The effect of multidimensional temperature distribution on the vibrational characteristics of a size-dependent thick bi-directional functionally graded microplate

Abstract

This article develops the modified couple stress theory to study the free vibration of bi-directional functionally graded microplates subjected to multidimensional temperature distribution. Third-order shear deformation and classical theories of plates are adapted for free vibration analysis of thick and thin microplates, respectively. Employing the third-order shear deformation theory, both normal and shear deformations are considered without the need for shear correction factor. Material of the bi-directional functionally graded microplate is graded smoothly through the length and thickness of the microplate. Gradient of the material is assumed to obey from the power law in terms of the volume fraction of the constituents. Assuming the uniform and nonuniform temperature distributions, the effect of thermal environment on dynamic behavior of the microplate is discussed in detail. Applying the Ritz method, the displacement field is expanded by admissible functions which satisfy the essential boundary conditions, and Hamilton principle is employed to determine the natural frequencies of the microplate. Developed model has been applied to determine the natural frequencies in problems of thin/thick, one-directional/bi-directional functionally graded, and homogeneous/nonhomogeneous microplates. Effects of parameters such as the thermal environment, power law indexes $(η_{x}, η_{z})$ and length scale parameter on free vibration of these problems are studied in detail. The results show that higher values of length scale parameter and temperature rise decrease the natural frequency of the bi-directional functionally graded microplate. According to results obtained by classical and third-order shear deformation theories, the third-order shear deformation theory is proposed for vibration analysis of microplates with thickness-to-length ratio less than five.

Keywords

Free vibration Two-directional functionally graded microplate third-order shear deformation theory modified couple stress theory sinusoidal temperature rise

Introduction

Functionally graded materials (FGMs) are made from a mixture of two constituent material and are characterized by a smooth and continuous change of the properties from one surface to another. FGMs were originally designed as thermal barrier for aerospace structural applications and fusion reactors. However, nonhomogeneous nature of many structures in microscale requires to model their mechanical properties as multi-directional FGMs. Systems such as thin films,¹ atomic force microscopes (AFMs),² microplate spectrophotometers hotplate, and micro-electromechanical systems (MEMS)/nano-electromechanical systems (NEMS),³ tunable micro/nanomechanical resonators, and paddle-like resonators⁴ are among the extensive applications of microstructures. Experimentally, the size effect in these structures has been investigated by many research scholars.^5–7

Well-known classical theories such as classical plate theory (CPT) based on Chauchy,⁸ Poisson⁹ or Kirchhoff¹⁰ assumptions types, first-order shear deformation theory (FSDT),¹¹ third-order Vlasov–Reddy theory (TSDT),¹² and high-order shear deformation theories (HSDT) have been used to analyze the dynamic and static behavior of FG plates. While the classical theories may not predict the size effect in micro-structures, non-classical continuum theories were formed to model the size effect by applying material length scale parameters. To model the size effects in micro- and nano-scale, several size-dependent theories have been developed in continuum mechanics. Theories such as nonlocal elasticity theory,¹³ strain gradient theory,¹⁴ couple stress theory,^15,16 and its modified version¹⁷ have been developed to model the behavior of structures in nano- and micro-scale. The nonlocal elasticity theory¹³ includes two material length scale parameters and the strain gradient theory¹⁴ includes three material length scale parameters. Toupin,¹⁶ Mindlin and Tiersten,¹⁵ and Koiter¹⁸ were the first research scholars who introduced the couple stress theory with two material length scale parameters. Afterward, Yang et al.¹⁷ presented the modified couple stress theory in which one material length scale parameter is employed to analyze the micro-structures. Due to difficulties in determining the material length scale parameters, the modified couple stress theory with one material length scale parameter has been attracted many research scholars in the past years. Lei et al.¹⁹ presented a size-dependent FG microplate model based on a modified couple stress theory requiring only one material length scale parameter. Based on the modified couple stress theory, Tsiatas²⁰ developed a new Kirchhoff theory for the static analysis of a microplate. Using the classical plate theory (CPT), Jomehzadeh et al.²¹ employed the modified couple stress theory to study the vibrational behavior of microplates. Ma et al.²² applied modified couple stress theory to analyze the vibrating microplate using FSDT. Ke et al.²³ developed a size-dependent model for Mindlin microplates based on the modified couple stress theory.

Displacement field based on the third-order shear deformation theory requires complex analysis for free vibration analysis of plates. However, since the theory predicts accurate results without employing the shear correction factor, it is popular in free vibration analysis of thick plates. Reddy¹² proposed a simple higher order shear deformation theory for laminated composite plates and reformulated it for bending and vibration of plates using the TSDT.²⁴ Using the third-order plate model and a modified couple stress theory, Gao et al.²⁵ developed a microplate model which contains one material length scale parameter. Furthermore, Ghayesh et al.²⁶ developed the size-dependent nonlinear third-order shear deformation for dynamic analysis of microplates.

Employed the modified couple stress theory, the majority of research works on FG microplates are limited to material grading in thickness direction (one-directional (1D)-FG). Thai and Choi²⁷ developed the size-dependent model for free vibration and buckling analysis of FG Kirchhoff and Mindlin plate. In their research, the equations of motion were derived from Hamilton’s principle based on a modified couple stress theory. Also, Thai and Kim²⁸ obtained more accurate results for the size-dependent thick FG plates, using Reddy’s plate theory. Recently, Thai and Vo²⁹ have used sinusoidal shear deformation theory based on the modified couple stress theory to determine the fundamental frequency of FG microplates. Reddy and Kim³⁰ presented a general nonlinear modified couple stress-based third-order plate theory and used the Hamilton principle to derive the equations of motion. He et al.³¹ developed a four variable refined plate theory for size-dependent FG microplates based on the modified couple stress theory and obtained closed form solutions for bending, buckling, and free vibration responses of the microplate. Salehipour et al.³² used the modified couple stress and three-dimensional elasticity theories for vibration analysis of FG microplates. Ansari et al.³³ utilized a size-dependent model for free vibration of Mindlin FG microplates within the framework of modified couple stress theory. Lou and He³⁴ studied the free vibration Kirchhoff, Mindlin, and Von karman FG microplates on elastic foundation using the modified couple stress theory.

New research works in the field of FG materials indicate that variation of volume fractions in two directions optimizes the mechanical and thermal tresses compared to conventional 1D-FGM. Some studies have been carried out about static, dynamic, and free vibration of two-directional (2D)-FG plate.^35–39 The size-dependent models are not employed in these works, and mechanical behavior of the plate in microscale may not be captured, consequently. Studies in the field of size-dependent 2D-FG microstructures are limited to the beam. Trinh et al.⁴⁰ employed a size-dependent model for vibrational behavior analysis of bi-directional FG microbeams with arbitrary boundary conditions. Based on the modified couple stress theory and a quasi-3D theory, they developed governing equations of motion for bi-directional FG microbeams. Yu et al.⁴¹ presented the formulation of a non-classical quasi-3D theory for 2D-FG microbeams. In their research, the small-scale effects were interpreted by the modified couple stress theory. Utilizing the modified couple stress theory and third-order shear deformation beam theory, Chen et al.⁴² derived the governing equations of motion for the bi-directional FG microbeams. Finally, they extracted the natural frequency of the beam by differential quadrature method (DQM). Shafiei et al.⁴³ studied free vibration of two-directional imperfect FG (2D-FG) porous nano-/microbeams. In their work, governing equations of motion were solved using generalized differential quadrature method (GDQM). Mirjavadi et al.⁴⁴ analyzed the vibrational behavior of 2D-FG porous microbeams based on the couple stress theory in thermal environment. The generalized differential quadrature method is used to solve the equations. The temperature gradient is also considered to be uniform and nonuniform across the thickness of the microbeam. Howevere, a two-dimensional temperature distribution exists for the case where the material is graded in two directions and the beam is subjected to nonunifrom temperature distribution.

As the literature review shows, 1D-FG material is mainly considered in most of the works regarding to vibration of microplates. While the third-order plate theory takes account both normal and shear deformations without the shear correction factor, the authors could not find any research on free vibration of 2D-FG microplates using this theory.^45–48 Moreover, the assumption of multidimensional temperature distribution is required to study effect of thermal environment in 2D-FGMs. So, the innovation of this article is application of the third-order plate theory with size-dependent model for free vibration analysis of bi-directional FG microplate. This article investigates the free vibration of a (2D-FG) microplate subjected to multidimensional temperature distribution. Applying the Hamilton principle and the Ritz method, the Lagrange function is minimized to obtain the natural frequencies of the microplate. The third-order and classical theories of plates with modified stress couple theory is used to model the size-dependent behavior of thick/thin microplates. Gradient of material through the thickness and length of the microplate is considered by volume fraction of constituent materials. A parameter study is performed to investigate effects of thermal environment, gradient of material in two directions, size-dependent behavior of microplate, and application of third-order theory on natural frequencies of the microplate.

Theoretical formulation

Formulation of 2D-FG plate

Assume a rectangular 2D-FGM plate of length a, width b, and thickness h, so that 0 ⩽ x ⩽ a, 0 ⩽ y ⩽ b, and h/2 ⩽ z ⩽−h/2. The x-axis is aligned with the longitudinal direction of the plate and the z-axis is aligned with the thickness direction of the plate. Figure 1 shows geometry and coordinate system of the 2D-FG rectangular plate which is considered in this article.

Figure 1.

Geometry and coordinate system of the 2D-FG rectangular plate with coordinate convention.

Two-dimensional FGMs are usually made by continuous gradation of three or four different material phases. One or two of these phases are ceramics and the others are metal alloy. The volume fractions of the materials vary in a predetermined composition profile. In this article, the lower surface of the plate is assumed to be made of first ceramic in the left corner and it is made of first metal in the right corner. The upper surface of the plate is made of second ceramic and second metal in the left and right corners, respectively. Using the linear rule of mixtures, material properties at each point may be determined. Therefore, linear combination of volume fractions and material properties of the basic materials may be used to determine the distribution of material property P (e.g. modulus of elasticity and mass density) through the 2D-FGM plate.⁴⁷ That is

P (x, z) = P_{m_{1}} V_{m_{1}} + P_{m_{2}} V_{m_{2}} + P_{c_{1}} V_{c_{1}} + P_{c_{2}} V_{c_{2}}

(1)

In equation (1), $c_{1}$ , $c_{2}$ , $m_{1}$ , and $m_{2}$ denote first ceramic, second ceramic, first metal, and second metal, respectively. The volume fraction distribution function of each material can be expressed as

V_{m_{1}} = [{(\frac{x}{a})}^{η_{x}}] [1 - {(\frac{2 z + h}{2 h})}^{η_{z}}]

(2a)

V_{m_{2}} = [{(\frac{x}{a})}^{η_{x}}] [{(\frac{2 z + h}{2 h})}^{η_{z}}]

(2b)

V_{c_{1}} = [1 - {(\frac{x}{a})}^{η_{x}}] [1 - {(\frac{2 z + h}{2 h})}^{η_{z}}]

(2c)

V_{c_{2}} = [1 - {(\frac{x}{a})}^{η_{x}}] [{(\frac{2 z + h}{2 h})}^{η_{z}}]

(2d)

where $η_{x}$ and $η_{z}$ are power law indexes in x and z directions.

