A refined sinusoidal model for functionally graded plates subjected to thermomechanical loading

Abstract

A refined sinusoidal model considering transverse normal strain has been developed for thermoelastic analysis of functionally graded material plate. Although transverse normal strain has been considered, the additional displacement parameters are not increased as transverse normal strain only includes the thermal expansion coefficient and thermal loading. Moreover, the merit of the previous sinusoidal model satisfying tangential stress-free boundary conditions on the surfaces can be maintained. It is important that the effects of transverse normal thermal deformation are incorporated in the in-plane displacement field, which can actively influence the accuracy of in-plane stresses. To assess the performance of the proposed model, the thermoelastic behaviors of functionally graded material plates with various configurations have been analyzed. Without increase of displacement variables, accuracy of the proposed model can be significantly improved by comparing to the previous sinusoidal model. Agreement between the present results and quasi-dimensional solutions are very good, and the proposed model only includes the five displacement variables which can illustrate the accuracy and effectiveness of the present model. In addition, new results using several models considered in this paper have been presented, which can serve as a reference for future investigations.

Keywords

Refined sinusoidal model thermoelastic analysis transverse normal strain analytical solution functionally graded plates

Introduction

Functionally graded materials (FGMs) are inhomogeneous materials with a gradually changing composition, which are widely used in many engineering as they can maintain structural integrity in high thermal gradient environment. To use them effectively, it is necessary to well understand deformation and stresses of FGM plates under thermal environment. Thus, various plate models have been proposed for the thermo-mechanical analysis of FGM structures.

The research works on the static behaviors of FGM structures have been studied earlier using the classical plate theory.¹ The classical plate theory neglecting transverse shear strain is expected to produce accurate results as the thickness-to-length ratio is small. Nevertheless, application of such model to thick or moderately thick FGM plates will result in serious errors in the values of deflection and stresses.² In order to consider transverse shear strain, the first-order shear deformation theory has been employed for static and dynamic analysis of FGM and composite plates.^3–9 However, the accuracy of results obtained from the first-order theory is strongly influenced by the shear correction factors¹⁰ as transverse shear strains in the first-order theory are assumed to be constant through the thickness direction. To overcome the lacuna of the first-order theory, the higher-order models have been developed to study the static and the dynamic response of FGM plates.

Based on the third-order shear deformation plate theory,¹¹ Reddy presented Navier's solutions and finite element models for analysis of FGM plates.¹² This model assumes a constant transverse displacement through the thickness, and in-plane displacement is taken as cubic functions of the thickness coordinate. The displacement field leads to parabolic distribution of transverse shear strains, so the shear correction factors are not required. In terms of the third-order model,¹¹ Cheng and Batra¹³ studied the buckling and the steady-state vibration of the simply-supported functionally graded plates. Oktem and Mantari¹⁴ extended the third-order theory to study the static response of functionally graded plates and doubly-curved shells. Taj et al.¹⁵ studied the static behaviors of skew FGM plates subjected to mechanical and thermal loading. Satouri et al.² employed the third-order model for buckling analysis of a two-dimensional functionally graded cylindrical shell reinforced by axial stiffeners. In addition, the non-polynomial shear deformation theories have been developed for analysis of FGM plates. Based on a higher-order theory,¹⁶ Mantari et al.¹⁷ presented Navier-type analytical solution of FGM plates subjected to mechanical loading. In-plane displacement field in the model was expressed as a combination of exponential and trigonometric functions of the thickness coordinate. Thai et al.¹⁸ employed the generalized shear deformation theories for static and dynamic analysis of FGM sandwich plates, in which in-plane displacements may be exponential function¹⁹ and trigonometric functions.^20,21 Mahi et al.²² proposed a hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded sandwich and composite plates. In addition, Pradhan and Chakraverty²³ developed a generalized power-law exponent-based shear deformation theory for dynamic analysis of FGM beams subjected to different sets of boundary conditions. In terms of the inverse trigonometric shear deformation theory, Kulkarni et al.²⁴ studied the bending and buckling behaviors of functionally graded plates.

Employing a sinusoidal theory proposed by Touratier,^25,26 Zenkour and Alghamdi^27,28 investigated the thermomechanical bending response of symmetric and nonsymmetric sandwich plates of uniform thickness. The sinusoidal theory satisfies the traction-free boundary conditions at plate surfaces, and transverse shear strains show the cosine-law distribution through the thickness of plate. Subsequently, Zenkour²⁹ used the sinusoidal theory for hygrothermal bending analysis of FGM plates resting on elastic foundations. Based on the sinusoidal theory, Zenkour et al.³⁰ investigated the bending response of the simply-supported functionally graded viscoelastic sandwich beams resting on Pasternak's elastic foundations. Moreover, effects of material distribution and time parameter on displacement and stresses were also studied. In addition, Zenkour and Sobby³¹ analyzed the dynamic displacement and stresses of functionally graded plates subjected to time harmonic thermal load by employing the sinusoidal theory. The sinusoidal theory was also extended to study hygrothermal behaviors of composite plates.^32–36 Various higher-order models developed by the investigators for FGM plates have been reported in review paper.³⁷

By literature review, it is found that the sinusoidal theory has been widely used to study thermoelastic behaviors of functionally graded and composite plates. However, transverse normal strain has been generally neglected in the sinusoidal model. Cho and Oh³⁸ showed that for thermomechanical problem even in moderate thick plate, transverse normal strain cannot be neglected since effect of out-of-plane thermal deformation is equally important in comparison with those of in-plane thermal deformation. In order to extend the sinusoidal model for thermoelastic analysis of thick plate, the sinusoidal models considering transverse normal strain have been developed. Sayyad and Ghugal³⁹ proposed a shear and normal deformation theory for bending analysis of isotropic, transversely isotropic, laminated composite and sandwich plates. Recently, Zenkour⁴⁰ proposed a four-unknown shear and normal deformations theory for thermal bending of layered composite plates resting on elastic foundations. However, the proposed sinusoidal models^39,40 will encounter difficulty for the thermal expansion problems of isotropic and FGM plates. In view of this situation, it is desirable to present a sinusoidal model that can accurately analyze the thermoelastic behaviors of the isotropic and FGM plates. To this end, a refined sinusoidal model considering transverse normal thermal deformation will be developed for the functionally graded plates subjected to thermomechanical load. In the proposed model, transverse normal strain has been taken into account while additional displacement parameters in the proposed model is not increased as transverse normal deformations due to thermal loading have been absorbed in the generalized force vector. The governing equations of the FGM plates are derived by using the principle of virtual displacement, which can be solved applying Navier solution method.⁴¹ Performance of the present model is verified by comparing to the results computed from available models in the literature.

