Thermal postbuckling of shear deformable CNT-reinforced composite plates with tangentially restrained edges and temperature-dependent properties

Abstract

Buckling and postbuckling behaviors of moderately thick composite plates reinforced by single-walled carbon nanotubes (SWCNTs), rested on elastic foundations and subjected to two types of thermal loading are investigated in this article. Carbon nanotubes (CNTs) are reinforced into isotropic polymer matrix according to functional rules in which volume fractions of constituents are graded in the thickness direction. Material properties of constituents are assumed to be temperature-dependent and effective properties of nanocomposite are estimated by extended rule of mixture. Formulations are based on first-order shear deformation theory taking von Karman nonlinearity, initial geometrical imperfection, tangential constraints of edges, and two-parameter elastic foundation into consideration. Approximate solutions are assumed to satisfy simply supported boundary conditions and Galerkin method is applied to derive nonlinear temperature–deflection relations from which buckling temperatures and postbuckling equilibrium paths are determined by an iteration algorithm. Novel findings of the present study are that deteriorative influences of temperature-dependent properties on the postbuckling behavior become more serious as plate edges are partially movable, CNT volume fraction is higher, elastic foundations are stiffer, plates are thicker, and/or temperature linearly changed across the thickness.

Keywords

CNT-reinforced composite thermal buckling and postbuckling tangential edge constraints temperature-dependent properties elastic foundations

Introduction

Since their excellent properties, carbon nanotubes (CNTs) have attracted attention of many researchers in many different fields. CNTs are obtained by rolling sheets of graphite and have single-walled (SWCNT) or multi-walled (MWCNT) structures. The reports of material scientists proven that CNTs possess extremely high elastic moduli and very superior electrical and thermal properties with very large aspect ratio.^1
–3 There is no previous material displayed extraordinary characteristics as CNTs. Due to unprecedentedly wonderful mechanical, electrical, and thermal properties, CNTs are used as excellent fillers in polymer matrix to obtain carbon nanotube–reinforced composite (CNTRC) known as advanced nanocomposite.^1,4 Nanocomposites with CNT fillers are widely used in industrial applications⁵ and have promising potential for numerous applications in the future.^6,7

The concept of functionally graded carbon nanotube–reinforced composite (FG-CNTRC) is first proposed by Shen⁸ in which volume fraction of CNTs is varied in the thickness direction according to functional rules. Shen’s this work on the nonlinear bending of FG-CNTRC plates in thermal environments has motivated subsequent investigations. Zhang et al.⁹ conducted the results of nonlinear bending analysis of thick FG-CNTRC plates resting on Pasternak foundations using a numerical method. Hajmohammad et al.¹⁰ used differential quadrature method (DQM) to study dynamic response of CNTRC cylindrical shells submerged in an incompressible fluid subjected to earthquake, thermal, and moisture loads. Hajmohammad et al.¹¹ analyzed vibration of nanocomposite sandwich conical shells with piezoelectric layers. Kolahchi and coworkers^12,13 employed numerical approaches based on different plate theories to investigate the bending, vibration, and buckling of viscoelastic sandwich nanoplates and wave propagation of FG-CNTRC sandwich plates. Vibration analysis of FG-CNTRC piezoelectric cylindrical shells subjected to temperature distributions has been carried out by Madani et al.¹⁴ utilizing differential cubature method. Buckling behavior of concrete columns armed with CNTs has been analyzed by Arani and Kolahchi.¹⁵ Basing on Reddy plate theory, Shokravi¹⁶ treated linear buckling problem of sandwich plates with FG-CNTRC layers resting on orthotropic elastic foundations. Linear stability of rectangular and skew FG-CNTRC plates in single layer and laminated forms under mechanical loadings has been analyzed in works^17

–20 making use of a numerical approach based on the first-order shear deformation theory (FSDT). Macias et al.²¹ used numerical simulations based on shell finite elements to analyze the linear buckling of FG-CNTRC cylindrical panels under axial compression and shear load. Basing on DQM and the FSDT, the results of linear buckling analysis for quadrilateral laminated FG-CNTRC plates under mechanical load have been presented by Malekzadeh and Shojaee.²² Wattanasakulpong and Chaikittiratana²³ analyzed the bending, mechanical buckling, and vibration of FG-CNTRC plates utilizing Navier series solution. Kiani^24,25 dealt with linear buckling problems of FG-CNTRC rectangular plates under shear and parabolic loadings using Chebyshev–Ritz methods. The shear buckling behavior of FG-CNTRC skew plates has been studied by Kiani and Mirzaei²⁶ making use of Ritz method whose shape functions are constructed according to the Gram–Schmidt process.

Thermal buckling and postbuckling behaviors of composite and nanocomposite structures are problems of considerable importance in the design of these structures. Rafiee et al.²⁷ examined thermal bifurcation buckling of piezoelectric FG-CNTRC beams. Mirzaei and Kiani^28,29 made use of Chebyshev–Ritz and discrete singular convolution methods to study the linear buckling of FG-CNTRC rectangular plates and conical shells subjected to uniform temperature rise, respectively. Based on the FSDT and Ritz method with Gram–Schmidt shape functions, thermal buckling behavior of FG-CNTRC skew plates has been addressed in work of Kiani.³⁰ Shen and Zhang³¹ presented the first investigation on the subject of nonlinear stability of FG-CNTRC rectangular plates subjected to uniform temperature rise and in-plane temperature variation. In their work, basic equations are established within the framework of a higher order shear deformation plate theory and thermal postbuckling behavior of simply supported FG-CNTRC plates are determined by means of a two-step perturbation technique. Recently, Kiani³² dealt with the postbuckling problem of FG-CNTRC plates with arbitrary combination of boundary conditions under uniform temperature rise employing Ritz method whose shape functions are selected as the Chebyshev polynomials. Utilizing a two-step perturbation technique, thermal postbuckling of FG-CNTRC hybrid laminated plates and postbuckling of sandwich plates with FG-CNTRC face sheets resting on elastic foundations subjected to compressive loading and uniform temperature rise have been treated in works,^33,34 respectively. Utilizing Ritz method, Kiani³⁵ has conducted results of postbuckling analysis for sandwich plates with FG-CNTRC face sheets exposed to uniform temperature rise. Kiani and coworker^36,37 made use of a nonuniform rational B-spline-based isogeometric finite element method to study the thermal buckling and postbuckling responses of graphene-reinforced laminated plates. Basing on numerical methods, dynamic stability of nanocomposite plates has been addressed in works of Kolahchi and collaborators^38

–43 including the effects of viscidity, piezoelectric layers, CNT agglomeration, orthotropic elastomeric medium, and temperature-dependent properties. Making use of asymptotic solutions and a perturbation technique, Shen and Xiang⁴⁴ and Shen⁴⁵ investigated thermal postbuckling behaviors of FG-CNTRC cylindrical panels and circular cylindrical shells subjected to uniform temperature rise, respectively.

