Thermal postbuckling behavior of CNT-reinforced composite sandwich plate models resting on elastic foundations with tangentially restrained edges and temperature-dependent properties

Abstract

Buckling and postbuckling behaviors of sandwich plates reinforced by single-walled carbon nanotube (CNT), rested on elastic foundations and subjected to uniform temperature rise, are investigated in this article. CNT is embedded into matrix phase through uniform or functionally graded distributions. The properties of constituent materials are assumed to be temperature-dependent, and effective properties of nanocomposite are determined by extended rule of mixture. Two models of sandwich plates with face sheets and core layer reinforced by CNTs are presented. Formulations are based on the first-order shear deformation theory taking geometrical nonlinearity, initial geometrical imperfection, plate-foundation interaction, and elasticity of tangential edge constraints into consideration. Analytical solutions of deflection and stress function are assumed, and Galerkin method is applied to derive nonlinear temperature–deflection relation from which buckling temperatures and thermal postbuckling paths are obtained through an iteration algorithm. Numerical examples show the effects of CNT volume fraction, distribution patterns, in-plane edge constraint, elastic foundations, geometrical ratios, initial imperfection, and temperature dependence of properties on thermal postbuckling behavior of nanocomposite sandwich plates. The most important finding is that sandwich plate constructed from CNT-poor nanocomposite core layer and thin homogeneous face sheets with partially movable edges bring the best capacities of thermal buckling resistance and postbuckling load carrying.

Keywords

CNT-reinforced composite sandwich plate thermal load buckling and postbuckling elastic edge constraint

Introduction

Due to unprecedentedly excellent mechanical, thermal, and electrical properties, and extremely large aspect ratio, carbon nanotube (CNT) is ideal filler into polymer matrix to form carbon nanotube–reinforced composite (CNTRC) known as an advanced nanocomposite.^1
–3 Idea of optimal distribution of CNTs motivated the concept of functionally graded CNTRC (FG-CNTRC)⁴ in which CNTs are reinforced into isotropic polymer matrix such that their volume fraction is varied along the plate thickness through functional rules in order to obtain desired responses of nanocomposite structures. This propositional work of Shen⁴ stimulated subsequent investigations on static and dynamic responses of FG-CNTRC structures. Tornabene et al.⁵ studied the linear static response of CNTRC plates and shells accounting for agglomeration of CNTs. Free vibration of FG-CNTRC plates with arbitrary shapes has been analyzed by Fantuzzi et al.⁶ using the generalized differential quadrature method. Based on a numerical approach, Mehar et al.⁷ examined the free vibration behavior of FG-CNTRC plates under elevated thermal environment. Fazzolari⁸ employed Ritz method to investigate the thermoelastic vibration and stability characteristics of FG-CNTRC plates. Bakhadda et al.⁹ analyzed the dynamic and bending behavior of CNTRC plates on elastic foundations. Free vibration response of functionally graded nanoscale plates has been considered in works.^10,11

Buckling and postbuckling behaviors of plates and shells made of composite materials have attracted attention of many researchers.^{12

–19} Numerous investigations have been conducted on the subject of stability behavior of FG-CNTRC plates and shells under mechanical loads. Based on a numerical approach with element-free methods, Liew and coworkers^20

–23 investigated the linear buckling of FG-CNTRC rectangular and skew plates with transverse shear deformation under compressive loads. Using Ritz method with different shape functions, Kiani and collaborator^24
–26 dealt with linear buckling problem for FG-CNTRC rectangular and skew plates with various boundary conditions subjected to shear, compressive, and parabolic mechanical loadings. Linear buckling behavior of FG-CNTRC cylindrical panels under axial compression and shear loads has been analyzed by Macias et al.²⁷ utilizing shell finite elements. Zhang and Liew²⁸ used element-free approach to study postbuckling response of axially compressed FG-CNTRC plates resting on Pasternak foundations. Making use of asymptotic solutions and a perturbation technique, Shen and co-operators^29

–34 presented the results of postbuckling analysis for higher order shear deformable FG-CNTRC cylindrical panels and shells subjected to axial compression, external pressure, and combined loads in thermal environments. Recently, Hieu and Tung³⁵ employed an analytical approach to examine the nonlinear stability of FG-CNTRC circular cylindrical shells surrounded by elastic media and simultaneously loaded by axial compression and lateral pressure.

Since composite and nanocomposite structures are usually exposed to severe temperature conditions, the stability of nanocomposite beams, plates, and shells under thermal loadings is a problem with considerable importance. Based on some different plate theories and approaches, Tounsi and coworkers^36

–39 dealt with the stability problem of single layer and sandwich plates made of functionally graded materials and subjected to thermal loads. Mirzaei and Kiani^40,41 treated the linear buckling problem of rectangular plates and conical shells reinforced by CNTs and exposed to uniform temperature rise. Based on different plate theories and various approaches, the postbuckling problem of FG-CNTRC rectangular plates subjected to thermal loadings has been dealt with in works of Shen and Zhang,⁴² Kiani,⁴³ and Tung.⁴⁴ Postbuckling analyses for FG-CNTRC cylindrical panels and shells under elevated temperatures have been carried out by Shen and Xiang⁴⁵ and Shen,⁴⁶ respectively, making use of a higher order shear deformation theory and a perturbation technique. Thermal buckling and postbuckling behaviors of functionally graded graphene-reinforced laminated plates have been addressed in works of Mirzaei and Kiani^47,48 using B-spline-based isogeometric finite-element method, Wu et al.⁴⁹ employing a differential quadrature-based iteration technique, and Shen et al.⁵⁰ utilizing a two-step perturbation technique.