Theoretical formulation

A sinusoidal shear deformation plate theory considering transverse normal strain

The classical sinusoidal shear deformation plate theory (SPT) proposed by Touratier^25,26 can be given by

\begin{matrix} u (x, y, z) = u_{0} (x, y) - z \frac{\partial w_{0} (x, y)}{\partial x} + Ψ (z) u_{1} (x, y) \\ v (x, y, z) = v_{0} (x, y) - z \frac{\partial w_{0} (x, y)}{\partial y} + Ψ (z) v_{1} (x, y) \\ w (x, y, z) = w_{0} (x, y) \end{matrix}

(1)

where

Ψ (z) = \frac{h}{π} sin (\frac{π z}{h})

In equation (1), it is found that while satisfying the zero shear traction conditions on the bounding surfaces of the plate, the model SPT does not account for transverse normal strain. Therefore, the model SPT will encounter difficulty for the analysis of thermal expansion problem of FGM plates. In order to extend the model SPT for thermoelastic analysis of FGM plates, the thermal deformations through the thickness and along in-plane directions due to thermal loads are respectively introduced in the displacement field of the model SPT. Thus, a sinusoidal shear deformation plate theory considering transverse normal strain (SPTC) can be obtained, which can be written as

\begin{matrix} u (x, y, z) = u_{0} (x, y) - z \frac{\partial w_{0} (x, y)}{\partial x} \\ + Ψ (z) u_{1} (x, y) + Ψ_{T} (z) \frac{\partial \bar{T} (x, y)}{\partial x} \\ v (x, y, z) = v_{0} (x, y) - z \frac{\partial w_{0} (x, y)}{\partial y} \\ + Ψ (z) v_{1} (x, y) + Ψ_{T} (z) \frac{\partial \bar{T} (x, y)}{\partial y} \\ w (x, y, z) = w_{0} (x, y) + F (z) \bar{T} (x, y) \end{matrix}

(2)

where

\bar{T} (x, y)

describes the distributions of temperature along in-plane direction, and the function

Ψ_{T} (z)

can describe the influence of temperature on in-plane displacement yet to be defined

F (z) = \int α_{z} f (z) d z

(3)

where

α_{z}

is the transverse normal thermal expansion coefficient, and

f (z)

denotes the temperature profiles through the thickness direction.

Once the thermal deformations due to thermal loads are introduced in the transverse displacement field, the free conditions of the transverse shear stresses on the upper and the lower surfaces will be violated. The function $Ψ_{T} (z)$ can be determined by enforcing the zero shear traction conditions at the top and the bottom plate surfaces. Employing the zero shear traction conditions on the bounding surfaces of the plate, the function $Ψ_{T} (z)$ can be given by

Ψ_{T} = \frac{z^{2}}{h} (\frac{F (z_{1}) - F (z_{2})}{2}) - \frac{4 z^{3}}{3 h^{2}} (\frac{F (z_{1}) + F (z_{2})}{2})

(4)

where z₁ and z₂ are, respectively, the z-coordinate at the bottom and the top of the FGM plate as shown in Figure 1.

Figure 1.

An FGM plate subjected to various loading. FGM: functionally graded material.

In order to verify the performance of the proposed model SPTC, a ninth-order theory HSDT-98 proposed by Matsunaga⁴² will be employed for thermoelastic analysis of FGM plate. The in-plane displacement field of the model HSDT-98 consists of ninth-order polynomial in the global thickness coordinate z and the transverse deflection is represented by an eighth-order polynomial in z. Performance of the model HSDT-98 has been verified,⁴² so that results of HSDT-98 will be used as the reference solutions to assess the performance of all other models. The ninth-order theory HSDT-98 is expressed as

\begin{matrix} u (x, y, z) = u_{0} (x, y) + \sum_{i = 1}^{9} z^{i} u_{i} \\ v (x, y, z) = v_{0} (x, y) + \sum_{i = 1}^{9} z^{i} v_{i} \\ w (x, y, z) = w_{0} (x, y) + \sum_{i = 1}^{8} z^{i} w_{i} \end{matrix}

(5)

In equation (5), there are 29 displacement variables in the displacement field of the model HSDT-98.

In addition, the higher-order model proposed by Reddy¹¹ has been also chosen for comparison, which can be written as

\begin{matrix} u (x, y, z) = u_{0} (x, y) - z \frac{\partial w_{0}}{\partial x} + z [1 - \frac{4}{3} (\frac{z}{h}) 2] u_{1} (x, y) \\ v (x, y, z) = v_{0} (x, y) - z \frac{\partial w_{0}}{\partial y} + z [1 - \frac{4}{3} (\frac{z}{h}) 2] v_{1} (x, y) \\ w (x, y, z) = w_{0} (x, y) \end{matrix}

(6)

Constitutive equations

For linear elasticity, the strain components of the model SPTC can be given by

\begin{matrix} ɛ_{x} = \frac{\partial u_{0}}{\partial x} + Ψ (z) \frac{\partial u_{1}}{\partial x} - z \frac{\partial^{2} w_{0}}{\partial x^{2}} + Ψ_{T} (z) \frac{\partial^{2} \bar{T}}{\partial x^{2}} \end{matrix}