Kinematics

The displacement field based on the TSDT is

\begin{array}{l} u_{1} (x, y, z, t) = u (x, y, t) + z ϕ_{x} (x, y, t) - \\ \frac{4}{3 h^{2}} z^{3} (ϕ_{x} (x, y, t) + \frac{\partial w (x, y, t)}{\partial x}) \end{array}

(3a)

\begin{array}{l} u_{2} (x, y, z, t) = v (x, y, t) + z ϕ_{y} (x, y, t) - \\ \frac{4}{3 h^{2}} z^{3} (ϕ_{y} (x, y, t) + \frac{\partial w (x, y, t)}{\partial y}) \end{array}

(3b)

u_{3} (x, y, z, t) = w (x, y, t)

(3c)

where t is the time, u, v, and w are displacements of a point on the middle plane in x, y, and z directions, respectively. Parameters $ϕ_{x}$ and $ϕ_{y}$ are rotations of the transverse normals about y and x axis, respectively. The strain–displacement relations are

ε_{i j} = \frac{1}{2} [u_{i, j} + u_{j, i}]

(4)

Displacement field and strain–displacement relations lead to the following equations for the strain tensor

ε_{x} = ε_{x}^{(0)} + z k_{x}^{(0)} + z^{3} k_{x}^{(2)}

(5a)

ε_{y} = ε_{y}^{(0)} + z k_{y}^{(0)} + z^{3} k_{y}^{(2)}

(5b)

γ_{x y} = γ_{x y}^{(0)} + z k_{x y}^{(0)} + z^{3} k_{x y}^{(2)}

(5c)

γ_{x z} = γ_{x z}^{(0)} + z^{2} k_{x z}^{(1)}

(5d)

γ_{y z} = γ_{y z}^{(0)} + z^{2} k_{y z}^{(1)}

(5e)

where

ε_{x}^{(0)} = u_{, x}, ε_{y}^{(0)} = v_{, y}

(6a)

γ_{x y}^{(0)} = u_{, y} + v_{, x}

(6b)

γ_{x z}^{(0)} = ϕ_{x} + w_{, x}, γ_{y z}^{(0)} = ϕ_{y} + w_{, y}

(6c)

k_{x}^{(0)} = ϕ_{x, x}, k_{y}^{(0)} = ϕ_{y, y}

(6d)

k_{x y}^{(0)} = ϕ_{x, y} + ϕ_{y, x}, k_{x}^{(2)} = - κ (ϕ_{x, x} + w_{, x x})

(6e)

k_{y}^{(2)} = - κ (ϕ_{y, y} + w_{, y y})

(6f)

k_{x y}^{(2)} = - κ (ϕ_{x, y} + ϕ_{y, x} + 2 w_{, x y})

(6g)

k_{x z}^{(1)} = - \frac{4}{h^{2}} γ_{x z}^{(0)}, k_{y z}^{(1)} = - \frac{4}{h^{2}} γ_{y z}^{(0)}

(6h)

In above equation, $κ = 4 / 3 h^{2}$ . The constitutive equations are

(\begin{matrix} σ_{x x} \\ σ_{y y} \\ σ_{x y} \\ σ_{x z} \\ σ_{y z} \end{matrix}) = [\begin{matrix} Q_{1} & Q_{2} & 0 & 0 & 0 \\ Q_{2} & Q_{1} & 0 & 0 & 0 \\ 0 & 0 & Q_{3} & 0 & 0 \\ 0 & 0 & 0 & Q_{3} & 0 \\ 0 & 0 & 0 & 0 & Q_{3} \end{matrix}] (\begin{matrix} ε_{x} \\ ε_{y} \\ ε_{x y} \\ ε_{x z} \\ ε_{y z} \end{matrix})

(7)

where $Q_{1} = E (z, x) / (1 - υ {(z, x)}^{2}),$ Q₂ = υ(z,x) E(z,x) / (1– υ(z,x)²), and $Q_{3} = E (z, x) / (2 (1 + υ (z, x)))$ . Suppose that the plate is stress free at initial temperature $T_{0}$ . The stress components $(σ_{x x}^{T}, σ_{y y}^{T})$ due to temperature rise ∆ $T (z)$ are

(\begin{matrix} σ_{x x}^{T} \\ σ_{y y}^{T} \end{matrix}) = (\begin{matrix} Q_{1} & Q_{2} \\ Q_{2} & Q_{1} \end{matrix}) [\begin{matrix} α (z, x) \\ α (z, x) \end{matrix}] Δ T (z)

(8)

In equation (8), the term $α$ represents longitudinal coefficient of thermal expansion. Based on the modified couple stress theory, symmetric curvature tensor $χ_{i j}$ and deviatoric part of stress couple tensor $m_{i j}$ are

χ_{i j} = \frac{1}{2} [θ_{i, j} + θ_{j, i}]

(9)

m_{i j} = \frac{E (z, x)}{1 + υ (z, x)} l^{2} χ_{i j}

(10)

where l is the material length scale parameter. The rotation vector $θ_{i}$ is

θ_{i} = \frac{1}{2} ϵ_{i j k} \times u_{k, j}

(11)

where $u_{i}$ is the displacement field. Substituting equation (3) into equation (11), components of the rotation vector are

θ_{1} = \frac{1}{2} [(w_{, y} - ϕ_{y}) + 3 κ z^{2} (ϕ_{y} + w_{, y})]

(12a)

θ_{2} = - \frac{1}{2} [(w_{, x} - ϕ_{x}) + 3 κ z^{2} (ϕ_{x} + w_{, x})]

(12b)

θ_{3} = \frac{1}{2} [(v_{, x} - u_{, y}) + (z - κ z^{3}) (ϕ_{y, x} - ϕ_{x, y})]

(12c)

According to the third-order shear deformation theory, components of the symmetric curvature tensor are

χ_{x x} = \frac{1}{2} (w_{, y x} - ϕ_{y, x}) + \frac{3}{2} κ z^{2} (ϕ_{y, x} + w_{, y x})

(13a)

χ_{y y} = - \frac{1}{2} (w_{, y x} - ϕ_{x, y}) - \frac{3}{2} κ z^{2} (ϕ_{x, y} + w_{, y x})

(13b)

χ_{x y} = \frac{1 + 3 κ z^{2}}{4} (w_{, y y} - w_{, x x}) + \frac{1 - 3 κ z^{2}}{4} (ϕ_{x, x} - ϕ_{y, y})

(13c)

\begin{array}{l} χ_{x z} = \frac{1}{4} (v_{, x x} - u_{, x y}) + \frac{z - κ z^{3}}{4} (ϕ_{y, x x} - ϕ_{x, x y}) \\ + \frac{3 κ z}{2} (w_{, y} + ϕ_{y}) \end{array}

(13d)

\begin{array}{l} χ_{y z} = \frac{1}{4} (v_{, x y} - u_{, y y}) + \frac{z - κ z^{3}}{4} (ϕ_{y, x y} - ϕ_{x, y y}) - \\ \frac{3 κ z}{2} (w_{, x} + ϕ_{x}) \end{array}

(13e)

χ_{z z} = \frac{1 - 3 κ z^{2}}{2} (ϕ_{y, x} - ϕ_{x, y})

(13f)

Strain energy and kinetic energy

The strain energy of the system U_S is equal to the sum of the energy of the classical-elasticity strain energy U_SC, the strain energy from the initial stresses due to temperature rise U_T, and couple–stress strain energy U_SNC. That is

U_{S} = U_{S C} + U_{T} + U_{S N C}

(14)

The strain energy associated with classical elastic energy may be written as

\begin{array}{l} U_{S C} = \frac{1}{2} \int_{Λ} (σ_{i j} ε_{i j}) d Λ = \\ \frac{1}{2} \int_{Λ} [\begin{array}{l} σ_{x x} ε_{x x} + σ_{y y} ε_{y y} + σ_{x z} ε_{x z} \\ + σ_{x y} ε_{x y} + σ_{y z} ε_{y z} \end{array}] d Λ \end{array}

(15)

Substituting equations (5)–(7) into equation (15), the classical elastic energy is

U_{S C} = \int_{0}^{a} \int_{0}^{b} [\begin{array}{l} \frac{1}{2} [A_{1} (ε_{x}^{{(0)}^{2}} + ε_{y}^{{(0)}^{2}}) + 2 A_{2} ε_{x}^{(0)} ε_{y}^{(0)} + A_{3} (γ_{x y}^{{(0)}^{2}} + γ_{x z}^{{(0)}^{2}} + γ_{y z}^{{(0)}^{2}})] \\ + B_{1} ε_{x}^{(0)} k_{x}^{(0)} + B_{1} ε_{y}^{(0)} k_{y}^{(0)} + B_{2} (ε_{x}^{(0)} k_{y}^{(0)} + ε_{y}^{(0)} k_{x}^{(0)}) + B_{3} γ_{x y}^{(0)} k_{x y}^{(0)} \\ + \frac{1}{2} (D_{1} k_{x}^{{(0)}^{2}} + D_{1} k_{y}^{{(0)}^{2}} + 2 D_{2} k_{x}^{(0)} k_{y}^{(0)} + D_{3} (k_{x y}^{{(0)}^{2}} + 2 k_{x z}^{(1)} γ_{x z}^{(0)} + 2 k_{y z}^{(1)} γ_{x z}^{(0)}) \\ + [F_{1} ε_{x}^{(0)} k_{x}^{(2)} + F_{1} ε_{y}^{(0)} k_{y}^{(2)} + F_{2} (ε_{x}^{(0)} k_{y}^{(2)} + ε_{y}^{(0)} k_{x}^{(2)}) + F_{3} γ_{x y}^{(0)} k_{x y}^{(2)}] \\ + [G_{1} (k_{x}^{(0)} k_{x}^{(2)} + G_{1} k_{y}^{(0)} k_{y}^{(2)}) + G_{2} (k_{x}^{(0)} k_{y}^{(2)} + k_{y}^{(0)} k_{x}^{(2)}) + G_{3} (k_{x y}^{(0)} k_{x y}^{(2)} + \frac{1}{2} (k_{x z}^{{(1)}^{2}} + k_{y z}^{{(1)}^{2}})] \\ + \frac{1}{2} [H_{1} k_{x}^{{(2)}^{2}} + H_{1} k_{y}^{{(2)}^{2}} + 2 H_{2} k_{x}^{(2)} k_{y}^{(2)} + H_{3} k_{x y}^{{(2)}^{2}}] \end{array}] d y d x

(16)

Constants $A_{i}, B_{i}, D_{i}, F_{i}, G_{i}, H_{i}$ , and $R_{i}$ in equation (16) are determined by the following integrals

\begin{array}{l} (A_{1}, B_{1}, D_{1}, F_{1}, G_{1}, H_{1}) = \\ \int_{- h / 2}^{h / 2} \frac{E (z, x)}{1 - υ {(z, x)}^{2}} {1, z, z^{2}, z^{3}, z^{4}, z^{6}} d z \end{array}

(17a)

\begin{array}{l} (A_{2}, B_{2}, D_{2}, F_{2}, G_{2}, H_{2}) = \\ \int_{- h / 2}^{h / 2} \frac{υ (z, x) E (z, x)}{1 - υ {(z, x)}^{2}} {1, z, z^{2}, z^{3}, z^{4}, z^{6}} d z \end{array}

