Free vibration analyses of functionally graded plates based on improved refined shear and normal deformation theory considering thickness stretching effect using Airy stress function

Abstract

In this paper, an improved refined shear and normal deformation theory is used in order to investigate the vibration behavior of functionally graded rectangular plates. In this theory, displacements of various points of plate are assumed to be due to in-plane displacements of the middle plane and transverse displacement. Transverse displacement is divided into three parts: bending, shear, and thickness stretching. Using the Airy stress function, corresponding to the compatibility equation, and employing the extended Hamilton’s principle, in-plane displacements are omitted from the equations of motions. Thus, the proposed approach uses only three-unknowns in the displacement field. The results of vibration analysis using the proposed approach are in excellent agreement with three-dimensional and quasi-three-dimensional solutions containing a greater number of unknowns to consider the thickness stretching effect. Static and dynamic behavior of wide variety of thin and thick functionally graded plates can be studied using the proposed approach in which not only the number of variables is reduced, but also the contribution of bending, shear, and thickness stretching are completely clarified.

Keywords

improved refined shear and normal deformation theory thickness stretching airy stress function linear vibration functionally graded plates

1. Introduction

In uniform composite materials, delamination failure at high temperatures is observed. Functionally graded (FG) materials as a class of composites have continuous properties from one point to another, leading to lower thermal and residual stresses compared to the conventional homogenous and composite materials. Since FG materials are high thermal, wear, and corrosion resistant, they are widely used in many engineering applications, such as mechanical, aerospace, fire retardants, nuclear, civil, biomaterials, and automotive industries.

Various theories can be used to analyze FG plates. In classical plate theory (CPT), both transverse shear and normal strain effects are ignored. Mindlin (1951) developed the first order shear deformation theory (FSDT) accounting for the effect of shear deformation based on a linear variation of in-plane displacements along the thickness direction. Unlike FSDT, higher-order shear deformation theories (HSDTs) do not need shear correction factor, while they lead to more accurate shear stress distribution results. Higher-order variations of in-plane displacements or both in-plane and transverse displacements (i.e., quasi-three-dimensional (quasi-3D) theories) along the thickness direction are considered in them as well.

Based on the first order and third-order shear deformation theory (TSDT), Ferreira et al. (2006) investigated free vibration behavior of FG plates. Shahrjerdi et al. (2011) studied natural frequencies of FG rectangular plates using a second order shear deformation theory (SSDT). Yaghoobi and Taheri (2021) investigated free vibration, buckling, and static responses of sandwich plates using the refined shear deformation theory.

In order to study hygrothermal environment effects, Hellal et al. (2019) used a novel four-unknown HSDT considering a trigonometric variation of transverse shear stress. The effects of temperature rise, moisture condition, elastic foundation coefficients, and power law index on vibration and buckling of FG sandwich plates supported by elastic foundations were investigated. In other study, Mudhaffar et al. (2021) used an integral HSDT with a sine shape function for the displacement field to study hygrothermal and hygromechanical bending behavior of FG plates.

The idea of partitioning the transverse displacement into bending and shear components was first proposed by Huffington (1963) and subsequently adopted by Murty (1987) and Senthilnathan et al. (1987). Using this concept, the number of unknowns is reduced and furthermore the contributions due to shear and bending to the total displacement are clarified. By implementing this method, Shimpi (2002) developed a refined plate theory (RPT) for isotropic plates.

Based on RPT, Thai and Choi (2011) investigated bending, buckling, and free vibration behaviors of FG plates resting on elastic foundations using Navier and Levy approaches. Using various HSDTs neglecting the thickness stretching effect, Thai and Choi (2013) studied bending and free vibration behaviors of FG plates as well.

Vu et al. (2018) studied static deflection, free vibration, and buckling of FG plates based on a refined third-order shear deformation theory using a meshfree method. Rouzegar et al. (2020) investigated dynamic behavior of composite laminates integrated with a piezoelectric fiber-reinforced composite actuator layer using a four variable refined theory.

It should be noted that the above-mentioned studies neglected the thickness stretching effect (i.e., $ε_{33} = 0$ ) by assuming a constant transverse displacement throughout the thickness. The effect of thickness stretching in FG plates which is significant in thick ones was studied by Carrera et al. (2011). It was concluded that increasing the order of expansion of the in-plane displacements is meaningless unless transverse normal strain effects are also considered.

Vel and Batra (2004) studied vibration of rectangular FG plates based on an exact (3D) method. Kant et al. (2014) analyzed static and vibration behaviors of FG elastic plates using a higher-order shear and normal deformation plate theory (HOSNT). Jha et al. (2013) investigated free vibration behavior of thick FG elastic plates using various higher-order shear/shear-normal deformation theories (HOSTs/HOSNTs).

Based on a quasi-3D hyperbolic sine shear deformation theory and using the radial basis function method, Neves et al. (2012) investigated static and free vibration behaviors of FG plates. In another work, Neves et al. (2013) studied the static, free vibration, and buckling behaviors of FG plates using a quasi-3D higher-order plate theory.

Using a quasi-3D hyperbolic shear deformation theory, Thai et al. (2014) studied bending and free vibration of FG plates. Mashat et al. (2020) studied bending of FG porous plates laying on elastic foundations based on a quasi-3D higher-order plate theory including five number of unknowns. Based on a quasi-3D trigonometric plate theory, Abualnour et al. (2018) studied free vibration behaviors of advanced composite plates. It was concluded that considering the thickness stretching in thick and even moderately thick plates is vital.

To account for the thickness stretching effect, Thai and Choi (2014) proposed an improved refined plate theory (IRPT), to study static and linear vibration behaviors of FG plates. Bennoun et al. (2016) studied free vibration behavior of FG sandwich plates considering five variables in the RPT to include the thickness stretching effect.

Based on a hybrid quasi-3D theory, Van Vinh (2021) studied deflections, stresses, and free vibration of FG sandwich plates considering five variables. Shahsavari et al. (2018) studied free vibration of FG porous plates resting on Winkler/Pasternak/Kerr foundation based on a quasi-3D hyperbolic theory using a Galerkin method. Other kinds of HSDTs were also proposed to study different parameters of FG plates under various conditions (Bouafia et al., 2021; Mellal et al., 2021).

In this paper, a method based on the improved refined shear and normal deformation theory (IRSNDT) using Airy stress function is proposed. Since this approach excludes the in-plane variables, it contains only three state variables, considering transverse bending deformation, transverse shear deformation and the thickness stretching effect which is more pronounced in studying thick plate behaviors. The accuracy and effectiveness of the current model is shown by comparing the results with those of exact 3D and quasi-3D solutions containing a greater number of unknowns to express the displacement field.

Moreover, the contribution of bending, shear, and thickness stretching parameters to the total displacement in thin and thick FG plates are studied. Thus, using Airy stress function leads to reduction of equations of motions. Therefore, it is an effective instrument for studying dynamical behavior of FG plates.

2. Improved refined shear and normal deformation theory approach

The geometry of a rectangular FG plate is shown in Figure 1. Top and bottom surfaces of the plate are assumed to be ceramic and metal rich, respectively. $x_{1}$ and $x_{2}$ are the in-plane coordinates and $z$ is the out of plane coordinate on the thickness direction, with zero being in the middle of the plate. H is the full thickness of the plate, while $l_{1}$ and $l_{2}$ are the plate lengths in $x_{1}$ and $x_{2}$ directions, respectively.

Figure 1.

Rectangular FG plate and coordinate system.

