Natural frequency and stability analysis of axially moving functionally graded carbon nanotube-reinforced composite thin plates

Abstract

This paper deals with the natural frequency and stability of axially moving functionally graded carbon nanotube-reinforced composite (FG-CNTRC) rectangular thin plates. Based on Reddy’s first-order shear deformation theory, the partial differential equations of motion for axially moving FG-CNTRC plates in thermal environment are derived by Hamilton’s principle and discretized by the Galerkin method. The in-plane displacement and inertia of the plate are retained in the dynamic model. By solving the complex eigenvalues of the gyroscope system, the variations of natural frequencies of the axially moving FG-CNTRC plates with axial moving velocity are obtained, and the stability of the plates is analyzed. The validity of the present model is verified by comparing the natural frequencies and the critical divergence velocity of the axially moving plate with the existing results in literature. The divergence and flutter type of instability, critical divergence velocity, and critical flutter velocity are studied. The numerical results reveal the effects of CNT distribution, volume fraction, aspect ratio, width-to-thickness ratio, and temperature rise on the natural frequencies of the plates. Additionally, the effects of geometry parameters on the critical divergence velocity and critical flutter velocity of axially moving FG-CNTRC plates are studied by plotting the instability regions.

1 Introduction

Axially moving structures are widely used in many engineering equipment and mechanical components, such as band saw blades, conveyor belts for underground coal mine, magnetic tape films, and wire cutting tools. These axially moving structures can be modeled as one-dimensional strings and beams, or two-dimensional plates and membranes. As a typical kind of gyroscopic system, axially moving structures possess rich dynamic characteristics, which have been a research hotspot in the field of structural dynamics and control. The axially velocity plays a key role in the dynamic behaviors of such systems. When the axial velocity exceeds a certain critical velocity, dynamic instabilities will occur in axially moving systems, such as divergence or flutter types of instability, and even lead to structural failures.

For the axially moving strings and beams, a large number of researches is reported. Zhang and Chen [1] used the complex modal method to analyze the transverse vibration of an axially moving string resting on a viscoelastic foundation. Kesimli et al. [2] investigated the nonlinear vibrations of an axially moving string with multiple supports by using the method of multiple scales. Ghayesh [3, 4] studied the parametric resonance and stability of an axially moving string, in which the moving velocity was assumed to be time-dependent, including a mean velocity and harmonic variations. Yang et al. [5] carried out the nonlinear vibrations of axially moving strings by applying the multiple scale analysis. Marynowski [6] analyzed the dynamic behavior of an axially moving sandwich beam with viscoelastic core in undercritical and supercritical range of speed. Li et al. [7] used the method of multiple scales to investigate the nonlinear vibration response and stability of an axially moving viscoelastic sandwich beam under low- and high-frequency principle resonances. Huang and coauthors [8,9,10] employed the incremental harmonic balance (IHB) method to investigate the nonlinear vibration of axially moving beams. They also combined the IHB method with the fast Fourier transform (FFT) to calculate the periodic solutions of high dimensional nonlinear models of the axially moving beams with in-plane and out-of-plane vibration. Mao et al. [11, 12] successively studied the nonlinear dynamics of a super-critically axially moving beam subjected to a harmonic exciting force and the pulsating speed by using the method of multiple scales. And the effects of moving speed and the pulsating speed frequency on the 3:1 internal resonance were discussed. Using the dynamic stiffness method to model the generalized boundary conditions, Ding et al. [13] discussed the influence of constrained spring stiffness on the free vibration characteristics of the axially moving beams. Considering the harmonic fluctuations of the axial moving speed, Yan et al. [14] presented the nonlinear dynamic analysis of an axially moving viscoelastic beam under parametric and external excitations. Karličić et al. [15] presented a novel model of axially moving beam with an attached nonlinear energy sink and analyzed the vibration suppression and energy harvesting capabilities of the system. Zhang et al. [16,17,18] also used the attached nonlinear energy sink to study the vibration suppression of axially moving beams. Khaniki et al. [19] performed the experimental and theoretical investigations on the coupled nonlinear dynamics of axially moving hyperelastic beams.

For the two-dimensional model, Hatami et al. [20] presented an exact finite strip model for the free vibration analysis of axially moving viscoelastic plates. Yang et al. [21] researched the vibrations and stability of an axially moving rectangular antisymmetric cross-ply composite plate using the differential quadrature method. Yang et al. [22] calculated the low-velocity impact response of axially moving graphene platelet-reinforced metal foam plates by using the Duhamel integration. Yang and Zhang [23] used the method of multiple scales to study the nonlinear dynamic response of an axially moving beam caused by the internal–external resonance, in which the coupling of the longitudinal and transversal motion are considered. Lu et al. [24] studied the dynamic stability of an axially moving graphene-reinforced laminated composite plate. Zhou and Wang [25] adopted the normalized power series method to analyze the dynamic instability of an axially moving viscoelastic plate by considering the Kelvin–Voigt model of viscoelasticity. Ghayesh and Amabili [26, 27] successively investigated the nonlinear forced vibrations of axially moving plates subjected to a concentrated harmonic force and a distributed harmonic force. The time histories, bifurcation diagrams, phase-plane portraits, Poincaré sections and frequency response curves are constructed by using direct time integration and the pseudo-arclength continuation technique. Tang and Chen [28] researched the stability in parametric resonance of axially moving viscoelastic plates with time-dependent speed by employing the method of multiple scales. Li et al. [29] studied the combination resonances of an axially moving plate partially immersed in fluid subjected to two frequency excitation based on classical thin plate theory. Li et al. [30] studied the characteristics of 1:3 internal resonance and its bifurcations for an axially moving unidirectional plate partially immersed in a fluid under foundation displacement excitation. Yang et al. [31] employed the finite difference method to investigate the linear free vibrations and nonlinear forced vibrations of axially moving viscoelastic plates. Yao et al. [32, 33] studied the stability and nonlinear primary resonances of an axially moving plate in aero-thermal environment. Despite abundant investigations were carried out for the vibration characteristics of axially moving beams and plates. However, to our best knowledge, very few studies reported the stability boundary and the instability region of divergence or flutter of axially moving structures.

Functionally graded materials (FGMs) are a kind of inhomogeneous material developed to meet the requirements of aerospace field for materials to work normally in extremely harsh environments. FGMs are superior to the traditional laminated composites, such as high strength, high stiffness, enhanced thermal resistance, and improved stress distribution. FGMs have become a promising field of material science in the present century. They have been widely used in aerospace, marine, vehicle and other fields, and have attracted the attentions of many researchers [34,35,36,37]. Carbon-based nano-reinforcements, such as carbon nanotubes (CNTs) and graphene platelets (GPLs), are extensively used as the reinforcements due to their excellent performances, to further improve the mechanical properties of composite structures. Wang et al. [38, 39] investigated the nonlinear bending of a sandwich beam with functionally graded graphene platelet-reinforced composite face-sheets, and the thermal buckling and postbuckling behaviors of graphene platelet-reinforced porous nanocomposite beams. Carbon nanotubes (CNTs) have become an ideal reinforcement for advanced composites because of their superior mechanical, electrical, and thermodynamic properties compared with carbon fibers [40, 41]. Functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) are a new generation of FGMs. In the past decade, investigations related to FG-CNTRC structures have mushroomed [42,43,44,45,46,47]. Zhu et al. [48] studied the free vibration and bending response of FG-CNTRC rectangular plates under uniform transverse load by using finite element method. Alibeigloo and Emtehani [49] also presented the bending and free vibration behaviors of FG-CNTRC plates based on differential quadrature method. Mirzaei and Kiani [50] used the Ritz method and an iterative process to investigate the thermal buckling of FG-CNTRC rectangular plates under the action of uniform temperature rise. In the analysis, Chebyshev polynomials suitable for different boundary conditions were used as the basis functions of Ritz method. Ansari and Gholami [51] investigated the nonlinear primary resonance of FG-CNTRC rectangular plates subjected to a harmonic transverse load based on Reddy’s third-order shear deformation theory. Besides, the effect of initial geometric imperfection on the nonlinear resonance response of FG-CNTRC rectangular plates was presented by Gholami and Ansari [52]. Wu et al. [53, 54] studied the effects of initial geometric imperfections on the nonlinear vibrations of FG-CNTRC beams, and the imperfection sensitivity of thermal post-buckling behavior of FG-CNTRC beams under the action of uniform and non-uniform temperature variations. Zhang et al. [55] used Reddy's third-order shear deformation theory and state-space Levy method to analyze the natural frequencies and vibration mode shapes of FG-CNTRC rectangular plates subjected to in-plane loads. Song et al. [56] used the assumed mode method and velocity feedback control method to study the active vibration control of FG-CNTRC plates with piezoelectric actuators and sensors on the top and bottom surfaces. Wu et al. [57,58,59] investigated the nonlinear dynamics of FG-CNTRC beams and cylindrical shells by using the IHB method. The 3/2 superharmonic resonance and 1/2 subharmonic resonance were first observed in the FG-CNTRC beams with asymmetric CNT distribution.