(17b)

\begin{array}{l} (A_{3}, B_{3}, D_{3}, F_{3}, G_{3}, H_{3}) = \\ \int_{- h / 2}^{h / 2} \frac{E (z, x)}{2 [1 + υ (z, x)]} {1, z, z^{2}, z^{3}, z^{4}, z^{6}} d z \end{array}

(17c)

The strain energy from the initial stresses due to the temperature rise U_T is

U_{T} = \frac{1}{2} \int_{Λ} [σ_{x x}^{T} d_{x x} + σ_{y y}^{T} d_{y y}] d Λ

(18)

where $d_{i j}$ are nonlinear strain–displacement components and are calculated from the following equation

d_{i j} = u_{, i} u_{, j} + v_{, i} v_{, j} + w_{, i} w_{, j} i, j = x, y

(19)

By substituting $d_{i j}$ into equation (18), the strain energy due to the temperature rise is

U_{T} = \frac{1}{2} \int_{Λ} [\begin{array}{l} σ_{x x}^{T} [{(u_{1, x})}^{2} + {(u_{2, x})}^{2} + {(u_{3, x})}^{2}] + \\ σ_{y y}^{T} [{(u_{1, y})}^{2} + {(u_{2, y})}^{2} + {(u_{3, y})}^{2}] \end{array}] d Λ

(20)

where $σ_{x x}^{T}, σ_{y y}^{T}$ are the stress components due to temperature rise. The strain energy associated with the couple stress theory $U_{S N C}$ is

\begin{array}{l} U_{S N C} = \frac{1}{2} \int_{Λ} (m_{i j} χ_{i j}) d Λ = \\ \frac{1}{2} \int_{Λ} (\begin{array}{l} m_{x x} χ_{x x} + m_{y y} χ_{y y} + 2 m_{x z} χ_{x z} + \\ 2 m_{x y} χ_{x y} + 2 m_{y z} χ_{y z} + m_{z z} χ_{z z} \end{array}) d Λ \end{array}

(21)

Equation (22) is rewritten as follows

U_{S N C} = \frac{1}{2} \int_{0}^{a} \int_{0}^{b} [\begin{array}{l} \frac{Y_{x}}{2} (w_{, y x} - ϕ_{y, x}) + \frac{3 κ T_{x}}{2} (ϕ_{y, x} + w_{, y x}) \\ - \frac{Y_{y}}{2} (w_{, y x} - ϕ_{x, y}) - \frac{3 κ T_{y}}{2} (ϕ_{x, y} + w_{, y x}) \\ + \frac{Y_{z} - 3 κ T_{z}}{2} (ϕ_{y, x} - ϕ_{x, y}) + \\ \frac{Y_{x y} + 3 κ T_{x y}}{2} (w_{, y y} - w_{, x x}) \\ + \frac{Y_{x y} - 3 κ T_{x y}}{2} (ϕ_{x, x} - ϕ_{y, y}) \\ + \frac{Y_{x z}}{2} (v_{, x x} - u_{, x y}) + \\ \frac{H_{x z} - κ L_{x z}}{2} (ϕ_{y, x x} - ϕ_{x, x y}) \\ + 3 κ H_{x z} (w_{, y} + ϕ_{y}) + \frac{Y_{y z}}{2} (v_{, x y} - u_{, y y}) \\ + \frac{H_{y z} - κ L_{y z}}{2} (ϕ_{y, x y} - ϕ_{x, y y}) \\ - 3 κ H_{y z} (w_{, x} + ϕ_{x}) \end{array}] d y d x

(22)

Following integrals are applied to define ${A^{'}}_{i}, {B^{'}}_{i}, {D^{'}}_{i}, {F^{'}}_{i}, {G^{'}}_{i}$ and ${H^{'}}_{i}$

{\begin{matrix} N_{i j} \\ M_{i j} \\ R_{i j} \\ P_{i j} \end{matrix}} = \int_{- h / 2}^{h / 2} σ_{i j} {\begin{matrix} 1 \\ z \\ z^{2} \\ z^{3} \end{matrix}} d z, {\begin{matrix} Y_{i j} \\ H_{i j} \\ T_{i j} \\ L_{i j} \end{matrix}} = \int_{- h / 2}^{h / 2} m_{i j} {\begin{matrix} 1 \\ z \\ z^{2} \\ z^{3} \end{matrix}} d z

(23)

Strain energy for FG microplate using classical plate theory is included in Appendix 2. The kinetic energy of the FG microplate is

T_{P} = \frac{1}{2} \int_{0}^{a} \int_{0}^{b} \int_{0 - h / 2}^{h / 2} ρ (z, x) ({\overset{\cdot}{u}}_{1}^{2} + {\overset{\cdot}{u}}_{2}^{2} + {\overset{\cdot}{u}}_{3}^{2}) d z d y d x

(24)

Temperature distribution

Two types of temperature distribution are assumed through the microplate. In the first type, a uniform temperature distribution is assumed through the body. In the second type, a sinusoidal temperature distribution is assumed through the thickness and length of the plate.

Uniform temperature rise

The uniform temperature rise though the microplate is represented as

T = Δ T + T_{0}

(25)

where ∆T denotes the uniform temperature change and $T_{0}$ is initial temperature of the structure. At room temperature, $T_{0} = 300 ° K$ .

Sinusoidal temperature rise

The temperature distribution of the plate should be determined by solving the heat transfer equation. However, in this case the heat transfer coefficient varies along the two directions, and the heat transfer equation changes to a differential equation with variable coefficients. As the result, no analytical solution could be found for the temperature distribution, and approximate temperature distribution which satisfies the thermal boundary conditions is employed. Generally, the form of this approximate solution is sinusoidal.^46,49,50 The temperature field for the second type (i.e. the sinusoidal temperature distribution) is represented as

T (z, x) = T_{b} + (T_{c} - T_{b}) \times \sin (\frac{π}{a} x) \times \sin (\frac{π z}{h} + \frac{π}{2})

(26)

where T_b and T_c are constant values. T_b is the temperature of the four surfaces of the plate which are perpendicular to x and z axis. Parameter T_c is the temperature of the central line of the plate along the y axis which is located at $(z = 0, x = a / 2, 0 \leq y \leq b)$ .

Mechanical boundary conditions

Assuming the simply supported conditions for all edges of the plate, the mechanical boundary conditions are

u_{2} = u_{3} = ϕ_{y} = N_{x} = M_{x} = 0 at x = 0, a

(27a)

u_{1} = u_{3} = ϕ_{x} = N_{y} = M_{y} = 0 at y = 0, b

(27b)

The following displacement functions satisfy the simply supported boundary conditions given in equation (27)

u_{1} = \sum_{m = 1}^{N} \sum_{n = 1}^{N} A_{m, n} \cos (\frac{m π x}{a}) \sin (\frac{n π y}{b}) e^{i ω t}

(28a)

u_{2} = \sum_{m = 1}^{N} \sum_{n = 1}^{N} B_{m, n} \sin (\frac{m π x}{a}) \cos (\frac{n π y}{b}) e^{i ω t}

(28b)

u_{3} = \sum_{m = 1}^{N} \sum_{n = 1}^{N} C_{m, n} \sin (\frac{m π x}{a}) \sin (\frac{n π y}{b}) e^{i ω t}

(28c)

ϕ_{x} = \sum_{m = 1}^{N} \sum_{n = 1}^{N} D_{m, n} \cos (\frac{m π x}{a}) \sin (\frac{n π y}{b}) e^{i ω t}

(28d)

ϕ_{y} = \sum_{m = 1}^{N} \sum_{n = 1}^{N} E_{m, n} \sin (\frac{m π x}{a}) \cos (\frac{n π y}{b}) e^{i ω t}

(28e)

where t is the time, $ω$ denotes the natural frequency of vibration, $i = \sqrt{- 1}$ , A_m,n, B_m,n, C_m,n, D_m,n, and E_m,n are unknown time coefficients. Parameters m and n are the number of half-waves in x and y directions, respectively.

Application of Rayleigh–Ritz method

In equation (28), the multipliers A_m,n, B_m,n, C_m,n, D_m,n, and E_m,n are calculated using the Ritz method. To this aim, the Lagrangian function is defined as

Π = \sum U_{\max} - \sum T_{\max}

(29)

Applying the Rayleigh–Ritz method and substitution of admissible displacements given in equation (28), the Lagrangian functional is determined. Now, we extermize the Lagrangian with respect to the coefficients A_m,n, B_m,n, C_m,n, D_m,n, and E_m,n. That is

\begin{array}{l} \frac{\partial Π}{\partial A_{m, n}} = 0, \frac{\partial Π}{\partial B_{m, n}} = 0, \\ \frac{\partial Π}{\partial C_{m, n}} = 0, \frac{\partial Π}{\partial D_{m, n}} = 0, \frac{\partial Π}{\partial E_{m, n}} = 0 \end{array}

(30)

Substituting equation (28) into equations (16), (20), (22), and (24), the frequency equation is obtained as follows

([K] - ω^{2} [M]) q = {0}

(31)

In this equation, ω is the natural frequency, [M] is the mass matrix and [K] is the stiffness matrix. Elements of these sub-matrices are given in Appendices 1 and 2 for TSDT and CPT respectively. Solution of this eigenvalue problem results to the natural frequency of the FG microplate.

Validation

To best of our knowledge, no other authors have considered all conditions considered in this work simultaneously. So to validate the results, we have reproduced some data of other authors in several steps. In first step, the size-dependent behavior of the microplate is verified with results of Ke et al.²³ To this aim, the nonhomogeneous microplate is simplified to a homogeneous one by setting both power indexes to zero. Then, based on the classical and third-order plate theories, fundamental natural frequencies of this work are compared with results of Ke et al.²³ The fundamental frequencies in Ref. [23] are based on the Mindlin plate theory. Material properties for data given in Table 1 are $E = 1.44 Gpa, ρ = 1220 {kg / m}^{3}, l = 17.6 \times 10^{- 6} m$ and $υ = 0.38$ . Table 1 shows that all results are almost identical for a thin homogeneous microplate. However, deviation between the natural frequencies of this work and those of Ke et al.²³ increases by thickening of the homogeneous microplate. In the second step, effect of temperature on natural frequencies is validated with Shahrjerdi et al.⁴⁶ To this aim, we set the length scale parameter and power index $η_{x}$ to zero and the 2D-FG microplate problem simplifies to 1D-FG plate. Figure 2 shows the effect of uniform temperature rise on natural frequencies of the plate, compared with Shahrjerdi et al.⁴⁶ These natural frequencies are obtained for a simply supported 1D-FG plate made of ZrO2/Ti–6Al–4V. Application of second-order plate theory in Shahrjerdi et al.⁴⁶ is the reason for slight differences between this work and Shahrjerdi et al.⁴⁶ Compared with 3D vibration analysis of Li et al.,⁴⁷ effects of through thickness gradient and length-to-thickness ratio on natural frequencies of a simply supported 1D-FG plate are demonstrated in Tables 2 and 3. Table 2 shows the natural frequencies for various uniform temperature rise and Table 3 shows the natural frequencies for various power law indexes. Close agreement is obtained between this work and Li et al.⁴⁷ In these tables, it is assumed that the plate is made of Si3N4/SUS304 FGM material. The frequency parameter Ω in these tables is defined as $Ω = \frac{ω b^{2}}{π^{2}} \sqrt{\frac{I_{0}}{D_{0}}},$ where $I_{0} = h ρ$ and $D_{0} = E h^{3} / 12 (1 - ν^{2})$ . The parameters $ρ$ , $E,$ and $ν$ are given in Table 4 for materials which are introduced in this article.