Considering the displacement field used by Senthilnathan et al. (1987) we have

\begin{array}{l} u_{1} (x_{1}, x_{2}, z, t) = u_{0} (x_{1}, x_{2}, t) - z \frac{\partial w_{b}}{\partial x_{1}} - \frac{4 z^{3}}{3 H^{2}} \frac{\partial w_{s}}{\partial x_{1}} \\ u_{2} (x_{1}, x_{2}, z, t) = v_{0} (x_{1}, x_{2}, t) - z \frac{\partial w_{b}}{\partial x_{2}} - \frac{4 z^{3}}{3 H^{2}} \frac{\partial w_{s}}{\partial x_{2}} \\ u_{3} (x_{1}, x_{2}, z, t) = w_{b} (x_{1}, x_{2}, t) + w_{s} (x_{1}, x_{2}, t) \end{array}

(1)

$u_{1}$ , $u_{2}$ , and $u_{3}$ are the displacements in $x_{1}$ , $x_{2}$ , and $z$ directions. $t$ is the time variable. $u_{0}$ , $v_{0}$ are the in-plane displacements of the middle plane in $x_{1}$ and $x_{2}$ directions. The bending components $w_{b}$ , defined by the second terms in $u_{1}$ and $u_{2}$ are used as in the CPT, while the third term containing $w_{s}$ , are due to the shearing effect. $u_{3}$ has bending and shear components to show the participation of each in the total transverse displacement.

To consider the thickness stretching effect $w_{z}$ , a new displacement field can be proposed (IRSNDT)

\begin{array}{l} u_{1} (x_{1}, x_{2}, z, t) = u_{0} (x_{1}, x_{2}, t) + g_{u} (z) \frac{\partial w_{b}}{\partial x_{1}} + f_{u} (z) \frac{\partial w_{s}}{\partial x_{1}} \\ u_{2} (x_{1}, x_{2}, z, t) = v_{0} (x_{1}, x_{2}, t) + g_{v} (z) \frac{\partial w_{b}}{\partial x_{2}} + f_{v} (z) \frac{\partial w_{s}}{\partial x_{2}} \\ u_{3} (x_{1}, x_{2}, z, t) = w_{b} (x_{1}, x_{2}, t) + s (z) w_{s} (x_{1}, x_{2}, t) + h (z) w_{z} (x_{1}, x_{2}, t) \end{array}

(2)

Functions $g_{u} (z), f_{u} (z), g_{v} (z), f_{v} (z), s (z) and h (z)$ should satisfy zero transverse shear stresses on free surfaces presented by

σ_{13} | z = \pm \frac{H}{2} = 0, σ_{23} | z = \pm \frac{H}{2} = 0

(3)

It should also be mentioned that omitting the thickness stretching effect $w_{z}$ and the transverse shear deformation $w_{s}$ in the displacement field equation (2), leads to theories, namely, RPT and CPT, respectively.

3. Material properties

Based on a simple power law distribution, functionally graded material (FGM) properties vary continuously along the thickness direction in the following form

P_{e f f} (z) = P_{b} + (P_{t} - P_{b}) {(\frac{1}{2} + \frac{z}{H})}^{p} - \frac{H}{2} \leq z \leq \frac{H}{2} 0 \leq p \leq \infty

(4)

where p is the power law index demonstrating the profile of material changing along the thickness direction.

P_{e f f}

is the effective material property, and

P_{t}

and

P_{b}

are properties of the top and bottom surfaces of the plate, respectively.

4. Strain-displacement relations

Linear strains according to the displacement field equation (2) are

\begin{array}{l} ε_{11} = \frac{\partial u_{0}}{\partial x_{1}} + g_{u} (z) \frac{\partial^{2} w_{b}}{\partial x_{1}^{2}} + f_{u} (z) \frac{\partial^{2} w_{s}}{\partial x_{1}^{2}} \\ ε_{22} = \frac{\partial v_{0}}{\partial x_{2}} + g_{v} (z) \frac{\partial^{2} w_{b}}{\partial x_{2}^{2}} + f_{v} (z) \frac{\partial^{2} w_{s}}{\partial x_{2}^{2}} \\ ε_{33} = \frac{d s (z)}{d z} w_{s} + \frac{d h (z)}{d z} w_{z} \\ γ_{12} = \frac{\partial u_{0}}{\partial x_{2}} + \frac{\partial v_{0}}{\partial x_{1}} + (g_{u} (z) + g_{v} (z)) \frac{\partial^{2} w_{b}}{\partial x_{1} \partial x_{2}} + (f_{u} (z) + f_{v} (z)) \frac{\partial^{2} w_{s}}{\partial x_{1} \partial x_{2}} \\ γ_{13} = (1 + \frac{d g_{u} (z)}{d z}) \frac{\partial w_{b}}{\partial x_{1}} + (s (z) + \frac{d f_{u} (z)}{d z}) \frac{\partial w_{s}}{\partial x_{1}} + h (z) \frac{\partial w_{z}}{\partial x_{1}} \\ γ_{23} = (1 + \frac{d g_{v} (z)}{d z}) \frac{\partial w_{b}}{\partial x_{2}} + (s (z) + \frac{d f_{v} (z)}{d z}) \frac{\partial w_{s}}{\partial x_{2}} + h (z) \frac{\partial w_{z}}{\partial x_{2}} \end{array}

(5)

5. Constitutive equations

Three-dimensional stress–strain relations can be presented as

{\begin{matrix} \begin{matrix} \begin{matrix} σ_{11} \\ σ_{22} \\ \begin{matrix} σ_{33} \\ σ_{12} \end{matrix} \end{matrix} \\ σ_{13} \end{matrix} \\ σ_{23} \end{matrix}} = [\begin{matrix} \begin{matrix} \begin{matrix} C_{11} & C_{12} & C_{13} \\ C_{21} & C_{22} & C_{23} \\ C_{31} & C_{32} & C_{33} \end{matrix} \\ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} \\ \begin{matrix} C_{44} & 0 & 0 \\ 0 & C_{55} & 0 \\ 0 & 0 & C_{66} \end{matrix} \end{matrix} \end{matrix}] {\begin{matrix} \begin{matrix} \begin{matrix} ε_{11} \\ ε_{22} \\ \begin{matrix} ε_{33} \\ γ_{12} \end{matrix} \end{matrix} \\ γ_{13} \end{matrix} \\ γ_{23} \end{matrix}}

(6)

For elastic isotropic FGMs, the elastic constants $(C_{i j})$ are given by

\begin{array}{l} C_{11} (z) = C_{22} (z) = C_{33} (z) = \frac{(1 - ν) E (z)}{(1 - 2 ν) (1 + ν)} \\ C_{12} (z) = C_{21} (z) = C_{13} (z) = C_{31} (z) = C_{23} (z) = C_{32} (z) = \frac{ν E (z)}{(1 - 2 ν) (1 + ν)} \\ C_{44} (z) = C_{55} (z) = C_{66} (z) = \frac{E (z)}{2 (1 + ν)} \end{array}

(7)

where

E (z)

and

ν

are Young’s modulus and Poison ratio. Since the value of Poison ratios of ceramic and metal are very close, variation of Poison ratio is not included for FG materials.