Above mentioned works only considered thermal buckling and postbuckling of FG-CNTRC plates and shells as in-plane condition of boundary edges is immovable. However, in practical situations, boundary edges of plates and shells may be restrained elastically and partially movable only. Accordingly, elastic condition of boundary edges should be considered to account for various degrees of tangential constraints of edges. Previous works of Tung^46,47 relating to thermal postbuckling of ceramic-metal functionally graded (FG) plates and shells indicated that temperature dependence of material properties has more detrimental influences on the thermal postbuckling behavior of plates and shells as their edges are not fully immovable. There is a limited number of investigations on behavior of FG-CNTRC structures taking into account elastic condition of in-plane restraint of edges. Zhang et al.^48,49 employed a numerical approach based on the FSDT and IMLS-Ritz method to analyze geometrically nonlinear response of FG-CNTRC plates with column supports and elastically restrained edges and postbuckling behavior of FG-CNTRC plates with edges elastically restrained against translation and rotation under axial compression, respectively. Recently, Tung⁵⁰ investigated the thermal buckling and postbuckling of thin FG-CNTRC plates on elastic foundations with temperature-independent properties and tangentially restrained edges making use of Galerkin method and classical plate theory. More recently, Tung and Trang^51
–53 presented analytical investigations on the nonlinear stability of thin FG-CNTRC rectangular plates and cylindrical panels subjected to mechanical and thermomechanical loadings taking tangential constraints of boundary edges into consideration.

Motivated by previous study⁴⁷ and as an extension of work,⁵⁰ the present article investigates the buckling and postbuckling of moderately thick FG-CNTRC plates with elastically restrained edges resting on elastic foundations and subjected to uniform temperature rise and linear temperature change across the thickness. CNTs are reinforced into isotropic polymer matrix according to uniform distribution (UD) or FG distribution along the plate thickness. The properties of CNTs and matrix are assumed to be temperature-dependent, and effective properties of CNTRC are determined by extended rule of mixture. Governing equations are based on the FSDT taking von Karman nonlinearity, initial geometrical imperfection, elastic foundations, and tangential constraints of edges into consideration. Approximate solutions are assumed to satisfy simply supported boundary conditions and Galerkin method is applied to obtain nonlinear temperature–deflection relation from which thermal buckling and postbuckling behaviors are determined by means of an iteration algorithm. The study reveals that the deteriorative effects of temperature-dependent material properties on the thermal postbuckling behavior become more pronounced as edges are partially movable, elastic foundation is stiffer, and/or as temperature is linearly changed across the thickness. Separate and combined influences of CNT volume fraction and distribution types, aspect ratios, elastic foundations, and various degrees of tangential edge constraints on the buckling temperatures and postbuckling loading capacity are analyzed through numerical examples.

CNT-reinforced composite plate resting on an elastic foundation

A rectangular nanocomposite plate of length a, width b, and thickness h is considered. The plate is reinforced by SWCNTs, rested on a Pasternak elastic foundation and defined in a Cartesian coordinate system $x y z$ origin of which is located on the middle surface at a corner, x and y are directed toward length and width dimensions, respectively, and z is perpendicular to the middle surface as shown in Figure 1. In this study, SWCNTs are reinforced into isotropic polymer matrix through UD or four different types of FG distributions named as FG-V, FG-Λ, FG-O, and FG-X types. The top and bottom surfaces of the plate are CNT-rich for FG-V and FG-Λ types, respectively. The middle surface of the plate is CNT-rich in case of FG-O type and both top and bottom surfaces are enriched by CNTs for FG-X type. Based on a micromechanical model, effective properties of composite materials may be determined by Mori–Tanaka scheme or linear rule of mixture efficiently applied to estimate overall properties of composites. An excellent agreement between results based on Mori–Tanaka approach and linear rule of mixture for buckling analysis of CNTRC cylindrical panels has been proven in the work of Macias et al.²¹ In the present study, effective Young moduli and shear modulus of the CNTRC plate are determined according to extended rule of mixture as⁸

E_{11} = η_{1} V_{CNT} E_{11}^{CNT} + V_{m} E^{m}

\frac{η_{2}}{E_{22}} = \frac{V_{CNT}}{E_{22}^{CNT}} + \frac{V_{m}}{E^{m}}

\frac{η_{3}}{G_{12}} = \frac{V_{CNT}}{G_{12}^{CNT}} + \frac{V_{m}}{G^{m}}

where $E_{11}^{CNT}$ , $E_{22}^{CNT}$ , and $G_{12}^{CNT}$ are the Young moduli and shear modulus, respectively, of the CNT, whereas E^m and G^m are Young and shear moduli, respectively, of the isotropic matrix. Coefficients η_j ( $j = 1, 2, 3$ ), called the CNT efficiency parameters, are introduced into equations (1 –3) to consider the size-dependent material properties, and $V_{CNT}$ and V_m are the volume fractions of CNTs and matrix, respectively, related by

V_{CNT} + V_{m} = 1

Figure 1.

Configuration and coordinate system of a rectangular plate resting on a Pasternak elastic foundation.

The volume fractions $V_{CNT}$ for five types of CNT distributions are assumed as

V_{CNT} = {\begin{matrix} V_{CNT}^{*} (UD) \\ (1 + \frac{2 z}{h}) V_{CNT}^{*} (FG- Λ) \\ (1 - \frac{2 z}{h}) V_{CNT}^{*} (FG-V) \\ 2 (1 - \frac{2 | z |}{h}) V_{CNT}^{*} (FG-O) \\ 2 (\frac{2 | z |}{h}) V_{CNT}^{*} (FG-X) \end{matrix}

where

V_{CNT}^{*} = \frac{w_{CNT}}{w_{CNT} + (ρ_{CNT} / ρ_{m}) (1 - w_{CNT})}

in which $w_{CNT}$ is the mass fraction of CNT in the CNTRC plate, $ρ_{CNT}$ and ρ_m are the densities of the CNTs and matrix, respectively. It is noted that, in such a way, the uniformly distributed (UD) CNTRC plate and FG-CNTRC plates have the same value of mass fraction of the CNT. Due to temperature dependence of constituent materials, effective properties of CNTRC plate are both temperature- and position-dependent.

Poisson ratio depending weakly on position and temperature can be determined by

ν_{12} = V_{CNT}^{*} ν_{12}^{CNT} + V_{m} ν^{m}

where $ν_{12}^{CNT}$ and $ν^{m}$ are Poisson ratios of the CNT and matrix, respectively.

The coefficients of thermal expansion of the CNTRC plate in the longitudinal and transverse directions have the form as^8,31

α_{11} = V_{CNT} α_{11}^{CNT} + V_{m} α^{m}

α_{22} = (1 + ν_{12}^{CNT}) V_{CNT} α_{22}^{CNT} + (1 + ν^{m}) V_{m} α^{m} - ν_{12} α_{11}

where $α_{11}^{C N T}$ , $α_{22}^{C N T}$ , and α_m are thermal expansion coefficients, respectively, of the CNT and isotropic matrix and, evidently, $α_{11}$ and $α_{22}$ are also temperature- and position-dependent.

Formulations

Based on the FSDT, strain components are expressed as

(\begin{matrix} ε_{x} \\ ε_{y} \\ γ_{x y} \end{matrix}) = (\begin{matrix} ε_{x}^{0} \\ ε_{y}^{0} \\ γ_{x y}^{0} \end{matrix}) + z (\begin{matrix} ε_{x}^{1} \\ ε_{y}^{1} \\ γ_{x y}^{1} \end{matrix}), (\begin{matrix} γ_{x z} \\ γ_{y z} \end{matrix}) = (\begin{matrix} ϕ_{x} + w_{, x} \\ ϕ_{y} + w_{, y} \end{matrix})

where

(\begin{matrix} ε_{x}^{0} \\ ε_{y}^{0} \\ γ_{x y}^{0} \end{matrix}) = (\begin{matrix} u_{, x} + w_{, x}^{2} / 2 \\ v_{, y} + w_{, y}^{2} / 2 \\ u_{, y} + v_{, x} + w_{, x} w_{, y} \end{matrix}), (\begin{matrix} ε_{x}^{1} \\ ε_{y}^{1} \\ γ_{x y}^{1} \end{matrix}) = (\begin{matrix} ϕ_{x, x} \\ ϕ_{y, y} \\ ϕ_{x, y} + ϕ_{y, x} \end{matrix})

in which $u, v, w$ are displacement components of middle surface along $x, y, z$ coordinate directions, respectively, and $ϕ_{x}, ϕ_{y}$ are rotations of normal to the middle surface with respect to y and x axes, respectively.

