Thermo-Mechanical Buckling Analysis of Functionally Graded Skew Laminated Plates with Initial Geometric Imperfections

Transcription

1 International Journal of Applied Mechanics Vol. 10, No. 2 (2018) (16 pages) c World Scientific Publishing Europe Ltd. DOI: /S X Thermo-Mechanical Buckling Analysis of Functionally Graded Skew Laminated Plates with Initial Geometric Imperfections Sanjay Singh Tomar and Mohammad Talha School of Engineering, Indian Institute of Technology Mandi Kamand , India talha@iitmandi.ac.in Received 21 August 2017 Revised 11 January 2018 Accepted 13 January 2018 Published 13 March 2018 The aim of the present study is to investigate thermo-mechanical buckling response of skew functionally graded laminated plates (FGLP) with initial geometric imperfections. The formulation has been performed using Reddy s higher order shear deformation theory (HSDT) with the C 0 continuous displacement field. A nine-noded isoparametric element has been employed to discretize the domain of the plate. Variational principle has been used to derive the governing differential equation of the problem. Several examples with various comparison and parametric studies have been shown to prove the efficiency and effectiveness of the present formulation. The numerical results have been highlighted with different system parameters and boundary conditions. Keywords: Functionally graded laminates; thermo-mechanical buckling analysis; imperfection sensitivity; finite element analysis; higher order shear deformation theory. 1. Introduction The use of traditional composite structures has increased drastically in the last few decades in the aerospace and space shuttle applications because of their excellent thermo-mechanical properties. The traditional composite material leads to the delamination at elevated temperature. This need led to the development of a special class of advanced composite materials known as functionally graded materials (FGM). FGM have excellent thermal properties which enable the structure to withstand its structural integrity at high thermal loading environment [Gupta and Talha, 2015]. These materials usually consist of metal and ceramic with smooth graded variation of the material properties in terms of the volume fraction of the constituent material. Graded variation prevents the structure from delamination. In certain cases, under high thermal loading, the metal ceramic interface leads to Corresponding author

2 S. S. Tomar & M. Talha some distortion, and develops micro cracks in the structure. In order to prevent this distortion, a metallic and ceramic sheet has been attached to the super structure [Kitipornchai et al., 2006], in which the metal providesbase to the structure whereas ceramic bears the thermal load. This construction is called as functionally graded laminates. Under inplane compressive loading, the structure leads to buckling at stress value less than its ultimate strength. These factors are quite dominant in space where the structure leads to high inplane loading conditions. Composite plate and shell structures have been commonly used in the aerospace and the space shuttle applications. Due to this, the buckling analyses of the functionally graded plate and shells structures have gained attention in recent years. Turvey and Marshall [1995] studied the buckling and post-buckling analysis of composite plate under various types of loading conditions. Ng et al. [2001] performed a study on the dynamic stability of FG shells under the harmonic type of axial loading. Liew et al. [2003] performed post-buckling analysis on the FGM plate embedded with piezoelectric layers. Authors used Galerkin s differential quadrature algorithm to solve nonlinear PDE. Plate was assumed to be under inplane loading, thermal loading and actuator voltage. Lanhe [2004] investigated the thermal buckling response of the FGM plate having the simply supported type of boundary conditions. He used first-order shear deformation plate theory to derive stability and equilibrium equations. Na and Kim [2006] studied the thermal buckling behavior of metal ceramic FGM plates using 18-node solid element. They assumed that material properties followed the simple power law distribution and Crank Nicolson method had been implemented for the time discretization. Shariat and Eslami [2007] employed thirdorder shear deformation theory to model the buckling response of the functionally graded plates. They performed a parametric study to measure the effect of uniform and nonuniform type of temperature distribution along with uniaxial and biaxial mechanical loading on the critical buckling load of the plate. Naderi and Saidi [2010] presented an analytical solution for the buckling response of FGM annular sector plate resting on the elastic foundation. Talha and Singh [2011] investigated the thermo-mechanical buckling behavior of metal ceramic FGM plate structure. They employed improved structural kinematics to account the effect of transverse shear deformation in the FGM plate structures. Taj and Chakrabarti [2013] performed a thermo-mechanical buckling analysis on the skew FGM plates. They used Voigt rule of mixtures and Mori Tanaka homogenization methodology to obtain the effective material properties of the plate. Ng et al. [2001] investigated the vibration response of the post-buckled cylindrical laminated shell under uniaxial and biaxial loading conditions. They used Galerkin s method to derive the governing differential equation of the problem and the N-R method in conjunction with the Riks approach for the solution purpose. Nejad et al. [2016] performed buckling analysis on the two-directional FGM nanobeams using nonlocal elasticity theory. They employed generalized differential quadrature method to obtain the critical buckling load of the beam. Nejati