Table 1.

First natural frequency (MHz) of Simply supported homogeneous microplates (a/b = 1 and h = 2l).

Method	a/h
Method	5	10	20	30
Ke et al.²³	1.4701	0.4042	0.1040	0.0465
This study (CPT)	1.6288	0.4160	0.1048	0.0466
This study (TSDT)	1.4381	0.4014	0.1038	0.0464

CPT: classical plate theory; TSDT: third-order Vlasov–Reddy theory.

Table 2.

Verification of the fundamental frequency parameter Ω for 1D-FG plate (Si₃N₄/ SUS304) (a/b = 1 and a/h = 10).

Method	η_z	T (K)
Method	η_z	300	800	Reduce (%)
Li et al.⁴⁷	1	2.7222	2.3850	12.39
Present (TSDT)	1	2.6942	2.3496	12.78
Li et al.⁴⁷	10	2.0867	1.7654	15.39
Present (TSDT)	10	2.0501	1.7472	14.77

TSDT: third-order Vlasov–Reddy theory.

Table 3.

Verification of the fundamental frequency parameter Ω for 1D-FG plate (Si₃N₄/SUS304) (∆T = 300°K and a/b = 1).

h/b	Method	Power law index (η_z)
h/b	Method	1	2	5	10
0.1	This study (TSDT)	2.4932	2.2220	2.0051	1.9012
0.1	Li et al.⁴⁷	2.5511	2.2690	2.0433	1.9323
0.2	This study (TSDT)	2.3922	2.1405	1.9421	1.8345
0.2	Li et al.⁴⁷	2.3902	2.1289	1.9182	1.8160

TSDT: third-order Vlasov–Reddy theory.

Table 4.

Material properties of FG plate in T₀ = 300°K.

Material	Property
Material	E (GPa)	α (1/K) × 10⁻⁶	ρ (kg/m³)
Al₂O₃	393	7.4	3960
SiC	427	4.3	3100
ZrO₂	168.063	18.591	3657
Si₃N₄	322.2715	7.474	2370
Ti–6Al–4 V	105.698	6.941	4420
Al	70	23.4	2702
SUS304	207.7877	15.321	8166

Figure 2.

Verification of the first two normalized frequencies Ω for 1D-FG microplate (ZrO₂/Ti–6Al–4V) (a/b = 1, a = 0.2 m, a/h = 10, η_x = 0, and η_z = 1).

Result and discussion

Natural frequencies of 2D-FG microplates are investigated in this work. The material of microplate is assumed to be Ti–6Al–4V, Al, SiC, and Si₃N₄ for metal type 1 (m₁), metal type 2 (m₂), ceramic type 1 (c₁), and ceramic type 2 (c₂), respectively. In Table 4, mechanical properties of FG plate at 300°K are specified. The simply supported boundary condition is assumed for all edges of the microplate and the length scale parameter is $l = 17.6 \times 10^{- 6} m$ . For sinusoidal temperature rise, T_b is 300°K. Using equation (24), the temperature values on perpendicular planes to x and z axis are

\begin{array}{l} T_{t o p} (\frac{h}{2}, x) = T_{b o t t o m} (- \frac{h}{2}, x) = \\ T_{r i g h t} (z, 0) = T_{l e f t} (z, a) = 300 ° K \end{array}

(32)

In Figure 3, the volume fraction of the first type of ceramic $V_{c_{1}}$ in x and z directions is plotted. This figure is plotted for a = 1 m and h = 0.2 m. As it can be seen from this figure, the value of $V_{c_{1}}$ is equal to unity at corner which is located at (z, x) = (1, 0.1) and it is zero at other corners, which means that the material at this coordinate is made of ceramic of type 1 and this ceramic type does not exist at other three corners.

Figure 3.

Variations of the volume fraction $V_{c_{1}}$ through the thickness and length directions (η_x = η_z = 8).

Figure 4 shows the plot of sinusoidal temperature distribution in x and z directions for $T_{c} = 800 ° K$ , a = 1 m, h = 0.2 m. As the figure shows, the temperature is minimum and equal to 300°K at four edges of the plate. The temperature is maximum and equal to 800 K at center of the plate. Natural frequencies of a 2D-FG microplate in thermal environment are listed in Tables 5 –7 for variations of a/b, a/h, T, η_x, and η_z. In the first column of these tables, the first number in parentheses represents η_x and the second number represents η_z. For the case which these numer are (0, 0), all volume fractions excluding the second metal volume fraction $(V_{m_{2}})$ , become zero. It means that the microplate is made of pure second metal. These tables show that higher natural frequencies are obtained by increasing the length-to-width and length-to-thickness ratios. Also, the temperature rise both in uniform and sinusoidal distributions decreases the natural frequencies. Since higher values of power indexes $η_{x}$ and $η_{z}$ are interpreted as higher volume fractions of metal and ceramic, higher values of $η_{x}$ and $η_{z}$ reduce the natural frequencies of the microplate, respectively. This is due to the fact that ceramic materials are brittle and their natural frequencies are higher, and the higher value of $η_{x}$ is identical with higher natural frequencies. However, increasing $η_{z}$ for the case which $η_{x} = 0$ results to increase in percent of m₁ (first metal) and decrease in percent of m₂ (second metal). Increasing $η_{z}$ for the case which $η_{x} = \infty$ results to increase in percent of c₁ and decrease in percent of c₂. Since brittleness of the first material type (i.e. first metal or first ceramic) is higher than the second material type, higher natural frequencies are obtained by increasing $η_{z}$ for both cases of $η_{x} = 0$ and $η_{x} = \infty$ .

Figure 4.

Sinusoidal temperature distribution T(x, z) along the thickness and length directions (T_c = 800°K).

Table 5.

First natural frequency (MHz) of the microplate in thermal environment (a/b = 0.5, a/h = 10, and h = l).

(η_x, η_z)	Uniform temperature rise			Sinusoidal temperature rise
	T_c (K)
	300	600	800	300	600	800
(0, 0)	12.2348	11.7503	11.4159	12.2348	12.0799	14.2183
(0.1, 0)	14.3818	13.8552	13.4927	14.3818	14.2183	14.1082
(0.5, 0)	18.2754	17.7868	17.4535	18.2754	18.1128	18.0036
(0, 0.5)	12.2715	11.8968	11.6403	12.2715	12.1456	12.0637
(1, 0.5)	20.1479	19.7913	19.550	20.1479	20.0238	19.9406
(1, 1)	20.2182	19.9009	19.6865	20.2182	20.109	20.0359
(2, 2)	22.4162	22.1495	21.9699	22.4162	22.3269	22.2671

Table 6.

First three natural frequency (MHz) of the microplate in thermal environments (a/b = 1, a/h = 10, and h = l).

(η_x, η_z)	Frequency no.	Uniform temperature rise			Sinusoidal temperature rise
		T_c (K)
		300	600	800	300	600	800
(0, 0)	ω₁	5.71792	5.2926	4.98895	5.71792	5.54948	5.43428
	ω₂	12.2348	11.75030	11.41585	12.2348	12.0016	11.8436
	ω₃	12.2348	11.75030	11.41585	12.2348	12.0486	11.9229
(0.1, 0)	ω₁	6.78824	6.33983	6.02237	6.78824	6.61437	6.49587
	ω₂	14.3294	13.81231	13.45135	14.3294	14.087	13.923
	ω₃	14.3367	13.81987	13.4694	14.3367	14.1429	14.0122
(0.5, 0)	ω₁	8.61738	8.18517	7.88387	8.61738	8.44018	8.31995
	ω₂	18.2078	17.7029	17.3525	18.2078	17.9646	17.7952
	ω₃	18.2157	17.7128	17.3749	18.2157	18.011	17.8786
(0, 0.5)	ω₁	5.73129	5.40461	5.17537	5.73129	5.59898	5.50900
	ω₂	12.2449	11.871	11.615	12.2449	12.0544	11.9944
	ω₃	12.2449	11.871	11.615	12.2449	12.0544	11.9944
(1, 0.5)	ω₁	9.41908	9.09506	8.87248	9.41908	9.32742	9.24552
	ω₂	20.1479	19.7587	19.4949	20.1479	19.9297	19.8087
	ω₃	20.2533	19.8869	19.6389	20.2533	20.0728	19.9247
(1, 1)	ω₁	9.44894	9.16177	8.96522	9.44894	9.28049	9.18694
	ω₂	20.2182	19.8742	19.6414	20.2182	20.0463	19.9309
	ω₃	20.3743	20.0485	19.8283	20.3743	20.2399	20.1498
(2, 2)	ω₁	10.3782	10.136	9.9712	10.3782	10.2784	10.2113
	ω₂	22.5179	22.2324	22.04	22.5179	22.379	22.2859
	ω₃	22.6761	22.4065	22.2249	22.6761	22.5677	22.4952

Table 7.

Fundamental frequency (MHz) of the microplate for uniform temperature rise and various length-to-height ratios (T_c = 800°K and h = l).

(η_x, η_z)	a/b	a/h
		15	20	15	20
		Uniform temperature rise		Sinusoidal temperature rise
(0, 0)	0.5	5.08292	2.59451	5.62924	3.16815
(0, 0)	1	4.98895	1.87154	2.36132	1.21719
(0.1, 0)	0.5	6.04668	3.12961	6.64532	3.75323
(0.1, 0)	1	6.02237	2.34216	2.85182	1.49695
(0.5, 0)	0.5	7.98787	4.30203	8.51646	4.84107
(0.5, 0)	1	7.88387	3.23928	3.69443	1.97924
(0, 0.5)	0.5	5.28707	2.80917	5.69000	3.23047
(0, 0.5)	1	5.17537	2.0782	2.43098	1.28923
(1, 0.5)	0.5	9.11124	5.05614	9.48216	5.42774
(1, 0.5)	1	8.87248	3.80893	4.12849	2.25797
(1, 1)	0.5	9.20406	5.13631	9.53528	5.46697
(1, 1)	1	8.96522	3.88669	4.16968	2.29424
(2, 2)	0.5	10.3422	5.83256	10.6226	6.1106
(2, 2)	1	9.9712	4.39060	4.63052	2.57266

Compared the classical and the modified stress couple theories (MSCT), Figure 5 represents variation of first two natural frequency of the 2D-FG microplate versus height-to-length ratio for uniform temperature rise. The figure shows that higher height-to-length ratio decreases the natural frequencies of the microplate. The slope of the curve is high for small values of height-to-length ratio, and the difference between the classical theory and stress couple theory is considerable for $h / l$ ≤ 5. The neglected length scale parameter in classical theory is responsible for this difference.

Figure 5.

Comparison of the first two natural frequency (MHz) of the microplate between the classic and stress couple theories for uniform temperature rise (η_x = 2, η_y = 2, a/h = 10, a/b = 1, and T = 600°K).

Figure 6 compares the natural frequency of 2D-FG microplate between the TSDT and CPT. As the figure shows, difference between the natural frequencies obtained by these theories is increased for $h / l$ ≤ 5. This is due to the fact that the effects of shear forces are predicted more accurately by TSDT, and this effect is significant for $h / l$ ≤ 5.