Due to outstanding properties, sandwich-type structures play an important role in engineering applications, and appearance of advanced nanocomposites motivates investigations on behavior of sandwich structures comprising of CNTRCs. Wang and Shen⁵¹ and Shen et al.⁵² investigated nonlinear vibration and bending behavior of sandwich plates with CNTRC face sheets on elastic foundations. Free vibration of sandwich plates with CNTRC face sheets has been analyzed by Wang et al.⁵³ employing multi-term Kantorovich–Galerkin method. The vibration characteristics of CNT-reinforced sandwich plates and curved panels exposed to elevated temperature have been considered in works of Mehar and coworkers^54,55 using the finite element method. Based on some different plate theories and differential cubature method, the bending, vibration, static buckling, and dynamic buckling responses of sandwich nanocomposite plates have been examined by Kolahchi⁵⁶ and Kolahchi et al.⁵⁷ taking into account piezoelectric layers and viscoelastic effects. Using a numerical approach, Natarajan et al.⁵⁸ studied the bending and free vibration of sandwich plates with CNTRC face sheets. Employing the shooting and pseudospectral methods, Jalali and Heshmati⁵⁹ analyzed the linear buckling behavior of circular sandwich plates with tapered cores and FG-CNTRC face sheets. Shokravi⁶⁰ used Navier’s solution method to calculate buckling load of sandwich plates with FG-CNTRC layers resting on orthotropic elastic foundations subjected to magnetic field. Shen and Zhu⁶¹ investigated the postbuckling behavior of sandwich plates with FG-CNTRC face sheets resting on elastic foundations subjected to compressive load and uniform temperature rise. The thermal postbuckling behavior of sandwich plates with FG-CNTRC face sheets and various boundary conditions has been considered in work of Kiani⁶² using Ritz method with Chebyshev polynomials as shape functions.

Due to elasticity of in-plane constraints of boundary edges, the effects of tangential edge restraints on the response of structures should be considered. Previous works of Tung^63

–66 indicated that tangential restraints of boundary edges have dominant influences on buckling load and postbuckling strength of ceramic-metal functionally graded plates and shells. Based on the first-order shear deformation theory (FSDT) and element-free improved moving least-squares Ritz method, Zhang et al.^67
–69 presented results of geometrically nonlinear, free vibration, and compressive postbuckling analyses for FG-CNTRC plates with edges elastically restrained against translation and rotation. Recently, Tung and Trang^{44,70

–74} considered the effects of tangential edge constraints on the nonlinear stability of FG-CNTRC plates and cylindrical panels resting on elastic foundations and subjected to thermal, mechanical, and thermomechanical loadings. More recently, Hieu and Tung⁷⁵ analyzed the effects of tangential edge constraints on the thermomechanical nonlinear buckling behavior of FG-CNTRC toroidal shell segments.

Investigations on CNTRC sandwich structures in the literature only considered sandwich plates and shells with CNTRC face sheets. To the best of authors’ knowledge, there is no study on thermal postbuckling behavior of CNTRC sandwich plates resting on elastic foundations with elastically restrained edges. This article investigates the buckling and postbuckling behaviors of sandwich plates reinforced by CNTs, rested on elastic foundations and exposed to uniform temperature rise. Besides sandwich plate model with CNTRC face sheets, another sandwich plate model with CNTRC core layer and homogeneous face sheets is suggested in this article. The material properties of CNTs, matrix, and homogeneous layer are assumed to be temperature-dependent, and the effective properties of nanocomposite layer are estimated by extended rule of mixture. Formulations for geometrically imperfect CNTRC sandwich plates are based on the FSDT accounting for Von Karman nonlinearity, plate-foundation interaction, and elasticity of in-plane constraints of edges. Analytical solutions are assumed, and Galerkin method is applied to obtain nonlinear temperature-deflection relation from which buckling temperatures and thermal postbuckling paths are determined through an iteration process. Many various effects including beneficial and detrimental influences on the thermal postbuckling response of CNTRC sandwich plates are analyzed through numerical examples, and concluding remarks are suggested. The most important finding is that sandwich plate model with CNTRC core layer and thin homogenous face sheets can bring the best capabilities of thermal buckling resistance and postbuckling load carrying.

Two models of sandwich plates reinforced by CNTs

Consider a rectangular sandwich plate of length a, width b, and total thickness h reinforced by CNTs and rested on an elastic foundation. The plate is defined in a Cartesian coordinate system xyz origin of which is located at a corner on the middle surface, x and y axes are directed toward length and width edges, respectively, and z-axis is perpendicular to the middle surface of the plate ( $- h / 2 \leq z \leq h / 2$ ). The sandwich plate is constructed from two face sheets separated by a thicker core layer made of homogeneous or CNTRC materials. It is assumed that core layer and face sheets are perfectly bonded, and thickness of each face sheet is h_f. The present study considers two different types of sandwich plates corresponding to CNTRC and homogeneous face sheets and referred to herein as sandwich plates of type A and type B, respectively.

Sandwich plate of type A: Homogeneous core layer and CNTRC face sheets

In this type of sandwich plate, the core layer is isotropic homogeneous, and face sheets are reinforced by CNTs as shown in Figure 1. The volume fractions V_CNT of CNTs in face sheets corresponding to five different types of distribution, named as uniform distribution (UD), FG-X, FG-Λ, FG-V, and FG-O, are given in Table 1 in which $h_{0} = - h / 2, h_{1} = - h / 2 + h_{f}, h_{2} = h / 2 - h_{f}, h_{3} = h / 2$ , and $V_{CNT}^{*}$ has its definition as in previous works.^4,42

Figure 1.

FG types of CNT distribution into CNTRC face sheets of sandwich plate of type A. FG: functionally graded; CNT: carbon nanotube; CNTRC: carbon nanotube–reinforced composite.

Table 1.

Volume fraction V_CNT(z) of CNTs in face sheets of sandwich plate of type A.

Distribution type	Top face sheet $(h_{0} \leq z \leq h_{1})$	Bottom face sheet $(h_{2} \leq z \leq h_{3})$
UD	$V_{CNT}^{*}$	$V_{CNT}^{*}$
FG-X	$2 \frac{\| h_{0} + h_{1} - 2 z \|}{h_{1} - h_{0}} V_{CNT}^{*}$	$2 \frac{\| h_{2} + h_{3} - 2 z \|}{h_{3} - h_{2}} V_{CNT}^{*}$
FG-Λ	$2 \frac{z - h_{0}}{h_{1} - h_{0}} V_{CNT}^{*}$	$2 \frac{h_{3} - z}{h_{3} - h_{2}} V_{CNT}^{*}$
FG-V	$2 \frac{h_{1} - z}{h_{1} - h_{0}} V_{CNT}^{*}$	$2 \frac{z - h_{2}}{h_{3} - h_{2}} V_{CNT}^{*}$
FG-O	$2 (1 - \frac{\| 2 z - h_{0} - h_{1} \|}{h_{1} - h_{0}}) V_{CNT}^{*}$	$2 (1 - \frac{\| 2 z - h_{2} - h_{3} \|}{h_{3} - h_{2}}) V_{CNT}^{*}$

CNT: carbon nanotube; UD: uniform distribution; FG: functionally graded.