If the thickness stretching effect is neglected $(ε_{33} = 0)$ , the constitutive relations can be rewritten as

{\begin{matrix} \begin{matrix} \begin{matrix} σ_{11} \\ σ_{22} \\ σ_{12} \end{matrix} \\ σ_{13} \end{matrix} \\ σ_{23} \end{matrix}} = \frac{E (z)}{(1 - ν^{2})} [\begin{matrix} 1 & ν & 0 & 0 & 0 \\ ν & 1 & 0 & 0 & 0 \\ 0 & 0 & \frac{1 - ν}{2} & 0 & 0 \\ 0 & 0 & 0 & \frac{1 - ν}{2} & 0 \\ 0 & 0 & 0 & 0 & \frac{1 - ν}{2} \end{matrix}] {\begin{matrix} \begin{matrix} \begin{matrix} ε_{11} \\ ε_{22} \\ γ_{12} \end{matrix} \\ γ_{13} \end{matrix} \\ γ_{23} \end{matrix}}

(8)

6. Governing equations

To derive the governing equations of motion, the extended Hamilton’s principle can be used as

\int_{0}^{T} (δ U + δ W_{e} - δ K) d t = 0

(9)

where

δ U

δ W_{e}

, and

δ K

are the strain energy variation, virtual work done by the externally applied loads, and the kinetic energy variation of the plate, respectively.

Variation of strain energy of the plate, $δ U$ , can be expressed as

δ U = \int_{v} (σ_{11} δ ε_{11} + σ_{22} δ ε_{22} + σ_{33} δ ε_{33} + σ_{12} δ γ_{12} + σ_{13} δ γ_{13} + σ_{23} δ γ_{23}) d v

(10)

Virtual work done by the externally transverse distributed applied load, q, acting on the top surface of the plate $(z = H / 2)$ is

δ W_{e} = \int_{A} - q (δ w_{b} + s (\frac{H}{2}) δ w_{s} + h (\frac{H}{2}) δ w_{z}) d A

(11)

Kinetic energy of the plate is given by

\begin{array}{l} K = \frac{1}{2} \int_{A} \int_{- \frac{H}{2}}^{\frac{H}{2}} ρ (z) ({\dot{u}}_{1}^{2} + {\dot{u}}_{2}^{2} + {\dot{u}}_{3}^{2}) d A d z = \\ \frac{1}{2} \int_{A} [\begin{array}{l} (I_{0} {({\dot{u}}_{0})}^{2} + {({\dot{v}}_{0})}^{2} + {({\dot{w}}_{b})}^{2}) + 2 I_{1} ({\dot{u}}_{0} \frac{\partial {\dot{w}}_{b}}{\partial x_{1}}) + 2 I_{2} ({\dot{u}}_{0} \frac{\partial {\dot{w}}_{s}}{\partial x_{1}}) + 2 I_{3} ({\dot{v}}_{0} \frac{\partial {\dot{w}}_{b}}{\partial x_{2}}) + \\ 2 I_{4} ({\dot{v}}_{0} \frac{\partial {\dot{w}}_{s}}{\partial x_{2}}) + 2 I_{5} ({\dot{w}}_{b} {\dot{w}}_{s}) + 2 I_{6} ({\dot{w}}_{b} {\dot{w}}_{z}) + 2 J_{1} (\frac{\partial {\dot{w}}_{b}}{\partial x_{1}} \frac{\partial {\dot{w}}_{s}}{\partial x_{1}}) + 2 J_{2} (\frac{\partial {\dot{w}}_{b}}{\partial x_{2}} \frac{\partial {\dot{w}}_{s}}{\partial x_{2}}) + \\ 2 J_{3} ({\dot{w}}_{s} {\dot{w}}_{z}) + K_{1} {(\frac{\partial {\dot{w}}_{b}}{\partial x_{1}})}^{2} + K_{2} {(\frac{\partial {\dot{w}}_{s}}{\partial x_{1}})}^{2} + K_{3} {(\frac{\partial {\dot{w}}_{b}}{\partial x_{2}})}^{2} + K_{4} {(\frac{\partial {\dot{w}}_{s}}{\partial x_{2}})}^{2} + K_{5} {({\dot{w}}_{s})}^{2} + K_{6} {({\dot{w}}_{z})}^{2} \end{array}] d A \end{array}

(12)

where dot convention represents differentiation with respect to time and

ρ (z)

is the effective mass density of the FG plate. Various inertias of the plate,

I_{i}

J_{i}

, and

K_{i}

, are defined by

\begin{array}{l} (I_{0}, I_{1}, I_{2}, I_{3}, I_{4}, I_{5}, I_{6}) = \int_{\frac{- H}{2}}^{\frac{H}{2}} ρ (z) (1, g_{u} (z), f_{u} (z), g_{v} (z), f_{v} (z), s (z), h (z)) d z \\ (J_{1}, J_{2}, J_{3}) = \int_{\frac{- H}{2}}^{\frac{H}{2}} ρ (z) (g_{u} (z) f_{u} (z), g_{v} (z) f_{v} (z), s (z) h (z)) d z \\ (K_{1}, K_{2}, K_{3}, K_{4}, K_{5}, K_{6}) = \int_{\frac{- H}{2}}^{\frac{H}{2}} ρ (z) (g_{u}^{2} (z), f_{u}^{2} (z), g_{u}^{2} (z), f_{v}^{2} (z), s^{2} (z), h^{2} (z)) d z \end{array}

(13)

Substituting equations (5)–(7) into the extended Hamilton’s principle equation (4), using the divergence theorem and collecting the coefficients of each variable, the equations of motion are obtained as

δ u_{0} \to \frac{\partial N_{1}^{0}}{\partial x_{1}} + \frac{\partial N_{12}^{0}}{\partial x_{2}} = I_{0} {\dot{u}}_{0} + I_{1} \frac{\partial {\ddot{w}}_{b}}{\partial x_{1}} + I_{2} \frac{\partial {\ddot{w}}_{s}}{\partial x_{1}}

(14)

δ v_{0} \to \frac{\partial N_{12}^{0}}{\partial x_{1}} + \frac{\partial N_{2}^{0}}{\partial x_{2}} = I_{0} {\dot{v}}_{0} + I_{3} \frac{\partial {\ddot{w}}_{b}}{\partial x_{2}} + I_{4} \frac{\partial {\ddot{w}}_{s}}{\partial x_{2}}

(15)

\begin{array}{l} δ w_{b} \to \frac{\partial^{2} N_{1}^{g}}{\partial x_{1}^{2}} + \frac{\partial^{2} N_{2}^{g}}{\partial x_{2}^{2}} + \frac{\partial^{2} (N_{12}^{g} + N_{21}^{g})}{\partial x_{1} \partial x_{2}} - \frac{\partial}{\partial x_{1}} (Q_{1}^{00} + Q_{1}^{g}) - \frac{\partial}{\partial x_{2}} (Q_{2}^{00} + Q_{2}^{g}) - q \\ = I_{1} \frac{\partial {\ddot{u}}_{0}}{\partial x_{1}} + I_{3} \frac{\partial {\ddot{v}}_{0}}{\partial x_{2}} - I_{0} {\ddot{w}}_{b} + K_{1} \frac{\partial^{2} {\ddot{w}}_{b}}{\partial x_{1}^{2}} + K_{3} \frac{\partial^{2} {\ddot{w}}_{b}}{\partial x_{2}^{2}} - I_{5} {\ddot{w}}_{s} + J_{1} \frac{\partial^{2} {\ddot{w}}_{s}}{\partial x_{1}^{2}} + J_{2} \frac{\partial^{2} {\ddot{w}}_{s}}{\partial x_{2}^{2}} - I_{6} {\ddot{w}}_{z} \end{array}