3 Thermo-Mechanical Buckling Analysis of Functionally Graded Skew Laminated Plates et al. [2016] studied the buckling and vibration response of the FGM cantilever beam reinforced with carbon nanotube (CNT). They employed Hamilton s principle with 2D elasticity theory to derive the governing equations. Pouresmaeeli and Fazelzadeh [2017] studied the uncertain buckling behavior of FG CNT composite beam. They employed the rule of mixtures to obtain effective material properties and Galerkin s method to investigate critical buckling load. Nejad et al. [2017] investigated the deflection response of piezolaminated composite plates having single wall CNT reinforcements. They employed first-order and third-order deformation theories to define the displacement kinematics of the plates, and employed Hamilton s principle to derive the governing differential equations. The rise of initial geometric imperfections in the structure usually happens during the manufacturing phase. It is an important consideration in the analysis because it cannot be fully removed in common practice. Several studies have been performed in the last couple of years in the field. Hui [1986] performed buckling and initial post-buckling analysis of laminated flat plate using the Koiters elasticity theory and paid special attention to investigate the effect of bending stretching coupling on the imperfection sensitivity of the plate. Kapania and Yang [1987] performed a thorough study on the isotropic and laminated composite thin plate having the initial geometric imperfection. Studies include the buckling, post-buckling and nonlinear vibration of the plate. Dawe et al. [1995] investigated the geometrical nonlinear response of the laminated composite plate using the finite strip method. Plate was assumed to be under pressure loading and having initial geometric imperfection. Featherston [2001] performed a nonlinear analysis on shells with geometric imperfections and obtained the buckling and post-buckling response of shells using the finite element analysis to check effectiveness of the methodology for such problems. Eslami and Shahsiah [2001] studied thermal buckling response of imperfect isotropic cylindrical shells. They used various stability equations and imperfection models to measure the effect of imperfection on buckling response. Shen [2004] investigated the post-buckling response of functionally graded cylindrical thin shells and considered the material properties to be temperature-dependent and assumed that they vary uniformly along the surface of the shell. Shariat et al. [2005] performed buckling analysis on FGM plate under inplane compressive loading and having initial geometric imperfection and used the classical plate theory to derive equilibrium equations. Yang et al. [2006] investigated a post-buckling response of FGM plate having geometric imperfections. They used higher order shear deformation theory (HSDT) for the formulation. Von-Karman assumptions have been employed to incorporate the geometric nonlinearity in the problem. Yang and Huang [2007] studied the transient behavior of FGM plates with initial geometric imperfections. Formulation of the problem has been done with the help of Reddy s HSDT, and the solution of the governing equation has been performed with the Galerkin method, Runge Kutta method and improved perturbation approach. Gupta and Talha [2016] proposed a nonpolynomial-based HSDT, and investigated the vibration response of geometrically imperfect FGM plate

4 S. S. Tomar & M. Talha This paper aims to investigate the thermo-mechanical buckling response of the skew functionally graded laminated plates (FGLP) with initial geometric imperfections. Theoretical formulation has been obtained using Reddy s HSDT [Reddy, 1984]. The material properties are assumed to be temperature-dependent and to vary linearly along the thickness of the plate. The effects of various system parameters on the critical buckling loads have been analyzed. Various comparison and parametric studies have been performed to demonstrate the effectiveness and accuracy of the present methodology. 2. Theoretical Formulation Consider a FGM laminated plate having dimensions axbxh with constant thickness h as shown in Fig. 1. The mid-plane of the plate is assumed to be at the origin of the rectangular coordinate. The cross-section of the plate shown in Fig. 2 consists of three layers i.e., metallic, FGM and ceramic layers, respectively. FGM zone is assumed to be graded from metal to ceramic from bottom to top surface of the plate in terms of the volume fractions of the constituent materials according to the power law. Ceramic and metallic layers are assumed to be fully isotropic and homogeneous. The effective material properties of the FGM layer have been obtained using the Fig. 1. Geometry of functionally graded laminated plate. Fig. 2. Cross section of functionally graded laminated plate