Figure 6.

Comparison of the first natural frequency (MHz) of the microplate between classic and stress couple theories for uniform temperature rise (η_x = 1, η_y = 1, a/h = 5, a/b = 1, and T = 600°K).

Figures 7 and 8 show the plot of natural frequencies versus η_x and η_z, repectively. Increasing η_x results to increase in percent of metal, while increase in η_z results to decrease in percent of first material (either ceramic or metal). However, due to the layout of material at four corners of the plate, the difference between the stiffness of metal and ceramic is more tangible in comparison with the stiffness of the first and the second materials (either ceramic or metal). Therefore, the slope of natural frequency versus η_x is higher. Figure 9 shows the plot of natural frequencies versus η_x and η_z. The figure shows that increasing η_x and η_z result to increase in natural frequency.

Figure 7.

First natural frequency (MHz) of the microplate versus power index η_x for various h/l and uniform tmperature rise (η_y = 2, T = 600°K, a/h = 10, and a/b = 1).

Figure 8.

First natural frequency (MHz) of the microplate versus the power index η_z for various h/l and uniform temperature rise (η_x = 4, T = 600°K, a/h = 10, and a/b = 1).

Figure 9.

First natural frequency (MHz) of the microplate versus η_x and η_z for uniform temperature rize (h/l = 1.5, T = 600°K, a/h = 10, and a/b = 1).

Figures 10 and 11 show the plot of natural frequency of the microplate versus uniform and sinusoidal temperature rise, respectively. As it can be seen from the figures, the tempreature rise decreases the natural frequencies for both cases. While increase in h/l result to decrease in natural frequency, it does not effect the slope of the curves. Figure 12 is plot of natural frequency versus a/h at T_c = 300 and 800°K. Higher value of a/h increases the flexibility of the microplate. Therefore, the natural frequency of the plate decreases. Furthermore, effect of temperature on natural frequency increases with reduction in stiffness of the plate.

Figure 10.

First natural frequency (MHz) of the microplate versus the uniform temperature rise for various h/l (η_x = 1, η_y = 1, a/h = 30, and a/b = 0.5).

Figure 11.

Variation of the first natural frequency (MHz) of the microplate for sinusoidal temperature rise for various h/l (η_x = 1, η_y = 1, a/h = 30, and a/b = 0.5).

Figure 12.

First natural frequency (MHz) of the microplate versus a/h for various T_c with sinosoidal temperature rise (η_x = 1, η_y = 1, h/l = 1, and a/b = 0.5).

Figure 13 compares variation of natural frequency with respect to the temperature for uniform and sinusoidal temperature rise. For the case of sinusoidal temparture rise, the temperature is $T_{c}$ . Since in sinusoidal temperature distribution, the temperature at edges of the plate is constant and varies sinusoidal through the thickness and length of the plate, reduction in stiffness of the plate is less tangible in comparison with the case of uniform temperature rise. In other words, the temperature in all regions of the plate for uniform temperature rise is higher than sinusoidal temperature rise.

Figure 13.

Comparison of the first natural frequency (MHz) of the microplate versus the temperature rise for uniform and sinusoidal condition when h/l = 1.5, a/b = 1, and a/h = 15.

Conclusion

In this article, thermoelastic vibration of a 2D-FG microplate based on the third-order shear deformation theory is investigated using the modified couple stress theory. The numerical Rayleigh–Ritz method is applied to obtain the natural frequencies. Analyzing the natural frequencies obtained by eigen value problem solution yields to the following conclusions.

Increasing the volume fraction powers in directions of x and z result to higher natural frequencies. Increasing the thickness-to-length ratio of the plate results to lower natural frequencies. For thickness-to-length ratio less than five, the difference between the classical theory and modified couple stress theory is significant. According to results obtained by classical and third-order shear deformation theories, the third-order shear deformation theory is proposed for vibration analysis of microplates with thickness-to-length ratio less than five. Increasing the geometric ratios such as length-to-thickness and length-to-width of the plate results to decrease in natural frequencies. Increasing the temperature reduces the natural frequencies, and this reduction of natural frequencies is more tangible for uniform temperature rise in comparison with sinusoidal temperature rise.

Footnotes

Appendix 1

The stiffness matrix is defined as the sum of matrices classical stiffness matrix [K]_SNC, couple–stress stiffness matrix [K]_SC, and thermal stiffness matrix [K]_T. The stiffness and mass matrix in equation (31) is

(33)

{[K]}_{S C} = \frac{\partial^{2} U_{S C}}{\partial q_{i} \partial q_{j}}, {[K]}_{T} = \frac{\partial^{2} U_{T}}{\partial q_{i} \partial q_{j}}, {[K]}_{S N C} = \frac{\partial^{2} U_{S N C}}{\partial q_{i} \partial q_{j}}, [M] = \frac{\partial^{2} T}{\partial q_{i} \partial q_{j}}

where $q = {A_{m, n}, B_{m, n}, C_{m, n}, D_{m, n}, E_{m, n}}$ includes five unknown time coefficients of the admissible trial functions for TSDT. Therefore, the stiffness matrices and mass matrix are composed of (5 × 5) sub-matrices for TSDT

\begin{array}{l} {[K]}_{S C} = [\begin{matrix} {[K_{m n}^{a a}]}_{S C} & {[K_{m n}^{a b}]}_{S C} & {[K_{m n}^{a c}]}_{S C} & {[K_{m n}^{a d}]}_{S C} & {[K_{m n}^{a e}]}_{S C} \\ {[K_{m n}^{b b}]}_{S C} & {[K_{m n}^{b c}]}_{S C} & {[K_{m n}^{b d}]}_{S C} & {[K_{m n}^{b e}]}_{S C} \\ {[K_{m n}^{c c}]}_{S C} & {[K_{m n}^{c d}]}_{S C} & {[K_{m n}^{c e}]}_{S C} \\ {[K_{m n}^{d d}]}_{S C} & {[K_{m n}^{d e}]}_{S C} \\ s y m & {[K_{m n}^{e e}]}_{S C} \end{matrix}] \\ {[K]}_{S N C} = [\begin{matrix} {[K_{m n}^{a a}]}_{S N C} & {[K_{m n}^{a b}]}_{S N C} & {[K_{m n}^{a c}]}_{S N C} & {[K_{m n}^{a d}]}_{S N C} & {[K_{m n}^{a e}]}_{S N C} \\ {[K_{m n}^{b b}]}_{S N C} & {[K_{m n}^{b c}]}_{S N C} & {[K_{m n}^{b d}]}_{S N C} & {[K_{m n}^{b e}]}_{S N C} \\ {[K_{m n}^{c c}]}_{S N C} & {[K_{m n}^{c d}]}_{S N C} & {[K_{m n}^{c e}]}_{S N C} \\ {[K_{m n}^{d d}]}_{S N C} & {[K_{m n}^{d e}]}_{S N C} \\ s y m & {[K_{m n}^{e e}]}_{S N C} \end{matrix}] \\ {[K]}_{T} = [\begin{matrix} {[K_{m n}^{a a}]}_{T} & {[K_{m n}^{a b}]}_{T} & {[K_{m n}^{a c}]}_{T} & {[K_{m n}^{a d}]}_{T} & {[K_{m n}^{a e}]}_{T} \\ {[K_{m n}^{b b}]}_{T} & {[K_{m n}^{b c}]}_{T} & {[K_{m n}^{b d}]}_{T} & {[K_{m n}^{b e}]}_{T} \\ {[K_{m n}^{c c}]}_{T} & {[K_{m n}^{c d}]}_{T} & {[K_{m n}^{c e}]}_{T} \\ {[K_{m n}^{d d}]}_{T} & {[K_{m n}^{d e}]}_{T} \\ s y m & {[K_{m n}^{e e}]}_{T} \end{matrix}] \end{array}

(34)

[M] = [\begin{matrix} [M_{m n}^{a a}] & [M_{m n}^{a b}] & [M_{m n}^{a c}] & [M_{m n}^{a d}] & [M_{m n}^{a e}] \\ [M_{m n}^{b b}] & [M_{m n}^{b c}] & [M_{m n}^{b d}] & [M_{m n}^{b e}] \\ [M_{m n}^{c c}] & [M_{m n}^{c d}] & [M_{m n}^{c e}] \\ [M_{m n}^{d d}] & [M_{m n}^{d e}] \\ s y m & [M_{m n}^{e e}] \end{matrix}]

The elements of the stiffness matrix and the mass matrix are given as follow

{[K_{m n}^{a a}]}_{S C} = \frac{1}{2} [A_{1} ℜ_{A A}^{{(u_{, x})}^{2}} + A_{3} ℜ_{A A}^{(u_{, y}) 2}]

{[K_{m n}^{b b}]}_{S C} = \frac{1}{2} [A_{1} ℜ_{B B}^{{(v_{, y})}^{2}} + A_{3} ℜ_{B B}^{{(v_{, x})}^{2}}]

\begin{array}{l} {[K_{m n}^{c c}]}_{S C} = \frac{1}{2} A_{3} [ℜ_{C C}^{{(w_{, x})}^{2}} + ℜ_{C C}^{{(w_{, y})}^{2}}] - D_{3} \frac{4}{h^{2}} [ℜ_{C C}^{{(w_{, y})}^{2}} + ℜ_{C C}^{{(w_{, y})}^{2}}] \\ + \frac{8}{h^{4}} G_{3} [ℜ_{C C}^{{(w_{, x})}^{2}} + ℜ_{C C}^{{(w_{, y})}^{2}}] + \frac{1}{2} κ^{2} [H_{1} ℜ_{C C}^{{(w_{, x x})}^{2}} + H_{1} ℜ_{C C}^{{(w_{, y y})}^{2}} + 2 H_{2} Ξ_{C C}^{(w_{, x x}) (w_{, y y})} + 2 H_{3} ℜ_{C C}^{{(w_{, x y})}^{2}})] \end{array}

\begin{array}{l} {[K_{m n}^{d d}]}_{S C} = \frac{1}{2} [A_{3} ℜ_{D D}^{{(ϕ_{x})}^{2}} + D_{1} ℜ_{D D}^{{(ϕ_{x, x})}^{2}} + D_{3} ℜ_{D D}^{{(ϕ_{x, y})}^{2}} - \frac{8}{h^{2}} F_{3} ℜ_{D D}^{{(ϕ_{x})}^{2}}] - κ G_{1} ℜ_{D D}^{{(ϕ_{x, x})}^{2}} - \\ G_{3} [κ ℜ_{D D}^{{(ϕ_{x, y})}^{2}} + \frac{2}{h^{2}} ℜ_{D D}^{{(ϕ_{x})}^{2}}] - \frac{1}{2} κ [H_{1} ℜ_{D D}^{{(ϕ_{x, x})}^{2}} + H_{3} ℜ_{D D}^{{(ϕ_{x, y})}^{2}}] \end{array}

\begin{array}{l} {[K_{m n}^{e e}]}_{S C} = \frac{1}{2} [A_{3} ℜ_{E E}^{{(ϕ_{y})}^{2}} + D_{1} ℜ_{E E}^{{(ϕ_{y, y})}^{2}} + D_{3} ℜ_{E E}^{{(ϕ_{y, x})}^{2}} - \frac{8}{h^{2}} F_{3} ℜ_{E E}^{{(ϕ_{y})}^{2}}] - κ G_{1} ℜ_{E E}^{{(ϕ_{y, y})}^{2}} \\ + G_{3} [- κ ℜ_{E E}^{{(ϕ_{y, x})}^{2}} - \frac{2}{h^{2}} ℜ_{E E}^{{(ϕ_{y})}^{2}}] + \frac{1}{2} [- κ H_{1} ℜ_{E E}^{{(ϕ_{y, y})}^{2}} - κ H_{3} ℜ_{E E}^{{(ϕ_{y, x})}^{2}}] \end{array}