(7)

\begin{matrix} ɛ_{y} = \frac{\partial v_{0}}{\partial y} + Ψ (z) \frac{\partial v_{1}}{\partial y} - z \frac{\partial^{2} w_{0}}{\partial y^{2}} + Ψ_{T} (z) \frac{\partial^{2} \bar{T}}{\partial y^{2}} \\ γ_{xy} = \frac{\partial u_{0}}{\partial y} + \frac{\partial v_{0}}{\partial x} + Ψ (z) (\frac{\partial u_{1}}{\partial y} + \frac{\partial v_{1}}{\partial x}) \\ - 2 z \frac{\partial^{2} w_{0}}{\partial x \partial y} + 2 Ψ_{T} (z) \frac{\partial^{2} \bar{T}}{\partial x \partial y} \\ γ_{xz} = \frac{\partial Ψ (z)}{\partial z} u_{1} + (F (z) + \frac{\partial Ψ_{T} (z)}{\partial z}) \frac{\partial \bar{T}}{\partial x} \\ γ_{yz} = \frac{\partial Ψ (z)}{\partial z} v_{1} + (F (z) + \frac{\partial Ψ_{T} (z)}{\partial z}) \frac{\partial \bar{T}}{\partial y} \end{matrix}

Equation (7) shows that transverse normal deformation caused by thermal loading has a significant impact on the in-plane and transverse shear strains.

The temperature distribution T through the thickness of plate is taken from Zenkour et al.²⁷

T (x, y, z) = {\bar{T}}_{0} (x, y) + (z / h) {\bar{T}}_{1} (x, y) + (f_{z} / h) {\bar{T}}_{2} (x, y)

(8)

where

f_{z} = (h / π) sin (π z / h)

The relationships between stresses and strains of the FGM plate subjected to thermomechanical load can be written as

\begin{matrix} {\begin{matrix} σ_{x} \\ σ_{y} \\ τ_{xy} \\ τ_{xz} \\ τ_{yz} \end{matrix}} = [\begin{matrix} Q_{11} (z) & Q_{12} (z) & 0 & 0 & 0 \\ Q_{21} (z) & Q_{22} (z) & 0 & 0 & 0 \\ 0 & 0 & Q_{33} (z) & 0 & 0 \\ 0 & 0 & 0 & Q_{44} (z) & 0 \\ 0 & 0 & 0 & 0 & Q_{55} (z) \end{matrix}] \\ \times {\begin{matrix} {\bar{ɛ}}_{x} \\ {\bar{ɛ}}_{y} \\ {\bar{γ}}_{xy} \\ {\bar{γ}}_{xz} \\ {\bar{γ}}_{yz} \end{matrix}} \end{matrix}

(9)

where

{\bar{ɛ}}_{x} = ɛ_{x} - α (z) Δ T

{\bar{ɛ}}_{y} = ɛ_{y} - α (z) Δ T

{\bar{γ}}_{xy} = γ_{xy}

{\bar{γ}}_{xz} = γ_{xz}

{\bar{γ}}_{yz} = γ_{yz}

;

Q_{11} (z) = \frac{E (z)}{1 - v^{2}}

Q_{12} (z) = Q_{21} (z) = \frac{vE (z)}{1 - υ^{2}}

Q_{22} (z) = \frac{E (z)}{1 - υ^{2}}

Q_{33} = Q_{44} = Q_{55} =

\frac{E (z)}{2 (1 + v)}

;

E (z)

is Young's modulus of FGM plate; α is the linear thermal expansion coefficients;

Δ T = T - \hat{T}

is the rise of temperature with respect to the reference temperature, where it needs to be shown that the initial temperature

\hat{T}

is taken as zero in the present work.

Equilibrium equations

The FGM plate with length a, width b, and thickness h is subjected to transverse loading q(x, y, ± h/2) on the surfaces and a temperature field T(x, y, z). Rectangular Cartesian coordinates (x, y, z) are used to describe the deformation of the FGM plate, where x ∈ [x₀, x_a] represents the plate longitudinal axis, y ∈ [y₀, y_b] denotes the plate width axis, and z ∈ [−h/2, h/2] is the thickness coordinate. The material properties $M$ of the FGM plate including Young's modulus E and thermal expansion coefficient α are taken from Zenkour⁴³

M (z) = M_{metal} + (M_{ceramic} - M_{metal}) (\frac{z}{h} + 0.5)^{k}

(10)

in which M_metal and M_ceramic denote the corresponding material properties of the metal and the ceramic; k represents the volume fraction exponent.

In terms of the model SPTC, the principle of virtual displacement is expressed as

δ U - δ W = 0

(11)

where the virtual strain energy

δ U

and virtual work

δ W

done by applied loads are respectively given by

δ U = \int A \int_{z_{1}}^{z_{2}} (σ_{x} δ ε_{x} + σ_{y} δ ε_{y} + τ_{x y} δ γ_{x y} + τ_{x z} δ γ_{x z} + τ_{y z} δ γ_{y z}) d z d x d y

(12)

δ W = δ W_{p x} + δ W_{p y} + δ W_{q} + δ W_{T x 0} + δ W_{T x a} + δ W_{T y 0} + δ W_{T y b}

(13)

where

\begin{array}{l} δ W_{p x} = - \iint_{A} [p_{x}^{b} δ u (x, y, z_{1}) + p_{x}^{t} δ u (x, y, z_{2})] d x d y \\ δ W_{p y} = - \iint_{A} [p_{y}^{b} δ v (x, y, z_{1}) + p_{y}^{t} δ v (x, y, z_{2})] d x d y \\ δ W_{q} = - \iint_{A} [q^{b} δ w (x, y, z_{1}) + q^{t} δ w (x, y, z_{2})] d x d y \\ δ W_{T x 0} = \int_{z_{1}}^{z_{2}} [T_{x 0} δ u (x_{0}, y, z) + T_{x z 0} δ w (x_{0}, y, z)] d z \\ δ W_{T y 0} = \int_{z_{1}}^{z_{2}} [T_{y 0} δ v (x, y_{0}, z) + T_{y z 0} δ w (x, y_{0}, z)] d z \\ δ W_{T x a} = - \int_{z_{1}}^{z_{2}} [T_{x a} δ u (x_{a}, y, z) + T_{x z a} δ w (x_{a}, y, z)] d z \\ δ W_{T y b} = - \int_{z_{1}}^{z_{2}} [T_{y b} δ v (x, y_{b}, z) + T_{y z b} δ w (x, y_{b}, z)] d z \end{array}

(14)

where p^b(x,y) and p^t(x,y) are the in-plane distributed loads on the bottom and the top surfaces of the plate; q^b(x,y) and q^t(x,y) are the transverse loads on the bottom and the top surfaces of the plate; T_x0, T_xa, T_y0, and T_yb are the axial loads at the boundary cross sections of the FGM plate; T_xz0, T_xza, T_yz0, and T_yzb are the transverse shear tractions at the boundary cross sections of the FGM plate.