(16)

\begin{array}{l} δ w_{s} \to \frac{\partial^{2} N_{1}^{f}}{\partial x_{1}^{2}} + \frac{\partial^{2} N_{2}^{f}}{\partial x_{2}^{2}} + \frac{\partial^{2} (N_{12}^{f} + N_{21}^{f})}{\partial x_{1} \partial x_{2}} + R^{s} - \frac{\partial}{\partial x_{1}} (Q_{1}^{f} + Q_{1}^{s 0}) - \frac{\partial}{\partial x_{2}} (Q_{2}^{f} + Q_{2}^{s 0}) - s (\frac{H}{2}) q \\ = I_{2} \frac{\partial {\ddot{u}}_{0}}{\partial x_{1}} + I_{4} \frac{\partial {\ddot{v}}_{0}}{\partial x_{2}} - I_{5} {\ddot{w}}_{b} + J_{1} \frac{\partial^{2} {\ddot{w}}_{b}}{\partial x_{1}^{2}} + J_{2} \frac{\partial^{2} {\ddot{w}}_{b}}{\partial x_{2}^{2}} - K_{5} {\ddot{w}}_{s} + K_{2} \frac{\partial^{2} {\ddot{w}}_{s}}{\partial x_{1}^{2}} + K_{4} \frac{\partial^{2} {\ddot{w}}_{s}}{\partial x_{2}^{2}} - J_{3} {\ddot{w}}_{z} \end{array}

(17)

δ w_{z} \to R^{h} - \frac{\partial}{\partial x_{1}} (Q_{1}^{h 0}) - \frac{\partial}{\partial x_{2}} (Q_{2}^{h 0}) - h (\frac{H}{2}) q = - I_{6} {\ddot{w}}_{b} - J_{3} {\ddot{w}}_{s} - K_{6} {\ddot{w}}_{z}

(18)

where stress resultants, are described in appendix A, equation (A-1).

7. Solution procedure

The compatibility equation of an FG plate can be expressed as

\frac{\partial^{2} ε_{11}}{\partial x_{2}^{2}} + \frac{\partial^{2} ε_{22}}{\partial x_{1}^{2}} - \frac{\partial^{2} γ_{12}}{\partial x_{1} \partial x_{2}} = 0

(19)

An Airy stress function $f$ , can be defined in such a way that the left sides of in-plane equations (9) and (10) become zero, meaning that

N_{1}^{0} = \frac{\partial^{2} f}{\partial x_{2}^{2}}, N_{2}^{0} = \frac{\partial^{2} f}{\partial x_{1}^{2}}, N_{12}^{0} = - \frac{\partial^{2} f}{\partial x_{1} \partial x_{2}}

(20)

Thus, the in-plane accelerations are obtained in terms of bending and shear components as

\begin{array}{l} {\ddot{u}}_{0} = - \frac{I_{1}}{I_{0}} \frac{\partial {\ddot{w}}_{b}}{\partial x_{1}} - \frac{I_{2}}{I_{0}} \frac{\partial {\ddot{w}}_{s}}{\partial x_{1}} \\ {\ddot{v}}_{0} = - \frac{I_{3}}{I_{0}} \frac{\partial {\ddot{w}}_{b}}{\partial x_{2}} - \frac{I_{4}}{I_{0}} \frac{\partial {\ddot{w}}_{s}}{\partial x_{2}} \end{array}

(21)

Therefore, the in-plane equations are omitted. Substituting equation (16) into equations 11–13, the in-plane accelerations are considered in those equations.

For movable simply supported boundary conditions, moments at the edges of the plate are zero and other boundary conditions are

\begin{array}{l} N_{1}^{0} = w_{b} = w_{s} = w_{z} = 0, a t x_{1} = 0, l_{1} \\ N_{2}^{0} = w_{b} = w_{s} = w_{z} = 0, a t x_{2} = 0, l_{2} \end{array}

(22)

The series solutions satisfying the aforementioned boundary conditions are

\begin{array}{l} w_{b} (x_{1}, x_{2}, t) = \sum_{m_{b} = 1}^{\infty} \sum_{n_{b} = 1}^{\infty} w b_{n_{b} m_{b}} (t) \sin (\frac{n_{b} π x_{1}}{l_{1}}) \sin (\frac{m_{b} π x_{2}}{l_{2}}) \\ w_{s} (x_{1}, x_{2}, t) = \sum_{m_{s} = 1}^{\infty} \sum_{n_{s} = 1}^{\infty} w s_{n_{s} m_{s}} (t) sin (\frac{n_{s} π x_{1}}{l_{1}}) sin (\frac{m_{s} π x_{2}}{l_{2}}) \\ w_{z} (x_{1}, x_{2}, t) = \sum_{m_{z} = 1}^{\infty} \sum_{n_{z} = 1}^{\infty} w z_{n_{z} m_{z}} (t) sin (\frac{n_{z} π x_{1}}{l_{1}}) sin (\frac{m_{z} π x_{2}}{l_{2}}) \end{array}

(23)

where

w b_{n_{b} m_{b}}

w s_{n_{s} m_{s}}

, and

w z_{n_{z} m_{z}}

are time dependent amplitudes.

By using definition of the stress function (15) and the expression of normal strains and in-plane shear strain in terms of $N_{1}^{0}$ , $N_{2}^{0}$ and $N_{12}^{0}$ defined in appendix A, equation (A-3-3), the compatibility equation (14) provides an equation in terms of the stress function $f$ , bending variable $w_{b}$ , shear variable $w_{s}$ , and thickness stretching effect $w_{z}$ . Using series solution of variables equation (18), the following form for the stress function can be obtained as

f (x_{1}, x_{2}, t) = \sum_{q^{s}} \sum_{p^{s}} f_{p q}^{s} (t) \sin (\frac{p^{s} π x_{1}}{l_{1}}) \sin (\frac{q^{s} π x_{2}}{l_{2}}) + \frac{1}{2} C_{x} x_{1}^{2} + \frac{1}{2} C_{y} x_{2}^{2}

(24)

where

f_{p q}^{s}

is a time dependent function in terms of variables defined in equation (18) as

\begin{array}{l} f_{p q}^{s} = - \frac{1}{e_{2,2} {(p^{s} π / l_{1})}^{4} + (e_{1,2} + e_{1} + e_{2,1}) {(p^{s} π / l_{1})}^{2} {(q^{s} π / l_{2})}^{2} + e_{1,1} {(q^{s} π / l_{2})}^{4}} \\ {\begin{cases} (e_{2,3} {(\frac{n_{b} π}{l_{1}})}^{4} + (e_{1,3} - e_{2} + e_{2,4}) {(\frac{n_{b} π}{l_{1}})}^{2} {(\frac{m_{b} π}{l_{2}})}^{2} + e_{1,4} {(\frac{m_{b} π}{l_{2}})}^{4}) w b_{n_{b} m_{b}} δ (n_{b} - p^{s}) δ (m_{b} - q^{s}) + \\ (e_{2,5} {(\frac{n_{s} π}{l_{1}})}^{4} + (e_{1,5} - e_{3} + e_{2,6}) {(\frac{n_{s} π}{l_{1}})}^{2} {(\frac{m_{s} π}{l_{2}})}^{2} + e_{1,6} {(\frac{m_{s} π}{l_{2}})}^{4} - e_{2,7} {(\frac{n_{s} π}{l_{1}})}^{2} - e_{1,7} {(\frac{m_{s} π}{l_{2}})}^{2}) \\ w s_{n_{s} m_{s}} δ (n_{s} - p^{s}) δ (m_{s} - q^{s}) - (e_{2,8} {(\frac{n_{z} π}{l_{1}})}^{2} + e_{1,8} {(\frac{m_{z} π}{l_{2}})}^{2}) w z_{n_{z} m_{z}} δ (n_{z} - p^{s}) δ (m_{z} - q^{s}) \end{cases}} \end{array}

(25)

where

e_{i, j}

and

e_{i}

are described in appendix A, equation (A-4-4).

