Composite Structures 128 (2015) 70–86

Contents lists available at ScienceDirect

Composite Structures
journal homepage: www.elsevier.com/locate/compstruct

Review

A review of theories for the modeling and analysis of functionally graded

plates and shells
Huu-Tai Thai, Seung-Eock Kim ⇑
Department of Civil and Environmental Engineering, Sejong University, 98 Gunja Dong Gwangjin Gu, Seoul 143-747, Republic of Korea

a r t i c l e i n f o a b s t r a c t

Article history: In this paper, a comprehensive review of various theories for the modeling and analysis of functionally
Available online 14 March 2015 graded plates and shells is presented. The review is devoted to theoretical models which were developed
to predict the global responses of functionally graded plates and shells under mechanical and thermal
Keywords: loadings. This review mainly focuses on the equivalent single layer theories including the classical plate
Functionally graded plate and shell theory, first-order shear deformation theory, higher-order shear deformation theories, simplified theories
Plate and shell theory and mixed theories since they were widely used in the modeling of functionally graded plates and shells.
Elasticity solution
In addition, a thorough review of the literature related to the development of three-dimensional elasticity
Unified formulation
solutions and a unified formulation is also presented.
Ó 2015 Elsevier Ltd. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2. ESL theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.1. CPT model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.2. FSDT model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.3. TSDT model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.4. HSDT models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.4.1. Polynomial function based-models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.4.2. Non-polynomial function based-models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.5. Simplified theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.6. Mixed theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3. 3D elasticity theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4. Unified formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5. Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

1. Introduction properties at an interface due to bonding of two discrete materials.

As a result, stress concentration usually occurs at the interface.
Multilayered composite materials are extensively used in aero- This can lead to damage in the form of delamination, matrix crack-
space, mechanics, civil engineering, nuclear and automotive due to ing and adhesive bond separation [1]. Functionally graded materi-
their outstanding features such as high ratio of stiffness- and als (FGMs) were therefore born to overcome these issues. The FGM
strength-to-weight and low maintenance cost. Conventional lami- is the advanced composite material which is made of two or more
nated composite materials exhibit a mismatch of mechanical constituent phases with a continuous variation of material proper-
ties from one surface to another, and thus eliminating the stress
concentration found in the conventional laminated composites.
⇑ Corresponding author. Tel.: +82 2 3408 3291; fax: +82 2 3408 3332. The concept of the FGM was proposed in 1984 by Japanese
E-mail address: sekim@sejong.ac.kr (S.-E. Kim).

http://dx.doi.org/10.1016/j.compstruct.2015.03.010
0263-8223/Ó 2015 Elsevier Ltd. All rights reserved.
 H.-T. Thai, S.-E. Kim / Composite Structures 128 (2015) 70–86 71

material scientists [2]. A typical FGM is made from a mixture of a analytical solutions for the critical buckling temperature of FG
ceramic and a metal. The history of the FGM as well as its applica- clamped plates resting on an elastic foundation under three differ-
tions can be found in the report by Jha et al. [3]. The modeling and ent types of thermal loadings. Ghannadpour et al. [15] also exam-
analysis of the FGM were also reviewed by Birman and Byrd [4]. ined the thermal buckling of FG plates using the CPT. However, the
In general, the behavior of functionally graded (FG) plates/shells buckling load was calculated using the finite strip method instead
under mechanical and thermal loadings can be predicted using of Navier solution in the work [12]. The buckling of FG plates sub-
either three-dimensional (3D) elasticity theory or equivalent- jected to non-uniform compression was examined by Mahdavian
single-layer (ESL) theories. The ESL models are derived from the [16] using the CPT and Fourier solutions. Mohammadi et al. [17]
3D elasticity theory by making suitable assumptions on the derived analytical solutions for the buckling load of FG plates with
kinematics of deformation or a tress state through the thickness two opposite edges simply supported and the other two edges hav-
of plates/shells [1]. These ESL theories may account for both shear ing arbitrary boundary conditions (i.e. Levy-type plate). The gov-
and normal deformation effects depending on the level of assump- erning equations derived from the CPT were analytically solved
tions. The simplest ESL model is the classical plate theory (CPT), using Levy-type solution approach.
also known as Kirchoff theory [5], which ignores both shear and Yang and Shen [18] employed the CPT to investigate the tran-
normal deformation effects. Thus it is only suitable for thin FG sient response of initially stressed FG plates resting on an elastic
plates/shells. The next theory in the hierarchy of ESL models is foundation subjected to impulsive lateral loadings. The semi-ana-
the first-order shear deformation theory (FSDT) developed by lytical differential quadrature method (DQM) and the modal super-
Mindlin [6]. The FSDT accounts for the shear deformation effect position approach were respectively employed to determine the
by the way of a linear variation of in-plane displacements through natural frequency and transient response of rectangular plates
the thickness. A shear correction factor is therefore required. The with two opposite edges clamped and the remaining edges having
shear correction factor is difficult to determine since it depends arbitrary boundary conditions. The nonlinear load–deflection and
not only on geometric parameters but also on the loading and postbuckling responses of FG plates resting on an elastic founda-
boundary conditions. To avoid the use of the shear correction tion under in-plane and transverse loadings were investigated by
factor, higher-order shear deformation theories (HSDTs) were Yanga and Shen [19] using the CPT with von Karman assumptions.
introduced. The HSDT can be developed by expanding the displace- A semi-analytical approach based on the DQM and Galerkin proce-
ment components in power series of the thickness coordinate. In dure was used to solve the governing equations. Alinia and
principle, the theories developed by this mean can be made as Ghannadpour [20] also used the CPT with von-Karman assump-
accurate as desired by including a sufficient number of terms in tions to study the nonlinear responses of FG plates under trans-
the series. Among the HSDTs, the third-order shear deformation verse pressure. However, they used the principle of minimum
theory (TSDT) of Reddy [7] is the most widely used one due to potential energy to obtain the analytical solutions of simply sup-
its simplicity and accuracy. A review of shear deformation theories ported plates.
for isotropic and laminated plates was carried out by Ghugal and Woo et al. [21] studied the nonlinear vibration of FG plates in
Shimpi [8] and Khandan et al. [9]. A comprehensive review of vari- thermal environments. The nonlinear equations derived from the
ous analytical and numerical models for predicting the bending, CPT with von Karman assumptions were solved for FG plates with
buckling and vibration responses of FG plates under mechanical arbitrary boundary conditions using a series method. Hu and
and thermal loadings was recently carried out by Swaminathan Zhang [22] also adopted the CPT with von Karman assumptions
et al. [10]. However, no literature has been reported for the review to perform vibration and stability analyses of FG plates under in-
of the development of various theories for the modeling and analy- plane excitation. Free vibration of FG plates with various boundary
sis of FG plates/shells. conditions resting on an elastic foundation was investigated by
The objective of this paper is to provide a comprehensive litera- Chakraverty and Pradhan [23] using the CPT and Rayleigh–Ritz
ture review of existing theories for the modeling and analysis of FG method. Chakraverty and Pradhan [24] improved their previous
plates/shells with the main emphasis on the ESL models such as work [23] by accounting for the effect of thermal environments.
the CPT, FSDT, TSDT, HSDTs, simplified theories, mixed theories. Ruan and Wang [25] investigate the vibration and stability of mov-
In addition, a detailed review of the literature related to the devel- ing FG skew plates using the CPT and DQM.
opment of 3D elasticity solutions and a unified formulation is also The CPT was also used to analyze circular plates. For example,
reported. Ma and Wang [26] investigated the nonlinear bending and thermal
postbuckling behaviors of FG circular plates under mechanical and
2. ESL theories thermal loadings. The governing equations derived in the frame-
work of the CPT and von Karman assumptions were numerically
2.1. CPT model solved using a shooting method. Li et al. [27] also studied the non-
linear postbuckling behavior of FG circular plate under mechanical
The CPT model is based on the Kirchhoff–Love hypothesis that and thermal loadings using the CPT with von Karman assumptions.
the straight lines remain straight and perpendicular to the mid- The initial geometric imperfections of FG plates were taken into
plane after deformation. These assumptions imply the vanishing account in their study. Allahverdizadeh et al. [28] studied the
of the shear and normal strains, and consequently, neglecting the steady-state vibration of FG circular plates in thermal environ-
shear and normal deformation effects. The CPT is the simplest ments using the CPT and a semi-analytical approach. Ghomshei
ESL model and it is only suitable for thin FG plates/shells where and Abbasi [29] studied the axisymmetric thermal buckling of FG
the shear and normal deformation effects are inconsiderable. annular plates with variable thickness subjected to thermal load-
Feldman and Aboudi [11] studied the elastic buckling of FG ings using the CPT and the finite element method.
plates under uniaxial compressive loading using a combination of In addition to FG plates, the CPT was also more preferably used
micromechanical and structural approaches. Governing equations for FG shells due to its simplicity. Loy et al. [30] studied the vibra-
derived from the CPT were analytically solved for the buckling load tion of FG cylindrical shells with simply supported boundary con-
of FG plates with various boundary conditions. Javaheri and Eslami ditions using the CPT and Rayleigh–Ritz method. A similar
[12,13] employed the CPT to investigate the buckling behavior of approach was adopted by Arshad et al. [31] to investigate the
FG plates under four types of thermal loadings [12] and compres- vibration characteristics of FG cylindrical shells under three differ-
sive loadings [13]. Based on the CPT, Kiani et al. [14] presented ent types of volume fraction laws. The vibration characteristics of
72 H.-T. Thai, S.-E. Kim / Composite Structures 128 (2015) 70–86