Integrating through the thickness of FGM plate, the principle of virtual displacement applied to the model SPTC can be rewritten as

\sum_{i = 1}^{5} δ {\bar{U}}_{i} + \sum_{i = 1}^{11} δ {\bar{W}}_{i} = 0

(15)

where

δ {\bar{U}}_{1} = \int \int_{A} N_{x} δ u_{0, x} + M_{x 1} δ u_{1, x} + M_{x 2} δ w_{0, xx} d x d y;

δ {\bar{U}}_{2} = \int \int_{A} N_{y} δ v_{0, y} + M_{y 1} δ v_{1, y} + M_{y 2} δ w_{0, yy} d x d y;

δ {\bar{U}}_{3} = \iint_{A} N_{x y} (δ u_{0, y} + δ v_{0, x}) + M_{x y 1} (δ u_{1, y} + δ v_{1, x}) + M_{x y 2} δ w_{0, x y} d x d y

δ {\bar{U}}_{4} = \int \int_{A} V_{x 1} δ u_{1} d x d y;

δ {\bar{U}}_{5} = \int \int_{A} V_{y 1} δ v_{1} d x d y;

δ {\bar{W}}_{1} = - \int \int_{A} p_{x 0} δ u_{0} + p_{x 1} δ u_{1} + p_{x 2} δ w_{0, x} d x d y;

δ {\bar{W}}_{2} = - \int \int_{A} p_{y 0} δ v_{0} + p_{y 1} δ v_{1} + p_{y 2} δ w_{0, y} d x d y;

δ {\bar{W}}_{3} = - \int \int_{A} q_{0} δ w_{0} d x d y;

δ {\bar{W}}_{4} (x_{0}) = {\bar{N}}_{x 0} δ u_{0} + {\bar{M}}_{x 10} δ u_{1} + {\bar{M}}_{x 20} δ w_{0, x};

δ {\bar{W}}_{5} (y_{0}) = {\bar{N}}_{y 0} δ v_{0} + {\bar{M}}_{y 10} δ v_{1} + {\bar{M}}_{y 20} δ w_{0, y};

δ {\bar{W}}_{6} (x_{a}) = - ({\bar{N}}_{xa} δ u_{0} + {\bar{M}}_{x 1 a} δ u_{1} + {\bar{M}}_{x 2 a} δ w_{0, x});

δ {\bar{W}}_{7} (y_{b}) = - ({\bar{N}}_{yb} δ v_{0} + {\bar{M}}_{y 1 b} δ v_{1} + {\bar{M}}_{y 2 b} δ w_{0, y});

δ {\bar{W}}_{8} (x_{0}) = {\bar{V}}_{xx 0} δ w_{0}; δ {\bar{W}}_{9} (y_{0}) = {\bar{V}}_{yx 0} δ w_{0};

δ {\bar{W}}_{10} (x_{a}) = {\bar{V}}_{xxa} δ w_{0}; δ {\bar{W}}_{11} (y_{a}) = {\bar{V}}_{yxa} δ w_{0} .

in which

[N_{x}, M_{x 1}, M_{x 2}] = \int_{z_{1}}^{z_{2}} [σ_{x}^{k}, Ψ (z) σ_{x}^{k}, - σ_{x}^{k}] d z;

[N_{y}, M_{y 1}, M_{y 2}] = \int_{z_{1}}^{z_{2}} [σ_{y}^{k}, Ψ (z) σ_{y}^{k}, - σ_{y}^{k}] d z;

[N_{xy}, M_{xy 1}, M_{xy 2}] = \int_{z_{1}}^{z_{2}} [τ_{xy}^{k}, Ψ (z) τ_{xy}^{k}, - τ_{xy}^{k}] d z;

V_{x 1} = \int_{z_{1}}^{z_{2}} \frac{\partial Ψ (z)}{\partial z} τ_{xz}^{k} d z;

V_{y 1} = \int_{z_{1}}^{z_{2}} \frac{\partial Ψ (z)}{\partial z} τ_{yz}^{k} d z;

[{\bar{N}}_{x β}, {\bar{M}}_{x 1 β}, {\bar{M}}_{x 2 β}] = \int_{z_{1}}^{z_{2}} [T_{x β}, Ψ (z) T_{x β}, - T_{x β}] d z;

[{\bar{V}}_{xx β}] = \int_{z_{1}}^{z_{2}} [T_{xz β}] d z, (β = 0, a);

[{\bar{N}}_{y λ}, {\bar{M}}_{y 1 λ}, {\bar{M}}_{y 2 λ}] = \int_{z_{1}}^{z_{2}} [T_{y λ}, Ψ (z) T_{y λ}, - T_{y λ}] d z;

[{\bar{V}}_{yx λ}] = \int_{z_{1}}^{z_{2}} [T_{yz λ}] d z, (λ = 0, b);

\begin{matrix} [p_{x 0}, p_{x 1}, p_{x 2}] = [p_{x}^{b} + p_{x}^{t}, Ψ (z_{1}) p_{x}^{b} + Ψ (z_{2}) p_{x}^{t}, \\ - z_{1} p_{x}^{b} - z_{2} Φ_{2}^{N} p_{x}^{t}]; \end{matrix}

\begin{matrix} [p_{y 0}, p_{y 1}, p_{y 2}] = [p_{x}^{b} + p_{x}^{t}, Ψ (z_{1}) p_{x}^{b} + Ψ (z_{2}) p_{x}^{t}, \\ - z_{1} p_{x}^{b} - z_{2} Ψ_{2}^{N} p_{x}^{t}]; \end{matrix}

[q_{0}] = [q^{b} + q^{t}] .