In the present study, the CNTRC plate is subjected to two types of thermal loading are that uniform temperature rise and linear temperature change across the thickness. In case of uniform temperature rise, the plate is exposed to an environment temperature of which is uniformly raised from thermal stress free initial state T ₀ to final value T and temperature change $Δ T = T - T_{0}$ is independent of thickness variable z. In the second case of thermal loading, temperature on a surface of the plate is maintained at reference value whereas temperature on other surface is elevated, and linear temperature change across the thickness is governed by equation

T (z) = Δ T (\frac{1}{2} - \frac{z}{h}) + T_{b}

where $Δ T = T_{t} - T_{b}$ with $T_{t} = T (- h / 2)$ and $T_{b} = T (h / 2)$ are temperatures at top and bottom surfaces of CNTRC plate, respectively.

Next, stress–strain relations of CNTRC plate are expressed as

(\begin{matrix} σ_{x} \\ σ_{y} \\ σ_{x y} \\ σ_{x z} \\ σ_{y z} \end{matrix}) = (\begin{matrix} Q_{11} & Q_{12} & 0 & 0 & 0 \\ Q_{12} & Q_{22} & 0 & 0 & 0 \\ 0 & 0 & Q_{66} & 0 & 0 \\ 0 & 0 & 0 & Q_{44} & 0 \\ 0 & 0 & 0 & 0 & Q_{55} \end{matrix}) ((\begin{matrix} ε_{x} \\ ε_{y} \\ γ_{x y} \\ γ_{x z} \\ γ_{y z} \end{matrix}) - (\begin{matrix} α_{11} \\ α_{22} \\ 0 \\ 0 \\ 0 \end{matrix}) Δ T)

where

Q_{11} = \frac{E_{11}}{1 - ν_{12} ν_{21}}, Q_{22} = \frac{E_{22}}{1 - ν_{12} ν_{21}}, Q_{12} = \frac{ν_{21} E_{11}}{1 - ν_{12} ν_{21}}, Q_{44} = G_{13}, Q_{55} = G_{23}, Q_{66} = G_{12}

and $Δ T$ is the temperature change with respect to a reference state in case of uniform temperature rise or temperature difference between two surfaces of CNTRC plate in case of linear temperature conduction across the thickness.

Based on the FSDT, force and moment resultants in the CNTRC plate are expressed as

\begin{array}{l} N_{x} = e_{11} ε_{x}^{0} + ν_{21} e_{11} ε_{y}^{0} + e_{12} ϕ_{x, x} + ν_{21} e_{12} ϕ_{y, y} - e_{11 T} Δ T \\ N_{y} = ν_{12} e_{21} ε_{x}^{0} + e_{21} ε_{y}^{0} + ν_{12} e_{22} ϕ_{x, x} + e_{22} ϕ_{y, y} - e_{21 T} Δ T \\ N_{x y} = e_{31} γ_{x y}^{0} + e_{32} (ϕ_{x, y} + ϕ_{y, x}) \\ Q_{x} = K_{S} e_{41} (ϕ_{x} + w_{, x}), Q_{y} = K_{S} e_{51} (ϕ_{y} + w_{, y}) \end{array}

\begin{array}{l} M_{x} = e_{12} ε_{x}^{0} + ν_{21} e_{12} ε_{y}^{0} + e_{13} ϕ_{x, x} + ν_{21} e_{13} ϕ_{y, y} - e_{12 T} Δ T \\ M_{y} = ν_{12} e_{22} ε_{x}^{0} + e_{22} ε_{y}^{0} + ν_{12} e_{23} ϕ_{x, x} + e_{23} ϕ_{y, y} - e_{22 T} Δ T \\ M_{x y} = e_{32} γ_{x y}^{0} + e_{33} (ϕ_{x, y} + ϕ_{y, x}) \end{array}

where K_S is shear correction coefficient assumed to be $5 / 6$ in the present study and

(e_{11}, e_{21}, e_{31}, e_{41}, e_{51}) = \int_{- h / 2}^{h / 2} (Q_{11}, Q_{22}, Q_{66}, Q_{44}, Q_{55}) d z

(e_{12}, e_{22}, e_{32}) = \int_{- h / 2}^{h / 2} (Q_{11}, Q_{22}, Q_{66}) z d z, (e_{13}, e_{23}, e_{33}) = \int_{- h / 2}^{h / 2} (Q_{11}, Q_{22}, Q_{66}) z^{2} d z

whereas thermal coefficients are defined as

\begin{array}{l} e_{11 T} = \int_{- h / 2}^{h / 2} Q_{11} (α_{11} + ν_{21} α_{22}) κ (z) d z, e_{21 T} = \int_{- h / 2}^{h / 2} Q_{22} (ν_{12} α_{11} + α_{22}) κ (z) d z \\ e_{12 T} = \int_{- h / 2}^{h / 2} Q_{11} (α_{11} + ν_{21} α_{22}) κ (z) z d z, e_{22 T} = \int_{- h / 2}^{h / 2} Q_{22} (ν_{12} α_{11} + α_{22}) κ (z) z d z \end{array}

in which $κ (z) = 1$ and $κ (z) = 1 / 2 - z / h$ for cases of uniform temperature rise and linear temperature change across the thickness, respectively.

Based on the FSDT, equilibrium equations of a plate resting on an elastic foundation and without transverse pressure are

N_{x, x} + N_{x y, y} = 0

N_{x y, x} + N_{y, y} = 0

Q_{x, x} + Q_{y, y} + N_{x} w_{, x x} + 2 N_{x y} w_{, x y} + N_{y} w_{, y y} - q_{f} = 0

M_{x, x} + M_{x y, y} - Q_{x} = 0

M_{x y, x} + M_{y, y} - Q_{y} = 0

where q_f is interaction pressure due to elastic foundation and represented by Pasternak model as

q_{f} = k_{1} w - k_{2} Δ w

in which k ₁ is Winkler foundation modulus, k ₂ is stiffness of shear layer of Pasternak foundation model, and $Δ = \partial^{2} / \partial x^{2} + \partial^{2} / \partial y^{2}$ is Laplace operator.

By virtue of constitutive equations (15, 16) and relations (11), and taking into account effects of initial geometrical imperfection, above equilibrium equations can be rewritten in a more compact form as

\begin{array}{l} a_{11} ϕ_{x, x x x} + a_{21} ϕ_{x, x y y} + a_{31} ϕ_{y, x x y} + a_{41} ϕ_{y, y y y} + a_{51} f_{, x x y y} + f_{, y y} (w_{, x x} + w_{, x x}^{*}) \\ - 2 f_{, x y} (w_{, x y} + w_{, x y}^{*}) + f_{, x x} (w_{, y y} + w_{, y y}^{*}) - k_{1} w + k_{2} Δ w = 0 \end{array}

where $f (x, y)$ is a stress function defined as

N_{x} = f_{, y y}, N_{y} = f_{, x x}, N_{x y} = - f_{, x y}

$w^{*} (x, y)$ is a known function representing initial imperfection of the plate, that is, small deviation of plate surfaces from perfectly flat configuration and detailed definitions of coefficients $a_{i 1}$ ( $i = 1 \div 5$ ) are given in equation (1A) in Appendix 1.