FG cylindrical shells under various boundary conditions were thickness. Therefore, the neutral surface of the FG plate does not
examined by Pradhan et al. [32] using the CPT and Rayleigh coincide with its middle one. This coupling could be eliminated if
method. This problem was reexamined by Naeem et al. [33] using the governing equations were derived based on the neutral surface.
Ritz method. Nonlinear forced vibrations of FG doubly curved shal- The statement was confirmed by Zhang and Zhou [54] by
low shells were investigated by Alijani et al. [34] using the CPT reformulating the CPT for FG plates based on the neutral surface,
with von Karman assumptions and the multi-modal Galerkin dis- and consequently, obtaining the governing equations of motion
cretization. Du et al. [35] studied the nonlinear vibration of FG in the form of isotropic plates. The neutral surface based-CPT
cylindrical shells under excitation based on the CPT with von was adopted by Bodaghi and Saidi [55] to study the buckling of
Karman assumptions in combination with a multiple scale method. FG plates resting on an elastic foundation under non-uniform com-
Du and Li [36] studied the nonlinear vibration response of FG cylin- pression. Damanpack et al. [56] used the neutral surface based-CPT
drical shells in thermal environments following a similar approach. model and the boundary element method to predict the bending
Ebrahimi and Najafizadeh [37] studied the free vibration of FG behavior of FG plates. Kowal-Michalska and Mania [57] also
cylindrical shells using the CPT in conjunction with the generalized adopted the neutral surface based-CPT model to study the static
differential quadrature and generalized integral quadrature and dynamic buckling of FG plates subjected to a simultaneous
methods. action of one directional compression and thermal loadings.
Shen [38–40] studied the postbuckling behavior of FG cylindri- Since the stretching–bending coupling produces the transverse
cal shells under axial compression [38] or lateral pressure [39] or a deflections and bending moments when a FG plate is subjected to
uniform temperature rise [40] using the CPT with von Karman in-plane compressive loadings. Hence, bifurcation-type buckling
assumptions. Both nonlinear prebuckling deformation and initial will not occur. The conditions for the bifurcation-type buckling
geometric imperfections were included in the postbuckling analy- to occur under the action of in-plane compressive loadings were
sis using a boundary layer theory of shell buckling. The postbuck- examined by Aydogdu [58]. It is observed that the bifurcation-type
ling equilibrium path and buckling load or temperature were buckling occurs when the plate is fully clamped. For a movable-
determined using a singular perturbation technique. Woo et al. edge plate, the bifurcation-type buckling occurs when the in-plane
[41] investigated the postbuckling behavior of FG plates and shal- loadings are applied at the neutral surface [58].
low cylindrical shells under mechanical and thermal loadings
using the CPT and von Karman assumptions. Analytical solutions 2.2. FSDT model
were obtained using a mixed series solution. Mirzavand and
Eslami [42] derived analytical solutions for the buckling load of The FSDT developed by Mindlin [6] accounts for the shear
imperfect FG cylindrical shells under axial compression in thermal deformation effect by the way of a linear variation of the in-plane
environments using two different models for initial geometrical displacements through the thickness. It is noted that the theory
imperfections. The governing equations derived from the CPT with developed by Reissner [59,60] also accounts for the shear deforma-
Sanders nonlinear kinematic relations were solved for the buckling tion effect. However, the Reissner theory is not similar with the
load using Galerkin method. The results indicate that the geo- Mindlin theory like erroneous perception of many researchers
metrical imperfections and the temperature dependency of mate- through the use of misleading descriptions such as ‘‘Reissner–
rial properties play major roles in dictating the bifurcation point Mindlin plates’’ and ‘‘FSDT of Reissner’’. The major difference
of the imperfect FG cylindrical shells under axial compression. between two theories was established by Wang et al. [61] by deriv-
The buckling of FG cylindrical shells under a combined action of ing the bending relationships between Mindlin and Reissner quan-
axial, lateral and torsional loadings was studied by Huang et al. tities for a general plate problem. Since the Reissner theory was
[43] using the CPT in conjunction with Ritz energy and finite ele- based on the assumption of a linear bending stress distribution
ment methods. It is observed that the contribution of the lateral and a parabolic shear stress distribution, its formulation will inevi-
pressure to buckling is more significant than that of axial compres- tably lead to the displacement variation being not necessarily lin-
sion or torsion, and the contributions of axial compression and tor- ear across the plate thickness [61]. Thus, it is incorrect to refer to
sion are almost the same. Sun et al. [44] presented analytical the Reissner theory as the FSDT which implies a linear variation
solutions for the buckling of FG imperfect cylindrical shells under of the displacements through the thickness. Another difference
thermal and mechanical loadings using the CPT and Galerkin between two theories is that the normal stress which was included
method. Cheng et al. [45] adopted the CPT to study the buckling in the Reissner theory was omitted in the Mindlin one [62].
of FG cylindrical shells under pure bending. Elastoplastic buckling The FSDT was widely used to model FG plates. Praveen and
behavior of FG cylindrical shells under a combination of axial com- Reddy [63] studied the nonlinear transient responses of FG plates
pression and external pressure was studied by Zhang et al. [46] under thermal and mechanical loadings using the finite element
using the CPT and J2 deformation theory. The buckling load of FG method and the FSDT with von Karman assumptions. Della Croce
simply supported shells was obtained using Galerkin method. and Venini [64] presented a hierarchic family of finite elements
Woo and Meguid [47] studied the nonlinear bending of FG shal- for the bending analysis of FG plates under mechanical and ther-
low shells under transverse loadings and a temperature field. The mal loadings using the FSDT and a variational formulation.
governing equations derived from the CPT with von Karman However, the stretching–bending coupling was ignored in their
assumptions were analytically solved for deflection, stresses and work. Kim et al. [65] investigated the nonlinear bending behavior
bending moments of a simply supported shell using Fourier series of FG plates and shells using the FSDT with a full definition of
method. The nonlinear behavior of imperfect eccentrically stiff- the Green strain tensor. Based on the quasi-conforming formula-
ened FG panels resting on an elastic foundation was studied by tion, a four-node shell element was developed and implemented
Nguyen [48], Nguyen and Tran [49,50] and Nguyen and Pham into the general purpose nonlinear dynamic finite element package
[51] using the CPT and Lekhnitsky smeared stiffener technique. XFINAS. Memar Ardestani et al. [66] used the FSDT and a reproduc-
Nguyen and Tran [52,53] performed the nonlinear dynamic analy- ing kernel particle method to study the bending behavior of con-
sis of imperfect FG doubly curved shallow shells resting on an elas- centrically and eccentrically FG stiffened plates under transverse
tic foundation subjected to mechanical and thermal loadings using loadings.
the CPT with von Karman assumptions. Chen [67] studied the nonlinear vibration of FG plates subjected
It is worth noting that the stretching–bending coupling exists in to a combined action of initial in-plane compressive and bending
FG plates due to the variation of material properties through the stresses using the FSDT with von Karman assumptions. The
 H.-T. Thai, S.-E. Kim / Composite Structures 128 (2015) 70–86 73