Employing integration by parts and collecting the variational coefficients $δ u_{0}$ , $δ u_{1}$ , $δ v_{0}$ , $δ v_{1}$ , and $δ w_{0}$ , the equilibrium equations for the model SPTC can be written as

\begin{matrix} δ u_{0} : N_{x, x} + N_{xy, y} + p_{x 0} = 0 \\ δ u_{1} : M_{x 1, x} + M_{xy 1, y} - V_{x 1} + p_{x 1} = 0 \\ δ v_{0} : N_{y, y} + N_{xy, x} + p_{y 0} = 0 \\ δ v_{1} : M_{y 1, y} + M_{xy 1, x} - V_{y 1} + p_{y 1} = 0 δ w_{0} : M_{x 2, xx} + M_{y 2, yy} + M_{xy 2, xy} + p_{x 2, x} + p_{y 2, y} - q_{0} = 0 \end{matrix}

(16)

A set of consistent geometric (kinematic-variable) and kinetic (stress-resultant) boundary conditions at the plate edges (x = x₀, x = x_a, y = y₀, and y = y_b) are written as

\begin{matrix} u_{0} (x_{β}, y) = {\bar{u}}_{0 β} or N_{x} (x_{β}, y) = {\bar{N}}_{x β} \\ u_{1} (x_{β}, y) = {\bar{u}}_{1 β} or M_{x 1} (x_{β}, y) = {\bar{M}}_{x 1 β} \\ v_{0} (x, y_{λ}) = {\bar{v}}_{0 λ} or N_{y} (x, y_{λ}) = {\bar{N}}_{y λ} \\ v_{1} (x, y_{β}) = {\bar{v}}_{1 λ} or M_{y 1} (x, y_{λ}) = {\bar{M}}_{y 1 λ} \\ w_{0, x} (x_{β}, y) = {\bar{w}}_{0, x β} or M_{x 2} (x_{β}, y) = {\bar{M}}_{x 2 β} \\ w_{0, y} (x, y_{λ}) = {\bar{w}}_{0, y λ} or M_{y 2} (x, y_{λ}) = {\bar{M}}_{y 2 λ} \\ w_{0} (x_{β}, y) = {\bar{w}}_{0 x β} or - {\bar{M}}_{x 2, x} (x_{β}, y) \\ - {\bar{M}}_{xy 2, y} (x_{β}, y) - p_{x 2} (x_{β}, y) = {\bar{V}}_{xx β} \\ w_{0} (x, y_{λ}) = {\bar{w}}_{0 y λ} or - {\bar{M}}_{y 2, y} (x, y_{λ}) \\ - {\bar{M}}_{xy 2, x} (x, y_{λ}) - p_{y 2} (x, y_{λ}) = {\bar{V}}_{yx λ} \end{matrix}

(17)

where

β = 0, a

;

λ = 0, b

Analytical solution

Analytical solutions in terms of trigonometric functions for the simply-supported FGM plates are taken into account, and the external force and the temperature loads are given by

\begin{matrix} q = q_{0} sin (α x) sin (β y) \\ {\bar{T}}_{i} = T_{i} sin (α x) sin (β y) \end{matrix}

(18)

where i = 0,1,2.

For simply-supported FGM plates, the tangential displacements on the boundary are admissible while the transverse displacements are inadmissible. Thus, the boundary conditions of simply-supported FGM plates⁴² are given by

At edges x = 0 and x = a

v_{0} = 0, v_{1} = 0, w_{0} = 0

(19)

At edges y = 0 and y = b

u_{0} = 0, u_{1} = 0, w_{0} = 0

(20)

Following the Navier's procedure,⁴¹ the displacement parameters satisfying the simply-supported boundary conditions are assumed to be a double trigonometric series as

\begin{matrix} u_{0} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} u_{0 mn} cos α x sin β y \\ u_{1} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} u_{1 mn} cos α x sin β y \\ v_{0} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} v_{0 mn} sin α x cos β y \\ v_{1} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} v_{1 mn} sin α x cos β y \\ w_{0} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} w_{0 mn} sin α x sin β y \end{matrix}

(21)

where

α = m π / a

;

β = n π / b

; a and b respectively represent the length and the width of FGM plate.

If equation (21) is substituted into equilibrium equations (16), the generalized displacement parameters can be obtained by collecting coefficients.

Results and discussion

In this section, numerical examples are presented and discussed to verify the performance of the proposed model by predicting the thermoelastic response of functionally graded plates. The FGM plates are taken to be made of Aluminum and Alumina, with the following material properties⁴²

\begin{matrix} Metal (Aluminum) : E_{M} = 70 GPa, \\ ν_{M} = 0.3, α_{M} = 23.0 \times 10^{- 6} / K; \\ Ceramic (Alumina) : E_{C} = 380 GPa, \\ v_{C} = 0.3, α_{C} = 7.4 \times 10^{- 6} / K \end{matrix}

The effective Young's modulus E(z) and thermal expansion coefficient α_z (z) in the thickness direction of FGM plate can be computed using equation (10).

Example 1

Thermal bending analysis of an FGM square (a = b) plate subjected to thermal loading $T (x, y, z) = T_{1} (2 z / h) sin (π x / a) sin (π y / b)$ .

Transverse displacement is expressed in terms of the following dimensionless parameter

\hat{w} = \frac{w (a / 2, b / 2, 0)}{h α_{c} T_{1}}, α_{c} = 7.4 \times 10^{- 6} / K

Table 1 presents transverse displacements computed from the proposed model SPTC and the higher order model HSDT-98 reported in literature.⁴² It needs to be shown that the results obtained from the model HSDT-98 are recalculated in this paper. In Table 1, it is found that transverse displacements obtained from the model HSDT-98 are in excellent agreement with those (HSDT-98*) obtained in Matsunaga⁴² using a similar analysis. Performance of the model HSDT-98 has been verified in Matsunaga,⁴² so that the model HSDT-98 will be employed to assess the performance of the proposed model SPTC.