{[K_{m n}^{a b}]}_{S C} = A_{2} Ξ_{A B}^{(u_{, x}) (v_{, y})} + A_{3} Ξ_{A B}^{(u_{, y}) (v_{, x})}

{[K_{m n}^{a c}]}_{S C} = - κ [F_{1} Ξ_{A C}^{(u_{, x}) (w_{, x x})} + F_{2} Ξ_{A C}^{(u_{, x}) (w_{, y y})} + 2 F_{3} Ξ_{A C}^{(u_{, y}) (w_{, x y})}]

{[K_{m n}^{a d}]}_{S C} = B_{1} Ξ_{A D}^{(u_{, x}) (ϕ_{x, x})} + B_{3} Ξ_{A D}^{(u_{, y}) (ϕ_{x, y})} - κ [F_{1} Ξ_{A D}^{(u_{, x}) (ϕ_{x, x})} + F_{3} Ξ_{A D}^{(u_{, y}) (ϕ_{x, y})}]

{[K_{m n}^{a e}]}_{S C} = B_{2} Ξ_{A D}^{(u_{, x}) (ϕ_{y, y})} + B_{3} Ξ_{A D}^{(u_{, y}) (ϕ_{y, x})} - κ [F_{2} Ξ_{A D}^{(u_{, x}) (ϕ_{y, y})} + F_{3} Ξ_{A D}^{(u_{, y}) (ϕ_{y, x})}]

{[K_{m n}^{b c}]}_{S C} = - κ [F_{1} Ξ_{B C}^{(v_{, y}) (w_{, y y})} + F_{2} Ξ_{B C}^{(v_{, y}) (w_{, x x})} + 2 F_{3} Ξ_{B C}^{(v_{, x}) (w_{, x y})}]

{[K_{m n}^{b d}]}_{S C} = B_{2} Ξ_{B D}^{(v_{, y}) (ϕ_{x, x})} + B_{3} Ξ_{B D}^{(v_{, x}) (ϕ_{x, y})} - κ [F_{2} Ξ_{B D}^{(v_{, y}) (ϕ_{x, x})} + F_{3} Ξ_{B D}^{(v_{, x}) (ϕ_{x, y})}]

{[K_{m n}^{b e}]}_{S C} = B_{1} Ξ_{B E}^{(v_{, y}) (ϕ_{y, y})} + B_{3} Ξ_{B E}^{(v_{, x}) (ϕ_{y, x})} - κ [F_{1} Ξ_{B E}^{(v_{, y}) (ϕ_{y, y})} + F_{3} Ξ_{B E}^{(v_{, x}) (ϕ_{y, x})}]

\begin{array}{l} {[K_{m n}^{c d}]}_{S C} = \frac{1}{2} A_{3} Ξ_{C D}^{(ϕ_{x}) (w_{, x})} - \frac{4}{h^{2}} D_{3} Ξ_{C D}^{(ϕ_{x}) (w_{, x})} - G_{1} κ Ξ_{C D}^{(ϕ_{x, x}) (w_{, x x})} - G_{2} κ Ξ_{C D}^{(ϕ_{x, x}) (w_{, y y})} \\ - G_{3} [2 κ Ξ_{C D}^{(ϕ_{x, y}) (w_{, x y})} + \frac{2}{h^{2}} Ξ_{C D}^{(ϕ_{x}) (w_{, x})}] - κ H_{1} Ξ_{C D}^{(ϕ_{x, x}) (w_{, x x})} - κ^{2} H_{2} Ξ_{C D}^{(ϕ_{x, x}) (w_{, y y})} - 2 κ H_{3} Ξ_{C D}^{(ϕ_{x, y}) (w_{, x y})} \end{array}

\begin{array}{l} {[K_{m n}^{c e}]}_{S C} = \frac{1}{2} A_{3} Ξ_{C E}^{(ϕ_{y}) (w_{, y})} - \frac{4}{h^{2}} D_{3} Ξ_{C E}^{(ϕ_{y}) (w_{, y})} - G_{1} κ Ξ_{C E}^{(ϕ_{y, y}) (w_{, y y})} - G_{2} κ Ξ_{C E}^{(ϕ_{y, y}) (w_{, x x})} \\ - 2 G_{3} [κ Ξ_{C E}^{(ϕ_{y, x}) (w_{, x y})} + \frac{1}{h^{2}} Ξ_{C E}^{(ϕ_{y}) (w_{, y})}] - κ H_{1} Ξ_{C E}^{(ϕ_{y, y}) (w_{, y y})} + κ^{2} H_{2} Ξ_{C E}^{(ϕ_{y, y}) (w_{, x x})} - 2 κ H_{3} Ξ_{C E}^{(ϕ_{y, x}) (w_{, x y})} \end{array}

{[K_{m n}^{a a}]}_{S N C} = \frac{1}{32} A_{4} [ℜ_{A A}^{{(u_{, x y})}^{2}} + ℜ_{A A}^{{(u_{, y y})}^{2}}]

{[K_{m n}^{b b}]}_{S N C} = \frac{1}{32} A_{4} [ℜ_{B B}^{{(v_{, x x})}^{2}} + ℜ_{B B}^{{(v_{, x y})}^{2}}]

\begin{array}{l} {[K_{m n}^{c c}]}_{S N C} = \frac{7}{15} F_{4} [160 ℜ_{C C}^{{(w_{, x})}^{2}} + 160 ℜ_{C C}^{{(w_{, y})}^{2}}] + \frac{5}{3} D_{4} [ℜ_{C C}^{{(w_{, x x})}^{2}} + 8 ℜ_{C C}^{{(w_{, x y})}^{2}}] \\ + \frac{24}{5} G_{4} [3 ℜ_{C C}^{{(w_{, x x})}^{2}} + 24 ℜ_{C C}^{{(w_{, x y})}^{2}} + 3 ℜ_{C C}^{{(w_{, y y})}^{2}}] + \frac{1}{32} A_{4} [ℜ_{C C}^{{(w_{, x x})}^{2}} + 8 ℜ_{C C}^{{(w_{, x y})}^{2}} + ℜ_{C C}^{{(w_{, y y})}^{2}}] \end{array}

\begin{array}{l} {[K_{m n}^{d d}]}_{S N C} = \frac{7}{15} F_{4} [160 ℜ_{D D}^{{(ϕ_{x})}^{2}} - ℜ_{D D}^{{(ϕ_{x, x y})}^{2}} - ℜ_{D D}^{{(ϕ_{x, y y})}^{2}} - ℜ_{D D}^{{(ϕ_{x, x x})}^{2}}] \\ + \frac{88}{63} H_{4} [ℜ_{D D}^{{(ϕ_{x, x y})}^{2}} + ℜ_{D D}^{{(ϕ_{x, y y})}^{2}} - 2 Ξ_{D D}^{(ϕ_{x, x y}) (ϕ_{y, x x})}] \\ + \frac{1}{192} B_{4} [ℜ_{D D}^{{(ϕ_{x, x y})}^{2}} + ℜ_{D D}^{{(ϕ_{x, y y})}^{2}}] + \frac{5}{3} D_{4} [- ℜ_{D D}^{{(ϕ_{x, x})}^{2}} - 8 ℜ_{D D}^{{(ϕ_{x, y})}^{2}} + Ξ_{D D}^{(ϕ_{x}) (ϕ_{x, y y})}] \\ + \frac{24}{5} G_{4} [3 ℜ_{D D}^{{(ϕ_{x, x})}^{2}} + 24 ℜ_{D D}^{{(ϕ_{x, y})}^{2}} - 4 Ξ_{D D}^{(ϕ_{x}) (ϕ_{x, y y})}] + \frac{1}{32} A_{4} [ℜ_{D D}^{{(ϕ_{x, x})}^{2}} + ℜ_{D D}^{{(ϕ_{x, y})}^{2}}] \end{array}

{[K_{m n}^{a b}]}_{S N C} = - \frac{1}{16} A_{4} [Ξ_{A B}^{(u_{, x y}) (v_{, x x})} + Ξ_{A B}^{(u_{, y y}) (v_{, x y})}]

{[K_{m n}^{a c}]}_{S N C} = 0, {[K_{m n}^{b c}]}_{S N C} = 0, {[K_{m n}^{a d}]}_{S N C} = 0, {[K_{m n}^{a e}]}_{S N C} = 0, {[K_{m n}^{b d}]}_{S N C} = 0

\begin{array}{l} {[K_{m n}^{c d}]}_{S N C} = \frac{7}{15} F_{4} [320 (Ξ_{C D}^{(w_{, x}) (ϕ_{x})})] + \frac{5}{3} D_{4} [- 2 (Ξ_{C D}^{(w_{, y}) (ϕ_{x, x y})}) + 2 (Ξ_{C D}^{(w_{, x}) (ϕ_{x, y y})})] + \frac{1}{32} A_{4} [- 8 (Ξ_{C D}^{(w_{, x y}) (ϕ_{x, y})})] \\ + \frac{24}{5} G_{4} [- 6 (Ξ_{C D}^{(w_{, y y}) (ϕ_{x, x})}) + 4 (Ξ_{C D}^{(w_{, y}) (ϕ_{x, x y})}) - 4 (Ξ_{C D}^{(w_{, x}) (ϕ_{x, y y})}) + 24 (Ξ_{C D}^{(w_{, x y}) (ϕ_{x, y})}) - 6 (Ξ_{C D}^{(w_{, x x}) (ϕ_{x, x})})] \end{array}

\begin{array}{l} {[K_{m n}^{c e}]}_{S N C} = \frac{7}{15} F_{4} [320 (Ξ_{C E}^{(w_{, y}) (ϕ_{y})})] + \frac{5}{3} D_{4} [2 (Ξ_{C E}^{(w_{, y}) (ϕ_{y, x x})}) - 2 (Ξ_{C E}^{(w_{, x}) (ϕ_{y, x y})})] \\ + \frac{24}{5} G_{4} [24 (Ξ_{C E}^{(w_{, x y}) (ϕ_{y, x})}) - 4 (Ξ_{C E}^{(w_{, y}) (ϕ_{y, x x})}) + 4 (ℜ_{C E}^{(ϕ_{y, x y}) (w_{, x})}) + 6 (Ξ_{C E}^{(ϕ_{y, y}) (w_{, y y})}) - 6 (Ξ_{C E}^{(w_{, x x}) (ϕ_{y, y})})] \\ + \frac{1}{32} A_{4} [- 2 (Ξ_{C E}^{(w_{, x x}) (ϕ_{y, x})}) + 2 (Ξ_{C E}^{(w_{, y y}) (ϕ_{y, x})}) - 8 (Ξ_{C E}^{(w_{, x y}) (ϕ_{y, x})}) + 2 (Ξ_{C E}^{(w_{, x x}) (ϕ_{y, y})}) - 2 (Ξ_{C E}^{(w_{, y y}) (ϕ_{y, y})})] \end{array}