nonlinear frequency of simply supported plates was obtained using transverse displacements. Mohammadi et al. [93] decoupled five
Galerkin method in combination with Runge–Kutta iterative pro- governing equations of the FSDT into two independent equations.
cedure. The FSDT with von Karman assumptions was also The obtained equations were then solved for the buckling load of
employed by Alijani et al. [68] to study the nonlinear vibration of FG plates under in-plane loadings using Levy-type solution
FG simply supported plates in thermal environments. Free vibra- approach. This solution approach was employed by Saidi and
tion of FG plates resting on an elastic foundation was studied by Jomehzadeh [94] to derive Levy-type solution for the deflection
Fallah et al. [69] using the FSDT and a semi-analytical approach and stresses of FG plates subjected to transverse loadings.
which is based on a combination of the infinite power series and Yaghoobi and Torabi [95] also follow a similar approach to derive
Kantorovich method. Levy-type solution for the buckling load of FG plates resting on
Lanhe [70] and Bouazza et al. [71] derived analytical solutions an elastic foundation under thermal loadings.
for the buckling temperature of simply supported FG plates under Yang et al. [96] studied the influence of the randomness of
two types of thermal loadings using the FSDT and Navier solution. material properties and foundation stiffness parameters on the
Ganapathi et al. [72] employed the FSDT and finite element buckling load of FG plates resting on an elastic foundation using
method to study the buckling of FG skew plates under compressive the FSDT and a perturbation technique. To avoid the use of the
loadings. Yaghoobi and Yaghoobi [73] studied the buckling of FG shear correction factor in the FSDT, Nguyen et al. [97] adopted
sandwich plates resting on an elastic foundation under thermal equilibrium equations in calculating the transverse shear stresses
and mechanical loadings. The FSDT and power series Frobenius and shear forces. Results of the static bending analysis of FG simply
method were adopted to calculate the critical buckling load of FG supported plates and FG sandwich clamped panels indicate that
plates under different boundary conditions. The buckling of FG the value of the shear correction factor of FG models is not the
plates under mechanical and thermal loadings was studied by same as that of the homogeneous ones. Prakash et al. [98] reformu-
Zhang et al. [74] using the FSDT and the local Kriging meshless lated the FSDT based on the neutral surface to investigate the effect
method which is based on the local Petrov–Galerkin weak-form of the neutral surface position on the nonlinear stability of FG skew
formulation and Kriging interpolation. plates under in-plane loadings. The governing equations based on
Park and Kim [75] adopted the FSDT with von Karman assump- von Karman assumptions were solved using an eight-node C0 con-
tions to study the postbuckling and vibration of FG plates under tinuous element. The neutral surface based-FSDT was also adopted
thermal loadings using the finite element method. The postbuck- by Singha et al. [99] to study the nonlinear bending behavior FG
ling behavior of FG plates under thermal and mechanical loadings plates under transverse pressure using the finite element method.
was studied by Wu et al. [76] using the FSDT and finite double The equilibrium equations were used to calculate the transverse
Chebyshev polynomials. This problem was also studied by shear stresses, while the energy method was adopted to derive
Nguyen and Hoang [77] using the FSDT and Galerkin method in the expressions for the shear correction factor.
combination with an iterative procedure. The buckling and post- In addition to the application to rectangular plates in the above-
buckling of FG sandwich plates resting on an elastic foundation mentioned works, the FSDT also applied to circular and annular
under mechanical loadings was studied by Kiani and Eslami [78] plates. Reddy et al. [100] performed the bending and stretching
using the FSDT with von Karman assumptions. The singlemode analysis of FG circular and annular plates using the FSDT. The ana-
approach combined with Galerkin technique was used to calculate lytical solutions for deflections, force and moment resultants were
the critical buckling temperature and postbuckling equilibrium expressed in terms of the corresponding quantities of isotropic
path of FG simply supported plates. plates based on the CPT. Najafizadeh and Eslami [101] derived ana-
Dai et al. [79] extended the element-free Galerkin method [80] lytical solutions for the buckling temperature of FG circular plates
to FG plates with piezoelectric layers under mechanical and ther- with clamped and simply supported boundary conditions sub-
mal loadings based on the FSDT. Results show that the element- jected to uniform temperature rise, gradient through the thickness
free Galerkin method has many attractive features compared to and linear temperature variation along the radius. Efraim and
the finite element method. Zhao et al. [81,82], Zhao and Liew Eisenberger [102] studied the free vibration of FG annular plates
[83,84] and Lee et al. [85,86] developed a meshless model based with variable thickness based on the FSDT. The exact solutions
on the FSDT and the element-free kp-Ritz method. This model for plates with various boundary conditions were obtained using
was applied to FG plates and shells through different problems, the exact element and dynamic stiffness methods. Bending analy-
e.g. geometrically nonlinear bending [84], vibration [81], thermal sis of a FG rotating disk was investigated by Bayat et al. [103] using
buckling [82], thermal bending [85], thermal postbuckling [86] the FSDT and a semi-analytical method. Based on the FSDT, Naderi
and thermo-mechanical buckling [83]. The transient response of and Saidi [104] derived analytical solutions for the buckling load of
FG plates and shells under transverse loadings was studied by FG sector and annular sector plates by decoupling governing equa-
Roque et al. [87] using the FSDT and a meshless method with radial tions. A similar approach was adopted by Saidi et al. [105] to derive
basis functions (RBFs). Based on the FSDT, Nguyen-Xuan et al. analytical solutions for the natural frequency of FG annular sector
[88,89] extended the edge-based smoothed finite element method plates on the basis of the FSDT. Amini et al. [106] studied the
(ES-FEM) [88] and the node-based smoothed finite element effects of geometric nonlinearity on the free and forced vibration
method (NS-FEM) [89] to the static bending, buckling and free of FG annular plates based on the FSDT with von Karman assump-
vibration analyses of FG plates. Valizadeh et al. [90] employed tions. Golmakani and Alamatian [107] and Alinaghizadeh and
the FSDT and an isogeometric approach (IGA) to study the bending, Kadkhodayan [108] performed the large deflection analysis of FG
free vibration, buckling and supersonic flutter responses of FG annular sector plates under transverse loadings using the FSDT
plates. The IGA utilizes the non-uniform rational B-spline with von Karman assumptions. Xie et al. [109] introduced an effi-
(NURBS) functions to exactly describe the complex geometry of cient solution for the free vibration of FG conical shells and annular
structures and easily achieve the smoothness with arbitrary plates using the FSDT and Haar wavelet method.
continuity order. The FSDT was also used to model FG shells. Reddy and Chin
Analytical solutions for the free vibration analysis of Levy-type [110] studied the dynamic response of FG cylinders and plates sub-
plates resting on an elastic foundation were given by jected to two different types of thermal loadings using the FSDT
Hosseini-Hashemi et al. [91] using the FSDT and Levy solution. and the finite element method. Shahsiah and Eslami [111,112]
Hosseini-Hashemi et al. [92] improved their previous work [91] derived analytical solutions for the buckling temperature of FG
by accounting for the coupling effect between the in-plane and cylindrical shells with simply supported boundary conditions
74 H.-T. Thai, S.-E. Kim / Composite Structures 128 (2015) 70–86