Sandwich plate of type B: CNTRC core layer and homogeneous face sheets

For sandwich plate of type B, the core layer is reinforced by CNTs, and face sheets are isotropic homogeneous as illustrated in Figure 2. The volume fractions V_CNT of CNTs in core layer ( $h_{1} \leq z \leq h_{2}$ ) corresponding to UD, FG-X, $FG-Λ$ , FG-V, and FG-O types of CNT distribution are given as follows

Figure 2.

FG types of CNT distribution into CNTRC core layer of sandwich plate of type B. FG: functionally graded; CNT: carbon nanotube; CNTRC: carbon nanotube–reinforced composite.

V_{CNT} = {\begin{array}{l} V_{CNT}^{*} UD \\ 4 \frac{| z |}{h_{2} - h_{1}} V_{CNT}^{*} FG-X \\ 2 \frac{z - h_{1}}{h_{2} - h_{1}} V_{CNT}^{*} FG-Λ \\ 2 \frac{h_{2} - z}{h_{2} - h_{1}} V_{CNT}^{*} FG-V \\ 2 (1 - \frac{2 | z |}{h_{2} - h_{1}}) V_{CNT}^{*} FG-O \end{array}

In this study, the properties of constituents are assumed to be temperature-dependent, and effective elastic moduli of CNTRC are determined according to extended rule of mixture as^4,42

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, G^m, and $V_{m} = 1 - V_{CNT}$ are Young modulus, shear modulus, and volume fraction of matrix phase, respectively. Due to imperfect interaction between CNTs and matrix phase, coefficients η_j ( $j = 1 \div 3$ ), called the CNT efficiency parameters, are introduced into equations (2)–(4) to consider the size-dependent material properties.

Effective Poisson ratio of CNTRC is assumed to be position- and temperature-independent and is determined as

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.

Subsequently, effective coefficients of thermal expansion of CNTRC layers in the longitudinal and transverse directions have the form as^45,46,62

α_{11} = \frac{V_{CNT} E_{11}^{CNT} α_{11}^{CNT} + V_{m} E^{m} α^{m}}{V_{CNT} E_{11}^{CNT} + V_{m} E^{m}}

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

where $α_{11}^{CNT}, α_{22}^{CNT}$ , and α^m are thermal expansion coefficients of the CNT and isotropic matrix, respectively. Evidently, effective Young moduli, shear modulus, and thermal expansion coefficients of CNTRC are both temperature- and position-dependent.

Formulations

The present study uses FSDT to formulate for moderately thick sandwich plates. Based on the FSDT, strain components at a z distance from middle surface 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, herein, subscript prime indicates partial derivative with respect to corresponding variable, for example, $w_{, x} = \partial w / \partial x$ and

(\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, and w denote the displacements of a point on the plane $z = 0$ in x, y, and z directions, respectively, whereas $ϕ_{x}$ and $ϕ_{y}$ are the rotations of a transverse normal about the y and x axes, respectively.

In this study, thermal loading is considered to be uniform temperature rise. Specifically, CNTRC sandwich plate is exposed to a thermal environment temperature of which is uniformly raised from thermal stress free initial state T₀ to final value T and temperature change $Δ T = T - T_{0}$ is independent of thickness variable z.

The stress components in CNTRC sandwich plate are determined 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} - α_{11} Δ T \\ ε_{y} - α_{22} Δ T \\ γ_{x y} \\ γ_{x z} \\ γ_{y z} \end{matrix})

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}

in CNTRC layers (i.e. $h_{0} \leq z \leq h_{1}$ and $h_{2} \leq z \leq h_{3}$ for sandwich plate of type A and $h_{1} \leq z \leq h_{2}$ for sandwich plate of type B), and

E_{11} = E_{22} = E_{H}, α_{11} = α_{22} = α_{H}, ν_{12} = ν_{21} = ν_{H}

Q_{11} = Q_{22} = \frac{E_{H}}{1 - ν_{H}^{2}}, Q_{12} = \frac{ν_{H} E_{H}}{1 - ν_{H}^{2}}, Q_{44} = Q_{55} = Q_{66} = \frac{E_{H}}{2 (1 + ν_{H})}

in isotropic homogeneous layers (i.e. $h_{0} \leq z \leq h_{1}$ and $h_{2} \leq z \leq h_{3}$ for sandwich plate of type B and $h_{1} \leq z \leq h_{2}$ for sandwich plate of type A) in which $E_{H}, α_{H}$ , and ν_H denote Young modulus, thermal expansion coefficient, and Poisson ratio of isotropic homogeneous material, respectively.

Force and moment resultants in the CNTRC sandwich plate are determined as

(N_{x}, N_{y}, N_{x y}) = \int_{- h / 2}^{h / 2} (σ_{x}, σ_{y}, σ_{x y}) d z, (M_{x}, M_{y}, M_{x y}) = \int_{- h / 2}^{h / 2} (σ_{x}, σ_{y}, σ_{x y}) z d z,

(Q_{x}, Q_{y}) = K_{S} \int_{- h / 2}^{h / 2} (σ_{x z}, σ_{y z}) d z

where $K_{S}$ is shear correction coefficient and is assumed to be 5/6 in the present study.

From equations (8), (10), and (13), the force and moment resultants are expressed as

(N_{x}, M_{x}) = (e_{11}, e_{12}) ε_{x}^{0} + ν_{21} (e_{11}, e_{12}) ε_{y}^{0} + (e_{12}, e_{13}) ϕ_{x, x} + ν_{21} (e_{12}, e_{13}) ϕ_{y, y} - (e_{11 T}, e_{12 T}) Δ T

(N_{y}, M_{y}) = ν_{12} (e_{21}, e_{22}) ε_{x}^{0} + (e_{21}, e_{22}) ε_{y}^{0} + ν_{12} (e_{22}, e_{23}) ϕ_{x, x} + (e_{22}, e_{23}) ϕ_{y, y} - (e_{21 T}, e_{22 T}) Δ T

(N_{x y}, M_{x y}) = (e_{31}, e_{32}) γ_{x y}^{0} + (e_{32}, e_{33}) (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})

where

(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

(e_{11 T}, e_{12 T}) = \int_{- h / 2}^{h / 2} Q_{11} (α_{11} + ν_{21} α_{22}) (1, z) d z, (e_{21 T}, e_{22 T}) = \int_{- h / 2}^{h / 2} Q_{22} (ν_{12} α_{11} + α_{22}) (1, z) d z

The sandwich plate is assumed to be rested on an elastic foundation, and plate-foundation interaction is represented according to Pasternak model as

q_{f} = k_{1} w - k_{2} (w_{, x x} + w_{, y y})

in which k₁ is the elastic modulus of Winkler springs, and k₂ is the stiffness of shear layer in Pasternak model.