All the aforementioned studies concerning the vibrations of FG-CNTRC structures have few reports on the axially moving plates. While, previous results revealed that CNT distribution and volume fraction play a large role on the vibration behaviors of FG-CNTRC structures. This prompted us to study the natural frequency and stability of axially moving FG-CNTRC plates and explore the influence of material properties and geometry parameters on the stability of the system.

In this paper, the natural frequency and stability of an axially moving FG-CNTRC rectangular thin plate are studied. The dynamic model of axially moving FG-CNTRC plate is established based on Reddy’s first-order shear deformation theory, Hamilton’s principle, and Galerkin method. The variations of natural frequencies and stabilities with the moving velocity are analyzed by solving the complex eigenvalue equations of the gyroscope system. The critical divergence velocity, boundary velocity of divergent instability and critical flutter velocity are obtained. Detailed parametric studies are performed to reveal the effects of CNT distribution, volume fraction, geometry parameter, and temperature rise on the instability regions of divergence and flutter.

2 Axially moving functionally graded carbon nanotube-reinforced composite plate

As shown in Fig. 1a, an axially moving FG-CNTRC rectangular thin plate with length a, width b, and thickness h is considered. A Cartesian coordinate system (O-xyz) is established. x and y axes are defined as the mid-plane coordinates of the plate, z represents the out-of-plane coordinate. The origin of the Cartesian coordinate system is located at a corner of the rectangular plate. The plate moves along the x-axis direction at a constant axial velocity V. The boundary condition of the rectangular plate is assumed as simply supported on four sides (x = 0, x = a, y = 0, y = b). The FG-CNTRC plate is made of a mixture of isotropic matrix and CNTs reinforcement. The CNTs are dispersed along the thickness direction of the plate in the patterns of uniform distribution (UD) or functionally graded distributions (FGA, FGO, FGX), as shown in Fig. 1b. The bottom surface of FGA-CNTRC and the mid-surface of FGO-CNTRC plate are CNT-rich. Both the bottom and top surfaces of FGX-CNTRC plate are CNT-rich.

Fig. 1

Schematic diagram of an axially moving FG-CNTRC rectangular plate with different CNT distribution patterns: a geometry and coordinate system, and b CNT distribution patterns in the thickness direction of plate

With the help of the extended rule of mixture [42], the material properties of the FG-CNTRC plate can be computed as:

$$ E_{11} = \eta_{1} V_{{{\text{cnt}}}} E_{11}^{{{\text{cnt}}}} + V_{{\text{m}}} E^{{\text{m}}} , $$

(1.1)

$$ \frac{{\eta_{2} }}{{E_{22} }} = \frac{{V_{{{\text{cnt}}}} }}{{E_{22}^{{{\text{cnt}}}} }} + \frac{{V_{{\text{m}}} }}{{E^{{\text{m}}} }}, $$

(1.2)

$$ \frac{{\eta_{3} }}{{G_{12} }} = \frac{{V_{{{\text{cnt}}}} }}{{G_{12}^{{{\text{cnt}}}} }} + \frac{{V_{{\text{m}}} }}{{G^{{\text{m}}} }}, $$

(1.3)

where $E_{{{11}}}^{{{\text{cnt}}}}$, $E_{22}^{{{\text{cnt}}}}$, and $G_{12}^{{{\text{cnt}}}}$ are the Young’s and shear modulus of CNTs, $E^{{\text{m}}}$ and G^m are the properties of the matrix, $\eta_{j} {(}j = {1, 2, 3)}$ denote the CNT efficiency parameters. $V_{{\text{m}}}$ and $V_{{{\text{cnt}}}}$ denote the volume fractions of matrix and CNTs, which are linear functions of z for each pattern of CNT distributions,

$$ V_{{{\text{cnt}}}} = \left\{ {\begin{array}{*{20}l} {V_{{{\text{cnt}}}}^{*} } \hfill & {\quad {\text{(UD)}}} \hfill \\ {\left( {1 - \frac{2z}{h}} \right)V_{{{\text{cnt}}}}^{*} } \hfill & {\quad {\text{(FGA)}}} \hfill \\ {2\left( {1 - \frac{2\left| z \right|}{h}} \right)V_{{{\text{cnt}}}}^{*} } \hfill & {\quad {\text{(FGO)}}} \hfill \\ {4\frac{\left| z \right|}{h}V_{{{\text{cnt}}}}^{*} } \hfill & {\quad {\text{(FGX)}}} \hfill \\ \end{array} } \right.,\quad V_{{\text{m}}} = {1} - V_{{{\text{cnt}}}} $$

(2)

where

$$ V_{{{\text{cnt}}}}^{*} = \frac{{\Lambda_{{{\text{cnt}}}} }}{{\Lambda_{{{\text{cnt}}}} + (\rho^{{{\text{cnt}}}} /\rho^{{\text{m}}} ) - (\rho^{{{\text{cnt}}}} /\rho^{{\text{m}}} )\Lambda_{{{\text{cnt}}}} }}, $$

(3)

where $\Lambda_{{{\text{cnt}}}}$ is the CNT mass fraction, $\rho^{{{\text{cnt}}}}$ and $\rho^{{\text{m}}}$ are the densities of CNTs and matrix.

The Poisson’s ratio and density of FG-CNTRCs could be obtained

$$ \nu_{12} = V_{{{\text{cnt}}}} \nu_{12}^{{{\text{cnt}}}} + V_{{\text{m}}} \nu^{{\text{m}}} ,\quad \rho = V_{{{\text{cnt}}}} \rho^{{{\text{cnt}}}} + V_{{\text{m}}} \rho^{{\text{m}}} , $$

(4)

where $\nu_{12}^{{{\text{cnt}}}}$ and $\nu^{{\text{m}}}$ are the Poisson’s ratios of CNTs and matrix.

The longitudinal and transverse thermal expansion coefficients of the FG-CNTRC plate also have a gradient distribution in the thickness direction, which can be expressed as:

$$ \alpha_{11} = \frac{{V_{{{\text{cnt}}}} E_{11}^{{{\text{cnt}}}} \alpha_{11}^{{{\text{cnt}}}} + V_{{\text{m}}} E^{{\text{m}}} \alpha^{{\text{m}}} }}{{V_{{{\text{cnt}}}} E_{11}^{{{\text{cnt}}}} + V_{{\text{m}}} E^{{\text{m}}} }}, $$

(5.1)

$$ \alpha_{22} = \left( {1 + \nu_{12}^{{{\text{cnt}}}} } \right)V_{{{\text{cnt}}}} \alpha_{22}^{{{\text{cnt}}}} + \left( {1 + \nu^{{\text{m}}} } \right)V_{{\text{m}}} \alpha^{{\text{m}}} - \nu_{12} \alpha_{11} $$

(5.2)

where $\alpha_{11}^{{{\text{cnt}}}}$, $\alpha_{22}^{{{\text{cnt}}}}$ and $\alpha^{{\text{m}}}$ are the thermal expansion coefficients of CNTs and matrix, respectively. The effective material properties of the FG-CNTRC plate are functions of temperature rise and position coordinates.

3 Equations of motion

On the basis of the Reddy’s first-order shear deformation theory, the displacement components at an any position of the thin plate are defined by

$$ u_{1} (x,y,z,t) = u(x,y,t) + z\varphi_{x} (x,y,t), $$

(6.1)

$$ v_{1} (x,y,z,t) = v(x,y,t) + z\varphi_{y} (x,y,t), $$

(6.2)

$$ w_{1} (x,y,z,t) = w(x,y,t). $$

(6.3)

where $u(x,y,t)$, $v(x,y,t)$, and $w(x,y,t)$ represent the in-plane (x–O–y) displacements and out-plane (z direction) transverse displacement at the mid-surface of the plate. $\varphi_{x} (x,y,t)$ and $\varphi_{y} (x,y,t)$ are the rotations of transverse normal around the y and x axes, respectively.