\begin{array}{l} {[K_{m n}^{d e}]}_{S N C} = \frac{7}{15} F_{4} [2 (Ξ_{D E}^{(ϕ_{x, x y}) (ϕ_{y, x x})}) + 2 (Ξ_{D E}^{(ϕ_{x, y y}) (ϕ_{y, x y})})] + \frac{88}{63} H_{4} [- 2 (Ξ_{D E}^{(ϕ_{x, x y}) (ϕ_{y, x x})}) - 2 (Ξ_{D E}^{(ϕ_{x, y y}) (ϕ_{y, x y})})] \\ + \frac{1}{192} B_{4} [- 2 (Ξ_{D E}^{(ϕ_{x, x y}) (ϕ_{y, x x})}) - 2 (Ξ_{D E}^{(ϕ_{x, y y}) (ϕ_{y, x y})})] + \frac{5}{3} D_{4} [- 2 (Ξ_{D E}^{(ϕ_{x, x y}) (ϕ_{y})}) + 8 (Ξ_{D E}^{(ϕ_{x, y}) (ϕ_{y, x})}) - 2 (Ξ_{D E}^{(ϕ_{x}) (ϕ_{y, x y})}) + 2 (Ξ_{D E}^{(ϕ_{x, x}) (ϕ_{y, y})})] \\ + \frac{24}{5} G_{4} [4 (Ξ_{D E}^{(ϕ_{x, x y}) (ϕ_{y})}) - 24 (Ξ_{D E}^{(ϕ_{x, y}) (ϕ_{y, x})}) + 4 (Ξ_{D E}^{(ϕ_{x}) (ϕ_{y, x y})}) - 6 (Ξ_{D E}^{(ϕ_{x, x}) (ϕ_{y, y})})] + \frac{1}{32} A_{4} [- 8 (Ξ_{D E}^{(ϕ_{x, y}) (ϕ_{y, x})}) - (Ξ_{D E}^{(ϕ_{x, x}) (ϕ_{y, y})})] \end{array}

{[K_{m n}^{a a}]}_{T} = \frac{1}{2} [A_{1}^{T} ℜ_{A A}^{{(u_{, x})}^{2}} + A_{3}^{T} ℜ_{A A}^{{(u_{, y})}^{2}}]

{[K_{m n}^{b b}]}_{T} = \frac{1}{2} [A_{1}^{T} ℜ_{B B}^{{(v_{, y})}^{2}} + A_{3}^{T} ℜ_{B B}^{{(v_{, x})}^{2}}]

{[K_{m n}^{c c}]}_{T} = \frac{1}{2} [A_{1}^{T} ℜ_{C C}^{{(w_{, x})}^{2}} + A_{3}^{T} ℜ_{C C}^{{(w_{, y})}^{2}} + κ^{2} H_{1}^{T} (ℜ_{C C}^{{(w_{, x x})}^{2}} + ℜ_{C C}^{{(w_{, y x})}^{2}}) + κ^{2} H_{3}^{T} (ℜ_{C C}^{{(w_{, y y})}^{2}} + ℜ_{C C}^{{(w_{, x y})}^{2}})]

{[K_{m n}^{d d}]}_{T} = \frac{1}{2} [[D_{1}^{T} ℜ_{D D}^{{(ϕ_{x, x})}^{2}} + D_{3}^{T} ℜ_{D D}^{{(ϕ_{x, y})}^{2}}] + κ^{2} [H_{1}^{T} ℜ_{D D}^{{(ϕ_{x, x})}^{2}} + H_{3}^{T} ℜ_{D D}^{{(ϕ_{x, y})}^{2}}]]

{[K_{m n}^{e e}]}_{T} = \frac{1}{2} [[D_{1}^{T} ℜ_{E E}^{{(ϕ_{y, x})}^{2}} + D_{3}^{T} ℜ_{E E}^{{(ϕ_{y, y})}^{2}}] + κ^{2} [H_{1}^{T} ℜ_{E E}^{{(ϕ_{y, x})}^{2}} + H_{3}^{T} ℜ_{E E}^{{(ϕ_{y, y})}^{2}}]]

{[K_{m n}^{a c}]}_{T} = \frac{1}{2} [- κ F_{1}^{T} Ξ_{A C}^{(u_{, x}) (w_{, x x})} - κ F_{3}^{T} Ξ_{A C}^{(u_{, y}) (w_{, x y})}]

{[K_{m n}^{a d}]}_{T} = B_{1} Ξ_{A D}^{(u_{, x}) (ϕ_{x, x})} + B_{3} Ξ_{A D}^{(u_{, y}) (ϕ_{x, y})} - κ [F_{1} Ξ_{A D}^{(u_{, x}) (ϕ_{x, x})} + F_{3} Ξ_{A D}^{(u_{, y}) (ϕ_{x, y})}]

{[K_{m n}^{a e}]}_{T} = 0

{[K_{m n}^{b c}]}_{T} = - \frac{1}{2} κ [2 F_{1}^{T} Ξ_{B C}^{(v_{, x}) (w_{, y x})} + 2 F_{3}^{T} Ξ_{B C}^{(v_{, y}) (w_{, y y})}]

{[K_{m n}^{b d}]}_{T} = 0

{[K_{m n}^{b e}]}_{T} = \frac{1}{2} [2 B_{1}^{T} Ξ_{B E}^{(v_{, x}) (ϕ_{y, x})} - 2 κ F_{1}^{T} Ξ_{B E}^{(v_{, x}) (ϕ_{y, x})} + (2 B_{3}^{T} - 2 κ F_{3}^{T}) Ξ_{B E}^{(v_{, y}) (ϕ_{y, y})}]

{[K_{m n}^{c d}]}_{T} = 0

{[K_{m n}^{c e}]}_{T} = \frac{1}{2} [2 κ^{2} H_{1}^{T} Ξ_{C E}^{(ϕ_{y, x}) (w_{, y x})} + 2 (κ^{2} H_{3}^{T} - κ G_{3}^{T}) κ Ξ_{C E}^{(ϕ_{y, y}) (w_{, y y})}]

[M_{m n}^{a a}] = I_{1} ℜ_{A A}^{{(u)}^{2}}

[M_{m n}^{b b}] = I_{1} ℜ_{B B}^{{(v)}^{2}}

[M_{m n}^{c c}] = I_{1} ℜ_{C C}^{{(w)}^{2}} + I_{6} κ^{2} [ℜ_{C C}^{{(w_{, x})}^{2}} + ℜ_{C C}^{{(w_{, y})}^{2}}]

[M_{m n}^{d d}] = I_{3} ℜ_{D D}^{{(ϕ_{x})}^{2}} - 2 I_{5} κ ℜ_{D D}^{{(ϕ_{x})}^{2}} + I_{6} κ^{2} ℜ_{D D}^{{(ϕ_{x})}^{2}}

[M_{m n}^{e e}] = I_{3} ℜ_{E E}^{{(ϕ_{y})}^{2}} - 2 I_{5} κ ℜ_{E E}^{{(ϕ_{y})}^{2}} + I_{6} κ^{2} ℜ_{E E}^{{(ϕ_{y})}^{2}}

[M_{m n}^{b c}] = - 2 I_{4} κ Ξ_{B C}^{(v) (w_{, y})}

[M_{m n}^{a c}] = - 2 I_{4} Ξ_{A C}^{(u) (w_{, x})}

[M_{m n}^{b e}] = 2 I_{2} Ξ_{B E}^{(v) (ϕ_{y})} - 2 I_{4} κ Ξ_{B E}^{(v) (ϕ_{y})}

[M_{m n}^{c d}] = - 2 I_{5} κ Ξ_{C D}^{(ϕ_{x}) (w_{, x})} + 2 I_{6} κ^{2} Ξ_{C D}^{(ϕ_{x}) (w_{, x})}

(35)

[M_{m n}^{d e}] = 0, [M_{m n}^{b d}] = 0, [M_{m n}^{a b}] = 0, [M_{m n}^{a d}] = 0

where

\begin{array}{l} ℜ_{Θ \tilde{Θ}}^{p} = \int_{0}^{a} \int_{0}^{b} \frac{\partial^{2} p}{\partial Θ_{m n} \partial {\tilde{Θ}}_{m n}} d y d x, Ξ_{Θ \tilde{Θ}}^{(p) (q)} = \int_{0}^{a} \int_{0}^{b} \frac{\partial p}{\partial Θ_{m n}} \frac{\partial q}{\partial {\tilde{Θ}}_{m n}} d y d x \\ (A_{4}, B_{4}, D_{4}, F_{4}, G_{4}, H_{4}) = 2 l^{2} μ \int_{- h / 2}^{h / 2} {1, z^{2}, z^{4}, z^{6}, z^{8}, z^{10}} d z \end{array}

(I_{1}, I_{2}, I_{3}, I_{4}, I_{5}, I_{6}) = \int_{- h / 2}^{h / 2} ρ (x, z) {1, z, z^{2}, z^{3}, z^{4}, z^{6}} d z

(A_{1}^{T}, B_{1}^{T}, D_{1}^{T}, F_{1}^{T}, G_{1}^{T}, H_{1}^{T}) = \int_{- h / 2}^{h / 2} σ_{x x}^{T} {1, z, z^{2}, z^{3}, z^{4}, z^{6}} d z

(36)

(A_{3}^{T}, B_{3}^{T}, D_{3}^{T}, F_{3}^{T}, G_{3}^{T}, H_{3}^{T}) = \int_{- h / 2}^{h / 2} σ_{y y}^{T} {1, z, z^{2}, z^{3}, z^{4}, z^{6}} d z

Appendix 2

The displacement field based on CPT is

u_{1} (x, y, z, t) = u_{0} (x, y, t) - z \frac{\partial w (x, y, t)}{\partial x}

u_{2} (x, y, z, t) = v_{0} (x, y, t) - z \frac{\partial w (x, y, t)}{\partial y}

(37)

u_{3} (x, y, z, t) = w (x, y, t)

The strain–displacement equations are

ε_{x} = ε_{x, 0} + z k_{x}^{(0)}

ε_{y} = ε_{y, 0} + z k_{y}^{(0)}

(38)

γ_{x y} = γ_{x y, 0} + z k_{x y}^{(0)}

where

ε_{x}^{(0)} = u_{, x} + \frac{1}{2} {(w_{, x})}^{2}

ε_{y}^{(0)} = v_{, y} + \frac{1}{2} {(w_{, y})}^{2}

γ_{x y}^{(0)} = u_{, y} + v_{, x} + w_{, x} w_{, y}

(39)

k_{x}^{(0)} = - w_{, x x}, k_{y}^{(0)} = - w_{, y y}, k_{x y}^{(0)} = - 2 w_{, x y}

The constitutive equations are

(40)

(\begin{matrix} σ_{x} \\ σ_{y} \\ σ_{x y} \end{matrix}) = [\begin{matrix} Q_{1} & Q_{2} & 0 \\ Q_{2} & Q_{1} & 0 \\ 0 & 0 & Q_{3} \end{matrix}] (\begin{matrix} ε_{x} \\ ε_{y} \\ ε_{x y} \end{matrix})

