thermal buckling, free vibration and dynamic stability analyses of FGM beams with piezoelectric layers and clamped ends. The nonlinear vibration, bending and thermal postbuckling of thick FGM beams have been dealt with by Shen and Wang ⁷ employing a higher order shear deformation beam theory (HSDBT) and perturbation technique. Based on an analytical approach, Trinh et al. ⁸ investigated the vibration and buckling of FGM beams under mechanical and thermal loads. The effects of nonlinear elastic foundations on the thermo-mechanical buckling and nonlinear free vibration response of FGM beams were analyzed in work of Fallah and Aghdam ⁹ using analytical solution. Thermal postbuckling analyses of single-layered and sandwich FGM plates with temperature-dependent properties were performed in works of Shen ¹⁰ and Tung, ¹¹ respectively.

Due to advanced properties, carbon nanotube (CNT) is used as filler into homogeneous matrix to constitute carbon nanotube-reinforced composite (CNTRC). Stimulated by concept of FGM, Shen ¹² proposed functionally graded carbon nanotube reinforced composite (FG-CNTRC) in which CNTs are reinforced into matrix according to functional rules of volume fraction. Yas and Samadi ¹³ employed Timoshenko beam theory and generalized differential quadrature method (GDQM) to study the mechanical buckling and linear vibration of CNTRC beams resting on elastic foundations. Large amplitude vibration, nonlinear bending and thermal postbuckling of FG-CNTRC beams have been analyzed by Shen and Xiang ¹⁴ using a HSDBT. Using shear deformation beam theory and Navier-type solutions, Wattanasakulpong and Ungbhakorn ¹⁵ executed the linear bending, vibration and buckling analyses of FG-CNTRC beams on elastic foundations with both simply supported ends. Linear buckling and vibration responses of cantilever CNTRC beams under axial load have been analyzed by Nejati et al. ¹⁶ making use of two-dimensional elasticity theory and GDQM. The GDQM based on various beam theories has been employed in work of Fattahi and Safaei ¹⁷ investigating the axial buckling characteristics of CNTRC beams with arbitrary boundary conditions. Linear free vibration and buckling analysis of sandwich beams with CNTRC face sheets were performed by Wu et al. ¹⁸ using Timoshenko theory and differential quadrature method. Dynamic buckling of thin CNTRC beams with piezoelectric layers under axial periodic force and applied voltage was treated by Yang et al. ¹⁹ Employing Timoshenko beam theory and GDQM, Mohseni and Shakouri ²⁰ presented a numerical investigation on free vibration and mechanical buckling of FG-CNTRC beams with variable thickness. The effects of initial geometric imperfections on the postbuckling of FG-CNTRC beams have been examined by Wu et al. ²¹ utilizing a first order shear deformation beam theory (FSDBT) and Newton-Raphson iterative technique. Recently, Karamanli and Vo ²² used finite element method (FEM) and shear deformation theory to explore the linear bending, vibration and buckling behaviors of composite beams reinforced by CNT and graphene nanoplatelet. More recently, Belarbi et al. ²³ employed FEM within the framework of a hyperbolic shear deformation theory to investigate the linear bending and mechanical buckling of FG-CNTRC beams.

Thermal instability of CNTRC structures is a problem of essential importance and should be addressed. Besides the work of Shen and Xiang, ¹⁴ thermal buckling and postbuckling of CNTRC beams have been considered in some works. Rafiee et al. ²⁴ used the Euler-Bernoulli beam theory and analytical solution to study the thermal buckling of FG-CNTRC beams with piezoelectric layers. The linear bending, buckling and free vibration characteristics of FG-CNTRC beams under different types of non-uniform thermal loadings have been analyzed by Mayandi and Jeyaraj ²⁵ employing FEM. The effects of geometrical imperfection on the thermal postbuckling of FG-CNTRC beams were considered in work of Wu et al. ²⁶ making use of differential quadrature method. Postbuckling response of CNTRC sandwich beams without and with elastic foundations under uniform temperature rise was dealt with by Kiani and coworker^27,28 using Ritz method within the framework of FSDBT. Khosravi et al. ²⁹ studied thermal buckling of rotating CNTRC beams using adjacent equilibrium criterion and GDQM. Thermal buckling and postbuckling behaviors of FG-CNTRC rectangular plates have been analyzed in works.^30,31 Thermal buckling and postbuckling of FG-CNTRC annular sector plates were numerically analyzed by Gholami and Ansari ³² using HSDBT and pseudo-arc-length method.