The strain–displacement relations of the plate are

$$ \left\{ {\begin{array}{*{20}c} {\varepsilon_{x} } \\ {\varepsilon_{y} } \\ {\gamma_{xy} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {\varepsilon_{x}^{0} } \\ {\varepsilon_{y}^{0} } \\ {\gamma_{xy}^{0} } \\ \end{array} } \right\} + z\left\{ {\begin{array}{*{20}c} {\kappa_{x} } \\ {\kappa_{y} } \\ {\kappa_{xy} } \\ \end{array} } \right\},\quad \left\{ {\begin{array}{*{20}c} {\gamma_{xz} } \\ {\gamma_{yz} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {\gamma_{xz}^{0} } \\ {\gamma_{yz}^{0} } \\ \end{array} } \right\}, $$

(7)

in which the normal strains and shear strains in the mid-plane are

$$ \left\{ {\begin{array}{*{20}c} {\varepsilon_{x}^{0} } \\ {\varepsilon_{y}^{0} } \\ {\gamma_{xy}^{0} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {\frac{\partial u}{{\partial x}}} \\ {\frac{\partial v}{{\partial y}}} \\ {\frac{\partial u}{{\partial y}} + \frac{\partial v}{{\partial x}}} \\ \end{array} } \right\},\quad \left\{ {\begin{array}{*{20}c} {\gamma_{xz}^{0} } \\ {\gamma_{yz}^{0} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {\varphi_{x} + \frac{\partial w}{{\partial x}}} \\ {\varphi_{y} + \frac{\partial w}{{\partial y}}} \\ \end{array} } \right\}, $$

(8)

and the changes of curvature and torsion are

$$ \left\{ {\begin{array}{*{20}c} {\kappa_{x} } \\ {\kappa_{y} } \\ {\kappa_{xy} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {\frac{{\partial \varphi_{x} }}{\partial x}} \\ {\frac{{\partial \varphi_{y} }}{\partial y}} \\ {\frac{{\partial \varphi_{x} }}{\partial y} + \frac{{\partial \varphi_{y} }}{\partial x}} \\ \end{array} } \right\}. $$

(9)

Based on the constitutive relations, the stress–strain relations are

$$ \left\{ {\begin{array}{*{20}c} {\sigma_{x} } \\ {\sigma_{y} } \\ {\sigma_{yz} } \\ {\sigma_{xz} } \\ {\sigma_{xy} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {Q_{11} } &\quad {Q_{12} } & 0&\quad & \quad 0 & \quad 0 \\ {Q_{12} } & \quad {Q_{22} } & \quad 0 & \quad 0 & \quad 0 \\ 0 & \quad 0 & \quad {Q_{44} } & \quad 0 & \quad 0 \\ 0 & \quad 0 & \quad 0 & \quad {Q_{55} } & \quad 0 \\ 0 & \quad 0 & \quad 0 & \quad 0 & \quad {Q_{66} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\varepsilon_{x} - \alpha_{11} \Delta T} \\ {\varepsilon_{y} - \alpha_{22} \Delta T} \\ {\gamma_{yz} } \\ {\gamma_{xz} } \\ {\gamma_{xy} } \\ \end{array} } \right\}, $$

(10)

where σ_x, σ_y, and σ_xy are the normal and shear stresses, α₁₁ and α₂₂ are the thermal expansion coefficients of the two directions, respectively. ΔT = T − T₀ is temperature rise, Q_ij is the reduced stiffness matrix, the effective elasticity coefficients are

$$ Q_{11} = \frac{{E_{11} }}{{1 - \nu_{12} \nu_{21} }},\quad Q_{22} = \frac{{E_{22} }}{{1 - \nu_{12} \nu_{21} }},\quad Q_{12} = \frac{{\nu_{21} E_{11} }}{{1 - \nu_{12} \nu_{21} }},\quad Q_{44} = G_{23} ,\quad Q_{55} = G_{13} ,\quad Q_{66} = G_{12} . $$

(11)

Then the stress components and their moments are integrated through the thickness of the plate, and the force and moment expressions could be acquired,

$$ \left\{ {\begin{array}{*{20}c} {N_{x} } \\ {N_{y} } \\ {N_{xy} } \\ \end{array} } \right\} = \int_{ - h/2}^{h/2} {\left\{ {\begin{array}{*{20}c} {\sigma_{x} } \\ {\sigma_{y} } \\ {\sigma_{xy} } \\ \end{array} } \right\}} {\text{d}}z,\quad \left\{ {\begin{array}{*{20}c} {M_{x} } \\ {M_{y} } \\ {M_{xy} } \\ \end{array} } \right\} = \int_{ - h/2}^{h/2} {\left\{ {\begin{array}{*{20}c} {\sigma_{x} z} \\ {\sigma_{y} z} \\ {\sigma_{xy} z} \\ \end{array} } \right\}} {\text{d}}z,\quad \left\{ {\begin{array}{*{20}c} {Q_{y} } \\ {Q_{x} } \\ \end{array} } \right\} = \int_{ - h/2}^{h/2} {\left\{ {\begin{array}{*{20}c} {\sigma_{yz} } \\ {\sigma_{xz} } \\ \end{array} } \right\}} {\text{d}}z. $$

(12)

Substituting Eqs. (7–9, 11) into Eq. (10) and then substituting the results into Eq. (12) give rise to the constitutive relations in matrix form as

$$ \left\{ {\begin{array}{*{20}c} {N_{x} } \\ {N_{y} } \\ {N_{xy} } \\ {M_{x} } \\ {M_{y} } \\ {M_{xy} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {A_{11} } & \quad {A_{12} } & \quad 0 & \quad {B_{11} } & \quad {B_{12} } & \quad 0 \\ {A_{12} } & \quad {A_{22} } & \quad 0 & \quad {B_{12} } & \quad {B_{22} } & \quad 0 \\ 0 & \quad 0 & \quad {A_{66} } & \quad 0 & \quad 0 & \quad {B_{66} } \\ {B_{11} } & \quad {B_{12} } & \quad 0 & \quad {D_{11} } & \quad {D_{12} } & \quad 0 \\ {B_{12} } & \quad {B_{22} } & \quad 0 & \quad {D_{12} } & \quad {D_{22} } & \quad 0 \\ 0 & \quad 0 & \quad {B_{66} } & \quad 0 & \quad 0 & \quad {D_{66} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\varepsilon_{x}^{0} } \\ {\varepsilon_{y}^{0} } \\ {\gamma_{xy}^{0} } \\ {\kappa_{x} } \\ {\kappa_{y} } \\ {\kappa_{xy} } \\ \end{array} } \right\} - \left\{ {\begin{array}{*{20}c} {N_{x}^{{\text{T}}} } \\ {N_{y}^{{\text{T}}} } \\ 0 \\ {M_{x}^{{\text{T}}} } \\ {M_{y}^{{\text{T}}} } \\ 0 \\ \end{array} } \right\},\quad \left\{ {\begin{array}{*{20}c} {Q_{y} } \\ {Q_{x} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {A_{44} } & \quad 0 \\ 0 & \quad {A_{55} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\gamma_{yz}^{0} } \\ {\gamma_{xz}^{0} } \\ \end{array} } \right\}, $$

(13)

where the extension, bending–extension coupling and bending stiffness coefficients are defined by

$$ (A_{ij} ,B_{ij} ,D_{ij} ) = \int_{ - h/2}^{h/2} {Q_{ij} } (1,z,z^{2} ){\text{d}}z\;\;\left( {i,j = 1,\;2,\;6} \right), $$

(14.1)

$$ A_{44} = k_{s} \int_{ - h/2}^{h/2} {Q_{44} } {\text{d}}z,\quad A_{55} = k_{s} \int_{ - h/2}^{h/2} {Q_{55} } {\text{d}}z, $$

(14.2)

in which k_s = 5/6 is the shear correction factor. The forces and moments caused by temperature rise are

$$ \left\{ {\begin{array}{*{20}c} {N_{x}^{{\text{T}}} } \\ {N_{y}^{{\text{T}}} } \\ \end{array} } \right\} = \int_{ - h/2}^{h/2} {\left[ {\begin{array}{*{20}c} {Q_{11} } & \quad {Q_{12} } \\ {Q_{12} } & \quad {Q_{22} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\alpha_{11} } \\ {\alpha_{22} } \\ \end{array} } \right\}} \Delta T{\text{d}}z,\quad \left\{ {\begin{array}{*{20}c} {M_{x}^{{\text{T}}} } \\ {M_{y}^{{\text{T}}} } \\ \end{array} } \right\} = \int_{ - h/2}^{h/2} {\left[ {\begin{array}{*{20}c} {Q_{11} } & \quad {Q_{12} } \\ {Q_{12} } & \quad {Q_{22} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\alpha_{11} } \\ {\alpha_{22} } \\ \end{array} } \right\}} z\Delta T{\text{d}}z. $$

(15)

It is noted that, under a uniform temperature rise, the system does not generate thermal bending moments for UD, FGO, and FGX-CNTRC plate with symmetric CNT distributions, i.e., $M_{x}^{{\text{T}}} = M_{y}^{{\text{T}}} = 0$. The bending–extension coupling stiffness components are all equal to zero (i.e., B_ij = 0) for these three types of CNTRC plates. Only the FGA-CNTRC beam with asymmetric CNT distribution has bending–extension coupling effect.