5 Thermo-Mechanical Buckling Analysis of Functionally Graded Skew Laminated Plates Voigt rule of mixture which is written as [Reddy and Chin, 1998] P eff = P c V c + P m V m. (1) Here, P represents the material properties such as Young s modulus, mass density, Poissons ratio, etc. and V is the volume fraction of the constituent materials. Subscripts c and m are used for ceramic and metal, respectively, where the volume fraction of ceramic (V c )isgivenbyneveset al. [2013] ( ) n zf h 2 V c =, h f = h 1 + h 2 (2) and h f V c + V m =1. (3) In order to incorporate the effect of the thermal environment, the material properties are assumed to be temperature-dependent which is written as [Reddy and Chin, 1998] P (t) =P 0 (P 1 T 1 +1+P 1 T + P 2 T 2 + P 3 T 3 ), (4) where P 0,P 1,P 1,P 2,P 3 represents the value of coefficient of temperature and these are specific for the various types of constituent materials and T represents temperature distribution, respectively. A linear temperature distribution across the thickness has been considered, ( z T (z) =T b + T h + 1 ), (5) 2 where T b represents the temperature at the bottom face of the plate and T = T t T b represents the temperature difference between the top and the bottom face of the plate Displacement field For the present analysis, variational principle has been adopted to derive the governing equation of the problem. Reddy s HSDT has been used for the formulation purpose. The displacement field for the current case can be written as [Reddy, 2000] U U 0 φ x ψ x ξ x V = V 0 + z φ y + z2 ψ y + z3 ξ y, (6) W W 0 where U, V represent the displacements in inplane directions whereas W represents displacements in the transverse directions, respectively. U 0,V 0,W 0 are the midplane displacements, φ x and φ y are the rotations about y and x axis, respectively. ψ x,ψ y,ξ x,ξ y represent the higher order terms of Taylor series expansion. The higher order terms are obtained by applying the transverse shear stress at top and bottom

6 S. S. Tomar & M. Talha face of the plate that is equal to zero. The modified displacement field can be written as W 0 U U 0 φ x x V = V 0 + f 1(z) φ y + f 2(z) W 0, (7) W W 0 0 y 0 where f 1 (z) =C 1 z C 2 z 3,f 2 (z) = C 3 z 3,C 1 =1,C 2 = C 3 =4/3h 2. In order to achieve the C 0 continuous displacement field, the penalty approach [Shankara and Iyengar, 1996] has been used. This imposes the addition of two degrees of freedom on the displacement field which can be written as W 0 x θ x =0, W 0 y θ y =0. (8) The C 0 continuous displacement fieldcanbewrittenas U U 0 φ x θ x V = V 0 + f 1(z) φ y + f 2(z) θ y. (9) W 0 0 W Strain displacement and constitutive relationship A linear relation among the strains and displacement has been considered which is written as ε x ε 0 1 κ κ 3 1 ε y ε 0 2 κ κ 3 2 ε = ε yz = ε z 0 + z 2 κ z 3 0. (10) ε zx ε κ ε xy 0 Here, ε 0 5 κ 1 3 ε 0 1 = U 0 x, ε0 2 = V 0 y, ε0 3 = φ y + W 0 y, ε0 4 = φ x + W 0 x, ε 0 5 = V 0 x + U 0 y, κ1 1 = φ x x, κ1 2 = φ y y, κ 2 1 = 3C 2(φ y + θ y ), κ 2 2 = 3C 2(φ x + θ x ), ( κ 3 1 = C φx 2 x + θ ) ( x, κ 3 2 x = C φy 2 ( κ 3 φy 3 = C 2 x + θ y x + φ x y + θ x y ). y + θ y y κ 3 3 κ1 3 = φ x y + φ y x, ),