Next, strain compatibility equation of a geometrically imperfect plate is defined as⁴⁷

ε_{x, y y}^{0} + ε_{y, x x}^{0} - γ_{x y, x y}^{0} = w_{, x y}^{2} - w_{, x x} w_{, y y} + 2 w_{, x y} w_{, x y}^{*} - w_{, x x} w_{, y y}^{*} - w_{, y y} w_{, x x}^{*}

Expressing $ε_{x}^{0}, ε_{y}^{0}, γ_{x y}^{0}$ in terms of $N_{x}, N_{y}, N_{x y}$ and using definition (26) in the obtained relations, above compatibility equation is rewritten in the form as

\begin{array}{l} a_{12} f_{, x x x x} + a_{22} f_{, x x y y} + a_{32} f_{, y y y y} + a_{42} ϕ_{x, x x x} + a_{52} ϕ_{y, x x y} + a_{62} ϕ_{y, y y y} + a_{72} ϕ_{x, x y y} \\ - w_{, x y}^{2} + w_{, x x} w_{, y y} - 2 w_{, x y} w_{, x y}^{*} + w_{, x x} w_{, y y}^{*} + w_{, y y} w_{, x x}^{*} = 0 \end{array}

where coefficients $a_{i 2} (i = 1 \div 7)$ can be found in equation (1B) in Appendix 1.

The present study considers simply supported plates and corresponding boundary conditions are expressed as⁴⁷

\begin{array}{l} w = ϕ_{y} = M_{x} = 0, N_{x} = N_{x 0} at x = 0, a \\ w = ϕ_{x} = M_{y} = 0, N_{y} = N_{y 0} at y = 0, b \end{array}

where $N_{x 0}, N_{y 0}$ are fictitious compressive force resultants related to average end-shortening displacements $Δ_{1}$ and $Δ_{2}$ between edges $x = 0, a$ and $y = 0, b$ , respectively, as

Δ_{1} c_{1} = N_{x 0}, Δ_{2} c_{2} = N_{y 0}

in which c ₁ and c ₂ are average tangential stiffness parameters in the x and y directions on each opposite edge, respectively. The definitions of the average end-shortening displacements are expressed as

Δ_{1} = - \frac{1}{a b} \int_{0}^{a} \int_{0}^{b} \frac{\partial u}{\partial x} d y d x, Δ_{2} = - \frac{1}{a b} \int_{0}^{a} \int_{0}^{b} \frac{\partial v}{\partial y} d y d x

It is noted from equation (30) that $Δ_{1} = 0$ and $Δ_{2} = 0$ correspond to immovable $x = 0, a$ and $y = 0, b$ edges, respectively, and are obtained by selecting $c_{1} \to \infty$ and $c_{2} \to \infty$ , respectively. Inversely, $c_{1} = 0$ and $c_{2} = 0$ imply cases of movable $x = 0, a$ and $y = 0, b$ edges, respectively, and corresponding fictitious compressive force resultants $N_{x 0}$ and $N_{y 0}$ are zero-valued in these cases.

The boundary conditions (29) are approximately satisfied by the following solutions

(w, w^{*}) = (W, μ h) sin β_{m} x sin δ_{n} y

f = A_{1} cos 2 β_{m} x + A_{2} cos 2 δ_{n} y + A_{3} sin β_{m} x sin δ_{n} y + \frac{1}{2} N_{x 0} y^{2} + \frac{1}{2} N_{y 0} x^{2}

ϕ_{x} = B_{1} cos β_{m} x sin δ_{n} y, ϕ_{y} = B_{2} sin β_{m} x cos δ_{n} y

where $β_{m} = m π / a$ , $δ_{n} = n π / b$ with $m, n$ are numbers of half waves in the $x, y$ directions, respectively, W is amplitude of the deflection, μ is a small value representing imperfection size, and $A_{1}, A_{2}, A_{3}, B_{1}, B_{2}$ are coefficients to be determined.

Now, substitution of solutions (32–34) into compatibility equation (28) yields the following results

\begin{array}{l} A_{1} = \frac{δ_{n}^{2}}{32 a_{12} β_{m}^{2}} (W^{2} + 2 μ h W), A_{2} = \frac{β_{m}^{2}}{32 a_{32} δ_{n}^{2}} (W^{2} + 2 μ h W) \\ (a_{12} β_{m}^{4} + a_{22} β_{m}^{2} δ_{n}^{2} + a_{32} δ_{n}^{4}) A_{3} + (a_{42} β_{m}^{3} + a_{72} β_{m} δ_{n}^{2}) B_{1} + (a_{52} β_{m}^{2} δ_{n} + a_{62} δ_{n}^{3}) B_{2} = 0 \end{array}

Subsequently, introduction of equations (11) and (15, 16) into the equations (22, 23), then placing solutions (32–34) into the obtained relations give a system of two equations relating to coefficients $A_{3}, B_{1}, B_{2}$ and a combination of these two equations and equation (35) results in

B_{1} = B_{1}^{*} W, B_{2} = B_{2}^{*} W, A_{3} = A_{3}^{*} W

where, for sake of brevity, coefficients $A_{3}^{*}, B_{1}^{*}$ , and $B_{2}^{*}$ are displayed in equation (1C) in Appendix 1.

Next, the solutions (32–34) are substituted into equilibrium equation (25) and applying Galerkin procedure for the resulting equation yield

\begin{array}{l} a_{13} \bar{W} + a_{23} \bar{W} (\bar{W} + μ) + a_{33} \bar{W} (\bar{W} + μ) (\bar{W} + 2 μ) - (m^{2} B_{a}^{2} {\bar{N}}_{x 0} + n^{2} {\bar{N}}_{y 0}) (\bar{W} + μ) = 0 \end{array}

where

\begin{array}{l} a_{13} = \frac{π}{B_{h}} ({\bar{a}}_{11} {\bar{B}}_{1}^{*} m^{3} B_{a}^{3} + {\bar{a}}_{21} {\bar{B}}_{1}^{*} m n^{2} B_{a} + {\bar{a}}_{31} {\bar{B}}_{2}^{*} m^{2} n B_{a}^{2} + {\bar{a}}_{41} {\bar{B}}_{2}^{*} n^{3}) + \frac{π^{2}}{B_{h}^{2}} {\bar{a}}_{51} {\bar{A}}_{3}^{*} m^{2} n^{2} B_{a}^{2} \\ - \frac{{\bar{a}}_{11} B_{a}^{4}}{π^{2} B_{h}^{2}} K_{1} - \frac{{\bar{a}}_{11} B_{a}^{2}}{B_{h}^{2}} K_{2} (m^{2} B_{a}^{2} + n^{2}), a_{23} = \frac{32 B_{a}^{2}}{3 B_{h}^{2}} {\bar{A}}_{3}^{*} m n γ_{m} γ_{n} \\ a_{33} = - \frac{m^{2} n^{2} π^{2}}{16 B_{h}^{2}} (\frac{n^{2}}{{\bar{a}}_{12} m^{2}} + \frac{m^{2} B_{a}^{4}}{{\bar{a}}_{32} n^{2}}), B_{a} = \frac{b}{a}, ({\bar{N}}_{x 0}, {\bar{N}}_{y 0}) = \frac{1}{h} (N_{x 0}, N_{y 0}), \bar{W} = W / h \end{array}

in which

\begin{array}{l} ({\bar{a}}_{11}, {\bar{a}}_{21}, {\bar{a}}_{31}, {\bar{a}}_{41}) = \frac{1}{h^{3}} (a_{11}, a_{21}, a_{31}, a_{41}), {\bar{a}}_{51} = \frac{a_{51}}{h}, ({\bar{a}}_{12}, {\bar{a}}_{32}, {\bar{B}}_{1}^{*}, {\bar{B}}_{2}^{*}) = (a_{12}, a_{32}, B_{1}^{*}, B_{2}^{*}) h \\ {\bar{A}}_{3}^{*} = \frac{A_{3}^{*}}{h^{2}}, (K_{1}, K_{2}) = \frac{a^{2}}{a_{11}} (k_{1} a^{2}, k_{2}), B_{h} = \frac{b}{h}, γ_{i} = \frac{1}{2} [1 - {(- 1)}^{i}] (i = m, n) \end{array}