The stress components due to temperature rise ∆ $T (z)$ are

(41)

(\begin{matrix} σ_{x x}^{T} \\ σ_{y y}^{T} \end{matrix}) = (\begin{matrix} Q_{1} & Q_{2} \\ Q_{2} & Q_{1} \end{matrix}) [\begin{matrix} α (z, x) \\ α (z, x) \end{matrix}] Δ T (z)

Components of the rotation vector for CPT are

θ_{1} = w_{, y}

θ_{2} = - w_{, x}

(42)

θ_{3} = \frac{1}{2} (v_{, x} - u_{, y})

Components of the symmetric curvature tensor according to the CPT are

χ_{x x} = w_{, y x}

χ_{y y} = - w_{, y x}

χ_{x y} = \frac{1}{2} (w_{, y y} - w_{, x x})

χ_{x z} = \frac{1}{4} (v_{, x x} - u_{, x y})

χ_{y z} = \frac{1}{4} (v_{, x y} - u_{, y y})

(43)

χ_{z z} = 0

The strain energy associated with the classical elasticity theory on CPT is

(44)

U_{SC} = \int_{0}^{a} \int_{0}^{b} [\begin{array}{l} \frac{1}{2} A_{1} {ε_{x}^{{(0)}^{2}} + A_{1} ε_{y}^{{(0)}^{2}} + 2 A_{2} ε_{x}^{(0)} ε_{y}^{(0)} + A_{3} (γ_{x y}^{{(0)}^{2}} + γ_{x z}^{{(0)}^{2}} + γ_{y z}^{{(0)}^{2}})} \\ + B_{1} ε_{x}^{(0)} k_{x}^{(0)} + B_{1} ε_{y}^{(0)} k_{y}^{(0)} + B_{2} (ε_{x}^{(0)} k_{y}^{(0)} + ε_{y}^{(0)} k_{x}^{(0)}) + B_{3} γ_{x y}^{(0)} k_{x y}^{(0)} \\ + \frac{1}{2} {D_{1} k_{x}^{{(0)}^{2}} + D_{1} k_{y}^{{(0)}^{2}} + 2 D_{2} k_{x}^{(0)} k_{y}^{(0)} + D_{3} (k_{x y}^{{(0)}^{2}} + 2 k_{x z}^{(1)} γ_{x z}^{(0)} + 2 k_{y z}^{(1)} γ_{x z}^{(0)}} \\ + [R_{1} (ε_{x}^{(0)} + ε_{y}^{(0)}) + R_{2} (k_{x}^{(0)} + k_{y}^{(0)}) + R_{3} (k_{x}^{(2)} + k_{y}^{(2)})] \end{array}] d y d x

The strain energy associated with couple stress theory based on CPT is

(45)

U_{S N C} = \frac{1}{2} \int_{0}^{a} \int_{0}^{b} [Y_{x} (w_{, y x}) - Y_{y} (w_{, y x}) + Y_{x y} (w_{, y y} - w_{, x x}) + \frac{Y_{x z}}{2} (v_{, x x} - u_{, x y}) + \frac{Y_{y z}}{2} (v_{, x y} - u_{, y y})] d y d x

The strain energy from the initial stresses due to the temperature rise U_T is

(46)

U_{T} = \frac{1}{2} \int_{Λ} [σ_{x x}^{T} d_{x x} + σ_{y y}^{T} d_{y y}] d Λ

where $d_{i j}$ are nonlinear strain–displacement components and are calculated from the following equation

(47)

d_{i j} = u_{, i} u_{, j} + v_{, i} v_{, j} + w_{, i} w_{, j} i, j = x, y

By substituting $d_{i j}$ into equation (46), the strain energy due to the temperature rise is

(48)

U_{T} = \frac{1}{2} \int_{Λ} [σ_{x x}^{T} [{(u_{1, x})}^{2} + {(u_{2, x})}^{2} + {(u_{3, x})}^{2}] + σ_{y y}^{T} [{(u_{1, y})}^{2} + {(u_{2, y})}^{2} + {(u_{3, y})}^{2}]] d Λ

Defining the unknown coefficients in CLPT by $q = {A_{m, n}, B_{m, n}, C_{m, n}}$ , the (3 × 3) sub-matrices corresponding to stiffness and mass matrixes are

\begin{array}{l} {[K]}_{S C} = [\begin{matrix} {[K_{m n}^{a a}]}_{S C} & {[K_{m n}^{a b}]}_{S C} & {[K_{m n}^{a c}]}_{S C} \\ {[K_{m n}^{b b}]}_{S C} & {[K_{m n}^{b c}]}_{S C} \\ s y m & {[K_{m n}^{c c}]}_{S C} \end{matrix}], {[K]}_{S N C} = [\begin{matrix} {[K_{m n}^{a a}]}_{S N C} & {[K_{m n}^{a b}]}_{S N C} & {[K_{m n}^{a c}]}_{S N C} \\ {[K_{m n}^{b b}]}_{S N C} & {[K_{m n}^{b c}]}_{S N C} \\ s y m & {[K_{m n}^{c c}]}_{S N C} \end{matrix}], \\ {[K]}_{T} = [\begin{matrix} {[K_{m n}^{a a}]}_{T} & {[K_{m n}^{a b}]}_{T} & {[K_{m n}^{a c}]}_{T} \\ {[K_{m n}^{b b}]}_{T} & {[K_{m n}^{b c}]}_{T} \\ s y m & {[K_{m n}^{c c}]}_{T} \end{matrix}] \end{array}

(49)

[M] = [\begin{matrix} [M_{m n}^{a a}] & [M_{m n}^{a b}] & [M_{m n}^{a c}] \\ [M_{m n}^{b b}] & [M_{m n}^{b c}] \\ s y m & [M_{m n}^{c c}] \end{matrix}]

Elements of the stiffness matrix and the mass matrix are given as follows

{[K_{m n}^{a a}]}_{S C} = \frac{1}{2} [A_{1} ℜ_{A A}^{{(u_{, x})}^{2}} + A_{3} ℜ_{A A}^{(u_{, y}) 2}]

{[K_{m n}^{b b}]}_{S C} = \frac{1}{2} [A_{1} ℜ_{B B}^{{(v_{, y})}^{2}} + A_{3} ℜ_{B B}^{{(v_{, x})}^{2}}]

{[K_{m n}^{c c}]}_{S C} = \frac{1}{2} [D_{1} ℜ_{C C}^{{(w_{, x x})}^{2}} + D_{1} ℜ_{C C}^{{(w_{, y y})}^{2}} + 2 D_{2} Ξ_{C C}^{(w_{, y y}) (w_{, x x})} + 4 D_{3} ℜ_{C C}^{{(w_{, x y})}^{2}}]

{[K_{m n}^{a b}]}_{S C} = A_{2} Ξ_{A B}^{(u_{, x}) (v_{, y})} + A_{3} Ξ_{A B}^{(u_{, y}) (v_{, x})}

{[K_{m n}^{a c}]}_{S C} = - B_{1} Ξ_{A C}^{(u_{, x}) (w_{, x x})} - B_{2} Ξ_{A C}^{(u_{, x}) (w_{, y y})} - 2 B_{3} Ξ_{A C}^{(u_{, y}) (w_{, x y})}

{[K_{m n}^{b c}]}_{S C} = - B_{1} Ξ_{B C}^{(v_{, y}) (w_{, y y})} - B_{2} Ξ_{B C}^{(v_{, y}) (w_{, x x})} - 2 B_{3} Ξ_{B C}^{(v_{, x}) (w_{, x y})}

{[K_{m n}^{a a}]}_{S N C} = \frac{1}{32} A_{4} [ℜ_{A A}^{{(u_{, x y})}^{2}} + ℜ_{A A}^{{(u_{, y y})}^{2}}]

{[K_{m n}^{b b}]}_{S N C} = \frac{1}{32} A_{4} [ℜ_{B B}^{{(v_{, x x})}^{2}} + ℜ_{B B}^{{(v_{, x y})}^{2}}]

{[K_{m n}^{c c}]}_{S N C} = \frac{1}{32} A_{4} [4 ℜ_{C C}^{{(w_{, x x})}^{2}} + 32 ℜ_{C C}^{{(w_{, x y})}^{2}} - 8 Ξ_{C C}^{(w_{, x x}) (w_{, y y})} + 4 ℜ_{C C}^{{(w_{, y y})}^{2}}]

{[K_{m n}^{a b}]}_{S N C} = - \frac{1}{16} A_{4} [Ξ_{A B}^{(u_{, x y}) (v_{, x x})} + Ξ_{A B}^{(u_{, y y}) (v_{, x y})}]

{[K_{m n}^{a c}]}_{S N C} = 0, {[K_{m n}^{b c}]}_{S N C} = 0

{[K_{m n}^{a a}]}_{T} = \frac{1}{2} [A_{1}^{T} ℜ_{A A}^{{(u_{, x})}^{2}} + A_{3}^{T} ℜ_{A A}^{{(u_{, y})}^{2}}]

{[K_{m n}^{b b}]}_{T} = \frac{1}{2} [A_{1}^{T} ℜ_{B B}^{{(v_{, y})}^{2}} + A_{3}^{T} ℜ_{B B}^{{(v_{, x})}^{2}}]

{[K_{m n}^{c c}]}_{T} = \frac{1}{2} [A_{1}^{T} ℜ_{C C}^{{(w_{, x})}^{2}} + A_{3}^{T} ℜ_{C C}^{{(w_{, y})}^{2}} + D_{1}^{T} (ℜ_{C C}^{{(w_{, x x})}^{2}} + ℜ_{C C}^{{(w_{, x y})}^{2}}) + D_{3}^{T} (ℜ_{C C}^{{(w_{, y y})}^{2}} + ℜ_{C C}^{{(w_{, x y})}^{2}})]

{[K_{m n}^{a c}]}_{T} = - \frac{1}{2} [2 B_{1}^{T} Ξ_{A C}^{(u_{, x}) (w_{, x x})} + 2 B_{3}^{T} Ξ_{A C}^{(u_{, y}) (w_{, x y})}]

{[K_{m n}^{b c}]}_{T} = - \frac{1}{2} [2 B_{1}^{T} Ξ_{B C}^{(v_{, x}) (w_{, x y})} + 2 B_{3}^{T} Ξ_{B C}^{(v_{, y}) (w_{, y y})}]

{[K_{m n}^{a b}]}_{T} = 0

[M_{m n}^{a a}] = I_{1} ℜ_{A A}^{{(u)}^{2}}

[M_{m n}^{b b}] = I_{1} ℜ_{B B}^{{(v)}^{2}}

[M_{m n}^{c c}] = I_{1} ℜ_{C C}^{{(w)}^{2}} + I_{3} (ℜ_{C C}^{{(w_{, x})}^{2}} + ℜ_{C C}^{{(w_{, y})}^{2}})

[M_{m n}^{a b}] = 0

[M_{m n}^{b c}] = - 2 I_{2} ℜ_{B C}^{(v) (w_{, y})}

(50)

[M_{m n}^{a c}] = - 2 I_{2} ℜ_{A C}^{(u) (w_{, x})}

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

Ali Bakhsheshy

Hossein Mahbadi

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