7 Thermo-Mechanical Buckling Analysis of Functionally Graded Skew Laminated Plates Fig ]. Functionally graded laminated plate having sine type imperfection [Yang and Huang, A sine type initial geometric imperfection has been considered in the study shown in Fig. 3, and the function is written as [ζ = ξh sin(πx/a)sin(πy/b)], which is included in the transverse direction through the strain displacement relationship, where ξ represents the maximum amplitude of the initially deflected geometry. The constitutive relationship can be written as Reddy [2000] σ xx Q 11 Q σ yy Q 12 Q σ yz = 0 0 Q {ε l ε t } or {σ i } =[ Q ij ]{ε i }, (11) σ zx Q 55 0 sσ xy Q 66 where σ, ɛ and [Q] represent the stress vector, strain vector and material property matrix, respectively. Elements of [Q] matrix are considered as Q 11 = Q 22 = Q 44 = Q 55 = Q 66 = E(z,T) 1 ν(z,t) 2, Q 12 = ν(z,t)e(z,t) 1 ν(z,t) 2, E(z,T) 2(1 + ν(z,t)) Finite element implementation A C 0 continuous finite element methodology has been adopted to discretize the plate. Displacement and geometric shape functions used in the formulation are as

8 S. S. Tomar & M. Talha Fig. 4. follows [Gupta and Talha, 2016]: {Γ} = Functionally graded laminated plate under in-plane loading. 9 N i {Γ} i ; x = i=1 9 N i x i ; y = i=1 9 N i y i, (12) where Γ represents the displacement vector whereas Ni represents the shape function in natural coordinate at ith node. {Γ i } = {U 0,V 0,W 0,φ x,φ y,θ x,θ y } T. (13) Strain energy calculation Strain energy of the plate after the implementation of the finite element method can be written as ne S = S (e) = 1 ne {Γ} e(t ) [K (e) ]{Γ} e. (14) 2 e=1 e= Work done due to inplane loading Buckling in the plate occurs as a result of compression of the plane due to inplane forces as shown in Fig. 4. The work done due to inplane force after the implementation of finite element method is given by ne U = U (e) = 1 ne {Γ} e(t ) λ cr [K (e) g ]{Γ} e, (15) 2 e=1 e=1 i=1 where [K g ] is the geometric stiffness matrix due to loading in the in-plane direction whereas λ cr is the critical buckling parameter

9 Thermo-Mechanical Buckling Analysis of Functionally Graded Skew Laminated Plates Fig Skew boundary transformation Functionally graded laminated plate having skew edges. Edges of Plate are assumed to be skewed at angle ψ as shown in Fig. 5. The nodal displacement has been transformed as [Garg et al., 2006] Γ i = T g Γ l i, (16) where T g is the transformation matrix consisting of matrix containing sine and cosine terms with skew angle (ψ), Γ i and Γ l i represent generalized and local displacement vectors at the respective ith node. c s s c T g = c s 0 0, (17) s c c s s c where c =cos(ψ) ands =sin(ψ), ψ represents the skew angle of the edge Eigenvalue problem Variational method has been adopted to derive the governing equation of thermomechanical buckling problem. The final eigenvalue problem is written as [ K]{q} = λ cr [ K g ]{q}, (18) where [ K], [ K g ] are the respective global transformed stiffness and geometric stiffness matrix, respectively. λ cr and q represent critical buckling parameter and global displacement vectors, respectively

10 S. S. Tomar & M. Talha 3. Numerical Results In order to define the accuracy and effectiveness of the present methodology, various numerical examples have been solved in this section. This section is mainly divided into two parts. The first part covers the convergence and validation studies, and the second covers the parametric studies. The material properties used in the analysis are tabulated in Table 1 [Talha and Singh, 2010]. The proportion of thickness in ceramic FGM and metal layer are considered in 1:1:2 ratio. Simply supported type of boundary conditions have been adopted throughout the study except in Table Convergence and validation studies This section covers the various convergence and comparison studies, which have been performed to prove the reliability of the present methodology. Example 1. Validation study for Al/Al 2 O 3 sandwich FGM plates with all edges simplysupportedhavebeenshownintable2,whichiscomparedwiththeresults from published literature [Neves et al., 2013]. Table shows that the results obtained from the present formulation agrees well with the reference results. Neves et al. [2013] used quasi 3D HSDT for formulation whereas principle of virtual work had been employed to derive the governing differential equations. The nondimensional buckling parameter used in the study as λ cr = λa2 100h 3, (19) E 0 where λ represents the critical buckling load and E 0 =1GPa. Table ]. Material properties of various materials (Temperature dependent) [Reddy and Chin, Material Properties P 1 P 0 P 1 P 2 P 3 P (T = 300 K) ZrO 2 E e e e e e9 α e e e e e 6 Ti 6Al 4V E e e e9 α e e e e 6 Table 2. Comparison of critical buckling parameter (λ cr) for sandwich FGM plate having homogeneous core. Mesh size Uniaxial Biaxial Neves et al. [2013] (%) Difference