The displacement vector of an arbitrary point of the plate can be given as

$$ {\mathbf{r}} = \left( {x + u_{1} } \right){\mathbf{i}} + \left( {y + v_{1} } \right){\mathbf{j}} + \left( {z + w_{1} } \right){\mathbf{k}}, $$

(16)

where i, j, and k are the unit vectors along the x, y, and z directions.

Substituting Eq. (6) into Eq. (16) and taking the total derivative of the displacement vector r with respect to time t, the velocity vector of an arbitrary point of the plate can be obtained,

$$ {\mathbf{V}} = \frac{{{\text{d}}{\mathbf{r}}}}{{{\text{d}}t}} = \left( {V + \frac{{{\text{d}}u}}{{{\text{d}}t}} + z\frac{{{\text{d}}\varphi_{x} }}{{{\text{d}}t}}} \right){\mathbf{i}} + \left( {\frac{{{\text{d}}v}}{{{\text{d}}t}} + z\frac{{{\text{d}}\varphi_{y} }}{{{\text{d}}t}}} \right){\mathbf{j}} + \frac{{{\text{d}}w}}{{{\text{d}}t}}{\mathbf{k}}, $$

(17)

where the total derivative operator with respect to time is defined as

$$ \frac{d( \bullet )}{{dt}} = \frac{\partial ( \bullet )}{{\partial x}}\frac{\partial x}{{\partial t}} + \frac{\partial ( \bullet )}{{\partial t}} = V\frac{\partial ( \bullet )}{{\partial x}} + \frac{\partial ( \bullet )}{{\partial t}}. $$

(18)

The kinetic energy T_k of the plate can be obtained as

$$ T_{{\text{k}}} = \frac{1}{2}\int_{0}^{b} {\int_{0}^{a} {\int_{ - h/2}^{h/2} {\rho (z)({\mathbf{V}} \cdot {\mathbf{V}})} } } {\text{d}}z{\text{d}}x{\text{d}}y. $$

(19)

Substituting Eq. (17) into Eq. (19) and retaining the in-plane displacement and inertia of the plate, the total kinetic energy of the plate can be written as

$$ \begin{aligned} T_{{\text{k}}} & \quad = \frac{1}{2}\int_{0}^{b} {\int_{0}^{a} {\int_{ - h/2}^{h/2} {\rho (z)\left[ \begin{gathered} \left( {\frac{\partial u}{{\partial t}}} \right)^{2} + \left( {\frac{\partial v}{{\partial t}}} \right)^{2} + \left( {\frac{\partial w}{{\partial t}}} \right)^{2} + V^{2} \left( {\left( {\frac{\partial u}{{\partial x}}} \right)^{2} + \left( {\frac{\partial v}{{\partial x}}} \right)^{2} + \left( {\frac{\partial w}{{\partial x}}} \right)^{2} } \right) \hfill \\ + 2V\left( {\frac{\partial u}{{\partial x}}\frac{\partial u}{{\partial t}} + \frac{\partial v}{{\partial x}}\frac{\partial v}{{\partial t}} + \frac{\partial w}{{\partial x}}\frac{\partial w}{{\partial t}}} \right) + \left( {V^{2} + 2V^{2} \frac{\partial u}{{\partial x}} + 2V\frac{\partial u}{{\partial t}}} \right) \hfill \\ \end{gathered} \right]} } } {\text{d}}z{\text{d}}x{\text{d}}y \\ & \quad + \frac{1}{2}\int_{0}^{b} {\int_{0}^{a} {\int_{ - h/2}^{h/2} {\rho (z)z\left[ \begin{gathered} 2\frac{\partial u}{{\partial t}}\frac{{\partial \varphi_{x} }}{\partial t} + 2\frac{\partial v}{{\partial t}}\frac{{\partial \varphi_{y} }}{\partial t} + 2V^{2} \left( {\frac{{\partial \varphi_{x} }}{\partial x} + \frac{\partial u}{{\partial x}}\frac{{\partial \varphi_{x} }}{\partial x} + \frac{\partial v}{{\partial x}}\frac{{\partial \varphi_{y} }}{\partial x}} \right) \hfill \\ + 2V\left( {\frac{{\partial \varphi_{x} }}{\partial t} + \frac{\partial u}{{\partial x}}\frac{{\partial \varphi_{x} }}{\partial t} + \frac{\partial u}{{\partial t}}\frac{{\partial \varphi_{x} }}{\partial x} + \frac{\partial v}{{\partial x}}\frac{{\partial \varphi_{y} }}{\partial t} + \frac{\partial v}{{\partial t}}\frac{{\partial \varphi_{y} }}{\partial x}} \right) \hfill \\ \end{gathered} \right]} } } {\text{d}}z{\text{d}}x{\text{d}}y \\ & \quad + \frac{1}{2}\int_{0}^{b} {\int_{0}^{a} {\int_{ - h/2}^{h/2} {\rho (z)z^{2} \left[ \begin{gathered} \left( {\frac{{\partial \varphi_{x} }}{\partial t}} \right)^{2} + \left( {\frac{{\partial \varphi_{y} }}{\partial t}} \right)^{2} + V^{2} \left( {\left( {\frac{{\partial \varphi_{x} }}{\partial x}} \right)^{2} + \left( {\frac{{\partial \varphi_{y} }}{\partial x}} \right)^{2} } \right) \hfill \\ + 2V\left( {\frac{{\partial \varphi_{x} }}{\partial x}\frac{{\partial \varphi_{x} }}{\partial t} + \frac{{\partial \varphi_{y} }}{\partial x}\frac{{\partial \varphi_{y} }}{\partial t}} \right) \hfill \\ \end{gathered} \right]} } } {\text{d}}z{\text{d}}x{\text{d}}y. \\ \end{aligned} $$

(20)

The potential energy of the plate is

$$ U_{{\text{p}}} = \frac{1}{2}\int_{0}^{b} {\int_{0}^{a} {\int_{ - h/2}^{h/2} {\left( {\sigma_{x} \varepsilon_{x} + \sigma_{y} \varepsilon_{y} + \sigma_{xy} \gamma_{xy} + \sigma_{xz} \gamma_{xz} + \sigma_{yz} \gamma_{yz} } \right)} } } {\text{d}}z{\text{d}}x{\text{d}}y. $$

(21)

Substituting Eqs. (20) and (21) into Hamilton’s principle,

$$ \int_{{t_{1} }}^{{t_{2} }} {(\delta T_{{\text{k}}} - \delta U_{{\text{p}}} )} \;{\text{d}}t = 0, $$

(22)

the equations of motion in terms of the force and moment could be obtained,

$$ \begin{array}{*{20}c} {\delta u:} & \quad {\frac{{\partial N_{x} }}{\partial x} + \frac{{\partial N_{xy} }}{\partial y} = I_{1} \left( {\frac{{\partial^{2} u}}{{\partial t^{2} }} + V^{2} \frac{{\partial^{2} u}}{{\partial x^{2} }} + 2V\frac{{\partial^{2} u}}{\partial x\partial t}} \right) + I_{2} \left( {\frac{{\partial^{2} \varphi_{x} }}{{\partial t^{2} }} + V^{2} \frac{{\partial^{2} \varphi_{x} }}{{\partial x^{2} }} + 2V\frac{{\partial^{2} \varphi_{x} }}{\partial x\partial t}} \right)} \\ \end{array} , $$

(23.1)

$$ \begin{array}{*{20}c} {\delta v:} & \quad {\frac{{\partial N_{xy} }}{\partial x} + \frac{{\partial N_{y} }}{\partial y} = I_{1} \left( {\frac{{\partial^{2} v}}{{\partial t^{2} }} + V^{2} \frac{{\partial^{2} v}}{{\partial x^{2} }} + 2V\frac{{\partial^{2} v}}{\partial x\partial t}} \right) + I_{2} \left( {\frac{{\partial^{2} \varphi_{y} }}{{\partial t^{2} }} + V^{2} \frac{{\partial^{2} \varphi_{y} }}{{\partial x^{2} }} + 2V\frac{{\partial^{2} \varphi_{y} }}{\partial x\partial t}} \right)} \\ \end{array} , $$

(23.2)

$$ \begin{array}{*{20}c} {\delta w:} \quad {\frac{{\partial Q_{x} }}{\partial x} + \frac{{\partial Q_{y} }}{\partial y} = I_{1} \left( {\frac{{\partial^{2} w}}{{\partial t^{2} }} + V^{2} \frac{{\partial^{2} w}}{{\partial x^{2} }} + 2V\frac{{\partial^{2} w}}{\partial x\partial t}} \right)} \\ \end{array} , $$