subjected to two types of thermal loadings using the FSDT and Karman-type geometric nonlinearity. Shen [129] presented the
Navier solution. Samsam Shariat and Eslami [113] and Mirzavand nonlinear analysis of FG plates under transverse loadings in ther-
et al. [114] investigated the influence of initial geometric imperfec- mal environments. The governing equations based on the TSDT
tions on the buckling of FG plates and cylindrical shells. Analytical with von Karman assumptions were solved for the load–deflection
solutions for the critical buckling temperature of FG plates and and load–bending moment curves of simply supported plates with
shells under different types of thermal loadings were obtained movable or immovable edges using a mixed Galerkin-perturbation
based on the FSDT. A geometrically nonlinear analysis of FG shells technique. Nonlinear bending analysis of FG plates under thermal
was performed by Arciniega and Reddy [115] using the FSDT and and mechanical loadings was performed by Yang and Shen [130]
the finite element method. Further studies on geometrically non- using the TSDT with von Karman assumptions. A multi-parameter
linear bending behavior of FSDT shells were carried out by perturbation approach was employed to obtain the bending
Barbosa and Ferreira [116] and Sheng and Wang [117] using responses of FG plates with two opposite edges clamped or simply
Marguerre shell element and four-order Runge–Kutta numerical supported and the remaining two edges having arbitrary boundary
method, respectively. Behjat et al. [118] studied the static bending, conditions. Yang et al. [131] investigated the buckling, free vibra-
free vibration and transient responses of FG piezoelectric cylindri- tion and dynamic stability of FG sandwich plates under a combined
cal panels subjected to mechanical, thermal and electrical loadings action of uniform temperature change and a periodic in-plane
using the FSDT and the finite element method. The static and compression using the TSDT and a semi-analytical method.
dynamic bending and free vibration problems of FG doubly curved Akbarzadeh et al. [132] studied the static and dynamic responses
panels under a combined action of mechanical and thermal load- of FG plates under lateral loadings using the TSDT and Fourier ser-
ings were addressed by Kiani et al. [119] using the FSDT and a ana- ies method. Zhang [133] employed the neutral surface based-TSDT
lytical hybrid Laplace–Fourier transformation. Xiang et al. [120] with von Karman assumption to study the nonlinear bending
adopted the meshless local collocation method and FSDT to predict response of FG plates resting on an elastic foundation in thermal
the natural frequency of FG cylindrical shells. environments. Analytical solutions for FG plates with six different
Sheng and Wang [121] investigated the effect of temperature on boundary conditions were obtained using Ritz method.
the vibration and buckling of FG cylindrical shells surrounded by Javaheri and Eslami [134] derived analytical solutions for the
an elastic medium using the FSDT in conjunction with a normal- critical buckling temperature of simply supported FG plates under
mode expansion and Bolotin method. Zhang and Hao [122] studied four types of thermal loadings using the TSDT and Navier solution.
the nonlinear vibration of FG cylindrical shells under a com- A similar work was carried out by Samsam Shariat and Eslami
bination of thermal loadings and external excitations using the [135] for FG plates subjected to three types of mechanical loadings
FSDT and Galerkin method. The buckling of FG cylindrical shells and two types of thermal loadings. Najafizadeh and Heydari [136]
under external pressure and axial compression was studied by derived analytical solutions for the buckling load of FG circular
Khazaeinejad et al. [123] using the FSDT and an analytical method. plates using the TSDT. Bodaghi and Saidi [137] derived analytical
Based on the FSDT with von Karman assumptions, Nguyen and solutions for the buckling load of FG Levy-type plates under in-
Pham [124] studied the nonlinear dynamic response and vibration plane loadings. By introducing four new functions, five governing
characteristics of imperfect FG cylindrical shells with eccentric stif- equations derived from the TSDT were converted into two
feners surrounded by an elastic medium. The Runge–Kutta independent equations. These equations were then solved for FG
numerical method was used to predict the dynamic response of rectangular plates using Levy solution. This solution approach
FG cylindrical shells subjected to axially and transversely mechani- was also adopted by Saidi et al. [138] to derive analytical solutions
cal and damping loadings. Isvandzibaei et al. [125] studied the for the deflection and stresses of FG Levy-type plates subjected to
vibration characteristics of FG cylindrical shells under pressure transverse loadings. Thai and Kim [139] reformulated the TSDT
loading. The governing equations derived from the FSDT are ana- based on the neutral surface and derived Levy-type solution for
lytically solved for the natural frequency of FG cylindrical shells the buckling load of FG plates resting on an elastic foundation.
under various boundary conditions using Ritz method. Yang et al. [140] investigated the sensitivity of initial geometric
Pradyumna and Nanda [126] investigated the geometrically non- imperfections on the postbuckling behavior of FG plates under
linear transient response of FG imperfect shell panels in thermal mechanical and thermal loadings. The initial geometric imperfec-
environments. The nonlinear governing equations derived from tion was assumed to be in the form of the product of trigonometric
the FSDT with von Karman assumptions were solved by using an and hyperbolic functions. The governing equations based on the
eight-node C0 continuous element with five degrees of freedom TSDT and von Karman assumptions were solved for the postbuck-
(DOF) per node. The transient response was obtained using ling equilibrium path of FG plates with different boundary condi-
Newmark integration scheme combined with the modified tions using a semi-analytical approach in combination with an
Newton–Raphson iteration method. iteration procedure. Shen [141] extended his previous work [129]
to the postbuckling analysis of simply supported FG plates under
2.3. TSDT model thermal loading. A two-step perturbation technique was used to
calculate buckling temperature and postbuckling equilibrium path.
The TSDT developed by Reddy [7] for laminated composite The results reveal that the temperature dependency has a signifi-
plates accounts for the transverse shear deformation effect and cant effect on the thermal postbuckling behavior of FG plates.
satisfies the zero-traction boundary conditions on the top and bot- The results also show that for the case of heat conduction, the post-
tom surfaces of a plate. A shear correction factor is therefore not buckling path for geometrically perfect plates is no longer of the
required. It is worth noting that the displacement field of Reddy bifurcation type. Nguyen and Pham [142] studied the postbuckling
theory is identical with that of Levinson theory [127]. However, behavior of FG plates resting on an elastic foundation. Analytical
the equations of motion of two theories are different each other. solutions for the buckling load and buckling temperature of FG
This is due to the fact that Levinson [127] used the equilibrium plates under mechanical and thermal loadings were obtained using
equations of the FSDT which are variationally inconsistent with the TSDT and Galerkin method.
those derived from the variational approach by Reddy [7]. Kim [143] investigated the vibration characteristics of initially
Reddy [128] presented both analytical and finite element stressed FG plates in thermal environments using the TSDT and
formulations based on the TSDT. The formulations account for Rayleigh–Ritz procedure. Yang and Shen [144] employed the
the thermo-mechanical coupling, time dependency and von TSDT with von Karman assumptions to study the vibration
 H.-T. Thai, S.-E. Kim / Composite Structures 128 (2015) 70–86 75