Elastic constraints of boundary edges are inherent in practical applications and should be considered in structural analyses. The mechanical buckling and vibration behavior of FG-CNTRC beams with elastic boundary conditions are analyzed by Wattanasakulpong and Mao ³³ using Chebyshev collocation method. Nonlinear vibration and dynamical response of FG-CNTRC plates with elastically restrained edges were presented in work of Anh et al. ³⁴ The influences of different boundary conditions on the nonlinear bending and dynamical characteristics of nanocomposite rectangular plates, cylindrical panels and circular cylindrical shells with CNT and graphene platelet reinforcements have been analyzed in works of Gholami and collaborators^35–39 using analytical and numerical approaches. The influences of tangentially elastic constraints of edges on the thermal buckling and postbuckling of plates and shells made of FG-CNTRC and porous FGM have been examined in works.^40–49 To the best of authors’ knowledge, there is no investigation on thermal postbuckling of CNTRC beams with elastically restrained ends.

In this paper, for the first time, the influences of tangentially elastic constraints of ends on the thermo-electrical postbuckling behavior of FG-CNTRC beams with surface-bonded piezoelectric layers under uniform temperature rise are investigated. CNTs are reinforced into matrix according to uniform and functionally graded distributions. The effective properties of CNTRC are determined using an extended rule of mixture. Formulations are based on Euler-Bernoulli beam theory including von Kármán nonlinearity. Analytical solutions of deflection are assumed to satisfy simply supported and clamped conditions of two ends and Galerkin method is used to derive the nonlinear relation of thermal load and deflection. Due to temperature dependence of material properties, critical buckling temperatures and postbuckling paths are determined using an iteration process. Parametric studies are carried out to examine the effects of CNT volume fraction and distributions, boundary condition, degree of in-plane end constraint, applied voltage, thickness of layers and slenderness ratio on thermal buckling and postbuckling of hybrid CNTRC beams.

Structural model and effective properties

Structural model of this study is a hybrid beam constituted from CNTRC host and two surface-bonded piezoelectric actuator layers. The length, width and total thickness of the beam are $L$ , $b$ and $H$ , respectively. The thickness of substrate CNTRC host and that of each piezoelectric layer are $h$ and $h_p$ , respectively, as shown in Figure 1. The beam is defined in a coordinate system $xz$ origin of which is located at one end, $x$ axis lies on the middle surface of the beam and $z$ axis is perpendicular to the mid-axis with downward positive points. It is assumed that CNTs are straight and aligned in the direction of $x$ axis, and adhesive thickness is neglected. CNTs are reinforced into matrix according to three mid-plane symmetric patterns, namely functionally graded distributions in the forms of X and O, which are respectively referred to as FG-X and FG-O distributions, along with uniform distribution (UD). The volume fraction $V_{CNT}$ of CNTs is varied across the thickness according to functional rules as ¹²

$V_{CNT}=\{\begin{array}{l}{V}_{CNT}^{*}\hspace{0.17em}\left(UD\right)\\ 2\left(1-2\frac{\left|z\right|}{h}\right)V_{CNT}^{*}\left(FG-O\right)\\ 2\left(2\frac{\left|z\right|}{h}\right)V_{CNT}^{*}\left(FG-X\right)\end{array}$

(1)

where total volume fraction $V_{CNT}^{*}$ of CNTs is defined as

$V_{CNT}^{*}=\frac{w_{CNT}}{w_{CNT}+\left({\rho }^{CNT}/{\rho }^m\right)\left(1-w_{CNT}\right)}$

(2)

in which $w_{CNT}$ is the mass fraction of CNTs, whereas ${\rho }^{CNT}$ and ${\rho }^m$ are the densities of CNT and matrix, respectively. In this study, the effective elastic modulus in the longitudinal direction of CNTRC is determined using an extended rule of mixture as ¹²

$E_{11}={\eta }_1{V}_{CNT}{E}_{11}^{CNT}+V_m{E}^m$

(3)

where $E_{11}^{CNT}$ and $E^m$ are elastic moduli of CNT and matrix, respectively, $V_m=1-V_{CNT}$ is volume fraction of matrix and ${\eta }_1$ is CNT efficiency parameter which account for size dependence of reinforcement on the effective elastic modulus. Effective Poisson’s ratio ${\nu }_{12}$ is assumed to be constant and evaluated using conventional rule of mixture as ¹²

${\nu }_{12}=V_{CNT}^{*}{\nu }_{12}^{CNT}+\left(1-V_{CNT}^{*}\right){\nu }^m$

(4)

where ${\nu }_{12}^{CNT}$ and ${\nu }^m$ are Poisson’s ratios of CNT and matrix, respectively. The effective coefficient of thermal expansion in the longitudinal direction of CNTRC is estimated basing on Schapery model as ¹⁴

${\alpha }_{11}=\frac{V_{CNT}{E}_{11}^{CNT}{\alpha }_{11}^{CNT}+V_m{E}^m{\alpha }^m}{V_{CNT}{E}_{11}^{CNT}+V_m{E}^m}$