(23.3)

$$ \begin{array}{*{20}c} {\delta \varphi_{x} :} & \quad {\frac{{\partial M_{x} }}{\partial x} + \frac{{\partial M_{xy} }}{\partial y} - Q_{x} = I_{2} \left( {\frac{{\partial^{2} u}}{{\partial t^{2} }} + V^{2} \frac{{\partial^{2} u}}{{\partial x^{2} }} + 2V\frac{{\partial^{2} u}}{\partial x\partial t}} \right) + I_{3} \left( {\frac{{\partial^{2} \varphi_{x} }}{{\partial t^{2} }} + V^{2} \frac{{\partial^{2} \varphi_{x} }}{{\partial x^{2} }} + 2V\frac{{\partial^{2} \varphi_{x} }}{\partial x\partial t}} \right)} \\ \end{array} , $$

(23.4)

$$ \begin{array}{*{20}c} {\delta \varphi_{\theta } :} & \quad {\frac{{\partial M_{y} }}{\partial y} + \frac{{\partial M_{xy} }}{\partial x} - Q_{y} = I_{2} \left( {\frac{{\partial^{2} v}}{{\partial t^{2} }} + V^{2} \frac{{\partial^{2} v}}{{\partial x^{2} }} + 2V\frac{{\partial^{2} v}}{\partial x\partial t}} \right) + I_{3} \left( {\frac{{\partial^{2} \varphi_{y} }}{{\partial t^{2} }} + V^{2} \frac{{\partial^{2} \varphi_{y} }}{{\partial x^{2} }} + 2V\frac{{\partial^{2} \varphi_{y} }}{\partial x\partial t}} \right)} \\ \end{array} , $$

(23.5)

in which the inertia terms are

$$ \left\{ {I_{1} ,I_{2} ,I_{3} } \right\} = \int_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-\nulldelimiterspace} 2}}} {\rho (z)\left\{ {1,z,z^{2} } \right\}} \;{\text{d}}z. $$

(24)

Substituting the results obtained from Eq. (12) into Eq. (23), the equations of motion in terms of displacements and rotations can be obtained. Meanwhile, the dimensionless parameters are introduced as follows:

$$ \left( {\overline{x},\overline{y},\overline{z}} \right) = \left( {\frac{x}{a},\frac{y}{b},\frac{z}{h}} \right),\quad \lambda_{1} = \frac{a}{b},\quad \lambda_{2} = \frac{a}{h},\quad \lambda_{3} = \frac{b}{h},\quad \left( {\overline{u},\overline{v},\overline{w}} \right) = \left( {\frac{u}{a},\frac{v}{b},\frac{w}{h}} \right),\quad \overline{t} = \frac{t}{a}\sqrt {\frac{{A_{110} }}{{I_{10} }}} , $$

(25.1)

$$ \overline{\Omega } = \Omega a\sqrt {\frac{{I_{10} }}{{A_{110} }}} ,\quad \overline{V} = V\sqrt {\frac{{I_{10} }}{{A_{110} }}} ,\quad \left( {\overline{N}_{x}^{{\text{T}}} ,\overline{N}_{y}^{{\text{T}}} } \right) = \left( {\frac{{N_{x}^{{\text{T}}} }}{{A_{110} }},\frac{{N_{y}^{{\text{T}}} }}{{A_{110} }}} \right),\quad \left( {\overline{M}_{x}^{{\text{T}}} ,\overline{M}_{y}^{{\text{T}}} } \right) = \left( {\frac{{M_{x}^{{\text{T}}} }}{{A_{110} h}},\frac{{M_{y}^{{\text{T}}} }}{{A_{110} h}}} \right) $$

(25.2)

$$ (a_{11} ,a_{12} ,a_{22} ,a_{44} ,a_{55} ,a_{66} ) = \left( {\frac{{A_{11} }}{{A_{110} }},\frac{{A_{12} }}{{A_{110} }},\frac{{A_{22} }}{{A_{110} }},\frac{{A_{44} }}{{A_{110} }},\frac{{A_{55} }}{{A_{110} }},\frac{{A_{66} }}{{A_{110} }}} \right), $$

(25.3)

$$ (b_{11} ,b_{12} ,b_{22} ,b_{66} ) = \left( {\frac{{B_{11} }}{{A_{110} h}},\frac{{B_{12} }}{{A_{110} h}},\frac{{B_{22} }}{{A_{110} h}},\frac{{B_{66} }}{{A_{110} h}}} \right), $$

(25.4)

$$ (d_{11} ,d_{12} ,d_{22} ,d_{66} ) = \left( {\frac{{D_{11} }}{{A_{110} h^{2} }},\frac{{D_{12} }}{{A_{110} h^{2} }},\frac{{D_{22} }}{{A_{110} h^{2} }},\frac{{D_{66} }}{{A_{110} h^{2} }}} \right),\quad \left( {\overline{I}_{1} ,\overline{I}_{2} ,\overline{I}_{3} } \right) = \left( {\frac{{I_{1} }}{{I_{10} }},\frac{{I_{2} }}{{I_{10} h}},\frac{{I_{3} }}{{I_{10} h^{2} }}} \right), $$

(25.5)

where A₁₁₀ and I₁₀ are the values of A₁₁ and I₁ for a homogeneous plate made of polymer matrix material at room temperature ΔT=0.

Omitting the superscript of the dimensionless parameter for brevity, the resulting dimensionless partial differential equations (PDEs) of motion are given as

$$ \delta u:\left\{ {\begin{array}{*{20}l} {a_{11} \lambda_{2}^{2} \frac{{\partial^{2} u}}{{\partial x^{2} }} + a_{66} \lambda_{1}^{2} \lambda_{2}^{2} \frac{{\partial^{2} u}}{{\partial y^{2} }} + \left( {a_{12} + a_{66} } \right)\lambda_{2}^{2} \frac{{\partial^{2} v}}{\partial x\partial y} + b_{11} \lambda_{2} \frac{{\partial^{2} \varphi_{x} }}{{\partial x^{2} }} + b_{66} \lambda_{1}^{2} \lambda_{2} \frac{{\partial^{2} \varphi_{x} }}{{\partial y^{2} }}} \hfill \\ { + \left( {b_{12} + b_{66} } \right)\lambda_{1} \lambda_{2} \frac{{\partial^{2} \varphi_{y} }}{\partial x\partial y}} \hfill \\ { = \lambda_{2}^{2} I_{1} \left( {\frac{{\partial^{2} u}}{{\partial t^{2} }} + V^{2} \frac{{\partial^{2} u}}{{\partial x^{2} }} + 2V\frac{{\partial^{2} u}}{\partial x\partial t}} \right) + \lambda_{2} I_{2} \left( {\frac{{\partial^{2} \varphi_{x} }}{{\partial t^{2} }} + V^{2} \frac{{\partial^{2} \varphi_{x} }}{{\partial x^{2} }} + 2V\frac{{\partial^{2} \varphi_{x} }}{\partial x\partial t}} \right)} \hfill \\ \end{array} } \right., $$

(26.1)

$$ \delta v:\left\{ \begin{gathered} \left( {a_{12} + a_{66} } \right)\lambda_{2}^{2} \frac{{\partial^{2} u}}{\partial x\partial y} + a_{66} \lambda_{3}^{2} \frac{{\partial^{2} v}}{{\partial x^{2} }} + a_{22} \lambda_{2}^{2} \frac{{\partial^{2} v}}{{\partial y^{2} }} + \left( {b_{12} + b_{66} } \right)\lambda_{2} \frac{{\partial^{2} \varphi_{x} }}{\partial x\partial y} + b_{66} \lambda_{3} \frac{{\partial^{2} \varphi_{y} }}{{\partial x^{2} }} \hfill \\ + b_{22} \lambda_{1} \lambda_{2} \frac{{\partial^{2} \varphi_{y} }}{{\partial y^{2} }} = \lambda_{3}^{2} I_{1} \left( {\frac{{\partial^{2} v}}{{\partial t^{2} }} + V^{2} \frac{{\partial^{2} v}}{{\partial x^{2} }} + 2V\frac{{\partial^{2} v}}{\partial x\partial t}} \right) + \lambda_{3} I_{2} \left( {\frac{{\partial^{2} \varphi_{y} }}{{\partial t^{2} }} + V^{2} \frac{{\partial^{2} \varphi_{y} }}{{\partial x^{2} }} + 2V\frac{{\partial^{2} \varphi_{y} }}{\partial x\partial t}} \right) \hfill \\ \end{gathered} \right., $$