characteristics and transient response of initially stressed FG plates et al. [169] extended the previous work [168] to FG cylindrical
in thermal environments. The DQM and the modal superposition shells surrounded by an elastic medium under mechanical and
approach were respectively used to determine the vibration char- thermal loadings. Kapuria et al. [170] developed a quadrilateral
acteristics and transient responses of FG plates with two opposite shallow shell element for the dynamic analysis of FG plates and
clamped and the remaining two edges having arbitrary boundary shells using the TSDT.
conditions. Huang and Shen [145] also examined the nonlinear
vibration and transient response of FG plates in thermal environ- 2.4. HSDT models
ments using the TSDT with von Karman assumptions. However,
they used an improved perturbation technique to derive analytical The HSDTs account for higher-order variations of the in-plane
solutions for simply supported plates. Based on the TSDT, Hosseini- displacements or both in-plane and transverse displacements (i.e.
Hashemi et al. [146] and Hasani Baferani et al. [147] derived Levy- quasi-3D theory) through the thickness, and consequently, captur-
type solution for the natural frequency of FG plates [146] and FG ing the effects of shear deformation or both shear and normal
plates resting on an elastic foundation [147] deformations. The HSDTs can be developed using polynomial
Ferreira et al. [148,149] employed the TSDT and the meshless shape functions or non-polynomial shape functions.
collocation method with multiquadric RBFs to study bending
behavior of FG plates. Ferreira et al. [150] extended their previous 2.4.1. Polynomial function based-models
works [148,149] to the free vibration of FG plates. Gulshan Taj et al. Qian et al. [171,172] and Gilhooley et al. [173] employed a
[151] developed a nine-node C0 continuous isoparametric element meshless local Petrov–Galerkin method and the quasi-3D of
with seven DOFs per node for the bending analysis of FG plates Batra and Vidoli [174] to study the bending and vibration of FG
under mechanical and thermal loadings using the TSDT. Foroughi plates. This quasi-3D theory was also used by Sheikholeslami and
and Azhari [152] used the TSDT and spline finite strip method to Saidi [175] to study the vibration of FG plates resting on an elastic
study the buckling and free vibration of FG plates resting on an foundation using Navier solution. Qian and Batra [176] extended
elastic foundation. Tran et al. [153] performed the static bending, their previous work [171] to the transient problems of FG plates
buckling and free vibration analyses of FG plates using the TSDT under thermal and mechanical loadings. Patel et al. [177] studied
and IGA. Tran et al. [154] extended their previous works [153] to the free vibration characteristics of FG elliptical cylindrical shells
thermal buckling of FG plates. The TSDT and IGA were also using a quasi-3D theory and the finite element method. The dis-
employed by Jari et al. [155] to study the linear and nonlinear placement field of the quasi-3D theory due to Lo et al. [178,179]
bending, buckling and free vibration behaviors of FG plates under has 11 unknowns and accounts for a cubic variation of the in-plane
thermal and mechanical loadings. displacements and a quadratic variation of the transverse displace-
The TSDT was also more preferably used for FG shells. Shen ment through the thickness. Roque et al. [180] studied the bending
[156] and Shen and Leung [157] performed the postbuckling analy- behavior of FG plates using a meshless collocation method with
sis of FG cylindrical panels in thermal environments subjected to multiquadric RBFs. The formulation was based on the HSDT of
axial compression [156] or lateral pressure [157]. The governing Pandya and Kant [181] with 7 unknowns and accounting for a
equations are derived from TSDT with von Karman assumptions. cubic variation of the in-plane displacements and a constant trans-
Both nonlinear prebuckling deformations and initial geometric verse displacement through the thickness.
imperfections of the FG cylindrical panels were included in the Matsunaga [182,183] developed a quasi-3D theory for the buck-
postbuckling analysis using a boundary layer theory of shell buck- ling and free vibration analyses of FG plates [183] and shallow
ling. The buckling load and postbuckling equilibrium path were shells [182]. The theory was based on the power series expansion
determined using a singular perturbation technique. Shen [158] of the in-plane and transverse displacements. Matsunaga further
and Shen and Noda [159] extended the previous works [156,157] extended his theory to thermal buckling problems [184] and ther-
to FG cylindrical shells with piezoelectric actuators. Shen and mal bending problems [185] of FG plates.
Liew [160] also extended the previous work [156] to FG cylindrical Pradyumna and Bandyopadhyay [186,187] developed a four-
panels with piezoelectric layers under a combined action of axial node C0 continuous shell element with nine DOFs per node for
compression, electrical and thermal loadings. Shen and Noda the free vibration [186] and dynamic instability [187] of FG curved
[161] provided analytical solutions for the postbuckling of FG panels. The formulation was based on the HSDT of Kant and Khare
cylindrical shells subjected to axial and radial loadings in thermal [188] in which the in-plane displacements are expanded as a cubic
environments. The formulation was based on the TSDT with von variation of the thickness coordinate while the transverse displace-
Karman assumptions and accounted for both nonlinear prebuck- ment is constant. Alijani et al. [189] studied the effect of tempera-
ling deformations and initial geometric imperfections. Shen [162] ture on the geometrically nonlinear vibration of FG doubly curved
provided analytical solutions for the thermal postbuckling of FG panels under thermal variation and harmonic excitation. The
cylindrical shells under heat conduction following a similar formulation was based on the HSDT of Amabili and Reddy [190]
approach. Shen and his colleagues [163–165] studied the post- and the multi-modal energy method. Chen et al. [191] studied
buckling behavior of FG cylindrical shells surrounded by an elastic the free vibration and buckling of FG plates under a combination
medium under mechanical loadings [163,164] or thermal loadings of a extensional stress and a pure bending stress. The governing
[165] using the TSDT and a singular perturbation technique. equations of motion based on the displacement field of Lo et al.
Bagherizadeh et al. [166] presented analytical solutions for the [178,179] are analytically solved for the natural frequency and
critical buckling load of FG simply supported cylindrical shells sur- buckling load of simply supported FG plates. The effects of various
rounded by an elastic medium using the TSDT and series method. parameters and initial stresses on the natural frequency and buck-
Oktem et al. [167] presented analytical solutions for the bending ling load of FG plates were studied. Talha and Singh [192] devel-
analysis of simply supported FG plates and doubly-curved shells oped a quasi-3D theory for static bending and free vibration
using the TSDT and boundary-discontinuous generalized double analyses of FG plates. The displacement field of their theory was
Fourier series approach. Hoang and Nguyen [168] investigated obtained by modifying the displacement field of Lo et al.
the nonlinear response of FG curved panels resting on an elastic [178,179] to satisfy the zero-traction boundary conditions of the
foundation. Analytical solutions for the load–deflection curve of top and bottom surfaces of the plate. A nine-node C0 continuous
simply supported panels under mechanical and thermal loadings isoparametric element with 13 DOFs per node was developed to
were provided using the TSDT and Galerkin method. Nguyen investigate the influences of aspect ratio, thickness ratio, volume
76 H.-T. Thai, S.-E. Kim / Composite Structures 128 (2015) 70–86