Based on the FSDT, nonlinear equilibrium equation of geometrically imperfect CNTRC sandwich plate resting on an elastic foundation has the form⁷³

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} (w_{, x x} + w_{, y y}) = 0

where $w^{*} (x, y)$ is a known function representing initial geometrical imperfection, $f (x, y)$ is a stress function defined such that $N_{x} = f_{, y y}, N_{y} = f_{, x x}$ , and $N_{x y} = - f_{, x y}$ , and coefficients $a_{i 1}$ ( $i = 1 \div 5$ ) are displayed in equation (A1) of Appendix 1.

Next, strain compatibility equation of moderately thick CNTRC sandwich plate with initial geometrical imperfection has the form⁷³

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

where coefficients $a_{i 2} (i = 1 \div 7)$ are given in equation (A2) of Appendix 1.

In the present study, all edges of the plate are assumed to be simply supported and tangentially restrained. The associated boundary conditions are expressed as

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

where $N_{x 0}$ and $N_{y 0}$ are fictitious force resultants generated due to heating and tangential restraint of boundary edges and are related to average end-shortening displacements $Δ_{1}$ and $Δ_{2}$ between edges $x = 0, a$ and $y = 0, b$ , respectively, as follows

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

in which c₁ and c₂ are average tangential stiffness parameters in the x and y directions on each opposite edge, respectively, and the average end-shortening displacements are defined 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

To satisfy boundary conditions (19) approximately, analytical solutions of the deflection, stress function, and rotations are assumed as

(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 $β_{m} = m π / a$ and $δ_{n} = n π / b$ $(m, n = 1, 2, \dots)$ , W and μ denote the amplitude of deflection and imperfection size, respectively, and $A_{i} (i = 1 \div 3)$ and $B_{j} (j = 1, 2)$ are coefficients to be determined. With the aid of compatibility equation (18) and two among five equilibrium equations, these coefficients A_i and B_j are determined according to procedure described in previous works^64,66,73 as

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),

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

where coefficients $A_{3}^{*}, B_{1}^{*}$ , and $B_{2}^{*}$ are displayed in equation (A3) of Appendix 1.

Next, solutions (22)–(24) are substituted into the equilibrium equation (17), and applying Galerkin method for the resulting equation yields the following expression

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

where

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{E_{0}^{m}}{π^{2} B_{h}^{2}} K_{1} - \frac{E_{0}^{m}}{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} = \frac{W}{h}

in which

({\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{b^{2}}{E_{0}^{m} h^{3}} (k_{1} b^{2}, k_{2}), B_{h} = \frac{b}{h}, γ_{i} = \frac{1}{2} [1 - {(- 1)}^{i}] (i = m, n)

and $E_{0}^{m}$ is value of E^m calculated at room temperature $T_{0} = 300 K$ .

From equations (9) and (14), $u_{, x}$ and $v_{, y}$ are expressed in terms of deflection, imperfection, stress function, and rotations. Then, introduction of the solutions (22)–(24) into expressions of $u_{, x}$ and $v_{, y}$ and placing the obtained relations into equation (21) yield

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

and a combination of equations (20), (29), and (30) leads to the following expressions of fictitious force resultants

{\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

in which coefficients $a_{k 4}$ , $a_{k 5}$ $(k = 1 \div 5)$ and $a_{j 6}$ , $a_{j 7}$ $(j = 1 \div 3)$ can be found in the work of Tung and Trang.⁷³

Now, placing ${\bar{N}}_{x 0}$ and ${\bar{N}}_{y 0}$ from equations (31) and (32) into equation (26) gives nonlinear relation between temperature change and nondimensional maximum deflection as

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)

Evidently, buckling temperature change of geometrically perfect CNTRC sandwich plate is determined as

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

and critical buckling temperature change $Δ T_{cr}$ is the smallest value among values of $Δ T_{b}$ with respect to numbers of half waves m and n.

Due to temperature dependence of material properties, critical buckling temperature changes and thermal postbuckling paths are determined through an iteration algorithm. Detailed steps of iteration process have been described in previous works^63,64 and are omitted here for the sake of brevity.

Results and discussion

This section presents numerical results of buckling and postbuckling analyses for CNTRC sandwich plates of types A and B constructed from CNTRC and homogeneous layers and subjected to uniform temperature rise. The CNTRC layers are made of poly_(methyl methacrylate), referred to as PMMA, as matrix material, and are reinforced by (10, 10) single-walled carbon nanotubes (SWCNTs). The isotropic homogeneous layers are made of Ti-6Al-4V temperature-dependent properties of which are⁶²

E_{H} = 122.56 (1 - 4.586 \times 10^{- 4} T) GPa

α_{H} = 7.5788 (1 + 6.638 \times 10^{- 4} T - 3.147 \times 10^{- 7} T^{2}) \times 10^{- 6} K^{- 1}

ν_{H} = 0.29

The temperature-dependent properties of the PMMA are assumed to be $ν^{m} = 0.34, α^{m} = 45 (1 + 0.0005 Δ T) \times 10^{- 6} K^{- 1}$ , 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 (10,10) SWCNTs are given at discrete values of temperature in works of Shen and coworker^42,46 and, by mathematical interpolation, are formulated as continuous functions of temperature in works of Kiani and collaborator^40,62 and Shen,³⁰ and are omitted here for the sake of brevity. The CNT efficiency parameters are given in works^32,42,46 as $(η_{1}, η_{2}, η_{3}) = (0.137, 1.022, 0.715)$ for the case of $V_{CNT}^{*} = 0.12, (η_{1}, η_{2}, η_{3}) = (0.142, 1.626, 1.138)$ for the case of $V_{CNT}^{*} = 0.17$ , and $(η_{1}, η_{2}, η_{3}) = (0.141, 1.585, 1.109)$ for the case of $V_{CNT}^{*} = 0.28$ . In addition, it is assumed that $G_{13} = G_{12}$ and $G_{23} = 1.2 G_{12}$ .^42,46

Verification

To verify the presented approach, the postbuckling behavior of a simply supported sandwich plate of type A with FG-CNTRC face sheets under uniform temperature rise is considered. The sandwich plate is assumed to be geometrically perfect, with immovable boundary edges and without elastic foundations. Postbuckling temperature-deflection curve of a sandwich plate with FG-CNTRC face sheets is given in Figure 3 in comparison with the result of Kiani⁶² using Ritz method, the shape functions of which are estimated according to the Chebyshev polynomials. As can be seen, a good agreement is achieved in this comparison.