In what follows, fictitious compressive force resultants $N_{x 0}$ and $N_{y 0}$ will be determined. The following expressions are deduced from equations (11), (15), and (26)

\begin{array}{l} \frac{\partial u}{\partial x} = \frac{1}{a_{82}} [e_{21} f_{, y y} - ν_{21} e_{11} f_{, x x} + (e_{11} e_{22} ν_{12} ν_{21} - e_{12} e_{21}) ϕ_{x, x} + (e_{11} e_{22} ν_{21} - e_{12} e_{21} ν_{21}) ϕ_{y, y}] \\ + \frac{1}{a_{82}} (e_{11 T} e_{21} - e_{11} e_{21 T} ν_{21}) Δ T - \frac{1}{2} w_{, x}^{2} - w_{, x} w_{, x}^{*} \end{array}

\begin{array}{l} \frac{\partial v}{\partial y} = \frac{1}{a_{82}} [e_{11} f_{, x x} - e_{21} ν_{12} f_{, y y} + (e_{12} e_{21} ν_{12} - e_{11} e_{22} ν_{12}) ϕ_{x, x} + (e_{12} e_{21} ν_{12} ν_{21} - e_{11} e_{22}) ϕ_{y, y}] \\ + \frac{1}{a_{82}} (e_{21 T} e_{11} - e_{21} e_{11 T} ν_{12}) Δ T - \frac{1}{2} w_{, y}^{2} - w_{, y} w_{, y}^{*} \end{array}

where $a_{82}$ is defined in equation (1B) of the Appendix 1.

By placing the solutions (32–34) into the equations (40, 41) and then putting the resulting expressions into equation (31), the average end-shortening displacements are obtained as

Δ_{1} = a_{14} {\bar{N}}_{x 0} + a_{24} {\bar{N}}_{y 0} + a_{34} \bar{W} + a_{44} \bar{W} (\bar{W} + 2 μ) + a_{54} Δ T

Δ_{2} = a_{15} {\bar{N}}_{x 0} + a_{25} {\bar{N}}_{y 0} + a_{35} \bar{W} + a_{45} \bar{W} (\bar{W} + 2 μ) + a_{55} Δ T

where $a_{k 4}$ and $a_{k 5}$ $(k = 1 \div 5$ ) are given in detail in equation (2A) of Appendix 2.

Subsequently, a combination of equations (42, 43) and (30) leads to the following expressions

{\bar{N}}_{x 0} = a_{16} \bar{W} + a_{26} \bar{W} (\bar{W} + 2 μ) + a_{36} Δ T

{\bar{N}}_{y 0} = a_{17} \bar{W} + a_{27} \bar{W} (\bar{W} + 2 μ) + a_{37} Δ T

where the detail of coefficients $a_{j 6}$ and $a_{j 7}$ $(j = 1 \div 3)$ are displayed in equation (2C) of Appendix 2.

Now, introduction of equations (44, 45) into equation (37) leads to the following expression of nonlinear relation between temperature change and nondimensional maximum deflection

\begin{array}{l} Δ T = \frac{a_{13}}{m^{2} B_{a}^{2} a_{36} + n^{2} a_{37}} \frac{\bar{W}}{\bar{W} + μ} + \frac{a_{23} - m^{2} B_{a}^{2} a_{16} - n^{2} a_{17}}{m^{2} B_{a}^{2} a_{36} + n^{2} a_{37}} \bar{W} \\ + \frac{a_{33} - m^{2} B_{a}^{2} a_{26} - n^{2} a_{27}}{m^{2} B_{a}^{2} a_{36} + n^{2} a_{37}} \bar{W} (\bar{W} + 2 μ) \end{array}

Equation (46) indicates that, for geometrically perfect CNTRC plates, bifurcation type buckling behavior can occur as temperature change reaches buckling point value predicted as

Δ T_{b} = \frac{a_{13}}{m^{2} B_{a}^{2} a_{36} + n^{2} a_{37}}

and critical buckling temperature change $Δ T_{c r}$ is the smallest value among values of $Δ T_{b}$ with respect to numbers of m, n half waves. Due to temperature dependence of material properties, an iteration procedure is adopted to determine critical buckling thermal loads and trace nonlinear temperature–deflection equilibrium paths for postbuckling analysis. Detailed steps of iteration process have been described in previous works^46,47 and are omitted here for sake of brevity.

Results and discussion

This section presents numerical results of buckling and postbuckling analysis for CNTRC plates subjected to two types of thermal loading. The plates are made of poly(methyl methacrylate), referred to as PMMA, as matrix material and are reinforced by (10,10) SWCNTs. The temperature-dependent properties of the PMMA is assumed to be $ν^{m} = 0.34$ , $α^{m} = 45 (1 + 0.0005 Δ T) \times 10^{- 6} / K$ , and $E^{m} = (3.52 - 0.0034 T) GPa$ in which $T = T_{0} + Δ T$ and $T_{0} = 300 K$ (room temperature). The temperature-dependent properties of the SWCNTs are given at discrete values of temperature in works of Shen and coworker^31,45 and, by mathematical interpolation, are formulated as continuous functions of temperature in the work of Shen⁵⁴ as follows

\begin{array}{l} E_{11}^{CNT} = (6.18387 - 0.00286 T + 4.22867 \times 10^{- 6} T^{2} - 2.2724 \times 10^{- 9} T^{3}) TPa \\ E_{22}^{CNT} = (7.75348 - 0.00358 T + 5.30057 \times 10^{- 6} T^{2} - 2.84868 \times 10^{- 9} T^{3}) TPa \\ G_{12}^{CNT} = (1.80126 + 7.7845 \times 10^{- 4} T - 1.1279 \times 10^{- 6} T^{2} + 4.93484 \times 10^{- 10} T^{3}) TPa \\ α_{11}^{CNT} = (- 1.12148 + 0.02289 T - 2.88155 \times 10^{- 5} T^{2} + 1.13253 \times 10^{- 8} T^{3}) \times 10^{- 6} / K \\ α_{22}^{CNT} = (5.43874 - 9.95498 \times 10^{- 4} T + 3.13525 \times 10^{- 7} T^{2} - 3.56332 \times 10^{- 12} T^{3}) \times 10^{- 6} / K \\ ν_{12}^{CNT} = 0.175 \end{array}

The CNT efficiency parameters $η_{j} (j = 1 \div 3)$ are estimated by matching the Young moduli E₁₁, E₂₂ and shear modulus G₁₂ of the CNTRC determined from the extended rule of mixture to those from the molecular dynamics simulations given by Han and Elliott.⁵⁵ As presented in the works,^31,45,54 these parameters are $(η_{1}, η_{2}, η_{3}) = (0.137, 1.022, 0.715)$ for case of $V_{CNT}^{*} = 0.12$ , $(η_{1}, η_{2}, η_{3}) = (0.142, 1.626, 1.138)$ for case of $V_{CNT}^{*} = 0.17$ , and $(η_{1}, η_{2}, η_{3}) = (0.141, 1.585, 1.109)$ for case of $V_{CNT}^{*} = 0.28$ . In addition, it is assumed that $G_{13} = G_{12}$ and $G_{23} = 1.2 G_{12}$ ^31,45 and temperature-dependent and -independent properties are referred to as T-D and T-ID, respectively, in the following numerical results, in which T-ID properties are those calculated at room temperature.