fraction index, boundary conditions on the bending and free vibra- Soldatos [233] first employed a hyperbolic function to develop a
tion responses of FG plates. Talha and Singh [193] extended their HSDT for laminated composite plates. Akavci [234,235] proposed a
previous work [191] to the thermo-mechanical vibration of FG new hyperbolic function to develop a HSDT for the bending analy-
plates. Gulshan Taj et al. [194] also extended the previous work sis of composite plates [234] and the free vibration analysis of FG
[191] to the bending analysis of FG skew sandwich plates. plates resting on an elastic foundation [235]. An inverse hyperbolic
Xiang et al. [195] proposed a nth-order shear deformation the- function was used by Grover et al. [236] to develop a HSDT for
ory for the free vibration analysis of FG and composite sandwich composite and sandwich plates. Mahi [237] recently developed a
plates. The displacement field of their theory was obtained by HSDT for FG sandwich and composite plates based on a new hyper-
modifying the displacement field of the TSDT to account for nth- bolic function.
order polynomial terms. The TSDT is therefore deduced as a speci- The exponential function was first used by Karama et al. [238]
fic case. This theory was extended to the bending problem of FG to develop a HSDT for composite beams. Aydogdu [239] extended
plates [196] and the free vibration problem of isotropic plates the previous work [238] to composite plates. Mantari et al. [240]
[197], FG sandwich plates [198] and FG plates resting on an elastic also employed the exponential function to develop a HSDT for
foundation [199]. Xiang and Kang [200] evaluated various sandwich and composite shells. This HSDT was adopted by
five-unknown shear deformation theories for the bending of FG Mantari and Guedes Soares [241] to study the bending behavior
plates using a meshless method with spline RBFs, while Sobhy of FG plates. Based on a new exponential function, Mantari et al.
[201] evaluated various five-unknown shear deformation theories [242] proposed a HSDT for the vibration analysis of FG plates rest-
for the buckling and free vibration of FG sandwich plates resting ing on an elastic foundation.
on an elastic foundation with various boundary conditions using A tangential function was employed by Mantari et al. [243] and
a series method. Wattanasakulpong et al. [202] used the improved Mantari and Guedes Soares [244] to develop a HSDT for isotropic,
TSDT of Shi [203] and Ritz method to analyze the free and forced composite and sandwich plates [243] and FG plates [244].
vibration problems of clamped FG plates under thermal loadings. Mantari et al. [245–247] combined exponential and trigonometric
Based on the displacement field of Lo et al. [178,179], Reddy functions to develop a HSDT for sandwich and composite plates
[204] developed a general HSDT with von Karman geometric non- [245], FG plates [246] and FG doubly curved shells [247]. Mantari
linearity for thermo-mechanical analysis of FG plates. The CPT, and Guedes Soares [248,249] combined exponential and hyper-
FSDT, TSDT and the HSDT with traction-free top and bottom sur- bolic functions to develop a HSDT for isotropic and multilayered
faces can be deduced from the general HSDT. This theory was also plates/shells [248] and FG plates [249]. A combination of tangential
employed by Kant et al. [205] to study the static bending and free and exponential functions was proposed by Mantari et al. [250] to
vibration responses of FG plates. Jha et al. [206–209] studied the develop a HSDT for FG plates. Nguyen et al. [251] combined inverse
static bending [206,209] and free vibration [207–209] of FG plates tangential and cubic functions for FG sandwich plates, while Thai
using a quasi-3D theory of Kant and Manjunatha [210] with 12 et al. [252] combined inverse tangential and linear functions for
unknowns. Analytical solutions were obtained for simply sup- composite and sandwich plates. Thai et al. [253] followed their
ported plates using Navier solution. The influence of higher-order previous work [252] to develop a HSDT for the IGA of FG sandwich
terms in the displacement field on the natural frequency of FG plates using two new trigonometric functions.
plates was also investigated [207]. This quasi-3D theory was also In addition, the non-polynomial functions were also employed
employed by Swaminathan and Naveenkumar [211] to study the to develop quasi-3D theories which account for both shear and
buckling of FG sandwich plates. Natarajan and Manickam [212] normal deformation effects. For example, Zenkour [254] employed
developed an eight-node C0 continuous serendipity element with the sinusoidal function to develop a quasi-3D theory for FG plates.
13 DOFs per node to investigate the bending and free vibration Mantari and Guedes Soares [255] presented a generalized formula-
of FG sandwich plates. The formulation was based on the quasi- tion in which many quasi-3D theories can be deduced by using
3D theory of Ali et al. [213]. They also studied the influence of polynomial or hybrid or trigonometric functions. Mantari and
higher-order terms in the displacement field on the accuracy of Guedes Soares [256] optimized the sinusoidal quasi-3D theory
the quasi-3D theory. Nguyen-Xuan et al. [214] presented a simple for the bending analysis of FG shells. Mantari and Guedes Soares
and effective formulation for composite sandwich plates using a [257] improved their previous work [243] by including the thick-
fifth-order shear deformation theory (FiSDT) in combination with ness stretching effect in FG plates.
the IGA. The static bending, buckling and free vibration behaviors
of rectangular and circular plates under different boundary condi- 2.5. Simplified theories
tions were studied.
It is well known that the HSDTs and quasi-3D theories devel-
2.4.2. Non-polynomial function based-models oped by expanding the displacements in power series of the thick-
The non-polynomial function was first used by Levy [215] with ness coordinate are more computationally expensive since each
a sinusoidal function to develop a refined theory for thick isotropic additional power of the thickness coordinate will induce an addi-
plates. The sinusoidal function was later adopted by Stein [216] tional unknown to the theory. Therefore, there is a need to simplify
and Touratier [217] to develop a five-unknown sinusoidal shear the existing HSDTs and quasi-3D theories or to develop simple
deformation theory (SSDT) for isotropic and laminated composite theories with fewer unknowns.
plates, respectively. The SSDT was extensively used to study the Senthilnathan et al. [258] simplified the TSDT by dividing the
thermal bending of composite plates [218–220], buckling of com- transverse displacement into the bending and shear components
posite plates [221], bending of FG sandwich plates [222,223], buck- and making further assumptions to the TSDT. Therefore, the num-
ling and vibration of FG sandwich plates [224,225], vibration of FG ber of unknowns is reduced by one. In fact, the idea of partitioning
plates [226], bending of FG plates [227], thermal bending of FG the transverse displacement into the bending and shear parts was
plates resting on an elastic foundation [228], thermal buckling of first proposed by Huffington [259] and later adopted by Krishna
FG plates resting on an elastic foundation [229], nanobeams Murty [260] (see [261]). The separation of the displacements into
[230] and nanoplates [231]. The bending relationships between the bending and shear parts not only reduces the number of
the SSDT and CPT quantities were derived by Zenkour [232] for unknowns but also helps one to see the contributions due to shear
FG Levy-type plates. and bending to the total displacements. Senthilnathan et al. [258]
 H.-T. Thai, S.-E. Kim / Composite Structures 128 (2015) 70–86 77

applied the simplified TSDT to laminated composite plates and sandwich plates [323] and laminated composite plates [324,325]
concluded that the simplified one is sufficient accuracy for predict- using a hyperbolic function. Al Khateeb and Zenkour [326] fol-
ing the buckling load of laminated composite plates. The simplified lowed the previous works [322–325] to propose a four-unknown
TSDT was later extended to FG plates [262] and FG sandwich plates quasi-3D theory for FG plates resting on an elastic foundation
[263]. using the sinusoidal function.
Shimpi [264] developed a refined plate theory (RPT) for isotro-
pic plates by dividing the displacements into the bending and 2.6. Mixed theories
shear components. The RPT contains only two unknowns com-
pared two three unknowns in the case of the FSDT and TSDT, but The above-mentioned ESL models are developed based on the
it is sufficient accuracy for predicting the global responses of iso- principle of virtual displacements (PVDs) where the displacement
tropic plates [264–267] and orthotropic plates [268–272]. Kim components are regarded as the primary variables and the stress
et al. [273] extended the RPT to laminated composite plates and components are calculated from the displacement components
modified the RPT by accounting the extension component of the using the strain–displacement and constitutive relationships. An
transverse displacement. It was concluded that the RPT can accu- alternative variational approach, namely the Reissner mixed varia-
rately predict the static bending, buckling and free vibration tional theorem (RMVT), was proposed by Reissner [327,328] by
behaviors of laminated composite plates [273–276]. The RPT was assuming two independent fields for the displacements and trans-
also extensively applied to FG plates [277–285], FG sandwich verse stresses. The advantage of the RMVT over the PVD is that the
plates [286–288], FG plates with piezoelectric layers [289], nano- compatibility of displacements and the equilibrium between two
plates [290,291] and nanobeams [292]. Thai and Uy [293] reformu- adjacent layers can be ‘‘naturally’’ satisfied. This is the main reason
lated the RPT based on the neutral surface and derived analytical why RMVT can be considered as a powerful tool for the analysis of
solutions for the buckling load of FG Levy-type plates. Thai and multilayered plates.
Choi [294] improved the RPT to account for the thickness stretch- Murakami [329] was the first to apply the RMVT to develop a
ing effect in FG plates. mixed laminate theory using a first-order zig-zag displacement
Using similar assumptions of Shimpi [264], many four-un- model. Based on the RMVT, Demasi [330–334] developed a variety
known shear deformation theories have been developed by using of mixed laminate theories in a series of five articles including
different shape functions. For example, Mechab et al. [295] and layerwise theories [331], HSDTs [332] and zig-zag models [333].
El Meiche et al. [296] proposed a four-unknown HSDT for FG plates Fares et al. [335] proposed a RMVT-based theory for the bending
[295] and FG sandwich plates [296] using hyperbolic functions. and free vibration analyses of FG plates. The theory is based on a
Merdaci et al. [297], Tounsi et al. [298], Ameur et al. [299] and displacement field which accounts for a linear variation of the in-
Thai and Vo [300] developed a four-unknown HSDT for FG sand- plane displacements and a quadratic variation of the transverse
wich plates [297,298] and FG plates [299,300] using the sinusoidal displacement through the thickness. Wu and Li [336] developed
function. Based on an inverse tangential function, Nguyen-Xuan a RMVT-based TSDT for the bending analysis of multilayer FG
et al. [301] proposed a four-unknown HSDT for IGA of FG plates. plates using the TSDT displacement model and the layerwise quad-
Thai and Choi [302,303] proposed various four-unknown HSDTs ratic distributions of transverse shear stresses. Therefore, the
for FG plates using different shape functions including cubic func- continuity conditions of both displacements and transverse shear
tions [7], sinusoidal functions [217], hyperbolic functions [233] stresses at the interfaces between adjacent layers are exactly satis-
and exponential functions [238]. The accuracy of these theories fied. A static condensation technique was used to reduce the num-
are evaluated for the bending and free vibration problems of FG ber of unknowns to five which is the same as that of the PVD-based
plates using both analytical solutions [302] and finite element TSDT. Analytical solutions were obtained for simply supported
solutions [303]. Yaghoobi and Fereidoon [304] simplified the nth- plates, and the obtained results were also compared with those
order shear deformation theory developed by Xiang et al. [195– predicted by the PVD-based model. It is found that the RMVT-
199] for the buckling of FG plates resting on an elastic foundation based model is slightly superior to the PVD-based one in predicting
subjected to thermal loadings. Thai and Choi [305,306] simplified the global responses of orthotropic, laminate and FG plates. Based
the FSDT for FG plates [305] and laminated composite plates on the RMVT and PVD, Wu and Li [337] developed finite layer mod-
[306]. This simplified FSDT was later employed by Yu et al. [307] els for 3D bending analysis of multilayered and FG plates. In these
and Yin et al. [308] for the IGA of FG plates accounting for geomet- models, the plate was divided into a number of finite layers in
ric nonlinearity. Thai et al. [309] also developed another simplified which trigonometric functions and Lagrange polynomials were
FSDT for FG sandwich plates using the assumptions from Shimpi respectively used to interpolate the in-plane and transverse varia-
et al. [310]. Analytical solutions for deflection, buckling load and tions of the field variables of each individual layer. These models
natural frequency were obtained for plates under arbitrary bound- were extended by Wu and Chang [338] and Wu et al. [339,340]
ary conditions using the solution method of Sobhy [201]. to the static bending [338], free vibration [339] and buckling
By dividing the transverse displacement into the bending, shear [340] of multilayered composite cylinders and sandwich cylinders.
and stretching parts, Thai and Kim [311] proposed a five-unknown A meshless collocation method and an element-free Galerkin
quasi-3D theory for FG plates using the sinusoidal function. Several method were developed by Wu et al. [341] and Wu and Chiu
similar five-unknown quasi-3D theories were also proposed using [342] using the differential reproducing kernel interpolation
different shape functions such as hyperbolic functions [312–315], [343] and the RMVT. These models were applied to the 3D bending
sinusoidal functions [316], combined hyperbolic and exponential [341] and 3D free vibration [342] analyses of multilayered compos-
functions [317,318] and combined hyperbolic and sinusoidal func- ite and FG plates. It is found that the meshless collocation method
tions [319]. Based on trigonometric functions, Mantari and Guedes gives slightly better performance than the element-free Galerkin
Soares [320,321] proposed simple four-unknown quasi-3D theo- method for the bending problem. However, for the free vibration
ries for FG plates by combining the shear and stretching parts of problems, the element-free Galerkin method gives better perfor-
the transverse displacement. By making further assumptions to mance than the meshless collocation method. Wu and Yang
the six-unknown quasi-3D theory, Zenkour [322–325] proposed [344,345] extended the previous works [341,342] to the multilay-
another four-unknown quasi-3D theory for FG plates [322], FG ered composite and FG cylindrical shells.
78 H.-T. Thai, S.-E. Kim / Composite Structures 128 (2015) 70–86