(5)

in which ${\alpha }_{11}^{CNT}$ and ${\alpha }^m$ are thermal expansion coefficients of CNT and matrix, respectively. In this study, the properties of CNT and matrix like $E_{11}^{CNT},{\alpha }_{11}^{CNT},E^m,{\alpha }^m$ are assumed to be temperature-dependent and, hence, the effective properties $E_{11},{\alpha }_{11}$ of CNTRC depend on both temperature and position. Alternatively, piezoelectric layers are made of isotropic homogeneous material with temperature-independent properties. Specifically, the elastic modulus, thermal expansion coefficient and Poisson’s ratio of piezoelectric material are denoted by $E_p$ , ${\alpha }_p$ and ${\nu }_p$ , respectively.

Figure 1.

Geometry of a piezoelectric CNTRC beam and three types of CNT distribution.

Mathematical formulations

In the present study, hybrid CNTRC beams are assumed to be thin and geometrically perfect. The classical beam theory (CBT) basing on Euler-Bernoulli assumptions and including von Kármán nonlinearity is used for mathematical formulations. Within the framework of the CBT, the displacement components $\overline{u}$ and $\overline{w}$ in the axial and lateral directions, respectively, are expressed in the form ²⁴

$\overline{u}\left(x,z\right)=u\left(x\right)-z{w}_{,x}\left(x\right)$

(6a)

$\overline{w}\left(x,z\right)=w\left(x\right)$

(6b)

where $u$ and $w$ are axial and lateral displacements of a point on the middle surface, respectively. Herein, subscript comma means the derivative with respect to the followed variable, e.g. $w_{,x}=dw/dx$ . Axial strain of the beam has the form

${\epsilon }_{xx}=u_{,x}-z{w}_{,xx}+\frac{1}{2}{w}_{,x}^2-{\alpha }_{11}\mathrm{\Delta }T$

(7)

where $\mathrm{\Delta }T=T-T_0$ is uniform temperature rise from reference temperature $T_0$ assumed to be room temperature in the present work. Axial stresses in the layers are determined using constitutive relations. Specifically, the axial stress in the CNTRC substrate is expressed as ²⁴

${\sigma }_{xx}=E_{11}\left(u_{,x}-z{w}_{,xx}+\frac{1}{2}{w}_{,x}^2-{\alpha }_{11}\mathrm{\Delta }T\right)$

(8)

Axial stresses in the piezoelectric layers are computed as ²⁴

${\sigma }_{xx}^p=E_p\left(u_{,x}-z{w}_{,xx}+\frac{1}{2}{w}_{,x}^2-{\alpha }_p\mathrm{\Delta }T-d_{31}{E}_z\right)$

(9)

where $d_{31}$ is piezoelectric strain constant and $E_z$ is thickness direction electric field defined as $E_z=-{\mathrm{\Phi }}_{,z}$ with $\mathrm{\Phi }$ is potential field. Since the thickness of piezoelectric layer is very thin, self-induced electric potential is marginal, and electric field within a piezoelectric actuator is defined as ²⁴

$E_z=\frac{V}{h_p}$

(10)

where $V$ is applied voltage across the piezoelectric layer. The force and moment resultants per unit length are computed in terms of stresses through the thickness as follows

$\left(N_x,M_x\right)=\underset{-H/2}{\overset{H/2}{{\displaystyle \int }}}{\sigma }_{xx}\left(1,z\right)dz$

(11)

Using equations (8) and (9) into equation (11), the force and moment resultants are expressed in the form

$N_x=A_{11}\left(u_{,x}+\frac{1}{2}{w}_{,x}^2\right)-B_{11}{w}_{,xx}-N^T-N^P$

(12a)

$M_x=B_{11}\left(u_{,x}+\frac{1}{2}{w}_{,x}^2\right)-D_{11}{w}_{,xx}-M^T-M^P$

(12b)

where

$\left(A_{11},B_{11},D_{11}\right)=\underset{-H/2}{\overset{H/2}{{\displaystyle \int }}}E\left(1,z,z^2\right)dz$

(13a)

$\left(N^T,M^T\right)=\underset{-H/2}{\overset{H/2}{{\displaystyle \int }}}E\alpha \mathrm{\Delta }T\left(1,z\right)dz,\hspace{0.17em}\left(N^P,M^P\right)=\underset{-H/2}{\overset{H/2}{{\displaystyle \int }}}E{d}_{31}{E}_z\left(1,z\right)dz$