(26.2)

$$ \delta w:\left\{ \begin{gathered} \left( {a_{55} - N_{x}^{T} } \right)\frac{{\partial^{2} w}}{{\partial x^{2} }} + \left( {a_{44} - N_{y}^{T} } \right)\lambda_{1}^{2} \frac{{\partial^{2} w}}{{\partial y^{2} }} + a_{55} \lambda_{2} \frac{{\partial \varphi_{x} }}{\partial x} + a_{44} \lambda_{1} \lambda_{2} \frac{{\partial \varphi_{y} }}{\partial y} \\ = I_{1} \left( {\frac{{\partial^{2} w}}{{\partial t^{2} }} + V^{2} \frac{{\partial^{2} w}}{{\partial x^{2} }} + 2V\frac{{\partial^{2} w}}{\partial x\partial t}} \right) \\ \end{gathered} \right.\;, $$

(26.3)

$$ \delta \varphi_{x} :\left\{ \begin{gathered} b_{11} \lambda_{2} \frac{{\partial^{2} u}}{{\partial x^{2} }} + b_{66} \lambda_{1}^{2} \lambda_{2} \frac{{\partial^{2} u}}{{\partial y^{2} }} + \left( {b_{12} + b_{66} } \right)\lambda_{2} \frac{{\partial^{2} v}}{\partial x\partial y} - a_{55} \lambda_{2} \frac{\partial w}{{\partial x}} + d_{11} \frac{{\partial^{2} \varphi_{x} }}{{\partial x^{2} }} + d_{66} \lambda_{1}^{2} \frac{{\partial^{2} \varphi_{x} }}{{\partial y^{2} }} \hfill \\ - a_{55} \lambda_{2}^{2} \varphi_{x} + \left( {d_{12} + d_{66} } \right)\lambda_{1} \frac{{\partial^{2} \varphi_{y} }}{\partial x\partial y} \hfill \\ = \lambda_{2} I_{2} \left( {\frac{{\partial^{2} u}}{{\partial t^{2} }} + V_{0}^{2} \frac{{\partial^{2} u}}{{\partial x^{2} }} + 2V\frac{{\partial^{2} u}}{\partial x\partial t}} \right) + I_{3} \left( {\frac{{\partial^{2} \varphi_{x} }}{{\partial t^{2} }} + V^{2} \frac{{\partial^{2} \varphi_{x} }}{{\partial x^{2} }} + 2V\frac{{\partial^{2} \varphi_{x} }}{\partial x\partial t}} \right) \hfill \\ \end{gathered} \right., $$

(26.4)

$$ \delta \varphi_{y} :\left\{ \begin{gathered} \left( {b_{12} + b_{66} } \right)\lambda_{1} \lambda_{2} \frac{{\partial^{2} u}}{\partial x\partial y} + b_{66} \lambda_{1} \lambda_{2} \frac{{\partial^{2} v}}{{\partial x^{2} }} + b_{22} \lambda_{1} \lambda_{2} \frac{{\partial^{2} v}}{{\partial y^{2} }} - a_{44} \lambda_{1} \lambda_{2} \frac{\partial w}{{\partial y}} + \left( {d_{12} + d_{66} } \right)\lambda_{1} \frac{{\partial^{2} \varphi_{x} }}{\partial x\partial y} \\ + d_{66} \frac{{\partial^{2} \varphi_{y} }}{{\partial x^{2} }} + d_{22} \lambda_{1}^{2} \frac{{\partial^{2} \varphi_{y} }}{{\partial y^{2} }} - a_{44} \lambda_{2}^{2} \varphi_{y} \\ = \lambda_{3} I_{2} \left( {\frac{{\partial^{2} v}}{{\partial t^{2} }} + V^{2} \frac{{\partial^{2} v}}{{\partial x^{2} }} + 2V\frac{{\partial^{2} v}}{\partial x\partial t}} \right) + I_{3} \left( {\frac{{\partial^{2} \varphi_{y} }}{{\partial t^{2} }} + V^{2} \frac{{\partial^{2} \varphi_{y} }}{{\partial x^{2} }} + 2V\frac{{\partial^{2} \varphi_{y} }}{\partial x\partial t}} \right) \\ \end{gathered} \right.. $$

(26.5)

4 Solution procedure and stability analysis

The in-plane and out-plane displacements for the simply supported plate with movable edges are approximated using the following series expansions:

$$ u(x,y,t) = \sum\limits_{m = 1}^{M} {\sum\limits_{n = 1}^{N} {\cos \left( {m\pi x} \right)\sin \left( {n\pi y} \right)u_{m,n} } (t)} = {\mathbf{U}}^{{\text{T}}} (x,y){\mathbf{q}}{}_{{^{u} }}(t), $$

(27.1)

$$ v(x,y,t) = \sum\limits_{m = 1}^{M} {\sum\limits_{n = 1}^{N} {\sin \left( {m\pi x} \right)\cos \left( {n\pi y} \right)v_{m,n} } (t)} = {\mathbf{V}}^{{\mathbf{T}}} (x,y){\mathbf{q}}_{v} (t), $$

(27.2)

$$ w(x,y,t) = \sum\limits_{m = 1}^{M} {\sum\limits_{n = 1}^{N} {\sin \left( {m\pi x} \right)\sin \left( {n\pi y} \right)w_{m,n} } (t)} = {\mathbf{W}}^{{\mathbf{T}}} (x,y){\mathbf{q}}_{w} (t), $$

(27.3)

$$ \varphi_{x} (x,y,t) = \sum\limits_{m = 1}^{M} {\sum\limits_{n = 1}^{N} {\cos \left( {m\pi x} \right)\sin \left( {n\pi y} \right)\varphi_{xm,n} } (t)} = {{\varvec{\Phi}}}_{x}^{{\text{T}}} (x,y){\mathbf{q}}_{{\varphi_{x} }} (t), $$

(27.4)

$$ \varphi_{y} (x,y,t) = \sum\limits_{m = 1}^{M} {\sum\limits_{n = 1}^{N} {\sin \left( {m\pi x} \right)\cos \left( {n\pi y} \right)\varphi_{ym,n} } (t)} = {{\varvec{\Phi}}}_{y}^{{\text{T}}} (x,y){\mathbf{q}}_{{\varphi_{y} }} (t), $$

(27.5)

where m and n denote the half wave numbers in x and y directions, respectively. M and N are the numbers of mode expansion. U(x, y), ${\mathbf{V}}(x,y)$, ${\mathbf{W}}(x,y)$, ${{\varvec{\Phi}}}_{x} (x,y)$, and ${{\varvec{\Phi}}}_{y} (x,y)$ are the vectors composed of mode shape functions. ${\mathbf{q}}_{u} (t)$, ${\mathbf{q}}_{v} (t)$, ${\mathbf{q}}_{w} (t)$, ${\mathbf{q}}_{{\varphi_{x} }} (t)$, and ${\mathbf{q}}_{{\varphi_{y} }} (t)$ are generalized coordinate vectors.

By introducing Eq. (27) into Eq. (26) and executing the Galerkin integration, the ordinary differential governing equations (ODEs) for free vibration of axially moving FG-CNTRC thin plates in matrix–vector form are obtained as follows:

$$ {\mathbf{M}}\ddot{{\mathbf{q}}} + {\mathbf{C}}_{V} {\dot{\mathbf{q}}} + ({\mathbf{K}} + {\mathbf{K}}_{V} + {\mathbf{K}}_{T} ){\mathbf{q}} = {\mathbf{0}} $$

(28)

where ${\mathbf{q}} = [{\mathbf{q}}_{u}^{{\text{T}}} ,{\mathbf{q}}_{v}^{{\text{T}}} ,{\mathbf{q}}_{w}^{{\text{T}}} ,{\mathbf{q}}_{{\varphi_{x} }}^{{\text{T}}} ,{\mathbf{q}}_{{\varphi_{y} }}^{{\text{T}}} ]^{{\text{T}}}$ is the generalized coordinate vector. M and K are mass and structural stiffness matrices, respectively. C_V is the gyroscopic matrix. K_V and K_T are, respectively, the stiffness matrices related to moving velocity and temperature rise. These matrices are in the form of:

$$ {\mathbf{M}} = \left[ {\begin{array}{*{20}c} {{\mathbf{M}}_{{{11}}} } & \quad {\mathbf{0}} & \quad {\mathbf{0}} & \quad {{\mathbf{M}}_{{{14}}} } & \quad {\mathbf{0}} \\ {\mathbf{0}} & \quad {{\mathbf{M}}_{{{22}}} } & \quad {\mathbf{0}} & \quad {\mathbf{0}} & \quad {{\mathbf{M}}_{{{25}}} } \\ {\mathbf{0}} & \quad {\mathbf{0}} & \quad {{\mathbf{M}}_{{{33}}} } & \quad {\mathbf{0}} & \quad {\mathbf{0}} \\ {{\mathbf{M}}_{{{41}}} } & \quad {\mathbf{0}} & \quad {\mathbf{0}} & \quad {{\mathbf{M}}_{{{44}}} } & \quad {\mathbf{0}} \\ {\mathbf{0}} & \quad {{\mathbf{M}}_{{{52}}} } & \quad {\mathbf{0}} & \quad {\mathbf{0}} & \quad {{\mathbf{M}}_{{{55}}} } \\ \end{array} } \right],\quad {\mathbf{C}}_{V} = \left[ {\begin{array}{*{20}c} {{\mathbf{C}}_{V11} } & \quad {\mathbf{0}} & \quad {\mathbf{0}} & \quad {{\mathbf{C}}_{V14} } & \quad {\mathbf{0}} \\ {\mathbf{0}} & \quad {{\mathbf{C}}_{{V{22}}} } & \quad {\mathbf{0}} & \quad {\mathbf{0}} & \quad {{\mathbf{C}}_{V25} } \\ {\mathbf{0}} & \quad {\mathbf{0}} & \quad {{\mathbf{C}}_{{V{33}}} } & \quad {\mathbf{0}} & \quad {\mathbf{0}} \\ {{\mathbf{C}}_{V41} } & \quad {\mathbf{0}} & \quad {\mathbf{0}} & \quad {{\mathbf{C}}_{{V{44}}} } & \quad {\mathbf{0}} \\ {\mathbf{0}} & \quad {{\mathbf{C}}_{V52} } & \quad {\mathbf{0}} & \quad {\mathbf{0}} & \quad {{\mathbf{C}}_{{V{55}}} } \\ \end{array} } \right], $$

(29.1)

$$ {\mathbf{K}} = \left[ {\begin{array}{*{20}c} {{\mathbf{K}}_{11} } & \quad {{\mathbf{K}}_{12} } & \quad {\mathbf{0}} & \quad {{\mathbf{K}}_{14} } & \quad {{\mathbf{K}}_{15} } \\ {{\mathbf{K}}_{21} } & \quad {{\mathbf{K}}_{{{22}}} } & \quad {\mathbf{0}} & \quad {{\mathbf{K}}_{{{24}}} } & \quad {{\mathbf{K}}_{{{25}}} } \\ {\mathbf{0}} & \quad {\mathbf{0}} & \quad {{\mathbf{K}}_{{{33}}} } & \quad {{\mathbf{K}}_{{{34}}} } & \quad {{\mathbf{K}}_{{{35}}} } \\ {{\mathbf{K}}_{{{41}}} } & \quad {{\mathbf{K}}_{{{42}}} } & \quad {{\mathbf{K}}_{{{43}}} } & \quad {{\mathbf{K}}_{{{44}}} } & \quad {{\mathbf{K}}_{{{45}}} } \\ {{\mathbf{K}}_{{{51}}} } & \quad {{\mathbf{K}}_{{{52}}} } & \quad {{\mathbf{K}}_{{{53}}} } & \quad {{\mathbf{K}}_{{{54}}} } & \quad {{\mathbf{K}}_{{{55}}} } \\ \end{array} } \right],\quad {\mathbf{K}}_{V} = \left[ {\begin{array}{*{20}c} {{\mathbf{K}}_{V11} } & \quad {\mathbf{0}} & \quad {\mathbf{0}} & \quad {{\mathbf{K}}_{V14} } & \quad {\mathbf{0}} \\ {\mathbf{0}} & \quad {{\mathbf{K}}_{{V{22}}} } & \quad {\mathbf{0}} & \quad {\mathbf{0}} & \quad {{\mathbf{K}}_{V25} } \\ {\mathbf{0}} & \quad {\mathbf{0}} & \quad {{\mathbf{K}}_{{V{33}}} } & \quad {\mathbf{0}} & \quad {\mathbf{0}} \\ {{\mathbf{K}}_{V41} } & \quad {\mathbf{0}} & \quad {\mathbf{0}} & \quad {{\mathbf{K}}_{V44} } & \quad {\mathbf{0}} \\ {\mathbf{0}} & \quad {{\mathbf{K}}_{V52} } & \quad {\mathbf{0}} & \quad {\mathbf{0}} & \quad {{\mathbf{K}}_{V55} } \\ \end{array} } \right], $$

(29.2)

$$ {\mathbf{K}}_{T} = \left[ {\begin{array}{*{20}c} {\mathbf{0}} & \quad {\mathbf{0}} & \quad {\mathbf{0}} & \quad {\mathbf{0}} & \quad {\mathbf{0}} \\ {\mathbf{0}} & \quad {\mathbf{0}} & \quad {\mathbf{0}} & \quad {\mathbf{0}} & \quad {\mathbf{0}} \\ {\mathbf{0}} & \quad {\mathbf{0}} & \quad {{\mathbf{K}}_{{T{33}}} } & \quad {\mathbf{0}} & \quad {\mathbf{0}} \\ {\mathbf{0}} & \quad {\mathbf{0}} & \quad {\mathbf{0}} & \quad {\mathbf{0}} & \quad {\mathbf{0}} \\ {\mathbf{0}} & \quad {\mathbf{0}} & \quad {\mathbf{0}} & \quad {\mathbf{0}} & \quad {\mathbf{0}} \\ \end{array} } \right]. $$

(29.3)

The elements of the above matrices are defined in the Appendix.

The complex eigenvalue problem of the system can be solved by applying the state space method. By introducing the state variable vector ${\mathbf{X}} = [{\mathbf{q}},{\dot{\mathbf{q}}}]^{{\text{T}}}$, the equations of motion are transformed into

$$ {\mathbf{AX}} = {\dot{\mathbf{X}}} $$

(30)

where the coefficient matrix is

$$ {\mathbf{A}} = \left[ {\begin{array}{*{20}c} {\mathbf{0}} & \quad {\mathbf{I}} \\ { - {\mathbf{M}}^{{ - {\mathbf{1}}}} \left( {{\mathbf{K}} + {\mathbf{K}}_{V} + {\mathbf{K}}{}_{T}} \right)} & \quad { - {\mathbf{M}}^{{ - {\mathbf{1}}}} {\mathbf{C}}_{V} } \\ \end{array} } \right]{\dot{\mathbf{X}}}. $$

(31)

On basis of the eigenvalue equations $\left| {{\mathbf{A}} - \lambda {\mathbf{I}}} \right| = {\mathbf{0}}$, the complex eigenvalues ω_i related to the axial velocity V can be obtained. When the axial velocity V increases gradually from zero, the stability of the system can be judged by the value of the complex eigenvalues ω_i. The imaginary Im(ω_i) and real Re(ω_i) parts of eigenvalues represent the natural frequency and stability, respectively. For different values of Im(ω_i) and Re(ω_i), the states of the system are as follows [21]:

• 1. For Im(ω_i) = 0, i.e., the natural frequency is equal to zero: If Re(ω_i) < 0, the system is static stable. If Re(ω_i) > 0, static divergence instability occurs. When Re(ω_i) = 0, the system is in the critical state of divergence instability, and the axial velocity is the critical velocity of divergence instability.

• 2. For Im(ω_i) > 0: If Re(ω_i) < 0, the vibration gradually attenuates and the system is in dynamic stable state. If Re(ω_i) > 0, the vibration is amplified continuously and dynamic flutter instability occurs. When Re(ω_i) = 0, the system is in the critical state of flutter instability and the critical velocity of flutter instability is obtained.

• 3. When the eigenvalues are conjugate complex with opposite real parts, the system generates coupled-mode flutter.

5 Numerical simulations and discussion

The natural frequencies of axially moving FG-CNTRC thin plates are numerically simulated, and the stabilities are analyzed by parameter studies in this Section. The effects of CNT distribution, volume fraction, aspect ratio, width-to-thickness ratio, and temperature rise on the natural frequencies, stabilities and critical velocity of instability are discussed by numerical examples.