3. 3D elasticity theory conditions using Ritz method with Chebyshev displacement func-
tions. Li et al. [366] performed the 3D free vibration analysis of
The development of exact solutions of 3D elasticity theory is FG sandwich plates with clamed and simply supported boundary
very useful in assessing the accuracy and validity of ESL models. conditions. Natural frequencies of two types of FG sandwich plates,
Mian and Spencer [346] established exact solutions for FG and namely the sandwich plate with FG faces and a homogeneous core
laminated composite plates. Ootao and Tanigawa [347] developed and the sandwich plate with homogeneous faces and a FG core, are
exact 3D solutions for thermal stress problems of FG simply sup- obtained using Ritz method with Chebyshev polynomials. Amini
ported plates under partial heating. Cheng and Batra [348] derived et al. [367] followed the same method of Li et al. [366] to derive
exact solutions for 3D bending analysis of FG clamped elliptic the natural frequency and mode shapes for the 3D free vibration
plates under thermal loadings using an asymptotic expansion analysis of FG plates resting on an elastic foundation with arbitrary
method. Reddy and Cheng [349] also adopted the asymptotic boundary conditions. Lu et al. [368] and Malekzadeh [369] also
expansion method to derive exact solutions for 3D bending analy- studied the 3D free vibration of FG plates resting on an elastic
sis of FG simply supported plates under thermal loadings. Instead foundation, but they used different solution approaches. Lu et al.
of using the asymptotic expansion method, Vel and Batra [350] [368] adopted the state-space method to derive exact solutions
adopted a power series method to derived exact solutions for the for the natural frequency of simply supported plates, while
3D bending analysis of FG simply supported plates subjected to Malekzadeh [369] used the DQM and series solution to derive
thermal and mechanical loadings. Vel and Batra [351] extended semi-analytical solutions for the natural frequency of Levy-type
their previous work [350] to analyze the transient heat conduction plates. The DQM was also adopted by Malekzadeh et al. [370] to
problems of FG simply supported plate subjected to either time- study the 3D free vibration response of FG annular plates account-
dependent temperature or hear flux on the top and bottom sur- ing for the effect of thermal environments. The exact 3D solutions
faces. Alibeigloo [352] performed the 3D bending analysis of FG for the free vibration of FG rectangular plates with general bound-
plates under thermal and mechanical loadings. Exact solutions ary conditions was also provided by Jin et al. [371] using Rayleigh–
for the temperature, stress and displacement are obtained for sim- Ritz method. Reddy and Kant [372] presented analytical solution
ply supported plates using the state-space method. for 3D free vibration analysis of FG simply supported plates using
The exact solutions for the 3D static bending analysis of FG power series method.
plates were derived by Kashtalyan [353] and Woodward and Yas and Tahouneh [373] investigated the 3D free vibration
Kashtalyan [354]. Exact solutions for the stresses and displace- responses of thick FG annular plates resting on an elastic foundation.
ments of simply supported plates under transverse pressure were The 3D elasticity theory and the DQM were used to obtain the natu-
obtained using Plevako displacement functions. Zenkour [254] ral frequency of FG annular plates with various boundary conditions.
adopted the state-space method to derive the exact solutions for Tahouneh and Yas [374] also adopted the 3D elasticity theory and
the 3D bending analysis of FG simply supported plates under trans- the DQM to study the 3D free vibration response of FG annular sector
verse pressure. Zhong and Shang [355] presented exact solutions plates under various boundary conditions. Sburlati and Bardella
for the 3D bending analysis of FG simply supported plates with [375] presented exact 3D solutions for the bending of FG circular
specific variations of material properties such as exponential plates subjected to axisymmetric transverse loadings using
model, linear model and reciprocal model. Semi-analytical solu- Plevako functions. The free vibration characteristics of FG cylindrical
tions for the 3D bending analysis FG plate with different boundary shells surrounded by an elastic medium was examined by Kamarian
conditions were provided by Zhang et al. [356] using the DQM and et al. [376] using 3D elasticity theory and the generalized DQM.
the state-space approach. Xu and Zhou [357] developed the exact Na and Kim [377] studied the 3D thermal buckling behavior of
solutions for the 3D bending analysis of FG plates with continu- FG plates under uniform, linear and sinusoidal temperature rise
ously varying thickness. The expressions for displacements and across the thickness using a 18-node solid element. To avoid the
stresses of simply supported plates subjected to transverse load- shear locking and keep kinematic stability for thin structures, an
ings were obtained using Fourier series method. Kashtalyan and assumed strain mixed formulation was employed. Na and Kim
Menshykova [358] performed 3D static bending analysis of simply [378] improved their previous work [377] by accounting for the
supported sandwich panels with a FG core under transverse time-dependent temperature rise. Na and Kim [379] extended
loadings. Woodward and Kashtalyan [359] extended the previous their previous work [378] to FG sandwich plates with homoge-
work [358] to simply supported sandwich panels subjected to neous faces and a FG core. Na and Kim [380] employed the finite
distributed and concentrated loadings. The 3D bending behavior element method to investigate the 3D thermal postbuckling of
of FG plates under point loading was investigated by Abali et al. FG plates under uniform or non-uniform temperature rise. A 18-
[360] using a combination of analytical and numerical approaches. node solid element based on the Green–Lagrange relation was
The analytical approach is based on the displacement function developed to account for large deflection. The Newton–Raphson
method, while the numerical modeling is based on Galerkin type iteration scheme was used to determine the postbuckling equilib-
finite element method. Yun et al. [361] investigated the 3D rium path. Na and Kim [381] extended their previous work [380] to
axisymmetric bending of FG circular plates subjected to arbitrarily investigate the 3D nonlinear bending of FG plates subjected to uni-
transverse loadings. Analytical solutions for the displacements, form pressure and thermal loadings. The postbuckling of FG annu-
stresses, axial forces and bending moments of FG simply supported lar sector plates was studied by Asemi et al. [382] using the 3D
or clamped plates were obtained using the direct displacement elasticity theory and the finite element method. An eight-node
method. Wen et al. [362] presented exact solutions for the 3D sta- brick element based on the complete Green strain tensor was
tic and dynamic bending analysis of FG plates using the RBFs. developed to account for geometric nonlinearity. The Newton–
Exact solutions for the 3D free and forced vibrations of FG sim- Raphson iteration scheme was used to determine the postbuckling
ply supported plates were provided by Vel and Batra [363] using equilibrium path. The postbuckling of FG annular sector plates was
the power series method. Exact solutions for natural frequencies, studied by Asemi et al. [382] using the 3D elasticity theory and the
displacements and stresses are compared with those predicted finite element method. An eight-node brick element based on the
by ESL models. Vel [364] extended the previous work [363] to FG complete Green strain tensor was developed to account for geo-
shells. Uymaz and Aydogdu [365] presented exact solutions for metric nonlinearity. Newton–Raphson iteration scheme was used
the 3D free vibration analysis of FG plates with various boundary to determine the postbuckling equilibrium path.
 H.-T. Thai, S.-E. Kim / Composite Structures 128 (2015) 70–86 79