(13b)

in which

$\left(E,\alpha \right)=\{\begin{array}{l}{E}_p\\ E_{11}\\ E_p\end{array},\begin{array}{l}{\alpha }_p\\ {\alpha }_{11}\\ {\alpha }_p\end{array}\begin{array}{l}-H/2\le z\le -h/2\\ -h/2\le z\le h/2\\ h/2\le z\le H/2\end{array}$

(14)

For hybrid CNTRC beams with mid-plane symmetric types of CNT distribution, $B_{11}$ , $M^T$ and $M^P$ are zero-valued. Non-zero quantities in equation (13a) are expressed in the form

$A_{11}=e_{11}+2E_p{h}_p,D_{11}=e_{13}+\frac{1}{2}{h}^2{h}_p+h{h}_p^2+\frac{2}{3}{h}_p^3$

(15a)

$N^T=\left(e_{11T}+2E_p{\alpha }_p{h}_p\right)\mathrm{\Delta }T,N^P=\left(e_{11}+2E_p{h}_p\right)d_{31}\frac{V}{h_p}$

(15b)

$\left(e_{11},e_{13}\right)=\underset{-h/2}{\overset{h/2}{{\displaystyle \int }}}{E}_{11}\left(1,z^2\right)dz,e_{11T}=\underset{-h/2}{\overset{h/2}{{\displaystyle \int }}}{E}_{11}{\alpha }_{11}dz$

(15c)

Based on the CBT, the equilibrium equations of hybrid CNTRC beams are written as^6,24

$N_{x,x}=0$

(16a)

$M_{x,xx}+{\left(N_x{w}_{,x}\right)}_{,x}=0$

(16b)

From equation (16a) it is deduced that $N_x=N_{x0}$ which is independent of $x$ . Using this expression into equation (12a), we obtain

$\frac{du}{dx}=\frac{1}{A_{11}}\left(N^T+N^P+N_{x0}\right)-\frac{1}{2}{\left(\frac{dw}{dx}\right)}^2$

(17)

Using equation (12b) into equation (16b) we receive equilibrium equation in the form

$D_{11}\frac{d^{4}w}{d{x}^4}-N_{x0}\frac{d^{2}w}{d{x}^2}=0$

(18)

where $N_{x0}$ is fictitious force resultant induced by thermo-electrical loads and tangential constraints of ends. In this study, two ends of the beam are assumed to be elastically restrained in tangential direction. Fictitious force resultant $N_{x0}$ is concerned with average end-shortening displacement according to the following expression

$N_{x0}=-\frac{c}{L}\underset{0}{\overset{L}{{\displaystyle \int }}}\frac{du}{dx}dx$

(19)

in which $c$ is average tangential stiffness of two ends. It is evident that movable, immovable and partially movable ends will be represented by $c=0$ , $c\to \infty$ and $0<c<\infty$ , respectively. Using equation (17) into equation (19), we obtain

$N_{x0}=-\frac{c}{L}\underset{0}{\overset{L}{{\displaystyle \int }}}\left[\frac{1}{A_{11}}\left(N^T+N^P+N_{x0}\right)-\frac{1}{2}{\left(\frac{dw}{dx}\right)}^2\right]dx$

(20)

Two ends of the beam are assumed to be simply supported or clamped. For simply supported ends, the deflection of beam is assumed in the form ⁵⁰

$w\left(x\right)=W\hspace{0.17em}\sin \frac{m\pi x}{L}$

(21)

where $W$ is amplitude of deflection and $m$ is number of half wave. For clamped ends, the deflection solution is assumed in the form ⁵⁰

$w\left(x\right)=W\left(1-\cos \frac{2m\pi x}{L}\right)$

(22)

By introducing the deflection solution into equilibrium equation (18) and applying Galerkin method, we obtain the following equation

${\overline{D}}_{11}\frac{\beta m^2{\pi }^2}{L_H^2{\left(1+2{\overline{h}}_p\right)}^2}\overline{W}+{\overline{N}}_{x0}\overline{W}=0$

(23)

where $\beta =1$ and $\beta =4$ for simply supported and clamped ends, respectively, and

$\overline{W}=\frac{W}{H},L_H=\frac{L}{H},{\overline{N}}_{x0}=\frac{N_{x0}}{h},{\overline{h}}_p=\frac{h_p}{h},{\overline{D}}_{11}=\frac{D_{11}}{h^3}$

(24)

Substituting the deflection solutions (21) and (22) into equation (20) yields the following expression of fictitious force resultant

${\overline{N}}_{x0}=-\lambda \left({\overline{N}}^T+{\overline{N}}^P\right)+\lambda {\overline{A}}_{11}\beta \frac{m^2{\pi }^2}{4L_H^2}{\overline{W}}^2$