C_{x}

and

C_{y}

in equation (19) are average membrane forces dependent to the boundary conditions of the plate. According to equation (17), we have the following relations

\begin{array}{l} N_{1}^{0} = 0, a t x_{1} = 0, l_{1} \\ N_{2}^{0} = 0, a t x_{2} = 0, l_{2} \end{array}

(26)

C_{x}

and

C_{y}

are obtained based on the average sense as

\begin{array}{l} C_{x} = \frac{1}{l_{1}} \int_{0}^{l_{1}} (N_{2}^{0}) d x_{1} = 0 \\ C_{y} = \frac{1}{l_{2}} \int_{0}^{l_{2}} (N_{1}^{0}) d x_{2} = 0 \end{array}

(27)

By using proper forms of functions in the displacement field equation (2), stress resultants defined in appendix A equation (A-1) are derived in terms of only $w b_{n_{b} m_{b}} (t)$ , $w s_{n_{s} m_{s}} (t)$ , and $w z_{n_{z} m_{z}} (t)$ . Thus, the system of equations of motion are obtained in the following matrix form

M {\begin{matrix} \ddot{w} b_{n_{b} m_{b}} \\ \begin{matrix} \ddot{w} s_{n_{s} m_{s}} \\ \ddot{w} z_{n_{z} m_{z}} \end{matrix} \end{matrix}} + K {\begin{matrix} w b_{n_{b} m_{b}} \\ w s_{n_{s} m_{s}} \\ w z_{n_{z} m_{z}} \end{matrix}} = F

(28)

where M, K, and F are the mass matrix, the stiffness matrix, and the force vector, respectively. Therefore, to investigate the behavior of thin and thick FG plates, one only needs to use three variables with arbitrary number of modes for each of them.

8. Validation

To show how well this approach works, three different FG material sets with the properties shown in Table 1 are used.

Table 1.

Material properties of FG plate constituents.

Material property	Material set-1		Material set-2		Material set-3
Material property	Al (metal layer)	ZrO₂ (ceramic layer)	Al (metal layer)	Al₂O₃ (ceramic layer)	Al (metal layer)	ZrO₂ (ceramic layer)
E (GPa)	70	200	70	380	68.9	211
$ρ$ (Kg/m³)	2702	5700	2702	3800	2700	4500
$ν$	0.3	0.3	0.3	0.3	0.33	0.33

Tables 2 and 3 compare dimensionless first natural frequencies of FG square plates with various p indices, found using the proposed approach using Airy stress function with those of Thai and Choi (2014) based on Navier solutions. The dimensionless form

\hat{ω} = ω H \sqrt{ρ_{b} / E_{b}}

is used to obtain the fundamental frequency where

ω

is the first natural frequency of the FG plate.

Table 2.

Dimensionless fundamental frequency $\hat{ω} = ω H \sqrt{ρ_{b} / E_{b}}$ of square Al/ZrO₂ FG plate, based on RPT.

Method	$ε_{33}$	No. of variables	$p = 0$		$p = 1$			$l_{1} / H = 5$
Method	$ε_{33}$	No. of variables	$l_{1} / H = \sqrt{10}$	$l_{1} / H = 10$	$l_{1} / H = 5$	$l_{1} / H = 10$	$l_{1} / H = 20$	$p = 2$	$p = 3$	$p = 5$
Thai and Choi (2014)	$= 0$	4	0.4623	0.0577	0.2169	0.0592	0.0152	0.2178	0.2193	0.2206
Proposed (RPT)	$= 0$	2	0.4623 (0%)	0.0577 (0%)	0.2169 (0%)	0.0592 (0%)	0.0152 (0%)	0.2178 (0%)	0.2193 (0%)	0.2206 (0%)

Note: RPT: refined plate theory.

Table 3.

Dimensionless fundamental frequency $\hat{ω} = ω H \sqrt{ρ_{b} / E_{b}}$ of square Al/ZrO₂ FG plate, based on IRPT.

Method	$ε_{33}$	No. of variables	$p = 0$		$p = 1$			$l_{1} / H = 5$
Method	$ε_{33}$	No. of variables	$l_{1} / H = \sqrt{10}$	$l_{1} / H = 10$	$l_{1} / H = 5$	$l_{1} / H = 10$	$l_{1} / H = 20$	$p = 2$	$p = 3$	$p = 5$
Thai and Choi (2014)	$\neq 0$	5	0.4661	0.0578	0.2192	0.0597	0.0153	0.2201	0.2214	0.2225
Proposed (IRPT)	$\neq 0$	3	0.4661 (0%)	0.0578 (0%)	0.2192 (0%)	0.0597 (0%)	0.0153 (0%)	0.2201 (0%)	0.2214 (0%)	0.2224 (0.045%)

Note: IRPT: improved refined plate theory.

Both RPT and IRPT are considered to obtain the results. Movable simply supported boundary conditions using Material Set-1 as shown in Table 1 are used for the FG plate. Numbers shown in parenthesis demonstrate the difference percentages of the results of the proposed approach in comparison with Thai and Choi (2014). To be able to compare the results, functions defined in the displacement field of equation (2) should be the same as those in the aforementioned reference. Thus, for IRPT, functions should be selected as

g_{u} (z) = - z, f_{u} (z) = \frac{z}{4} - \frac{5 z^{3}}{3 H^{2}}, g_{v} (z) = - z, f_{v} (z) = \frac{z}{4} - \frac{5 z^{3}}{3 H^{2}}, s (z) = 1, h (z) = \frac{5}{4} (1 - \frac{4 z^{2}}{H^{2}})

(29)

For RPT in which $ε_{33} = 0$ , functions, except $h (z) = 0$ , are exactly the same as those in equation (29).

Comparisons demonstrate excellent agreement of the proposed approach with those of Thai and Choi (2014) in which greater number of unknowns are used. Meaning that, implementing the stress function is not only effective, but also simplifies the model.

9. Results and discussions

To express the effectiveness of the proposed approach, natural frequencies of FG plates are compared with those of other references using HSDTs and neglecting the effect of thickness stretching. The natural frequencies are also compared with those of the mentioned theories containing through-the-thickness deformation effect. Material properties of the FG plates are tabulated in Table 1. In this study, the functions required for the IRSNDT approach, which are used in the displacement field equation (2) to satisfy equation (3), are

g_{u} (z) = - z, f_{u} (z) = - \frac{4 z^{3}}{3 H^{2}}, g_{v} (z) = - z, f_{v} (z) = - \frac{4 z^{3}}{3 H^{2}}, s (z) = 1, h (z) = \frac{5}{4} (1 - \frac{4 z^{2}}{H^{2}})

(30)

h (z)

must be zero when neglecting the through-the-thickness deformation effect in RPT.

9.1. Fundamental frequencies comparison

Dimensionless fundamental frequencies

(\hat{ω} = ω H \sqrt{ρ_{b} / E_{b}})

of a movable simply supported square plate made of Material Set-1 with various p indices, are obtained based on IRSNDT and RPT and shown in Table 4. Comparisons are done with 3D solutions of Vel and Batra (2004), quasi-3D solutions by Neves et al. (2012, 2013) and Thai et al. (2014), TSDT solutions by Ferreira et al. (2006) and HSDT solutions by Thai and Choi (2013). Number of variables used by different approaches are shown in the third column. Difference percentages of the results compared to 3D exact solutions (Vel and Batra (2004)), are shown in parenthesis.

Table 4.

Dimensionless fundamental frequency $\hat{ω} = ω H \sqrt{ρ_{b} / E_{b}}$ of square Al/ZrO₂ FG plate.