4. Unified formulation sandwich plates [415] and FG shells [416]. Both shear and normal
deformation effects are taken into account by using a quasi-3D the-
The unified formulation proposed by Carrera [383–398] for ory with a cubic variation of the in-plane displacements and a
multilayered composite structures is a hierarchical formulation quadratic variation of the transverse displacement through the
which offers a procedure to describe and implement numerous thickness.
plate/shell theories as well as finite elements in a unified manner Cinefra et al. [417] combined the CUF and the mixed interpola-
by referring to a few fundamental nuclei. All theories can be easily tion of tensorial components (MITC) technique to developed a
developed in the framework of the Carrera unified formulation nine-node shell element for the bending analysis of FG plates/
(CUF) by expanding the displacement variables in the thickness shells under transverse loadings. The MITC overcomes the locking
coordinate using Taylor’s expansions of N-order with N being a free phenomenon and all refined models contained in the CUF can be
parameter. More detailed information and applications of the CUF implemented in their proposed shell element. Dozio [418] pre-
can be found in the books authored by Carrera et al. [399–401]. sented a powerful modeling technique capable of accurately pre-
The CUF was extensively used for the analysis of FG plates based dicting the natural frequency of FG sandwich plates with
on the PVD or the RMVT. For example, Carrera et al. [402] arbitrary boundary conditions. The formulation was based on the
employed the CUF and the PVD to develop a variable kinematic proper combination of the classical Ritz method and the CUF.
model for the bending analysis of FG plates under mechanical load- Dozio [419] employed the CUF and the PVD to derive analytical
ings. The obtained analytical and finite element solutions are com- solutions for the natural frequency of Levy-type plates on the basis
pared well with existing 3D solutions. Brischetto et al. [403] of a family of quasi-3D theories with variable order. Fazzolari [420]
extended the CUF to the bending analysis of FG plates under ther- recently extended the CUF and the hierarchical trigonometric Ritz
mal and mechanical loadings. The governing equations derived formulation to the free vibration and thermal buckling of FG sand-
from the PVD are analytically solved for simply supported plates wich plates.
using Navier solution. Cinefra et al. [404] extended the application
of the CUF to the thermal analysis of FG shells, while Cinefra and 5. Conclusions
Soave [405] extended the application of the CUF to the free vibra-
tion analysis of FG sandwich plates. Brischetto and Carrera [406] The development of various models for the modeling and analy-
extended the previous works [402] in the framework of the sis of FG plates and shells has been comprehensively reviewed and
RMVT. Analytical solutions of simply supported plates are obtained discussed in this paper. Based on the ESL models, the 3D elasticity
using Navier solution. Compared to the PVD-based models, the theory and the unified formulation, a large number of com-
RMVT-based ones give a better prediction of the transverse stres- putational models have been proposed to predict the global
ses since they are considered as the primary variables in the response of FG plates and shells under mechanical and thermal
RMVT-based models. Brischetto [407] applied the PVD-based mod- loadings. The following points can be outlined from the present
els [402] and the RMVT-based models [406] to investigate the literature survey:
bending responses of FG sandwich plates with a FG core. Carrera
et al. [408] also studied the bending response of FG sandwich (1) Among the ESL models, the CPT is extensively used to pre-
plates and shells using the CUF with both PVD-based and RMVT- dict the nonlinear and postbuckling responses of FG thin
based models. The results of simply supported plates and shells plates/shells. All the effects of temperature, initial geometric
under transverse pressure indicate that the use of refined models imperfections and geometric nonlinearity can be easily
is mandatory for FG sandwich structures since the effects of shear included in the CPT model since it is the simplest one among
and normal deformations are significant. Carrera et al. [409] the ESL models. Although the CPT ignores the shear and nor-
adopted the CUF and the PVD to evaluate the effects of normal mal deformation effects, it can provide acceptable predic-
strain in FG plates and shells by comparing the theories containing tions for the thin plates/shells where the effects of the
the constant transverse displacement with the corresponding shear and normal deformations are insignificant.
models having linear to fourth order expansion terms in the thick- (2) Among the shear deformation theories, the FSDT and TSDT
ness direction. The importance of the thickness stretching effects in were widely used for the modeling and analysis of FG
FG plates/shells was pointed out in their work. It was concluded plates/shells. This might be due to the fact that both FSDT
that the increase in the order of expansion for the in-plane dis- and TSDT was developed long time ago compared with other
placements is meaningless if the transverse normal strains were HSDTs having the same number of unknowns.
not considered. (3) A large number of non-polynomial function based-HSDTs
Ferreira et al. [410] employed the CUF and the PVD to develop a have been developed recently. However, they are not widely
sinusoidal shear deformation theory for the static bending and free used compared with the polynomial function based-HSDTs
vibration analyses of laminated shells. The theory accounts for except for the case of the SSDT. Moreover, most of studies
both shear and normal deformation effects by considering a sinu- based on the non-polynomial functions are limited to ana-
soidal variation of all displacements through the thickness. The lytical solutions of linear problems. The development of
governing equations are solved for the bending and free vibration numerical models based on the non-polynomial functions
problems using the meshless collocation method with multiquad- is therefore necessary to fully assess the accuracy as well
ric RBFs. Neves et al. [411,412] used the similar approach of as efficiency of the non-polynomial function based-HSDTs.
Ferreira et al. [410] to study the effect of thickness stretching on (4) The computational models based on a combination of the
the static bending [411,412] and free vibration [412] responses IGA and the simplified HSDTs seem to be very attractive
of FG plates. However, their formulations are based on a hybrid and promising since the IGA can exactly describe the geome-
quasi-3D theory in which the in-plane and transverse displace- try of complex structures and, more importantly, the simpli-
ments are respectively assumed as sinusoidal and quadratic varia- fied HSDTs contains fewer unknowns. However, most of the
tions across the thickness. Neves et al. [413,414] also did a similar simplified HSDT-based models are limited to geometrically
work using a hyperbolic function instead of the sinusoidal function linear problems. Thus, further studies on the development
as in their previous work [412]. Neves et al. [415,416] extended the of geometrically nonlinear models should be done to fully
application of the CUF and the collocation method with RBFs to FG evaluate the benefits of the simplified HSDTs.
80 H.-T. Thai, S.-E. Kim / Composite Structures 128 (2015) 70–86

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