Verification

To validate the proposed approach, thermal buckling and postbuckling behaviors of a simply supported CNTRC plate with all immovable edges, without elastic foundation and subjected to uniform temperature rise are considered. The critical buckling temperatures of geometrically perfect CNTRC plates obtained from the present study are shown in Table 1 in comparison with results of Shen and Zhang³¹ making use of asymptotic solutions and a two-step perturbation technique and those of Mirzaei and Kiani²⁸ using Ritz method with Chebyshev polynomials as shape functions. Also, the present analysis of thermal postbuckling behavior of geometrically perfect CNTRC plates without elastic foundations ( $K_{1} = K_{2} = 0$ ) is compared in Figure 2 with results of Shen and Zhang.³¹ It is evident from Table 1 and Figure 2 that the present study excellently agree with works in the literature.

Table 1.

Comparisons of critical buckling temperatures $T_{c r} = T_{0} + Δ T_{c r} (K)$ of perfect CNTRC plates with immovable edges subjected to uniform temperature rise [ $V_{CNT}^{*} = 0.17$ , $T_{0} = 300 K$ , $(m, n) = (1, 1)$ , $(K_{1}, K_{2}) = (0, 0)$ ].

$b / h$	Source	$a / b = 1$		$a / b = 2$
$b / h$	Source	UD	FG-X	UD	FG-X
10	Shen and Zhang³¹	399.44^a (412.36^b)	419.09 (439.88)	363.67 (368.21)	381.51 (389.16)
	Present	399.01 (411.83)	419.50 (440.54)	363.82 (368.37)	381.60 (389.21)
	Mirzaei and Kiani²⁸	399.03	419.13	363.86	380.94
20	Shen and Zhang³¹	343.00 (344.54)	359.52 (362.79)	319.01 (319.34)	326.16 (326.77)
	Present	342.96 (344.52)	359.75 (363.00)	319.01 (319.36)	325.92 (326.60)
	Mirzaei and Kiani²⁸	343.00	359.43	319.03	325.68

T-D: temperature-dependent; T-ID: temperature-independent; CNT: carbon nanotube; CNTRC: carbon nanotube–reinforced composite; UD: uniform distribution; FG: functionally graded.

^a T-D.

^b T-ID.

Figure 2.

Comparison of the postbuckling behavior of CNTRC plates with all immovable edges under uniform temperature rise. CNTRC: carbon nanotube–reinforced composite.

The remainder of this section presents numerical results for buckling and postbuckling behaviors of the CNTRC plates with tangentially restrained edges subjected to two types of thermal loading. It is examined that the CNTRC plates buckle at fundamental mode $(m, n) = (1, 1)$ under thermal loadings for arbitrary degree of in-plane edge restraint and various ratios of geometrical parameters. To measure tangential constraints of edges in a more efficient way, the following nondimensional tangential stiffness parameters are defined

λ_{1} = \frac{c_{1}}{c_{1} + e_{11}}, λ_{2} = \frac{c_{2}}{c_{2} + e_{11}}

Evidently, $(λ_{1}, λ_{2}) = (0, 0)$ and $(λ_{1}, λ_{2}) = (1, 1)$ imply cases of movable and immovable $x = 0, a$ and $y = 0, b$ edges, respectively, and cases of partially movable edges correspond to intermediate values of tangential stiffness parameters, that is, $0 < λ_{1}, λ_{2} < 1$ . Moreover, for sake of brevity, uniform temperature rise and linear temperature change across the thickness will be mentioned as UTR and LTC, respectively, and the plates are assumed to be without foundation interaction and with T-D properties unless otherwise specified.

Thermal buckling analysis

The results of buckling analysis for geometrically perfect CNTRC plates under two types of thermal loading are presented in tabular form. Table 2 considers effects of geometrical ratios $a / b$ , $b / h$ and tangential edge constraints $(λ_{1}, λ_{2})$ on the critical buckling temperatures of FG-X CNTRC plates under uniform temperature rise. It is evident that critical buckling temperatures are considerably reduced as ratios $a / b$ , $b / h$ and/or tangential stiffness parameters $(λ_{1}, λ_{2})$ are increased. The critical buckling temperatures are quickly decreased as $(λ_{1}, λ_{2})$ are increased from about $(0.4, 0.4)$ to $(0.7, 0.7)$ and slowly decreased as $(λ_{1}, λ_{2})$ are increased from about $(0.7, 0.7)$ to $(1.0, 1.0)$ . It is also observed from Table 2 that the deteriorative influences of T-D properties on the critical temperatures are more pronounced for shorter, thicker plates and/or edges are more loosely restrained in tangential motion. In contrast, the influences of T-D properties on critical temperatures are negligibly small for longer, thinner plates and/or with immovable edges. Next, the effects of CNT volume fraction and distribution types and elastic foundations on the critical buckling loads of CNTRC plates with immovable edges subjected to the UTR are analyzed in Table 3. As can be seen, the CNT volume fraction and distribution type have relatively slight and remarkable effects, respectively, on the critical thermal loads. Specifically, the critical loads are slightly increased when $V_{CNT}^{*}$ is increased from 0.12 to 0.28, and critical thermal loads $Δ T_{c r}$ and difference between predictions for $Δ T_{c r}$ in cases of T-D and T-ID properties are the highest and lowest for the FG-X and FG-O types of CNT distribution, respectively. It is interesting that critical thermal loads are the same for $FG- Λ$ and FG-V types and these critical loads are smaller than those corresponding to the UD type of CNT distribution. Furthermore, both the critical load and difference between critical loads for two cases of T-D and T-ID properties are enhanced as CNTRC plates are supported by elastic foundations.

Table 2.

Effects of geometrical ratios and tangential edge restraints on the critical buckling temperatures $T_{c r} = T_{0} + Δ T_{c r} (K)$ of FG-X CNTRC plates under uniform temperature rise [ $V_{CNT}^{*} = 0.17$ , $T_{0} = 300 K$ , $(m, n) = (1, 1)$ , $(K_{1}, K_{2}) = (0, 0)$ ].