(25)

where

$\lambda =\frac{c}{c+A_{11}},{\overline{A}}_{11}={\overline{e}}_{11}+2E_p{\overline{h}}_p,{\overline{N}}^T=\left({\overline{e}}_{11T}+2E_p{\alpha }_p{\overline{h}}_p\right)\mathrm{\Delta }T,{\overline{N}}^P=d_{31}\frac{V}{h_p}\left({\overline{e}}_{11}+2E_p{\overline{h}}_p\right)$

(26a)

$\left({\overline{e}}_{11},{\overline{e}}_{11T}\right)=\frac{1}{h}\left(e_{11},e_{11T}\right)$

(26b)

Above-defined non-dimensional parameter $\lambda$ will be used to measure the degree of tangential constraints of ends in numerical results. It is readily recognized that movable ( $c=0$ ), immovable ( $c\to \infty$ ) and partially movable ( $0<c<\infty$ ) ends will be characterized by values of $\lambda =0$ , $\lambda =1$ and $0<\lambda <1$ , respectively. Introduction of ${\overline{N}}_{x0}$ from equation (25) into equation (23) leads to the following expression

$\mathrm{\Delta }T=\frac{1}{{\overline{e}}_{11T}+2E_p{\alpha }_p{\overline{h}}_p}\left[\frac{\beta {\overline{D}}_{11}{m}^2{\pi }^2}{\lambda L_H^2{\left(1+2{\overline{h}}_p\right)}^2}+\beta \frac{{\overline{A}}_{11}{m}^2{\pi }^2}{4L_H^2}{\overline{W}}^2-d_{31}\frac{V}{h_p}\left({\overline{e}}_{11}+2E_p{\overline{h}}_p\right)\right]$

(27)

Above expression is nonlinear relation between thermal load and non-dimensional maximum deflection of CNTRC beams integrated by surface-bonded piezoelectric layers with tangentially restrained ends. Obviously, buckling thermal loads are obtained from equation (27) by setting $m=1$ and $\overline{W}\to 0$ as the following

$\mathrm{\Delta }{T}_b=\frac{1}{{\overline{e}}_{11T}+2E_p{\alpha }_p{\overline{h}}_p}\left[\frac{\beta {\overline{D}}_{11}{\pi }^2}{\lambda L_H^2{\left(1+2{\overline{h}}_p\right)}^2}-d_{31}\frac{V}{h_p}\left({\overline{e}}_{11}+2E_p{\overline{h}}_p\right)\right]$

(28)

Since material properties of CNT and matrix are temperature dependent, critical thermal loads $\mathrm{\Delta }{T}_{cr}$ and postbuckling load-deflection paths are determined adopting an iteration process. Theoretically, it is evident from equation (28) that critical thermal loads can be reduced to zero in case the applied voltage $V$ is enough large. Nevertheless, in practical applications of piezo-CNTRC beams, the applied voltage can be not large and, as a result, electrically-induced buckling cannot occur. In the present study, the applied voltage is assumed to be pre-existing and non-destabilizing.

Numerical results and discussion

This section presents parametric studies of thermal buckling and postbuckling behaviors of CNTRC beams with piezoelectric actuators. The CNTRC material is made of Poly (methyl methacrylate), referred to as PMMA, as matrix and (10,10) single-walled carbon nanotubes (SWCNTs) as reinforcements. The properties of PMMA are ${\nu }^m=0.34$ , $E^m=\left(3.52-0.0034T\right)$ GPa, ${\alpha }^m=45\left(1+0.0005\mathrm{\Delta }T\right)\times {10}^{-6}/$ K in which $T=T_0+\mathrm{\Delta }T$ and $T_0=300$ K (room temperature). ¹⁴ The properties of (10,10) SWCNTs at specific temperatures are given in the work of Shen and Zhang. ³⁰ The properties of (10,10) SWCNTs are expressed as continuous functions of temperature in the works^27,31 adopting mathematical interpolation and omitted here for the sake of brevity. In numerical results, piezoelectric material is PZT-5A the temperature-independent properties of which are $E_p=63$ GPa, ${\alpha }_p=0.9\times {10}^{-6}/$ K, ${\nu }_p=0.3$ and $d_{31}=2.54\times {10}^{-10}$ m/V. ²⁴ In addition, CNT efficiency parameter ${\eta }_1$ is chosen as that given in works,^14,30 specifically ${\eta }_1=0.137$ , 0.142 and 0.141 for the values of $V_{CNT}^{*}=0.12$ , 0.17 and 0.28, respectively.