Figure 3.

Comparison of postbuckling curves of a sandwich plate with CNTRC face sheets and immovable edges under uniform temperature rise. CNTRC: carbon nanotube–reinforced composite.

As a second example for validation, thermal buckling behavior of a simply supported CNTRC plate with immovable edges, without elastic foundations and subjected to uniform temperature rise, is considered. Critical buckling temperatures of CNTRC plates obtained by the present work for a special case of CNTRC sandwich plate of type B as $h_{f} = 0$ are compared in Table 2 with the results of Mirzaei and Kiani⁴⁰ utilizing Ritz method and those of Shen and Zhang⁴² making use of asymptotic solutions and a two-step perturbation technique. For the sake of comparison, only results in Table 2 are calculated as longitudinal thermal expansion coefficient is estimated according to the rule of mixture as

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

Table 2.

Critical buckling temperatures $T_{cr} = T_{0} + Δ T_{cr} (K)$ of simply supported CNTRC plates with immovable edges and temperature-dependent properties under uniform temperature rise [ $V_{CNT}^{*} = 0.17, (m, n) = (1, 1)$ , $(K_{1}, K_{2}) = (0, 0)$ ].

$a / b, b / h$	Distribution	Mirzaei and Kiani⁴⁰	Shen and Zhang⁴²	Present
1.0, 20	UD	343.00	343.00	342.71
1.0, 20	FG-X	359.43	359.52	359.38
1.5, 20	UD	325.08	325.08	324.94
1.5, 20	FG-X	335.14	335.54	335.31
2.0, 10	UD	363.86	363.67	363.45
2.0, 10	FG-X	380.94	381.51	381.11

CNTRC: carbon nanotube–reinforced composite; CNT: carbon nanotube; UD: uniform distribution; FG: functionally graded.

Again, it is evident from Table 2 that our result excellently agrees with those in the literature.

In what follows, thermal buckling and postbuckling behaviors of CNTRC sandwich plates of types A and B will be analyzed. To measure degree of in-plane constraints of edges in a more convenient 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}}

In such a way, values of $λ_{1} = 0$ and $λ_{2} = 0$ represent movable edges $x = 0, a$ and $y = 0, b$ , respectively, whereas values of $λ_{1} = 1$ and $λ_{2} = 1$ describe immovable edges $x = 0, a$ and $y = 0, b$ , respectively. Intermediate levels of tangential edge constraints, known as partially movable edges, are determined by intermediate values of tangential stiffness parameters, that is, $0 < λ_{1}, λ_{2} < 1$ . For the sake of brief expression, the sandwich plates are assumed to be square ( $a / b = 1$ ), geometrically perfect, with all immovable edges and temperature-dependent properties, and without elastic foundations (i.e. $K_{1} = K_{2} = 0$ ), unless otherwise specified. Furthermore, temperature-dependent and independent material properties will be mentioned as T-D and T-ID properties, respectively. Previous works^42,44,64,73 on thermal buckling and postbuckling of composite plates demonstrated that plates are buckled at fundamental mode shape $(m, n) = (1, 1)$ under thermal loads. Therefore, numerical results presented in this section correspond to mode shape $(m, n) = (1, 1)$ of deflection.

Thermal buckling analysis

Results of thermal buckling analysis for geometrically perfect CNTRC sandwich plates of types A and B are given in this subsection. Table 3 indicates the effects of CNT volume fraction and distribution patterns in nanocomposite layer and ratio of homogeneous core thickness to face sheet thickness $h_{H} / h_{f}$ on the critical buckling temperature change $Δ T_{cr}$ of sandwich plates of type A with CNTRC face sheets and immovable edges. It is clear that, critical thermal load is the smallest and highest for FG-Λ and FG-V types of CNT distribution, respectively, whereas UD, FG-X, and FG-O types give close values of critical temperature change. Furthermore, critical temperature change is increased and decreased as CNT volume fraction and thickness of face sheets are increased, respectively. As a special case, $h_{H} / h_{f}$ ratio very slightly influences on critical temperature change of sandwich plate with FG-V type and $V_{CNT}^{*} = 0.28$ . Next, the effects of tangential edge constraints, elastic foundations, and aspect ratio on critical buckling temperature change of sandwich plates of type A with FG-V CNTRC face sheets are shown in Table 4. Obviously, tangential edge constraints have sensitive effects on thermal buckling behavior, and critical temperature change is significantly reduced due to increase in tangential stiffness parameters λ₁ and λ₂, and tangential constraint in x-direction (i.e. direction of aligned CNTs), characterized by $λ_{1}$ parameter, gives more pronounced effects. More specifically, critical temperature of plates with $(λ_{1}, λ_{2}) = (0.5, 0.5)$ is approximately twice that of plates with $(λ_{1}, λ_{2}) = (1.0, 1.0)$ and critical temperature of square plates with $(λ_{1}, λ_{2}) = (0.5, 1.0)$ is considerably higher than that of square plates with $(λ_{1}, λ_{2}) = (1.0, 0.5)$ . Moreover, critical temperature change of sandwich plates is enhanced and dropped due to increase in stiffness of elastic foundations and aspect ratio, respectively.

Table 3.

Comparisons of critical buckling temperature change $Δ T_{cr} (K)$ of sandwich plates of type A with CNTRC face sheets and immovable edges [ $a / b = 1$ , $b / h = 20, (K_{1}, K_{2}) = (0, 0), (λ_{1}, λ_{2}) = (1, 1), (m, n) = (1, 1)$ ].

$V_{CNT}^{*}$	$h_{H} / h_{f}$	UD	FG-X	FG-O	FG-V	$F G - Λ$
0.12	8	230.48	230.60	230.36	233.44	227.52
	6	216.53	216.78	216.41	221.35	211.84
	4	197.15	197.64	196.78	205.91	188.51
0.17	8	244.06	244.31	244.06	248.50	239.86
	6	233.56	233.94	233.44	240.48	226.90
	4	219.49	220.36	219.12	232.09	207.39
0.28	8	270.73	271.22	270.85	277.89	264.06
	6	266.41	267.15	266.41	277.64	255.91
	4	261.59	263.07	261.22	281.59	242.95

CNTRC: carbon nanotube–reinforced composite; CNT: carbon nanotube; UD: uniform distribution; FG: functionally graded.