Method	$ε_{33}$	No. of variables	$p = 0$		$p = 1$			$l_{1} / H = 5$
Method	$ε_{33}$	No. of variables	$l_{1} / H = \sqrt{10}$	$l_{1} / H = 10$	$l_{1} / H = 5$	$l_{1} / H = \sqrt{10}$	$l_{1} / H = 10$	$l_{1} / H = 5$	$l_{1} / H = \sqrt{10}$	$l_{1} / H = 10$
3D exact (Vel and Batra (2004))	$\neq$ 0	-	0.4658	0.0578	0.2192	0.0596	0.0153	0.2197	0.2211	0.2225
Quasi-3D (Neves et al. (2012))	$\neq$ 0	9	-	-	0.2193 (0.046%)	0.0596 (0.000%)	0.0153 (0.000%)	0.2201 (0.182%)	0.2216 (0.226%)	0.2230 (0.225%)
Quasi-3D (Neves et al. (2013))	$\neq$ 0	9	-	-	0.2193 (0.046%)	-	-	0.2200 (0.136%)	0.2215 (0.181%)	0.2230 (0.225%)
Quasi-3D (Thai et al. (2014))	$\neq$ 0	5	0.4661 (0.065%)	0.0578 (0.000%)	0.2192 (0.000%)	0.0597 (0.168%)	0.0153 (0.000%)	0.2201 (0.182%)	0.2214 (0.136%)	0.2225 (0.000%)
TSDT (Ferreira et al. (2006))	=0	5	-	-	0.2188 (0.182%)	0.0592 (0.671%)	0.0147 (3.921%)	0.2188 (0.409%)	0.2202 (0.407%)	0.2215 (0.449%)
HSDT (Thai and Choi (2013))	=0	4	0.4622 (0.773%)	0.0577 (0.173%)	0.2169 (1.049%)	0.0592 (0.671%)	0.0152 (0.653%)	0.2178 (0.865%)	0.2193 (0.814%)	0.2206 (0.854%)
Present, RPT $(w_{b}, w_{s})$	=0	2	0.4623 (0.751%)	0.0577 (0.173%)	0.2169 (1.049%)	0.0592 (0.671%)	0.0152 (0.653%)	0.2178 (0.865%)	0.2193 (0.814%)	0.2206 (0.854%)
Present, IRSNDT $(w_{b}, w_{s}, w_{z})$	$\neq$ 0	3	0.4661 (0.065%)	0.0578 (0.000%)	0.2192 (0.000%)	0.0597 (0.168%)	0.0153 (0.000%)	0.2201 (0.182%)	0.2214 (0.136%)	0.2224 (0.045%)

Note: HSDT: higher-order shear deformation theories; TSDT: third-order shear deformation theory; IRSNDT: improved refined shear and normal deformation theory.

By increasing the power law index $p$ and therefore softening the system, natural frequencies decrease. As can be seen, natural frequencies obtained by the present approach, including the effect of thickness stretching, match well with solutions achieved by 3D exact method. Whereas results obtained by Ferreira et al. (2006) and Thai and Choi (2013) and also proposed approach using RPT slightly underestimate the frequency due to ignoring the thickness stretching effect. Additionally, as the plate becomes thicker, the effect of $w_{z}$ becomes more crucial and the percentage differences of the results using RPT are increased.

9.2. Natural frequencies comparison

Table 5 compares dimensionless natural frequencies obtained by the present approach using IRSNDT and those of Shahrjerdi et al. (2011) using SSDT neglecting the effect of through-the-thickness deformation.

Table 5.

Dimensionless natural frequency $\bar{ω_{m n}} = ω_{m n} l_{2}^{2} / H \sqrt{ρ_{t} / E_{t}}$ of rectangular Al/ZrO₂ FG plate $(\frac{l_{1}}{l_{2}} = 2, \frac{l_{2}}{H} = 10)$ .

$m \times n$	Method	$ε_{33}$	No. of variables	$p = 0$	$p = 0.5$	$p = 1$	$p = 2$	Diff. %
$1 \times 1$	HOSNT12 (Jha et al. (2013))	$\neq$ 0	12	3.6911	3.3664	3.2179	3.1291	-
	HOSNT11 (Kant et al. (2014))	$\neq$ 0	11	3.691	3.366	3.218	3.129	0.005
	SSDT(Shahrjerdi et al. (2011))	=0	7	3.6983	3.3713	3.2225	3.1354	0.171
	Present, IRSNDT	=0	3	3.6911	3.3849	3.2512	3.1729	0.746
$1 \times 2$	HOSNT12(Jha et al. (2013))	$\neq$ 0	12	5.8323	5.3238	5.0886	4.9434	-
	HOSNT11 (Kant et al. (2014))	$\neq$ 0	11	5.832	5.324	5.089	4.943	0.006
	SSDT(Shahrjerdi et al. (2011))	=0	7	5.8498	5.3359	5.1002	4.9594	0.269
	Present, IRSNDT	$\neq$ 0	3	5.8323	5.3527	5.1406	5.0118	0.737
$2 \times 1$	HOSNT12(Jha et al. (2013))	$\neq$ 0	12	11.965	10.946	10.461	10.137	-
	HOSNT11(Kant et al. (2014))	$\neq$ 0	11	11.965	10.946	10.461	10.137	0.000
	SSDT(Shahrjerdi et al. (2011))	=0	7	12.0345	10.9940	10.5062	10.1985	0.514
	Present, IRSNDT	$\neq$ 0	3	11.9646	11.0033	10.5642	10.2731	0.714
$2 \times 2$	HOSNT12(Jha et al. (2013))	$\neq$ 0	12	13.921	12.745	12.180	11.794	-
	HOSNT11 (Kant et al. (2014))	$\neq$ 0	11	13.921	12.745	12.180	11.794	0.000
	SSDT(Shahrjerdi et al. (2011))	=0	7	14.0144	12.8103	12.2421	11.8784	0.602
	Present, IRSNDT	$\neq$ 0	3	13.9213	12.8106	12.2984	11.9503	0.703
$2 \times 3$	HOSNT12(Jha et al. (2013))	$\neq$ 0	12	17.096	15.668	14.973	14.481	-
	HOSNT11 (Kant et al. (2014))	$\neq$ 0	11	17.096	15.668	14.973	14.481	0.000
	SSDT(Shahrjerdi et al. (2011))	=0	7	17.2325	15.7660	15.0670	14.6092	0.734
	Present, IRSNDT	$\neq$ 0	3	17.0958	15.7468	15.1152	14.669	0.688
$3 \times 2$	HOSNT12(Jha et al. (2013))	$\neq$ 0	12	26.051	23.941	22.876	22.059	-
	HOSNT11 (Kant et al. (2014))	$\neq$ 0	11	26.051	23.941	22.876	22.059	0.000
	SSDT(Shahrjerdi et al. (2011))	=0	7	26.3462	24.1494	23.0749	22.3273	1.022
	Present, IRSNDT	$\neq$ 0	3	26.0509	24.0541	23.0817	22.334	0.654
$3 \times 3$	HOSNT12(Jha et al. (2013))	$\neq$ 0	12	28.871	26.554	25.372	24.446	-
	HOSNT11 (Kant et al. (2014))	$\neq$ 0	11	28.871	26.554	25.372	24.446	0.000
	SSDT(Shahrjerdi et al. (2011))	=0	7	29.2257	26.8100	25.6184	24.7781	1.130
	Present, IRSNDT	$\neq$ 0	3	28.8709	26.6767	25.596	24.745	0.642

Note: SSDT: second order shear deformation theory; HOSNT: higher-order shear and normal deformation plate theory.