Verification

To verify the proposed approach, three comparative studies are carried out for thermal buckling and postbuckling of CNTRC beams with and without piezoelectric layers. First, the thermal buckling of single-layer CNTRC beams with simply supported and immovable ends and temperature dependent properties is considered. This problem was also dealt with in works of Shen and Xiang ¹⁴ using asymptotic solutions and a HSDBT-based perturbation technique, Kiani ²⁷ employing FSDBT-based Ritz method, and Wu et al. ²⁶ making use of FSDBT along with differential quadrature method. Critical thermal loads obtained by the present study are compared in Table 1 with results presented in the works^14,26,27 for different values of slenderness ratio and volume fraction of CNTs. As can be seen, the present results are slightly higher than those presented in the works.^14,26,27 This difference is expected because the present study employs the classical beam theory overestimating critical loads of shear deformable beams.

Table 1.

Comparison of critical thermal loads of single-layer CNTRC beams with simply supported and immovable ends and temperature-dependent properties.

Load	$L/h$	Source	$V_{CNT}^{*}=0.12$		$V_{CNT}^{*}=0.17$		$V_{CNT}^{*}=0.28$
			UD	FG-X	UD	FG-X	UD	FG-X
$T_{cr}$ (K)	50	Shen & Xiang 14	370.16	398.86	370.73	400.32	373.42	402.59
		Kiani 27	375.46	404.54	376.64	406.82	375.93	405.59
		Present	379.09	412.79	379.83	414.02	380.56	415.25
$\mathrm{\Delta }{T}_{cr}$ (K)	100	Wu et al. 26	21.560	31.097	21.839	31.630	21.973	31.892
		Present	21.771	31.734	22.017	32.226	22.263	32.595

As the second comparative study, thermal buckling of CNTRC beams integrated with surface-bonded piezoelectric actuators with two clamped and immovable ends is investigated. The critical buckling temperatures computed by the present work are given in Table 2 in comparison with those reported in the work of Rafiee et al. ²⁴ using Euler-Bernoulli beam theory and eigenvalue-based approach. It is evident from Table 2 that a good agreement is achieved in this comparison. As final example for verification, the thermal postbuckling of a single-layer FG-X CNTRC beam with simply supported and immovable ends and temperature dependent properties under uniform temperature rise is considered. The postbuckling temperature-deflection path traced using the present approach is compared in Figure 2 with result obtained by Wu et al. ²⁶ employing FSDBT-based differential quadrature method. Within the Figure 2, $r=h/\sqrt{12}$ is the radius of gyration of the beam cross section. It is evident from Figure 2 that the present result very well agrees with that reported in the work. ²⁶

Table 2.

Comparison of critical temperatures $T_{cr}=T_0+\mathrm{\Delta }{T}_{cr}$ (K) of piezoelectric CNTRC beams with clamped and immovable ends ( $h_p/h=1/12$ , $V=0$ , $h=10$  mm, $L/H=100$ ).

Source	$V_{CNT}^{*}=0.12$			$V_{CNT}^{*}=0.17$			$V_{CNT}^{*}=0.28$
	UD	FG-O	FG-X	UD	FG-O	FG-X	UD	FG-O	FG-X
Ref. [24]	308.34	305.67	311.72	308.05	305.29	311.84	308.43	305.55	313.45
Present	306.06	303.26	309.37	306.39	303.48	310.12	307.25	304.15	312.23

Figure 2.

Comparison of thermal postbuckling paths of single-layer FG-X CNTRC beams with simply supported and immovable ends.

In what follows, parametric studies of thermal buckling and postbuckling of CNTRC with piezoelectric layers and tangentially restrained ends will be carried out. In the parametric studies, the thickness of CNTRC substrate is $h=10$  mm, buckling mode $m=1$ and the degree of tangential constraints of two ends is measured by non-dimensional tangential stiffness parameter $\lambda$ defined in equation (26a). Temperature-dependent and temperature-independent properties will be referred to as T-D and T-ID, respectively. Simply supported and clamped boundary conditions at two ends will be briefly mentioned as S-S and C-C, respectively. The material properties and two ends are assumed to be temperature-dependent (T-D) and immovable, respectively, unless otherwise specified.