$a / b$	$b / h$	$(λ_{1}, λ_{2})$
$a / b$	$b / h$	(0.4,0.4)	(0.7,0.7)	(1.0,1.0)
1.0	10	536.90^a (634.66^b, 18.21^c)	458.02 (497.84, 8.69)	418.76 (440.54, 5.2)
	20	432.71 (450.02, 4.00)	381.85 (388.69, 1.79)	359.38 (363.00, 1.01)
	40	345.68 (347.23, 0.45)	327.28 (327.93, 0.19)	319.51 (319.84, 0.11)
2.0	10	460.73 (489.63, 6.27)	407.40 (421.12, 3.37)	381.11 (389.20, 2.13)
	20	353.95 (356.55, 0.74)	334.94 (336.12, 0.35)	325.92 (326.60, 0.21)
	40	314.69 (314.91, 0.07)	309.51 (309.52, 0.01)	306.91 (307.02, 0.03)

T-D: temperature-dependent; T-ID: temperature-independent; CNT: carbon nanotube; CNTRC: carbon nanotube–reinforced composite; FG: functionally graded.

^a T-D.

^b T-ID.

^c Difference = 100%[ $T_{c r}^{T-ID} - T_{c r}^{T-D}$ ]/ $T_{c r}^{T-D}$ .

Table 3.

Effects of CNT volume fraction and distribution types on the critical thermal loading $Δ T_{c r} (K)$ for CNTRC plates on elastic foundations under uniform temperature rise [ $a / b = 1$ , $b / h = 20$ , $(m, n) = (1, 1)$ , $(λ_{1}, λ_{2}) = (1, 1)$ ].

$V_{CNT}^{*}$	$(K_{1}, K_{2})$	UD	FG-X	FG-O	$FG- Λ$	FG-V
0.12	(0,0)	39.13^a (4.02^b)	52.59 (5.72)	25.31 (2.34)	31.35 (3.13)	31.35 (3.13)
	(50,0)	62.59 (5.13)	88.02 (7.26)	38.02 (3.35)	48.15 (4.25)	48.15 (4.25)
	(50,5)	107.52 (7.26)	154.81 (10.48)	62.84 (4.62)	80.74 (5.83)	80.74 (5.83)
0.17	(0,0)	42.71 (4.24)	59.38 (6.10)	27.65 (2.92)	34.56 (3.54)	34.56 (3.54)
	(50,0)	67.16 (5.42)	96.91 (7.92)	41.23 (3.45)	52.47 (4.28)	52.47 (4.28)
	(50,5)	113.94 (7.63)	167.76 (11.26)	67.40 (4.85)	86.78 (6.09)	86.78 (6.09)
0.28	(0,0)	43.70 (4.92)	63.95 (7.12)	30.62 (3.07)	38.27 (3.96)	38.27 (3.96)
	(50,0)	71.72 (6.30)	109.01 (9.88)	46.91 (4.11)	59.63 (5.36)	59.63 (5.36)
	(50,5)	125.05 (8.94)	193.32 (14.12)	78.39 (5.82)	100.74 (7.52)	100.74 (7.52)

T-D: temperature-dependent; CNT: carbon nanotube; CNTRC: carbon nanotube–reinforced composite; UD: uniform distribution; FG: functionally graded.

^a T-D.

^b Difference = 100%[ $Δ T_{c r}^{T-ID} - Δ T_{c r}^{T-D}$ ]/ $Δ T_{c r}^{T-D}$ .

As final example of buckling analysis, Table 4 gives the critical buckling loads of CNTRC plates subjected to the LTC as bottom surface of the plate is maintained at room temperature $T_{b} = T_{0} = 300 K$ . Obviously, the critical buckling temperature change is significantly higher and T-D properties more detrimentally influence on the critical loads as the CNTRC plate is subjected to the LTC. Unlike the UTR case, tangential constraints of edges have more pronounced effects on the critical temperatures for case of the LTC. Specifically, the critical thermal loads for case of $(λ_{1}, λ_{2}) = (1, 1)$ almost equal half of those corresponding to case of $(λ_{1}, λ_{2}) = (0.5, 0.5)$ . It is worth to note that, in the present case of thermal loading, the critical loads of the FG-O plates are almost the same as those of the FG-Λ plates and nearly equal half of those of the FG-X plates. In addition, there is a relatively big difference between the critical temperatures corresponding to two cases of T-D and T-ID properties as the CNTRC plates are more loosely restrained at edges and/or rested on stiffer foundations, especially for FG-X distribution type. For instance, the critical temperature of FG-X plate with partially movable edges ( $λ_{1} = λ_{2} = 0.5$ ) resting on Pasternak foundations $(K_{1}, K_{2}) = (30, 2)$ in case of the T-D is smaller about 29.33 percent than that in case of the T-ID.

Table 4.

Effects of tangential edge constraints and elastic foundations on the critical thermal loading $Δ T_{c r} (K)$ of CNTRC plates subjected to linear temperature change across the thickness [ $a / b = 1$ , $b / h = 20$ , $T_{b} = T_{0} = 300 K$ , $(m, n) = (1, 1)$ , $V_{CNT}^{*} = 0.17$ ].

$(λ_{1}, λ_{2})$	$(K_{1}, K_{2})$	UD	FG-X	FG-O	FG-Λ
(0.5,0.5)	(0,0)	151.47^a (14.01^b)	201.10 (21.21)	101.23 (8.78)	100.73 (9.26)
	(30,0)	200.98 (16.34)	272.58 (24.84)	130.00 (10.06)	130.86 (10.80)
	(30,2)	263.56 (19.24)	361.34 (29.33)	166.90 (11.70)	169.50 (12.65)
(1.0,1.0)	(0,0)	82.59 (7.83)	112.83 (11.67)	54.19 (5.03)	53.58 (5.25)
	(30,0)	110.36 (9.25)	154.68 (13.71)	69.87 (5.84)	70.00 (6.12)
	(30,2)	146.16 (10.87)	207.76 (16.26)	90.12 (6.94)	91.23 (7.23)

T-D: temperature-dependent; CNT: carbon nanotube; CNTRC: carbon nanotube–reinforced composite; UD: uniform distribution; FG: functionally graded.

^a T-D.

^b Difference = 100%[ $Δ T_{c r}^{T-ID} - Δ T_{c r}^{T-D}$ ]/ $Δ T_{c r}^{T-D}$ .

Thermal postbuckling analysis

In what follows, the postbuckling behavior of geometrically perfect and imperfect CNTRC plates under two types of thermal loads are graphically analyzed. Figure 3 indicates the effects of five CNT distribution types on the postbuckling behavior of perfect CNTRC plates with immovable edges subjected to the UTR. In general, FG-X and FG-O types gives the highest and lowest equilibrium paths, respectively, in small region of postbuckling response. It is very worth to note from Figure 3 that the FG-Λ and FG-V plates have the strongest and weakest postbuckling strengths, respectively, in the deep region of postbuckling response (i.e. large deflection region), although the buckling temperatures of these plates are the same (as also shown in Table 3). Subsequently, the effects of CNT distribution types on the load–deflection equilibrium paths for perfect and imperfect CNTRC plates with immovable edges under the LTC are depicted in Figure 4. It is clear that the buckling temperatures and load carrying capability of CNTRC plates are the highest and lowest for FG-X and FG-O types, respectively, and variation trend of temperature–deflection paths is relatively regular.

Figure 3.

Effects of CNT distribution types on the postbuckling behavior of perfect CNTRC plates with immovable edges subjected to the UTR. CNTRC: carbon nanotube–reinforced composite; CNT: carbon nanotube.

Figure 4.

Effects of CNT distribution types on the postbuckling behavior of CNTRC plates with immovable edges subjected to the LTC. CNTRC: carbon nanotube–reinforced composite; CNT: carbon nanotube.