Example 2

Thermoelastic analysis of FGM plate under thermomechanical loading q = q₀sin(πx/a)sin(πy/b) and $T (x, y, z) = {\bar{T}}_{0} (x, y) + (z / h) {\bar{T}}_{1} (x, y) + (f_{z} / h) {\bar{T}}_{2} (x, y)$ , in which $[{\bar{T}}_{0} (x, y), {\bar{T}}_{1} (x, y), {\bar{T}}_{2} (x, y)] = (T_{0}, T_{1}, T_{2}) sin (π x / a) sin (π y / b)$ .

Table 1.

Transverse displacement $\hat{w}$ of the FGM plate subjected to thermal loading (q₀ = 0, T₁ = 100, T₀ = T₂ = 0).

a/h	Models	k = 0	k = 0.5	k = 1
5	SPTC	3.2929	5.3946	5.9894
	HSDT-98	3.2270	5.2855	5.8415
	HSDT-98*⁴²	3.227	5.286	5.842
10	SPTC	13.1717	21.5893	24.0086
	HSDT-98	13.1065	21.4803	23.8602
	HSDT-98*⁴²	13.11	21.48	23.86

FGM: functionally graded material. *denotes the results obtained by Matsunaga.⁴²

The dimensionless parameters of displacements and stresses are expressed as follows

\begin{matrix} \hat{u} = u (0, b / 2, z) / h α_{c} \tilde{T} a, \hat{w} = w (a / 2, b / 2, z) / h α_{c} \tilde{T}, \\ {\hat{σ}}_{x} = σ_{x} (a / 2, b / 2, z) / E_{c} α_{c} \tilde{T}, {\hat{τ}}_{xy} = τ_{xy} (0, 0, z) / E_{c} α_{c} \tilde{T}, \\ \tilde{T} = T_{0} or \tilde{T} = T_{1} / 2; \end{matrix}

\begin{matrix} \bar{u} = u (a, b / 2, z) / h α_{0} T_{0} a, \bar{w} = w (a / 2, b / 2, z) / h α_{0} T_{0}, \\ {\bar{σ}}_{x} = σ_{x} (a / 2, b / 2, z) / E_{c} α_{0} T_{0}, {\bar{τ}}_{xy} = τ_{xy} (0, 0, z) / E_{c} α_{0} T_{0}, \\ {\bar{τ}}_{xz} = τ_{xz} (0, b / 2, z) / E_{c} α_{0} T_{0}, α_{0} = 10^{- 6} /^{\circ} C \end{matrix}

In order to study the effect of transverse normal deformation on displacement and stresses, thermal bending behaviors of an isotropic plate are first analyzed (q₀ = 0, T₀ = 0, T₁ = 100, T₂ = 0, a/h = 5). Distribution of displacements and stresses through the thickness are shown in Figure 2. It is seen that results of the present model SPTC agree well with those obtained from the model HSDT-98. By adding the function (h/π)cos(πz/h)w₁(x,y) to transverse displacement field of the model SPT,²⁵ Sayyad and Ghugal³⁹ proposed a new sinusoidal model considering transverse normal deformation (SPTWCOS). Although transverse normal deformation is considered, the model SPTWCOS overestimates the in-plane normal stress. By adding the function 0.25(cosh(z/h)-4z²cosh(0.5)/h²)w₁(x,y) to transverse displacement field of the model SPT, Zenkour⁴⁰ presented a shear and normal deformation model (SPTWCOSH). Nevertheless, the model SPTWCOSH is even less accurate in comparison with the model SPT neglecting transverse normal deformation. In addition, distributions of displacements and stresses through the thickness of an isotropic plate under thermal load (q₀ = 0, T₀ = 100, T₁ = 0, T₂ = 0, a/h = 5) are presented in Figure 3. It is found that the models SPTWCOS and SPTWCOSH are still unable to predict accurately displacements and stresses for thermal expansion problems. Moreover, the maximum percentage errors of displacements and stresses computed from the models SPTWCOS and SPTWCOSH relative to those obtained from the model HSDT-98 are more than 30%, as the unbefitting transverse displacement field is used. In order to verify the viewpoint of authors, a sinusoidal model SPTWZ is presented for comparison by adding the function zw₁(x,y) to transverse displacement field of the model SPT. Numerical results show that the model SPTWZ can only calculate accurately transverse displacement. The proposed model SPTC can compute accurately displacements and stresses of plates subjected to different thermal loadings, so the model SPTC will be employed to study thermoelastic behaviors of FGM plates.

Figure 2.

Comparison of displacements and stresses through thickness of a plate subjected to thermal loading (q₀ = 0, T₀ = 0, T₁ = 100, T₂ = 0, a/h = 5).

Figure 3.

Comparison of displacements and stresses through thickness of a plate subjected to thermal loading (q₀ = 0, T₀ = 100, T₁ = 0, T₂ = 0, a/h = 5).

Figure 4 depicts the effect of volume fraction exponent k on the displacements and stresses of the FGM plate subjected to thermomechanical load (q₀ = 100, T₀ = 100, T₁ = T₂ = 0, a/h = 2.5). It is found that the FGM plates are very sensitive to the change of volume fraction exponent k. Numerical results show that it is evident that rise in volume fraction k tends to elevate the maximum absolute values of all results. Moreover, the results are compared to those computed from the models SPT²⁵ and Reddy.¹¹ It is observed that the distribution of in-plane displacements obtained from the models SPT and Reddy is linear through the thickness of plate for the different k. Nevertheless, in-plane displacements obtained from the proposed model SPTC nonlinearly distribute across the thickness of the FGM plate. In addition, results obtained from the model SPTC are in good agreement with those computed from the model HSDT-98. Figure 5 shows the distributions of in-plane displacement $\bar{u} (a, y, z)$ at the section A of FGM plate (q₀ = 100, T₀ = 100, T₁ = T₂ = 0, a/h = 2.5). It can be easily observed that profiles of in-plane displacement distribution obtained from the proposed model SPTC completely differ from those computed from the model SPT. It is because of the fact that transverse normal strain has been neglected in the model SPT, so that such model is not suitable for three-dimensional thermoelastic analysis of FGM plates. Subsequently, distributions of in-plane stresses ${\bar{σ}}_{x} (a / 2, y, z)$ at the section B of FGM plate (q₀ = 100, T₀ = 100, T₁ = T₂ = 0, a/h = 2.5) are shown in Figure 6. It is again found that profiles of in-plane stress distribution obtained from the proposed model SPTC are different from those computed from the model SPT.