Thermal buckling analysis

Different influences on the critical thermal loads $\mathrm{\Delta }{T}_{cr}$ of CNTRC beams with piezoelectric layers are analyzed in this subsection. First, the effects of boundary condition (B.C), CNT volume fraction $V_{CNT}^{*}$ , CNT distribution patterns, and $h_p/h$ ratio of piezoelectric layer-to-CNTRC host thicknesses on the thermal buckling behavior under a grounding condition ( $V=0$ ) are given in Table 3. Equation (28) and Table 3 indicate that when the properties are T-ID, critical thermal loads of C-C beams are about four times those of S-S beams under a grounding condition. Additionally, the temperature dependence of material properties renders critical thermal loads lower, and detrimental influence of T-D properties on thermal buckling resistance capability is more significant when two ends are clamped. It is demonstrated in Table 3 that CNT distribution pronouncedly affects thermal buckling behavior and critical thermal loads of FG-O and FG-X beams are lowest and highest, respectively. Alternatively, volume percentage $V_{CNT}^{*}$ of CNTs slightly influence the thermal buckling of CNTRC beams and critical thermal loads are benignly larger as $V_{CNT}^{*}$ is increased. Furthermore, critical thermal loads are enhanced when $h_p/h$ ratio becomes smaller under a grounding condition. Next, the effects of slenderness ratio $L/H$ and applied voltage $V$ on the thermal buckling behavior of FG-X CNTRC beams with clamped and immovable ends are shown in Table 4. Obviously, the critical thermal loads are considerably dropped when enhancing $L/H$ ratio. In addition, minus and plus voltages increase and decrease the critical thermal loads of CNTRC beams with piezoelectric actuators, respectively. This fact can be interpreted that the minus and plus voltages induce axial tensile and compressive forces prior to application of thermal loading, respectively.

Table 3.

Critical thermal loads $\mathrm{\Delta }{T}_{cr}$ (K) of CNTRC beams with piezoelectric layers and immovable ends ( $h=10$  mm, $L/H=80$ , $\lambda =1$ , $V=0$ ).

	B.C	$V_{CNT}^{*}$	$h_p/h=1/10$			$h_p/h=1/20$
			UD	FG-O	FG-X	UD	FG-O	FG-X
T-ID	S-S	0.12	24.228	12.497	35.836	29.183	14.994	43.221
		0.17	24.718	12.632	36.737	29.667	15.120	44.134
		0.28	25.181	12.765	37.569	30.128	15.246	44.976
	C-C	0.12	96.911	49.988	143.343	116.731	59.975	172.885
		0.17	98.871	50.528	146.949	118.668	60.480	176.537
		0.28	100.722	51.059	150.274	120.512	60.986	179.904
T-D	S-S	0.12	23.001	12.177	33.456	27.552	14.514	39.852
		0.17	23.493	12.300	34.194	27.921	14.637	40.590
		0.28	23.862	12.423	34.809	28.290	14.760	41.205
	C-C	0.12	83.517	45.633	118.818	98.523	53.874	140.097
		0.17	84.747	46.002	120.909	99.753	54.243	142.065
		0.28	85.854	46.248	122.754	100.737	54.489	143.787

Table 4.

Effects of slenderness ratio and applied voltage on critical thermal loads $\mathrm{\Delta }{T}_{cr}$ (K) of piezoelectric FG-X CNTRC beams with clamped and immovable ends ( $h_p/h=1/12$ , $V_{CNT}^{*}=0.17$ , $\lambda =1$ ).

	$L/H$	$V$ (V)
		$-500$	$-300$	$-200$	0	200	300	500
T-ID	50	444.506	426.393	417.337	399.224	381.112	372.055	353.943
	80	201.229	183.116	174.060	155.947	137.834	128.778	110.665
	100	145.088	126.975	117.919	99.806	81.693	72.637	54.524
T-D	50	334.191	320.538	313.773	300.366	287.205	280.563	267.525
	80	159.900	146.985	140.466	127.428	114.144	107.502	93.849
	100	119.433	106.149	99.384	85.608	71.463	64.206	49.323

The ultimate study in this subsection is displayed in Figure 3 considering the effects of in-plane and out-of-plane boundary conditions along with applied voltage on the thermal buckling of FG-X CNTRC beams with piezoelectric layers. In general, both in-plane and out-of-plane boundary conditions substantially affect thermal buckling behavior and the critical thermal loads are pronouncedly decreased when ends are more rigorously restrained in tangential direction, specifically $\lambda$ parameter is increased. In addition, it is recognized from Figure 3 that effect of applied voltage on S-S beams is more significant and difference between critical loads of C-C and S-S beams is smaller as $\lambda$ parameter become larger.

Figure 3.

Effects of boundary conditions and applied voltage on critical thermal loads of FG-X CNTRC beams with piezoelectric actuators.

Thermal postbuckling analysis

Numerous influences on thermal load-deflection response in the postbuckling region of CNTRC beams integrated with piezoelectric actuators are studied in this subsection. As the first analysis, the effects of CNT distribution on thermal postbuckling of piezoelectric CNTRC beams with simply supported and clamped ends undergoing positive voltage ( $V=100$  V) are shown in Figures 4 and 5, respectively. As can be observed, FG-X beams in which CNTs are more densely reinforced at near two surfaces possess the strongest capacity of loading carrying, and uniform distribution brings to an intermediate capability of thermal loading bearing of CNTRC beams. Besides, temperature dependence of material properties negatively influences the thermal postbuckling behavior of the beams, especially in the region of large deflection. The remaining analyses are performed for FG-X CNTRC beams with temperature-dependent (T-D) properties. Unlike CNT distribution, volume percentage of CNTs has slight effects on the postbuckling response of hybrid beams undergoing thermo-electrical loads, as demonstrated in Figure 6. More concretely, increase in CNT volume fraction leads to slight decrease and increase in postbuckling deflection of the beam when the deflection is small and large, respectively.