Table 4.

Effects of tangential edge constraints, elastic foundations, and aspect ratio on critical buckling temperature change $Δ T_{cr} (K)$ of sandwich plates of type A with FG-V CNTRC face sheets [ $b / h = 20, h_{H} / h_{f} = 6, V_{CNT}^{*} = 0.17, (m, n) = (1, 1)$ ].

$(K_{1}, K_{2})$	$a / b$	$(λ_{1}, λ_{2})$
$(K_{1}, K_{2})$	$a / b$	$(0.5, 0.5)$	$(0.8, 0.8)$	$(0.5, 1.0)$	$(1.0, 0.5)$	$(1.0, 1.0)$
$(0, 0)$	1	468.4 (9.5%)^a	297.4 (8.3%)	352.9 (7.2%)	296.6 (9.6%)	240.5 (7.4%)
	1.5	277.5 (10.7%)	180.1 (7.4%)	192.2 (6.9%)	199.6 (9.1%)	146.3 (6.3%)
	2	226.5 (10.0%)	147.3 (6.7%)	149.3 (6.1%)	174.3 (8.6%)	119.7 (5.5%)
$(50, 0)$	1	502.44 (7.8%)	316.3 (7.5%)	376.6 (6.1%)	315.3 (8.9%)	255.3 (6.8%)
	1.5	318.0 (9.3%)	205.1 (6.7%)	219.2 (6.0%)	227.3 (8.4%)	166.3 (5.7%)
	2	272.7 (8.5%)	176.3 (5.9%)	178.6 (5.2%)	208.8 (7.7%)	143.0 (5.0%)
$(50, 2)$	1	530.7 (6.4%)	331.5 (6.9%)	395.8 (5.2%)	330.4 (8.3%)	267.0 (6.4%)
	1.5	341.8 (8.4%)	219.5 (6.3%)	234.9 (5.5%)	243.3 (7.9%)	177.8 (5.4%)
	2	296.2 (7.7%)	190.7 (5.5%)	193.4 (4.7%)	226.0 (7.2%)	154.7 (4.6%)

CNTRC: carbon nanotube–reinforced composite; CNT: carbon nanotube.

^aDifference = $\frac{100 % [Δ T_{cr}^{T-ID} - Δ T_{cr}^{T-D}]}{Δ T_{cr}^{T-D}}$ .

The effects of CNT volume fraction $V_{CNT}^{*}$ , CNT distribution types, and ratio of face sheet thickness to total thickness $h_{f} / h$ on the critical buckling temperature change $Δ T_{cr}$ of sandwich plates of type B with CNTRC core layer and immovable edges are indicated in Table 5. There are some interesting remarks from observing Table 5. Firstly, critical temperatures of pure CNTRC plates (i.e. $h_{f} = 0$ ) are slightly increased when CNT volume percentage increases from 0.12 to 0.28, whereas critical temperatures of CNTRC sandwich plates (i.e. $h_{f} \neq 0$ ) are decreased as CNT volume percentage is increased. Secondly, critical temperatures of CNTRC sandwich plates corresponding to FG-X and FG-O distribution types are the highest and lowest, respectively, and critical temperatures of FG-O plates are strongly increased when $h_{f} / h$ increases from 0 to 0.1, then slowly enhanced as $h_{f} / h$ increases from 0.1 to 0.2. Thirdly, for FG-X and UD types of CNT distribution, critical thermal loads of pure CNTRC plates (i.e. $h_{f} / h = 0$ ) are remarkably lower than those of CNTRC sandwich plates (i.e. $h_{f} / h \neq 0$ ), especially for smaller values of $V_{CNT}^{*}$ , but critical buckling temperatures of CNTRC sandwich plates are slightly reduced (even extremely slightly reduced in the case of UD type and higher values of $V_{CNT}^{*}$ ) as $h_{f} / h$ increases from 0.1 to 0.2.

Table 5.

Comparisons of critical temperature change $Δ T_{cr} (K)$ of sandwich plates of type B with CNTRC core layer and immovable edges [ $a / b = 1, b / h = 25, (K_{1}, K_{2}) = (0, 0)$ , $(λ_{1}, λ_{2}) = (1, 1)$ , $(m, n) = (1, 1)$ ].

$V_{CNT}^{*}$	$h_{f} / h$	FG-X	UD	FG-O
0.12	0	226.4^a (281.6^b)	174.1 (207.6)	108.5 (122.5)
	0.1	333.8 (382.3)	303.7 (350.8)	276.7 (319.8)
	0.2	306.2 (347.2)	297.0 (337.7)	290.2 (328.5)
0.17	0	233.7 (287.8)	176.8 (209.2)	108.6 (122.2)
	0.1	306.3 (350.9)	270.7 (312.5)	238.8 (275.5)
	0.2	284.6 (323.2)	272.6 (310.3)	263.1 (298.3)
0.28	0	237.3 (316.0)	182.6 (229.1)	115.4 (134.5)
	0.1	294.9 (351.5)	250.7 (299.5)	210.2 (249.5)
	0.2	266.5 (310.5)	250.0 (291.5)	236.1 (273.9)

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

^aT-D.

^bT-ID.

Final example for thermal buckling analysis is shown in Figure 4 considering the effects of $h_{f} / h$ thickness ratio on the critical buckling temperature change $Δ T_{cr}$ of sandwich plates of type B with FG-CNTRC core layer, immovable edges, and various values of volume percentage of CNTs. Obviously, there is an optimal $h_{f} / h$ ratio, about $0.05 \leq h_{f} / h \leq 0.15$ in this specific illustration, for which the critical thermal load of CNTRC sandwich plate is the highest, and for larger values of thickness of homogeneous face sheets, $Δ T_{cr}$ is decreased due to increase in $h_{f} / h$ ratio. It is very interesting that, unlike pure CNTRC plates (i.e. $h_{f} / h = 0$ ), critical thermal load of CNTRC sandwich plate with homogeneous face sheets is lowered as core layer is CNT-richer.

Figure 4.

Effects of thickness of face sheets on critical thermal load of geometrically perfect sandwich plates of type B with FG-CNTRC core layer and immovable edges. FG-CNTRC: functionally graded carbon nanotube–reinforced composite.