Figure 4.

Comparison of displacements and stresses through thickness of an FGM plate subjected to thermomechanical loading (q₀ = 100, T₀ = 100, T₁ = T₂ = 0, a/h = 2.5). FGM: functionally graded material.

Figure 5.

Distribution of in-plane displacement $\bar{u} (a, y, z)$ at the section A of FGM plate (q₀ = 100, T₀ = 100, T₁ = T₂ = 0, a/h = 2.5). FGM: functionally graded material.

Figure 6.

Distribution of in-plane stresses ${\bar{σ}}_{x} (a / 2, y, z)$ at the section B of FGM plate (q₀ = 100, T₀ = 100, T₁ = T₂ = 0, a/h = 2.5). FGM: functionally graded material.

In Figure 7, the effects of thermomechanical loading on the displacements and the stresses of the FGM plates (a/h = 4) are investigated in detail. For thermoelastic expansion of FGM plate (q₀ = 100, T₀ = 100, T₁ = T₂ = 0), the model SPT is less accurate in producing reliable displacements and stresses. Even if integration of three-dimensional equilibrium equations is adopted, transverse shear stress obtained from the model SPT is close to zero, which completely differs from those computed from the models SPTC and HSDT-98. It needs to be shown that there are 29 displacement parameters in the model HSDT-98. However, only five displacement parameters are included in the proposed model SPTC, so the proposed model SPTC is accurate and efficient. In Figures 8–10, effects of the volume fraction exponent k on in-plane displacement and in-plane stresses of the FGM plate (q₀ = 100, T₀ = T₁ = T₂ = 100, a/h = 4) are investigated in detail. It can be found that the volume fraction exponent k has a significant impact on the dimensionless in-plane displacement $\bar{u}$ and in-plane stress ${\bar{σ}}_{x}$ . In addition, effects of span-to-thickness ratio on the dimensionless in-plane displacement and in-plane stresses are also shown in Figures 11–13.

Figure 7.

Comparison of displacements and stresses through thickness of an FGM plate subjected to various loadings (k = 1, a/h = 4). FGM: functionally graded material.

Figure 8.

Effect of the volume fraction exponent k on in-plane displacement $\bar{u}$ (q₀=100, T₀ = T₁ = T₂ = 100, a/h = 4).

Figure 9.

Effect of the volume fraction exponent k on in-plane stress ${\bar{σ}}_{x}$ (q₀ = 100, T₀ = T₁ = T₂ = 100, a/h = 4).

Figure 10.

Effect of the volume fraction exponent k on in-plane stress ${\bar{τ}}_{xy}$ (q₀ = 100, T₀ = T₁ = T₂ = 100, a/h = 4).

Figure 11.

Effect of span to thickness ratio on in-plane displacement $\bar{u} (0, b / 2, h / 2)$ (q₀ = 100, T₀ = T₁ = T₂ = 100, a/h = 4).

Figure 12.

Effect of span to thickness ratio on in-plane stress ${\bar{σ}}_{x} (a / 2, b / 2, h / 2)$ (q₀ = 100, T₀ = T₁ = T₂ = 100, a/h = 4).

Figure 13.

Effect of span to thickness ratio on in-plane stress ${\bar{τ}}_{xy} (0, 0, h / 2)$ (q₀ = 100, T₀ = T₁ = T₂ = 100, a/h = 4).

Conclusions

Although transverse normal strain is taken into account, the existing sinusoidal models are still less accurate for thermoelastic analysis of isotropic and FGM plates. In view of this situation, a refined sinusoidal model for FGM plates considering transverse normal strain is presented. The in-plane displacements are expanded as a combination of sinusoidal functions and displacement components due to thermal loading. Displacement field of the proposed model contains five displacement parameters, while the present model can account for adequate distribution of transverse shear stains through the thickness of plate. Moreover, tangential stress-free boundary conditions at the surfaces can be satisfied. The performance of the proposed model is ascertained by studying the thermoelastic response of FGM plates. Numerical results show that under thermomechanical loading, through-the-thickness deformations in thick FGM plates are very significant. The proposed model SPTC performs better than the existing sinusoidal models for analyzing thermoelastic behaviors of FGM plates. Moreover, it is important that although transverse normal strain has been taken into account, displacement parameters in the proposed model SPTC are the same as those in the model SPT as transverse normal deformations due to thermal loading have been absorbed in the generalized force vector.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The National Natural Sciences Foundation of China [No. 11402152, 11272217, 11572204].

References

Yang

Shen

. Non-linear analysis of functionally graded plates under transverse and in-plane loads. Int J Nonlinear Mech 2003; 38: 467–482.

Satouri

Kargarnovin

Allahkarami

et al.

Application of third order deformation theory in buckling analysis of 2D-functionally graded cylindrical shell reinforced by axial stiffeners. Compos B 2015; 79: 236–253.

Dai

Liu

Lim

et al.

A meshfree radial point interpolation method for analysis of functionally graded material (FGM) plates. Comput Mech 2004; 34: 213–223.

Bohlooly

Mirzavand

. Thermomechanical buckling of hybrid cross-ply laminated rectangular plates. Adv Compos Mater 2017; 26: 407–426.

Han

Kim

Cho . New enhanced first-order shear deformation theory for thermomechanical analysis of laminated composite and sandwich plates. Compos B 2017; 116: 422–450.

Nguyen-Xuan

Tran

Nguyen-Thoi

et al.

Analysis of functionally graded plates using an edge-based smoothed finite element method. Compos Struct 2011; 93: 3019–3039.

Zhuang

Huang

Zhu

et al.

A new and simple locking free triangular thick plate element using independent shear degrees of freedom. Finite Elem Anal Des 2013; 75: 1–7.