11 Thermo-Mechanical Buckling Analysis of Functionally Graded Skew Laminated Plates Table 3. Comparison of critical buckling parameter (λ cr) for sandwich FGM plate having homogeneous core. a/h Plate model n = 1 (%) Difference 10 Present DET a Present DET a ESDT a a ED a ED a Ed z Note: a Fazzolari [2016]. Example 2. Comparison of buckling behavior of biaxially-loaded SUS304/Si 3 N 4 FGM plates obtained from the present methodology with standard threedimensional elasticity solution (3DET), Exponential shear deformation theory (ESDT) and Carrera s unified formulation (ED 1,ED 2,ED z ) have been presented in Table 3. Comparison studies have been presented at various values of thickness ratios (a/h), and volume fraction index (n) of the plate had been considered as 1. The material properties of constituent materials are taken as SUS304: Elastic Modulus (E m ) = e9, Poisson s ratio (ν m )= Si 3 N 4 : Elastic Modulus (E c ) = e9, Poisson s ratio (ν c )= Plates are assumed to be simply supported at all edges. It is evident from the table that results obtained with present methodology agree well with the standard close form solutions and can be employed for computing the new results for functionally graded skew laminated plates. Example 3. Table 4 shows the validation of critical buckling parameter for biaxially loaded Al/Al 2 O 3 skew FGM plate by Ganapathi et al. [2006]. Comparison studies have been performed at various volume fraction index (n) and skew angles (ψ). Thickness ratio (a/h) of skew FGM plates having all edges simply supported has been taken as 10. They used first-order shear deformation theory in conjunction Table 4. Comparison of critical buckling parameter for Al/Al 2 O 3 Skew FGM plate. n ψ Present Ganapathi et al. [2006]

12 S. S. Tomar & M. Talha with finite element methodology for formulation. Table 4 shows that results obtained with the present formulation agree fairly well with the literature Parametric studies The validation shows that the current formulation can be used for the analysis of functionally graded laminated skew plates. In this section, parametric studies have been performed to measure the effect of system parameters on the critical buckling response of functionally graded laminated skew plates under uniaxial and biaxial types of inplane loading conditions. The nondimensional buckling parameter used in the study is λ cr = λa2 100h 3, (20) E 0 where λ represents the critical buckling load and E 0 =1GPa,a, h are the length and thickness of plates, respectively. Table 5 shows the variation of critical buckling parameter of square Ti 6Al 4V/ZrO 2 functionally graded skew laminated plate with skew angle (ψ) andvolume fraction index (n). It can be observed that with the increase in volume fraction index (n), the value of critical buckling parameter decreases whereas with increase in the skew angle (ψ), the critical buckling parameter increases. Table 6 shows the variation of critical buckling parameter of Ti 6Al 4V/ZrO 2 functionally graded skew laminated plates with the variation of thickness ratio (a/h) and the volume fraction index (n). It can be easily observed that with the increase Table 5. Variation of critical buckling parameter (λ cr) of FGLP with volume fraction index (n) and skew angle (ψ) at T = 100 K, a/h= 10. ψ Uniaxial Biaxial n =1 n =2 n =5 n =1 n =2 n = Table 6. Variation of critical buckling parameter (λ cr) of FGLP with volume fraction index (n) and thickness ratio (a/h) at T = 100 K, ψ =10 0. a/h Uniaxial Biaxial n =1 n =2 n =5 n =1 n =2 n =

13 Thermo-Mechanical Buckling Analysis of Functionally Graded Skew Laminated Plates Table 7. Variation of critical buckling parameter (λ cr) of FGLP with temperature ( T ) and skew angle (ψ) atξ =0.2, n =1,a/h = 10. ψ Uniaxial Biaxial T =0 T = 100 T = 300 T =0 T = 100 T = in thickness ratio, the buckling parameter increases whereas with volume fraction index, it decreases. It is due to the fact that with increase in volume fraction the index material tends to change from ceramic to metal due to which less buckling load is required to buckle the structure. The fluctuation of critical buckling parameter of Ti 6Al 4V/ZrO 2 skew FGLP with skew angle and temperature difference ( T ) is shown in Table 7, where temperature difference represents the difference between the temperature of top and bottom surfaces of the plate. With increase in T, the buckling parameter increases. This can be understood as with the increase in the temperature, the stiffness of the plate decreases but the decrease in geometric stiffness [ K g ]isgreater than the stiffness [ K] ofplate. Figure 6 shows the variation of critical buckling parameter (λ cr )ofti 6Al 4V/ZrO 2 skew FGLP having all edges simply supported with the imperfection Fig. 6. Variation of critical buckling parameter (λ cr) of FGLP with imperfection amplitude (ξ)