Figure 4.

Effects of CNT distribution on thermal postbuckling of CNTRC beams with simply supported and immovable ends.

Figure 5.

Effects of CNT distribution on thermal postbuckling of CNTRC beams with clamped and immovable ends.

Figure 6.

Effects of CNT volume fraction on thermal postbuckling of hybrid CNTRC beams with simply supported and immovable ends.

The influences of in-plane constraints of two ends, i.e. $\lambda$ parameter, on the thermal postbuckling behavior of piezoelectric CNTRC beams under a grounding condition ( $V=0$ ) are depicted in Figures 7 and 8 for S-S and C-C cases of out-of-plane boundary condition, respectively. Obviously, like out-of-plane boundary condition, the in-plane boundary condition dramatically affects the thermal postbuckling of hybrid CNTRC beams, and postbuckling strength is significantly weakened when ends are more severely restrained in tangential direction. Next, the effects of applied voltage on the thermal postbuckling behavior of piezoelectric CNTRC beams with S-S and C-C ends are examined in Figures 9 and 10, respectively. Evidently, the postbuckling deflection is decreased and increased when piezoelectric layers are actuated by negative and positive voltages, respectively. Accordingly, the postbuckling loading capacity can be controlled by means of variation of applied voltage. It is also realized from Figures 9 and 10 that the effect of applied voltage is more pronounced when two ends are simply supported.

Figure 7.

Effects of tangential constraints of ends on thermal postbuckling of simply supported FG-CNTRC beams with piezoelectric layers.

Figure 8.

Effects of tangential constraints of ends on thermal postbuckling of clamped FG-CNTRC beams with piezoelectric layers.

Figure 9.

Effects of applied voltage on thermal postbuckling of hybrid FG-CNTRC beams with simply supported and immovable ends.

Figure 10.

Effects of applied voltage on thermal postbuckling of hybrid FG-CNTRC beams with clamped and immovable ends.

Subsequent study is sketched in Figure 11 examining the effects of $h_p/h$ ratio of piezoelectric layer-to-CNTRC host thicknesses on the thermal postbuckling paths of S-S beams with immovable ends under a positive voltage ( $V=200$  V). Contrary to situation in Table 3 where critical thermal loads are increased as a result of decreasing $h_p/h$ ratio under a grounding condition ( $V=0$ ), Figure 11 finds that the critical thermal loads and post-critical curves of the hybrid CNTRC beams under a positive voltage are lowered as a result of decrease in $h_p/h$ ratio. Finally, the effects of slenderness ratio $L/H$ on the thermal postbuckling behavior of S-S hybrid CNTRC beams with partially movable ends ( $\lambda =0.6$ ) under a positive voltage ( $V=200$ V) are plotted in Figure 12. As shown, the $L/H$ ratio strongly affects the thermal postbuckling of CNTRC beams with piezoelectric layers. Specifically, the postbuckling paths are lower and more gradually developed when $L/H$ ratio is larger.

Figure 11.

Effects of thickness of piezoelectric layers on thermal postbuckling of FG-CNTRC beams with simply supported and immovable ends.

Figure 12.

Effects of slenderness ratio on thermal postbuckling of hybrid FG-CNTRC beams with simply supported and partially movable ends.

Concluding remarks

Thermo-electrical buckling and postbuckling behaviors of CNTRC beams with surface-bonded piezoelectric layers and tangentially restrained ends under uniform temperature rise and applied voltage have been investigated. The results indicate that distribution patterns and volume percentage of CNTs have significant and slight influences on thermal postbuckling of the beams, respectively. Besides, the following remarks are concluded:

(i) In-plane constraint of ends substantially affects thermal stability of hybrid CNTRC beams. Critical thermal loads along with postbuckling paths are strongly dropped and difference between critical loads of S-S and C-C beams is narrowed when tangential end constraints are more rigorous.

(ii) The thermal postbuckling loading capacity of piezoelectric CNTRC beams is stronger and weaker when applied voltages are negative and positive, respectively. The effects of applied voltage is more significant when two ends are simply supported.

(iii) The influence of piezoelectric layer thickness depends on the sign of applied voltage. Specifically, increase in piezoelectric layer thickness enhances and lowers the thermal postbuckling curves of beams under positive and negative voltages, respectively.