Thermal postbuckling analysis

This subsection presents numerical illustrations of thermal postbuckling analysis for CNTRC sandwich plates in graphical form. Figure 5 shows the effects of CNT distribution types on the postbuckling behavior of sandwich plates of type A with CNTRC face sheets and immovable edges under uniform temperature rise. It is obvious that postbuckling paths of FG-V and FG-Λ plates are the highest and lowest, respectively, whereas postbuckling paths of FG-X, FG-O, and UD plates are very close to each other. Accordingly, for midplane symmetric model of CNTRC sandwich plate of type A, FG-V type of CNT distribution bring the best postbuckling response under thermal load. Next, the effects of tangential constraints of boundary edges on the thermal postbuckling behavior of CNTRC sandwich plates of type A are examined in Figure 6. This figure indicates that thermal load carrying capability of sandwich plates is significantly lowered as edges are prevented from in-plane displacement. More specifically, equilibrium paths are strongly dropped as $(λ_{1}, λ_{2})$ parameters increase from (0.5,0.5) to (0.8,0.8) and then gradually lowered as $(λ_{1}, λ_{2})$ parameters vary from (0.8,0.8) to (1.0,1.0). Moreover, detrimental effect of T-D properties on the postbuckling response is more slight when edges are more rigorously restrained.

Figure 5.

Effects of CNT distribution types on the thermal postbuckling behavior of sandwich plates with CNTRC face sheets and all immovable edges. CNT: carbon nanotube; CNTRC: carbon nanotube–reinforced composite.

Figure 6.

Effects of tangential edge constraints on the thermal postbuckling behavior of sandwich plates with FG-CNTRC face sheets. FG-CNTRC: functionally graded carbon nanotube–reinforced composite.

As a sequent example, Figure 7 considers the effects of homogeneous core thickness-to-face sheet thickness ratio $h_{H} / h_{f}$ on the thermal postbuckling behavior of geometrically perfect and imperfect CNTRC sandwich plates of type A. Evidently, the thermal load bearing capacity of CNTRC sandwich plates is improved as homogeneous core layer becomes thicker. Subsequently, the effects of initial geometrical imperfection on the nonlinear stability of CNTRC sandwich plates of type A with partially movable and immovable edges under thermal loading are analyzed in Figure 8. As shown, thermally loaded CNTRC sandwich plates are sensitive to initial geometrical imperfection, and temperature–deflection path is lowered due to increase in imperfection size μ in initial postbuckling region. Nevertheless, initial imperfection has beneficial influence on the load carrying capability of sandwich plates as deflection exceeds a definite value which is smaller for immovable case of in-plane boundary condition.

Figure 7.

Effects of thickness of face sheets and imperfection on the thermal postbuckling behavior of sandwich plates with FG-CNTRC face sheets and all immovable edges. FG-CNTRC: functionally graded carbon nanotube–reinforced composite.

Figure 8.

Effects of geometrical imperfection and degree of in-plane constraints on the thermal postbuckling behavior of sandwich plates with FG-CNTRC face sheets. FG-CNTRC: functionally graded carbon nanotube–reinforced composite.

Next, Figure 9 considers the effects of CNT volume fraction on the postbuckling behavior of sandwich plates of type A with FG-CNTRC face sheets and immovable edges. As shown, thermal postbuckling equilibrium paths are enhanced, and difference between these paths corresponding to cases of T-D and T-ID properties is bigger as CNT volume percentage in face sheets is increased. The effects of elastic foundations and side-to-thickness ratio on the thermal postbuckling behavior of sandwich plates with FG-CNTRC face sheets are depicted in Figure 10. Temperature–deflection equilibrium paths are considerably lower and more gradual as the plates become thinner. Furthermore, elastic foundations pronouncedly improve the load carrying capability of sandwich plates, and beneficial influences of elastic foundations are more remarked for thicker plates. Ultimate example for postbuckling analysis of CNTRC sandwich plates of type A is displayed in Figure 11 considering effects of elastic foundations and aspect ratio on the thermal postbuckling behavior of sandwich plates with FG-CNTRC face sheets. It is recognized that both buckling temperature and postbuckling temperature–deflection path are lowered due to increase in $a / b$ ratio, and the support of elastic foundation brings more efficiency for rectangular sandwich plates.

Figure 9.

Effects of CNT volume fraction on the thermal postbuckling behavior of sandwich plates with FG-CNTRC face sheets and all immovable edges. CNT: carbon nanotube; FG-CNTRC: functionally graded carbon nanotube–reinforced composite.

Figure 10.

Effects of elastic foundations and side-to-thickness ratio on the thermal postbuckling behavior of sandwich plates with FG-CNTRC face sheets and all immovable edges. FG-CNTRC: functionally graded carbon nanotube–reinforced composite.

Figure 11.

Effects of elastic foundation and aspect ratio on the thermal postbuckling behavior of sandwich plates with FG-CNTRC face sheets and all immovable edges. FG-CNTRC: functionally graded carbon nanotube–reinforced composite.

In what follows, thermal postbuckling behavior of sandwich plates of type B with CNTRC core layer will be analyzed. Figure 12 illustrates the effects of CNT distribution types on the postbuckling behavior of sandwich plates of type B with CNTRC core layer and immovable edges under uniform temperature rise. Among three types of CNT distribution, FG-X and FG-O types give the best and worst postbuckling responses, respectively, and UD pattern leads to intermediate postbuckling strength of thermally loaded CNTRC sandwich plates of type B. Accordingly, in most of the following numerical examples relating to CNTRC sandwich plates of type B, FG-X type of CNT distribution in core layer will be considered. Next, the effects of tangential edge constraints and temperature-dependent properties on thermal postbuckling behavior of sandwich plates with FG-CNTRC core layer are analyzed in Figure 13. As can be observed, in small region of deflection, postbuckling temperature–deflection paths are dropped due to rigorous constraints of edges and/or temperature dependence of material properties. It is unexpected that equilibrium paths are higher as T-D properties are accounted for in relatively deep region of postbuckling response.

Figure 12.

Effects of CNT distribution types on thermal postbuckling behavior of sandwich plates with CNTRC core layer and immovable edges. CNT: carbon nanotube; CNTRC: carbon nanotube–reinforced composite.

Figure 13.

Effects of tangential edge constraints and temperature-dependent properties on thermal postbuckling behavior of sandwich plates with FG-CNTRC core layer. FG-CNTRC: functionally graded carbon nanotube–reinforced composite.