14 S. S. Tomar & M. Talha Critical buckling parameter, λ cr Skew angle, Ψ SSSS CCCC CSCS CFCF SFSF Fig. 7. Variation of critical buckling parameter (λ cr) of biaxially loaded FGLP with boundary conditions and skew angle (ψ) atξ =0.4, n =1,a/h = 10, T = 300 K. amplitude (λ cr ) with imperfection amplitude at various skew angles (ψ). The volume fraction index (n) and thickness ratio (a/h) have been considered to be 1 and 10, respectively. It can be observed that the buckling parameter values are lower in case of biaxial loading condition as compared to uniaxial loading condition. Biaxial loading lines are closer as compared to the uniaxial lines as less load is required in case of biaxial loading condition. Value also increases with the increase in the skew angle. Figure 7 represents the deviation of the critical buckling parameter of imperfect functionally graded skew laminated plate with various boundary conditions. The plate is assumed to be under the biaxial type of loading whereas temperature difference was considered to be 300 K. It can be observed that the value of critical buckling parameter is maximum in case of fully clamped (CCCC) type of boundary conditions whereas it is minimum in case of (SFSF ) type of boundary condition, where S represents the simply supported edge whereas F represents the free edge. 4. Conclusion Thermo-mechanical buckling analyses of functionally graded skew laminated plate with initial geometric imperfection have been studied. The formulation has been performed using Reddy s HSDT. Plate consists of three layers i.e., metallic, FGM and Ceramic starting from bottom to top surface. Variational principle has been adopted to obtain the governing equations. A C 0 continuous finite element methodology has

15 Thermo-Mechanical Buckling Analysis of Functionally Graded Skew Laminated Plates been adopted to discretize the domain of the plate. Various examples have been shown to demonstrate the reliability of the present methodology. It can be concluded that the gradation of material properties, thickness ratio, thermal environment imperfection parameter and skew angle significantly affect the critical buckling parameter of the functionally graded laminated skew plate. References Dawe, D. J., Wang, S. and Lam, S. S. E. [1995] Finite strip analysis of imperfect laminated elates under end shortening and normal pressure, International Journal for Numerical Methods in Engineering 38(24), Eslami, M. R. and Shahsiah, R. [2001] Thermal buckling of imperfect cylindrical shells, Journal of Thermal Stresses 24(1), Fazzolari, F. A. [2016] Stability analysis of FGM sandwich plates by using variablekinematics ritz models, Mechanics of Advanced Materials and Structures 23(9), Featherston, C. [2001] Imperfection sensitivity of flat plates under combined compression and shear, International Journal of Non-Linear Mechanics 36(2), Ganapathi, M., Prakash, T. and Sundararajan, N. [2006] Influence of functionally graded material on buckling of skew plates under mechanical loads, Journal of Engineering Mechanics 132(8), Garg, A. K., Khare, R. K. and Kant, T. [2006] Free vibration of skew fiber-reinforced composite and sandwich laminates using a shear deformable finite element model, Journal of Sandwich and Structures 8(8), Gupta, A. and Talha, M. [2015] Recent development in modeling and analysis of functionally graded materials and structures, Progress in Aerospace Science 79, Gupta, A. and Talha, M. [2016] An assessment of a non-polynomial based higher order shear and normal deformation theory for vibration response of gradient plates with initial geometric imperfections, Composites Part B: Engineering 107, Hui, D. [1986] Imperfection sensitivity of axially compressed laminated flat plates due to bending-stretching coupling, International Journal of Solids and Structures 522(1), Kapania, R. K. and Yang, T. Y. [1987] Buckling, postbuckling, and nonlinear vibrations of imperfect plates, AIAA Journal 25(10), Kitipornchai, S., Yang, J. and Liew, K. M. [2006] Random vibration of the functionally graded laminates in thermal environments, Computer Methods in Applied Mechanics and Engineering 195(9 12), Lanhe, W. [2004] Thermal buckling of a simply supported moderately thick rectangular FGM plate, Composite Structures 64(2), Liew, K. M., Yang, J. and Kitipornchai, S. [2003] Postbuckling of piezoelectric FGM plates subject to thermo-electro-mechanical loading, International Journal of Solids and Structures 40(15), Naderi, A. and Saidi, A. R. [2010] Buckling analysis of functionally graded annular sector plates resting on elastic foundations, Proceedings of the Institution of Mechanical Engineers, Part C : Journal of Mechanical Engineering Science 1(1), Na, K. S. and Kim, J. H. [2006] Three-dimensional thermomechanical buckling analysis for functionally graded composite plates, Composite Structures 73(4), Nejad, M. Z., Hadi, A. and Rastgoo, A. [2016] Buckling analysis of arbitrary twodirectional functionally graded Euler Bernoulli nano-beams based on nonlocal elasticity theory, International Journal of Engineering Science 103,