Also, results of Kant et al. (2014) and Jha et al. (2013) based on HOSNT including 11 and 12 variables are compared with those of the proposed approach. Higher-order shear and normal deformation plate theory 11 contains the expansion of in-plane displacements up to order 3 and transverse displacement up to order 2 along the thickness direction. Higher-order shear and normal deformation plate theory 12 contains the expansion of in-plane displacement up to order 3 and transverse displacement up to order 3 in along the thickness direction.

For this case, a simply supported movable rectangular FG plate using Material Set-3 is investigated. Average of difference percentages between frequency values of Jha et al. (2013) with other studies have been shown in the last column. Dimensionless natural frequency is obtained using the relation $\bar{ω_{m n}} = ω_{m n} l_{2}^{2} / H \sqrt{ρ_{t} / E_{t}}$ . $ω_{m n}$ is the value of natural frequency for a selective mode of vibration in relation with the $m$ th mode in $x_{1}$ direction and $n$ th mode in $x_{2}$ direction.

Comparing values obtained by the proposed approach with only 3 variables with the aforementioned references shows that, natural frequencies of various modes are in excellent agreement. From the difference percentage column, it can be concluded that for higher modes of vibration, considering the thickness stretching effect becomes more important. This can be seen by comparing the results of the proposed approach using only 3 unknowns with those of Jha et al. (2013) using 12 unknowns, to account for thickness stretching effect.

Thus, this approach is not only accurate but also comparable with higher-order shear and normal deformation theories containing more variables (i.e., in this case 3 vs 12 DOFs).

9.3. Effect of transverse shear deformation and thickness stretching on natural frequencies

In this section, the importance of including transverse shear deformation and thickness stretching on the values of first four symmetric natural frequencies of FG square plates are investigated.

Material Set-2 is selected for the FG plate with movable simply supported boundary condition. Dimensionless natural frequency is obtained using the relation ${\hat{ω}}_{i} = ω_{i} H \sqrt{ρ_{b} / E_{b}}$ where $ω_{i}$ is the natural frequency of $i$ th mode of vibration.

Figures 2–4 show the first four natural frequencies of an FG square plate (with various power law indices $p$ ) in terms of the plate thickness ratio. It includes IRSNDT, considering transverse bending deformation $w_{b}$ , transverse shear deformation $w_{s}$ , and thickness stretching effect $w_{z}$ . It should be mentioned that four symmetric modes have been considered for each variable. The effects of considering only $w_{b}$ and $w_{s}$ in RPT and using only $w_{b}$ in CPT are also shown. Also, it should be noted that due to geometrical symmetry of the square plate, second and third natural frequencies ( ${\hat{ω}}_{2}$ and ${\hat{ω}}_{3}$ ) have the same values. This is why the third natural frequencies are not shown in the figures.

Figure 2.

Effects of $w_{s}$ and $w_{z}$ on values of natural frequencies of square Al/Al₂O₃ FG plate versus plate thickness ratio $(p = 0)$ , (a) ${\hat{ω}}_{1}$ , (b) ${\hat{ω}}_{2}$ , (c) ${\hat{ω}}_{4}$ .

Figure 3.

Effects of $w_{s}$ and $w_{z}$ on values of natural frequencies of square Al/Al₂O₃ FG plate versus plate thickness ratio $(p = 1)$ , (a) ${\hat{ω}}_{1}$ , (b) ${\hat{ω}}_{2}$ , (c) ${\hat{ω}}_{4}$ .

Figure 4.

Effects of $w_{s}$ and $w_{z}$ on values of natural frequencies of square Al/Al₂O₃ FG plate versus plate thickness ratio $(p = 10)$ , (a) ${\hat{ω}}_{1}$ , (b) ${\hat{ω}}_{2}$ , (c) ${\hat{ω}}_{4}$ .

As can be seen, by increasing the plate thickness ratio and therefore thinning the FG plate, all three curves converge demonstrating that the bending has the most influence in thin plates. As the thickness of the plate increases, the difference between the three curves will increase. Furthermore, by increasing mode numbers, not only frequency values increase, but also larger distinction among three curves considering IRSNDT, RPT, and CPT can be observed.

For plates thicker than $l_{1} / 20$ , considering only transverse bending deformation $w_{b}$ is not enough and transverse shear deformation $w_{s}$ should be considered, too. While, for plates thicker than $l_{1} / 5$ , the effect of thickness stretching $w_{z}$ also becomes important and should be considered.

The same results can be seen in Figure 5 by plotting frequency values versus power law index $(p)$ . Considering the thickness stretching effect will lead to higher frequency values compared to when it is neglected. By increasing $p$ and thus softening the system, values of natural frequencies for modes of vibrations are reduced.

Figure 5.

Effects of $w_{s}$ and $w_{z}$ on values of natural frequencies of square Al/Al₂O₃ FG plate versus power law index, (a) ${\hat{ω}}_{1}$ , (b) ${\hat{ω}}_{2}$ , (c) ${\hat{ω}}_{4}$

10. Conclusion

An IRSNDT approach including the effect of through-the-thickness deformation was proposed for vibration analyses of both thin and thick FG plates with movable simply supported boundaries. Through using the Airy stress function and omitting the in-plane displacements, number of unknowns was reduced to only three, including bending, shear, and thickness stretching effects. By using this approach, which simplifies the dynamic model, vibration behavior of FG rectangular plates, especially thick plates can be investigated. Validations and effectiveness of the proposed approach were shown by comparing the results with those of other theories whether including or excluding the thickness stretching effect. It was proven that this approach is capable of handling wide variety of other theories with selecting proper functions defined in the displacement field. Additionally, it was shown that natural frequencies obtained by the proposed approach are in excellent agreement with 3D and quasi-3D solutions containing more variables to consider the effect of thickness stretching.

The effect of transverse shear deformation and thickness stretching on values of first four symmetric natural frequencies of FG square plates were discussed. It was concluded that, for FG square plates thinner than $l_{1} / 20$ , considering only the transverse bending deformation is enough. While to investigate thicker plates, the effect of transverse shear deformation also becomes important. To obtain more accurate results for plates thicker than $l_{1} / 5$ , thickness stretching effect also must be considered.

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

ORCID iDs

Leila Monajati

Navid Farid

Appendix

Substituting strain-displacement relations equation (5) into the variation of strain energy equation (10), the stress resultants of the plate are expressed as (A-1)

\begin{array}{l} (N_{1}^{0}, N_{1}^{g}, N_{1}^{f}) = \int_{\frac{- H}{2}}^{\frac{H}{2}} (1, g_{u} (z), f_{u} (z)) σ_{11} d z \\ (N_{2}^{0}, N_{2}^{g}, N_{2}^{f}) = \int_{\frac{- H}{2}}^{\frac{H}{2}} (1, g_{v} (z), f_{v} (z)) σ_{22} d z \\ (R^{s}, R^{h}) = \int_{\frac{- H}{2}}^{\frac{H}{2}} (\frac{d s (z)}{d z}, \frac{d h (z)}{d z}) σ_{33} d z \\ (N_{12}^{0}, N_{12}^{g}, N_{12}^{f}, N_{21}^{g}, N_{21}^{f}) = \int_{\frac{- H}{2}}^{\frac{H}{2}} (1, g_{u} (z), f_{u} (z), g_{v} (z), f_{v} (z)) σ_{12} d z \\ (Q_{1}^{00}, Q_{1}^{s 0}, Q_{1}^{h 0}, Q_{1}^{g}, Q_{1}^{f}) = \int_{\frac{- H}{2}}^{\frac{H}{2}} (1, s (z), h (z), \frac{d g_{u} (z)}{d z}, \frac{d f_{u} (z)}{d z}) σ_{13} d z \\ (Q_{2}^{00}, Q_{2}^{s 0}, Q_{2}^{h 0}, Q_{2}^{g}, Q_{2}^{f}) = \int_{\frac{- H}{2}}^{\frac{H}{2}} (1, s (z), h (z), \frac{d g_{v} (z)}{d z}, \frac{d f_{v} (z)}{d z}) σ_{23} d z \end{array}