The effects of CNT distribution types and T-D properties on the postbuckling behavior of perfect CNTRC plates with nearly immovable edges ( $λ_{1} = λ_{2} = 0.9$ ) resting on Winkler foundation and subjected to the LTC are analyzed in Figure 5. Evidently, for this case of thermal loading, the FG-Λ plates have no the advantage of buckling resistance and postbuckling loading capacities, and the FG-X plates have the best postbuckling behavior in the whole region of deflection. However, the difference between temperature–deflection curves corresponding to two cases of the T-D and T-ID properties is the biggest for the FG-X plates, especially in the deep region of postbuckling response. Subsequently, Figure 6 compares the effects of the T-D properties on the postbuckling behavior of perfect FG-CNTRC plates with immovable edges subjected to two types of thermal loading. It is recognized from the Figure 6 that the FG-CNTRC plates are considerably more stable as loading condition is the LTC. Moreover, the detrimental effects of T-D properties on the buckling and postbuckling responses of the FG-CNTRC plates are more significant for the LTC case of thermal loading.

Figure 5.

Effects of CNT distribution and T-D properties on the postbuckling behavior of CNTRC plates on Winkler foundation under the LTC. CNTRC: carbon nanotube–reinforced composite; CNT: carbon nanotube; T-D: temperature-dependent.

Figure 6.

Effects of T-D properties on the postbuckling behavior of FG-CNTRC plates with immovable edges under two types of thermal loading. T-D: temperature-dependent; FG-CNTRC: functionally graded carbon nanotube–reinforced composite.

An analysis of thermal postbuckling behavior of perfect FG-CNTRC plates with various degrees of tangential edge constraints under the UTR is carried out in Figure 7 plotted with different three couples of values $(λ_{1}, λ_{2})$ . There is an obvious fact that the postbuckling strength of the plates is strongly reduced when the $(λ_{1}, λ_{2})$ vary from (0.4,0.4) to (0.7,0.7), then are gradually decreased as the $(λ_{1}, λ_{2})$ increase from (0.7,0.7) to (1.0,1.0). Furthermore, the looser tangential edge constraints are, the bigger difference between postbuckling curves corresponding to cases of the T-D and T-ID properties becomes. As a sequent example, the effects of CNT volume fraction on the postbuckling behavior of FG-CNTRC plates with partially movable edges ( $λ_{1} = λ_{2} = 0.8$ ) exposed to the UTR are considered in Figure 8. As can be seen, the influences of volume percentage of CNT reinforcement on the thermal postbuckling behavior of CNTRC plates are not impressive. Specifically, temperature–deflection equilibrium paths are slowly elevated as volume fraction $V_{C N T}^{*}$ is increased from 0.12 to 0.28.

Figure 7.

Effects of tangential edge constraints on the postbuckling behavior of FG-CNTRC plates under the UTR. FG-CNTRC: functionally graded carbon nanotube–reinforced composite.

Figure 8.

Effects of CNT volume fraction on the postbuckling behavior of FG-CNTRC plates with partially movable edges under the UTR. CNT: carbon nanotube; FG-CNTRC: functionally graded carbon nanotube–reinforced composite.

Next, Figure 9 shows the effects of elastic foundations on the postbuckling behavior of FG-CNTRC plates with partially movable edges under the LTC. It is evident that elastic foundations, especially Pasternak type foundations have very beneficial effects on the thermal postbuckling behavior of the plates. However, there is a considerable difference between temperature–deflection curves in two cases of T-D and T-ID properties due to the presence of elastic foundations.

Figure 9.

Effects of elastic foundations and T-D properties on the postbuckling behavior of FG-CNTRC plates with partially movable edges under the LTC. T-D: temperature-dependent; FG-CNTRC: functionally graded carbon nanotube–reinforced composite.

Finally, the effects of geometrical ratios $a / b$ and $b / h$ on the thermal postbuckling behavior of perfect FG-CNTRC plates with immovable edges, resting on Pasternak foundations and exposed to the UTR are examined in Figures 10 and 11, respectively. It is clear from these figures that the buckling temperatures and postbuckling load carrying capability of FG-CNTRC plates are substantially decreased as $a / b$ and $b / h$ ratios are enhanced. In addition, the Figures 10 and 11 also point out that the effects of temperature dependence of material properties on the thermal postbuckling behavior of CNTRC plates become immaterial for longer and thinner plates (i.e. larger values of $a / b$ and $b / h$ ratios).

Figure 10.

Effects of aspect ratio and T-D properties on the postbuckling behavior of FG-CNTRC plates on elastic foundation under the UTR. T-D: temperature-dependent; FG-CNTRC: functionally graded carbon nanotube–reinforced composite.

Figure 11.

Effects of side-to-thickness ratio and T-D properties on the postbuckling behavior of FG-CNTRC plates on elastic foundation under the UTR. T-D: temperature-dependent; FG-CNTRC: functionally graded carbon nanotube–reinforced composite.

Concluding remarks

Based on an analytical approach with Galerkin method and an iteration algorithm, the buckling and postbuckling behaviors of moderately thick CNTRC plates with tangentially restrained edges resting on elastic foundations and subjected to two types of thermal loading have been presented. From above analysis, the following remarks are reached:

Tangential constraints of boundary edges have the most pronounced influences on the thermal postbuckling behavior of the CNTRC plates, and the buckling temperatures and postbuckling strength of the plates are considerably reduced as tangential restraint of edges become more rigorous.

In general, the FG-X type of CNT distribution gives the best postbuckling behavior of thermally loaded CNTRC plates. In particular, under UTR, the FG-Λ and FG-V plates have the highest and lowest equilibrium paths, respectively, in the deep region of postbuckling response, although the buckling temperatures of these plates are the same.

The thermal postbuckling behavior of the CNTRC plates is much better in case of the LTC. Unlike in case of the UTR, the FG-Λ plates have no advantage of buckling resistance and postbuckling loading capacities under the LTC.

Detrimental influences of the T-D properties become more severe in the following conditions/situations, especially as these are combined: partially movable edges, linear temperature change across the thickness, FG-X type of CNT distribution, shorter and thicker plates, and stiffer elastic foundations. In contrast, the effects of T-D properties become immaterial for longer and thin CNTRC plates with immovable edges and without foundation interaction, especially in case of the UTR.

The critical buckling temperatures and temperature–deflection equilibrium paths are benignly enhanced as the volume percentage of CNT reinforcement is higher. In other words, the CNT volume fraction has no remarked effect on the thermal postbuckling behavior of the CNTRC plates. In another direction, the elastic foundations have sensitive and beneficial influences on the thermal postbuckling response of the CNTRC plates.

Accordingly, on the engineering point of view, CNTRC plates with intermediate value of CNT volume fraction, partially movable boundary edges, CNT-rich surfaces and elastic foundation interaction can exhibit optimal thermal postbuckling response.

Footnotes

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02-2017.11.

ORCID iD

Hoang Van Tung

Appendix 1

The coefficients $a_{i 1}$ ( $i = 1 \div 5$ ) in the equation (25) are defined as

The detailed definitions of the coefficients $a_{i 2} (i = 1 \div 7)$ in the equation (28) are

The coefficients $A_{3}^{*}, B_{1}^{*}$ , and $B_{2}^{*}$ in equation (36) are defined as follows

in which

Appendix 2

The coefficients $a_{k 4}$ and $a_{k 5}$ $(k = 1 \div 5$ ) in the equations (42, 43) are defined as

in which

The detail of coefficients $a_{j 6}$ and $a_{j 7}$ $(j = 1 \div 3)$ in the equations (44, 45) are

in which

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