16 S. S. Tomar & M. Talha Nejad, M. Z., Taghizadeh, T., Mehrabadi, S. J. and Herasati, H. [2017] Elastic analysis of carbon nanotube-reinforced composite plates with piezoelectric layers using shear deformation theory, International Journal of Applied Mechanics 09, Nejati, M., Amirhossein, E. and Mohammadmahdi, N. [2016] Buckling and vibration analysis of functionally graded carbon nanotube-reinforced beam under axial load, International Journal of Applied Mechanics 8(1), Neves, A. M. A., Ferreira, A. J. M., Carrera, E., Cinefra, M., Roque, C. M. C., Jorge, R. M. N. and Soares, C. M. M. [2013] Static, free vibration and buckling analysis of isotropic and sandwich functionally graded plates using a quasi-3d higher-order shear deformation theory and a meshless techniques, Composites Part B: Engineering 44(1), Ng, T. Y., Lam, K. Y., Liew, K. M. and Reddy, J. N. [2001] Dynamic stability analysis of functionally graded cylindrical shells under periodic axial loading, Int. J. Solids Struct. 38(8), Pouresmaeeli, S. and Fazelzadeh, S. A. [2017] Uncertain buckling and sensitivity analysis of functionally graded carbon nanotube-reinforced composite beam, International Journal of Applied Mechanics 9(5), Reddy, J. N. [1984] A simple higher-order theory for laminated composite plates, Journal of Applied Mechanics 51(4), Reddy, J. N. [2000] Analysis of functionally graded plates, International Journal for Numerical Methods in Engineering 47(4), Reddy, J. N. and Chin, C. D. [1998] Thermo-mechanical analysis of functionally graded cylinders and plates, Journal of Thermal Stresses 21(6), Shankara, C. A. and Iyengar, N. G. R. [1996] A C0 element for the free vibration analysis of laminated composite plates, Journal of Sound and Vibration 191(5), Shariat, B. A. S. and Eslami, M. R. [2007] Buckling of thick functionally graded plates under mechanical and thermal loads, Composite Structures 78(3), Shariat, B. A. S., Javaheri, R. and Eslami, M. R. [2005] Buckling of imperfect functionally graded plates under in-plane compressive loading, Thin-Walled Structures 3(7), Shen, H.-S. [2004] Thermal postbuckling behavior of functionally graded cylindrical shells with temperature-dependent properties, International Journal of Solids and Structures 41(7), Taj, M. N. A. G. and Chakrabarti, A. [2013] Buckling analysis of functionally graded skew plates: An efficient finite element approach, International Journal of Applied Mechanics 5(4), Talha, M. and Singh, B. N. [2010] Thermo-mechanical induced vibration characteristics of shear deformable functionally graded ceramic-metal plates using finite element method, Proceedings of the Institution of Mechanical Engineers, Part C : Journal of Mechanical Engineering Science 225(1), Talha, M. and Singh, B. N. [2011] Thermo-mechanical buckling analysis of finite element modeled functionally graded ceramic-metal plates, International Journal of Applied Mechanics 3(4), Turvey, G. J. and Marshall, I. H. [1995] Buckling and Postbuckling of Composite Plates (Springer, Netherlands). Yang, J. and Huang, X. L. [2007] Nonlinear transient response of functionally graded plates with general imperfections in thermal environments, Computer Methods in Applied Mechanics and Engineering 196, Yang, J., Liew, K. M. and Kitipornchai, S. [2006] Imperfection sensitivity of the postbuckling behavior of higher-order shear deformable functionally graded plates, International Journal of Solids and Structures 43(17),