Thai

Choi

. A simple first-order shear deformation theory for the bending and free vibration analysis of functionally graded plates. Compos Struct 2013; 101: 332–340.

Lee

Kim

. Aero-hygrothermal effects on stability regions for functionally graded panels. Compos Struct 2015; 133: 257–264.

10.

Matsunaga

. Assessment of a global higher-order deformation theory for laminated composite and sandwich plates. Compos Struct 2002; 56: 279–291.

11.

Reddy

. A simple higher-order theory for laminated composite plates. J Appl Mech 1984; 51: 745–752.

12.

Reddy

. Analysis of functionally graded plates. Int J Numer Meth Eng 2000; 47: 663–684.

13.

Cheng

Batra

. Exact correspondence between eigenvalues of membranes and functionally graded simply supported polygonal plates. J Sound Vib 2000; 229: 879–895.

14.

Oktem

Mantari

Soares

. Static response of functionally graded plates and doubly-curved shells based on a higher order shear deformation theory. Eur J Mech A/Sol 2012; 36: 163–172.

15.

Taj

MNAG

Chakrabarti

Sheikh

. Analysis of functionally graded plates using higher-order shear deformation theory. Appl Math Modell 2013; 37: 8484–8494.

16.

Mantari

Oktem

Soares

. A new higher-order shear deformation theory for sandwich and composite laminated plates. Compos B 2012; 43: 1489–1499.

17.

Mantari

Oktem

Soares

. Bending response of functionally graded plates by using a new higher-order shear deformation theory. Compos Struct 2012; 94: 714–723.

18.

Thai

Kulasegaram

Tran

et al.

Generalized shear deformation theory for functionally graded isotropic and sandwich plates based on isogeometric approach. Comput Struct 2014; 141: 94–112.

19.

Karama

Afaq

Mistou

. Mechanical behavior of laminated composite beam by new multi-layered laminated composite structures model with transverse shear stress continuity. Int J Solids Struct 2003; 40: 1525–1546.

20.

Arya

Shimpi

Naik

. A zigzag model for laminated composite beams. Compos Struct 2002; 56: 21–24.

21.

Thai

Ferreira

AJM

Rabczuk

et al.

Isogeometric analysis of laminated composite and sandwich plates using a new inverse trigonometric shear deformation theory. Eur J Mech A/Sol 2014; 43: 89–108.

22.

Mahi

Bedia

EAA

Tounsi

. A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates. Appl Math Modell 2015; 39: 2489–2508.

23.

Pradhan

Chakraverty

. Generalized power-law exponent based shear deformation theory for free vibration of functionally graded beams. Appl Math Computat 2015; 268: 1240–1258.

24.

Kulkarni

Singh

Maiti

. Analytical solution for bending and buckling analysis of functionally graded plates using inverse trigonometric shear deformation theory. Compos Struct 2015; 134: 147–157.

25.

Touratier

. An efficient standard plate theory. Int J Eng Sci 1991; 29: 601–916.

26.

Touratier

. A refined theory of laminated shallow shells. Int J Solids Struct 1992; 29: 1401–1415.

27.

Zenkour

Alghamdi

. Bending analysis of functionally graded sandwich plates under the effect of mechanical and thermal loads. Mech Adv Mater Struct 2010; 17: 419–432.

28.

Zenkour

Alghamdi

. Thermomechanical bending response of functionally graded nonsymmetric sandwich plates. J Sandwich Struct Mater 2010; 12: 7–46.

29.

Zenkour

. Hygro-thermo-mechanical effects on FGM plates resting on elastic foundations. Compos Struct 2010; 93: 234–238.

30.

Zenkour

Allam

MNM

Sobby

. Bending analysis of FG viscoelastic sandwich beams with elastic cores resting on Pasternak's elastic foundations. Acta Mech 2010; 212: 233–252.

31.

Zenkour

Sobby

. Dynamic bending response of thermoelastic functionally graded plates resting on elastic foundations. Aerosp Sci Technol 2013; 29: 7–17.

32.

Zenkour

. Hygrothermal effects on the bending of angle-ply composite plates using a sinusoidal theory. Compos Struct 2012; 94: 3685–3696.

33.

Zenkour

Allam

MNM

Radwan

. Bending of cross-ply laminated plates resting on elastic foundations under thermo-mechanical loading. Int J Mech Mater Des 2013; 9: 239–251.

34.

Zenkour

Allam

MNM

Radwan

. Effects of hygrothermal conditions on cross-ply laminated plates resting on elastic foundations. Arch Civil Mech Eng 2014; 14: 144–159.

35.

Zenkour

. Simplified theory for hygrothermal response of angle-ply composite plates. AIAA J 2014; 52: 1466–1473.

36.

Zenkour

Mashat

Alghanmi

. Hygrothermal analysis of antisymmetric cross-ply laminates using a refined plate theory. Int J Mech Mater Des 2014; 10: 213–226.

37.

Gupta

Talha

. Recent development in modeling and analysis of functionally graded materials and structures. Progr Aerosp Sci 2015; 79: 1–15.

38.

Cho

. Higher order zig-zag theory for fully coupled thermo-electric-mechanical smart composite plates. Int J Solids Struct 2004; 41: 1331–1356.

39.

Sayyad

Ghugal

. A new shear and normal deformation theory for isotropic, transversely isotropic, laminated composite and sandwich plates. Int J Mech Mater Des 2014; 10: 247–267.

40.

Zenkour

. Thermal bending of layered composite plates resting on elastic foundations using four-unknown shear and normal deformations theory. Compos Struct 2015; 122: 260–270.

41.

Kant

Swaminathan

. Analytical solutions for the static analysis of laminated composite and sandwich plates based on a higher order refined theory. Compos Struct 2002; 56: 329–344.

42.

Matsunaga

. Stress analysis of functionally graded plates subjected to thermal and mechanical loadings. Compos Struct 2009; 87: 344–357. .

43.

Zenkour

. Generalized shear deformation theory for bending analysis of functionally graded plates. Appl Math Modell 2006; 30: 67–84.