To access the effects of $h_{f} / h$ ratio on the thermal postbuckling behavior of sandwich plates of type B, Figure 14 depicts the thermal load–deflection curves of geometrically perfect and imperfect CNTRC sandwich plates with homogeneous face sheets and immovable edges corresponding to three various values of $h_{f} / h$ ratio. Evidently, pure FG-CNTRC plate (i.e. $h_{f} / h = 0$ ) has the lowest equilibrium paths in shallow region of deflection and equilibrium paths of sandwich plates become lower when $h_{f} / h$ ratio is increased from 0.1 to 0.2. Therefore, probably, a sandwich plate with CNTRC core layer covered by relatively thin homogeneous face sheets has the highest capacity of thermal load carrying. As a next illustration, the effects of CNT volume fraction on the thermal postbuckling behavior of pure FG-CNTRC plates ( $h_{f} / h = 0$ ) and sandwich plates ( $h_{f} / h = 0.2$ ) with FG-CNTRC core layer are examined in Figure 15. It is clear that critical temperature and thermal postbuckling path of pure CNTRC plates are slightly enhanced as $V_{CNT}^{*}$ is increased from 0.12 to 0.28. In contrast, the postbuckling strength of sandwich plates with homogenous face sheets is evidently reduced as CNT volume percentage of core layer is increased. Then, Figure 16 displays the thermal postbuckling curves of geometrically perfect sandwich plates with CNTRC core layer and different values of $a / b$ and $b / h$ geometrical ratios. Obviously, both buckling temperature and postbuckling strength of sandwich plates are significantly decreased as $a / b$ and $b / h$ ratios are increased. Furthermore, postbuckling equilibrium path is very slowly developed for larger values of $a / b$ and $b / h$ ratios, that is, for longer and thinner sandwich plate.

Figure 14.

Effects of thickness of face sheets on thermal postbuckling behavior of sandwich plates with FG-CNTRC core layer and immovable edges. FG-CNTRC: functionally graded carbon nanotube–reinforced composite.

Figure 15.

Effects of CNT volume fraction on thermal postbuckling behavior of pure FG-CNTRC plates and sandwich plates with FG-CNTRC core layer. CNT: carbon nanotube; FG-CNTRC: functionally graded carbon nanotube–reinforced composite.

Figure 16.

Effects of geometrical parameters on the thermal postbuckling behavior of sandwich plates with FG-CNTRC core layer and immovable edges. FG-CNTRC: functionally graded carbon nanotube–reinforced composite.

Subsequently, Figure 17 demonstrates that thermal buckling resistance and postbuckling load carrying capacities of sandwich plates with CNTRC core layer are fundamentally improved due to support of elastic foundations, especially Pasternak-type foundations. Finally, thermal postbuckling behavior of two sandwich plate models, specifically type A with UD or FG-V CNTRC face sheets and type B with UD or FG-X CNTRC core layer, with $h_{f} = h / 4$ are compared in Figure 18. It is noted that volume fractions of CNT reinforcement, matrix material, and homogeneous material in these two models are the same. It is worth to note that temperature–deflection curves of sandwich plate of type B are considerably higher than those of sandwich plate of type A for both UD and FG distributions of CNTs in nanocomposite layers. In other words, with the same volume fractions of constituents, thermal buckling and postbuckling responses of CNTRC sandwich plate are significantly better as CNTRC core layer is covered by homogeneous face sheets.

Figure 17.

Effects of elastic foundations on the thermal postbuckling behavior of sandwich plates with CNTRC core layer and immovable edges. CNTRC: carbon nanotube–reinforced composite.

Figure 18.

Comparison of thermal postbuckling behavior of sandwich plates with CNTRC face sheets (type A) and CNTRC core layer (type B). CNTRC: carbon nanotube–reinforced composite.

Concluding remarks

Based on an analytical approach and an iteration algorithm, buckling and postbuckling behaviors of moderately thick CNTRC sandwich plates resting on elastic foundations with initial geometrical imperfection, tangentially restrained edges and temperature-dependent properties, and subjected to uniform temperature rise have been analyzed. Abovementioned numerical results suggest the following remarks.

For sandwich plate of type A with CNTRC face sheets and homogeneous core layer:

1. Type of CNT distribution in face sheets has slight effects on buckling temperatures and thermal postbuckling response of sandwich plate. Specifically, critical thermal loads and postbuckling curves of FG-V and FG-Λ plates are slightly higher and lower, respectively, whereas those of UD, FG-X, and FG-O plates are almost the same.

2. Critical buckling temperature and thermal postbuckling path of CNTRC sandwich plates are increased and decreased as CNT volume fraction and thickness of face sheets h_f are increased, respectively.

For sandwich plate of type B with homogeneous face sheets and CNTRC core layer:

3. Thermal postbuckling curves of CNTRC sandwich plates ( $h_{f} \neq 0$ ) are pronouncedly higher than those of pure CNTRC plates ( $h_{f} = 0$ ); however, these curves are lowered as thickness of face sheets is increased. Accordingly, relatively thin homogeneous face sheets can bring the best postbuckling response of CNTRC sandwich plate under thermal loads.

4. Unlike pure CNTRC plates, thermal buckling and postbuckling behaviors of CNTRC sandwich plates with homogeneous face sheets become worse as volume fraction of CNT reinforcement in core layer is increased.

5. Type of CNT distribution in core layer has significant effects on critical buckling temperature and postbuckling strength of CNTRC sandwich plates, and FG-X type of CNT distribution gives the best thermal postbuckling response.

For both CNTRC sandwich plates of types A and B:

6. Critical buckling temperature and thermal postbuckling strength of CNTRC sandwich plates are strongly reduced when boundary edges are more rigorously restrained in tangential displacement.

7. Thermal postbuckling strength of CNTRC sandwich plates is evidently enhanced and dropped as stiffness of elastic foundation and $a / b, b / h$ ratios are increased, respectively.

8. Thermally loaded CNTRC sandwich plate is sensitive to initial geometrical imperfection, and equilibrium path of the plate in deep region of deflection is higher due to the presence of initial imperfection.

As a final remark, a sandwich plate with CNTRC core layer and thin homogeneous face sheets resting on a Pasternak elastic foundation and with partially movable edges can bring the best thermal postbuckling response.

Footnotes

Funding

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

ORCID iD

Hoang Van Tung

Appendix 1

The coefficients $a_{i 1}$ ( $i = 1 \div 5$ ) in equation (17) are defined as

The detailed definitions of the coefficients $a_{i 2} (i = 1 \div 7)$ in equation (18) are

The coefficients $A_{3}^{*}, B_{1}^{*}$ , and $B_{2}^{*}$ in equation (25) are defined as follows

in which

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