Substituting strain-displacement relations equation (5) and stress–strain relations equation (6) in definitions of stress resultants $N_{1}^{0}$ , $N_{2}^{0}$ , and $N_{12}^{0}$ described in equation (A-1), various coefficients are defined as (A-2)

\begin{array}{l} (A_{11}^{0}, A_{11}^{g}, A_{11}^{f}) = \int_{\frac{- H}{2}}^{\frac{H}{2}} C_{11} (1, g_{u} (z), f_{u} (z)) d z \\ (A_{22}^{0}, A_{22}^{g}, A_{22}^{f}) = \int_{\frac{- H}{2}}^{\frac{H}{2}} C_{22} (1, g_{v} (z), f_{v} (z)) d z \\ [\begin{matrix} \begin{matrix} \begin{matrix} A_{12}^{0} & A_{12}^{g} & A_{12}^{f} \end{matrix} & A_{12}^{g'} & A_{12}^{f'} \end{matrix} \\ \begin{matrix} \begin{matrix} A_{44}^{0} & A_{44}^{g} & A_{44}^{f} \end{matrix} & A_{44}^{g'} & A_{44}^{f'} \end{matrix} \end{matrix}] = \int_{\frac{- H}{2}}^{\frac{H}{2}} {\begin{matrix} C_{12} \\ C_{44} \end{matrix}} (1, g_{u} (z), f_{u} (z), g_{v} (z), f_{v} (z)) d z \\ [\begin{matrix} D_{13}^{s} & D_{13}^{h} \\ D_{23}^{s} & D_{23}^{h} \end{matrix}] = \int_{\frac{- H}{2}}^{\frac{H}{2}} {\begin{matrix} C_{13} \\ C_{23} \end{matrix}} (\frac{d s (z)}{d z}, \frac{d h (z)}{d z}) d z \end{array}

According to definitions of $N_{1}^{0}$ , $N_{2}^{0}$ , and $N_{12}^{0}$ in equation (A-1) and by using coefficients defined in equation (A-2), derivatives of the in-plane variables $u_{0}$ and $v_{0}$ are obtained based on the stress resultants $N_{1}^{0}$ , $N_{2}^{0}$ , and $N_{12}^{0}$ derivatives of variables of the problem. Substituting derivatives of in-plane variables into the strain-displacement relations equation (5), normal strains along $x_{1}$ and $x_{2}$ directions and in-plane shear strain are obtained as (A-3)

\begin{array}{l} ε_{i j} = e_{i, 1} N_{1}^{0} + e_{i, 2} N_{2}^{0} + e_{i, 3} \frac{\partial^{2} w_{b}}{\partial x_{1}^{2}} + e_{i, 4} \frac{\partial^{2} w_{b}}{\partial x_{2}^{2}} + e_{i, 5} \frac{\partial^{2} w_{s}}{\partial x_{1}^{2}} \\ + e_{i, 6} \frac{\partial^{2} w_{s}}{\partial x_{2}^{2}} + e_{i, 7} w_{s} + e_{i, 8} w_{z} (i = j = 1,2) \\ γ_{12} = e_{1} N_{12}^{0} + e_{2} \frac{\partial^{2} w_{b}}{\partial x_{1} \partial x_{2}} + e_{3} \frac{\partial^{2} w_{s}}{\partial x_{1} \partial x_{2}} \end{array}

Where coefficients

e_{i, j}

and

e_{i}

are obtained as (A-4)

\begin{array}{l} e_{1,1} = \frac{A_{22}^{0}}{A_{11}^{0} A_{22}^{0} - A_{12}^{0}^{2}}, e_{1,2} = - \frac{A_{12}^{0}}{A_{11}^{0} A_{22}^{0} - A_{12}^{0}^{2}} \\ e_{1,3} = \frac{A_{12}^{0} A_{12}^{g} - A_{22}^{0} A_{11}^{g}}{A_{11}^{0} A_{22}^{0} - A_{12}^{0}^{2}} + g_{u} (z), e_{1,4} = \frac{A_{12}^{0} A_{22}^{g} - A_{22}^{0} A_{12}^{g'}}{A_{11}^{0} A_{22}^{0} - A_{12}^{0}^{2}} \\ e_{1,5} = \frac{A_{12}^{0} A_{12}^{f} - A_{22}^{0} A_{11}^{f}}{A_{11}^{0} A_{22}^{0} - A_{12}^{0}^{2}} + f_{u} (z), e_{1,6} = \frac{A_{12}^{0} A_{22}^{f} - A_{22}^{0} A_{12}^{f'}}{A_{11}^{0} A_{22}^{0} - A_{12}^{0}^{2}} \\ e_{1,7} = \frac{A_{12}^{0} D_{23}^{s} - A_{22}^{0} D_{13}^{s}}{A_{11}^{0} A_{22}^{0} - A_{12}^{0}^{2}}, e_{1,8} = \frac{A_{12}^{0} D_{23}^{h} - A_{22}^{0} D_{13}^{h}}{A_{11}^{0} A_{22}^{0} - A_{12}^{0}^{2}} \\ e_{2,1} = - \frac{A_{12}^{0}}{A_{11}^{0} A_{22}^{0} - A_{12}^{0}^{2}}, e_{2,2} = \frac{A_{11}^{0}}{A_{11}^{0} A_{22}^{0} - A_{12}^{0}^{2}} \\ e_{2,3} = \frac{A_{12}^{0} A_{11}^{g} - A_{11}^{0} A_{12}^{g}}{A_{11}^{0} A_{22}^{0} - A_{12}^{0}^{2}}, e_{2,4} = \frac{A_{12}^{0} A_{12}^{g'} - A_{11}^{0} A_{22}^{g}}{A_{11}^{0} A_{22}^{0} - A_{12}^{0}^{2}} + g_{v} (z) \\ e_{2,5} = \frac{A_{12}^{0} A_{11}^{f} - A_{11}^{0} A_{12}^{f}}{A_{11}^{0} A_{22}^{0} - A_{12}^{0}^{2}}, e_{2,6} = \frac{A_{12}^{0} A_{12}^{f'} - A_{11}^{0} A_{22}^{f}}{A_{11}^{0} A_{22}^{0} - A_{12}^{0}^{2}} + f_{v} (z) \\ e_{2,7} = \frac{A_{12}^{0} D_{13}^{s} - A_{11}^{0} D_{23}^{s}}{A_{11}^{0} A_{22}^{0} - A_{12}^{0}^{2}}, e_{2,8} = \frac{A_{12}^{0} D_{13}^{h} - A_{11}^{0} D_{23}^{h}}{A_{11}^{0} A_{22}^{0} - A_{12}^{0}^{2}} \\ e_{1} = \frac{1}{A_{44}^{0}}, e_{2} = g_{u} (z) + g_{v} (z) - \frac{A_{44}^{g} + A_{44}^{g'}}{A_{44}^{0}} \\ e_{3} = f_{u} (z) + f_{v} (z) - \frac{A_{44}^{f} + A_{44}^{f'}}{A_{44}^{0}} \end{array}

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