Poly methyl methacrylate (PMMA) is chosen as the matrix with ρ^m = 1150 kg/m³, E^m = (3.52–0.0034 T)Gpa, ν^m = 0.34, α^m = 45(1 + 0.0005ΔT) × 10^–6/K, where ΔT = T − T₀ is the temperature rise, T₀ = 300 K (E^m = 2.5 Gpa and α^m = 45 × 10^–6/K for ΔT = 0). The (10, 10) single-walled carbon nanotubes (SWCNTs) are chosen as reinforcements, and the material properties are related to the temperature. Table 1 gives the temperature-dependent material properties of SWCNTs at four specific temperatures [44], namely: T = 300 K, 400 K, 500 K, and 700 K. In order to obtain the continuous change of material properties of SWCNTs with temperature, a cubic interpolation is performed to obtain the thermophysical parameters. In the range of 300 K ≤ T ≤ 700 K, the functions of material properties of SWCNTs with respect to temperature are as follows [50]:

$$ \begin{aligned} E_{11}^{{{\text{cnt}}}} \left( T \right) & = \left( {6.3998 - 4.338417 \times 10^{ - 3} T + 7.43 \times 10^{ - 6} T^{2} - 4.458333 \times 10^{ - 9} T^{3} } \right)\;\left( {{\text{Tpa}}} \right), \\ E_{22}^{{{\text{cnt}}}} \left( T \right) & = \left( {8.02155 - 5.420375 \times 10^{ - 3} T + 9.275 \times 10^{ - 6} T^{2} - 5.5625 \times 10^{ - 9} T^{3} } \right)\;\left( {{\text{Tpa}}} \right), \\ G_{12}^{{{\text{cnt}}}} \left( T \right) & = \left( {1.40755 + 3.476208 \times 10^{ - 3} T - 6.965 \times 10^{ - 6} T^{2} + 4.479167 \times 10^{ - 9} T^{3} } \right)\;\left( {{\text{Tpa}}} \right), \\ \alpha_{11}^{{{\text{cnt}}}} \left( T \right) & = \left( { - 1.12515 + 0.02291688T - 2.887 \times 10^{ - 5} T^{2} + 1.13625 \times 10^{ - 8} T^{3} } \right) \times {10}^{ - 6} {\text{/K}}, \\ \alpha_{22}^{{{\text{cnt}}}} \left( T \right) & = \left( {5.43715 - 9.84625 \times 10^{ - 4} T + 2.9 \times 10^{ - 7} T^{2} + 1.25 \times 10^{ - 11} T^{3} } \right) \times {10}^{ - 6} {\text{/K}}, \\ \rho^{{{\text{cnt}}}} & = 1400\;{\text{kg}}/{\text{m}}^{{3}}, \\ \nu_{12}^{{{\text{cnt}}}} & = 0.175. \\ \end{aligned} $$

(32)

Table 1 Temperature-dependent material properties of SWCNTs [44] (tube length = 9.26 nm, tube mean radius = 0.68 nm, tube thickness = 0.067 nm, $\nu_{12}^{{{\text{cnt}}}}$ = 0.175)

The CNT efficiency parameters used in the numerical simulations are [47]: η₁ = 0.137, η₂ = 1.022, and η₃ = 0.715 for $V_{{{\text{cnt}}}}^{*} = 0.12$, η₁ = 0.142, η₂ = 1.626, and η₃ = 1.138 for $V_{{{\text{cnt}}}}^{*} = 0.17$, and η₁ = 0.141, η₂ = 1.585 and η₃ = 1.109 for $V_{{{\text{cnt}}}}^{*} = 0.28$. Unless otherwise stated, the material parameters used in the numerical simulations are as described above.

5.1 Validation study

In this article, the Galerkin method is adopted to establish the discrete model, and the truncation numbers are selected by the frequency convergence analysis. In Table 2, the natural frequencies are compared for different truncation numbers. It can be seen that the natural frequency will gradually converge with the increase of the truncation number M, while the increase of N has no effect on the frequency convergence. It is found that for the same half wave number n, modes (m, n) with different m are coupled, while the modes (m, n) with the same m and different n are uncoupled. For instance, $\omega_{1,1}$ and $\omega_{2,1}$ are coupled, and ω_1,2 and $\omega_{2,2}$ are coupled, while $\omega_{1,1}$ and $\omega_{1,2}$ are uncoupled [60]. In the subsequent analysis, this phenomenon can also be observed in Fig. 2. According to the above result, the assumed truncation numbers are chosen as M = 2, N = 2 in the following numerical simulations.

Table 2 Comparisons of the natural frequency of axially moving FGA-CNTRC plates for different truncation numbers ($V_{{{\text{cnt}}}}^{*}$ = 0.17, a/b = 1, b/h = 100, ΔT = 0, V = 0.02)

Fig. 2

Variations of the complex natural frequencies of axially moving FGA-CNTRC plate with respect to the moving velocity ($V_{{{\text{cnt}}}}^{*}$ = 0.17, a/b = 1, b/h = 100, ΔT = 0): a imaginary parts of eigenvalues; b real parts of eigenvalues

In order to verify the validity of the structural model and solution method, the results obtained in this paper are compared with those in literature. Firstly, the critical divergence velocity and natural frequency of an axially moving isotropic plate are calculated and compared with those of Hatami et al. [20] and Yao et al. [33]. Table 3 shows the comparisons of dimensionless critical divergence velocity of the axially moving isotropic plate with different aspect ratios (a/b = 3/10, 1). The material properties considered are: ρ = 7600 kg/m³, E = 70GPa, ν = 0.3. The dimensionless critical divergence velocity is defined as $\overline{V}_{{{\text{cr}}}} = V_{{{\text{cr}}}} \sqrt {\rho hb^{2} /D}$, where $D = \frac{{Eh^{3} }}{{12(1 - \mu^{2} )}}$ is the bending stiffness. When the plate is static (V = 0) or the moving speed is half of the critical divergence speed (V = 0.5V_cr), the dimensionless natural frequencies $\overline{\omega } = \omega \sqrt {\rho hb^{4} /D}$ are computed and compared in Table 4. It can be seen from Tables 3 and 4 that the dimensionless natural frequency and critical divergence velocity obtained in this paper are in good agreement with the results in literature.

Table 3 Comparisons of dimensionless critical divergence velocity ($\overline{V}_{{{\text{cr}}}} = V_{{{\text{cr}}}} \sqrt {{{\rho hb^{2} } \mathord{\left/ {\vphantom {{\rho hb^{2} } D}} \right. \kern-\nulldelimiterspace} D}}$) for an axially moving isotropic plate (ρ = 7600 kg/m³, E = 70GPa, ν = 0.3)

Table 4 Comparisons of dimensionless fundamental frequency ($\overline{\omega } = \omega \sqrt {{{\rho hb^{4} } \mathord{\left/ {\vphantom {{\rho hb^{4} } D}} \right. \kern-\nulldelimiterspace} D}}$) of an axially moving isotropic plate under different moving velocities

Secondly, the dimensionless natural frequencies of static FG-CNTRC plates are calculated and compared with the results of Zhu et al. [48], Alibeigloo and Emtehani [49]. The geometric parameters of the thin plate are: a/b = 1, b/h = 50, and the temperature is T = 300 K. The elastic modulus of the matrix (PMMA) is selected as E^m = 2.1 Gpa. The CNT efficiency parameters are: η₁ = 0.149, η₂ = η₃ = 0.934 for $V_{{{\text{cnt}}}}^{*} = 0.11$, η₁ = 0.150, η₂ = η₃ = 0.941 for $V_{{{\text{cnt}}}}^{*} = 0.14$, and η₁ = 0.149, η₂ = η₃ = 1.381 for $V_{{{\text{cnt}}}}^{*} = 0.17$. Considering four CNT distributions (UD, FGV, FGO, FGX), the dimensionless natural frequencies $\overline{\omega } = \omega (a^{2} /h)\sqrt {\rho^{{\text{m}}} /E^{{\text{m}}} }$ of modes (m = 1, n = 1) and (m = 1, n = 2) for simply supported FG-CNTRC square plates are compared in Table 5. It can be seen that the natural frequencies of FG-CNTRC plates calculated in this paper are consistent with the existing results in literature, which verifies the effectiveness of the present structural model and solution procedure.

Table 5 Comparisons of dimensionless natural frequency ($\overline{\omega } = \omega ({{a^{2} } \mathord{\left/ {\vphantom {{a^{2} } h}} \right. \kern-\nulldelimiterspace} h})\sqrt {{{\rho^{{\text{m}}} } \mathord{\left/ {\vphantom {{\rho^{{\text{m}}} } {E^{{\text{m}}} }}} \right. \kern-\nulldelimiterspace} {E^{{\text{m}}} }}}$ ) for simply supported FG-CNTRC square plates (a/b = 1, b/h = 50, T = 300 K)

5.2 The variations of natural frequency and stability

Figure 2 illustrates the variations of natural frequency and stability of axially moving FG-CNTRC plates with the dimensionless moving velocity. Considering FGA-CNTRC plates with asymmetric CNT distribution, the simulation parameters are: $V_{{{\text{cnt}}}}^{*} = 0.17$, a/b = 1, b/h = 100, ΔT = 0. The numbers of modal expansions used in Galerkin discretization are M = 2 and N = 2. The complex natural frequencies ω_{m, n} are obtained by solving the complex eigenvalue problem of the system, where the subscripts m and n are the half wave numbers in x and y directions.

As shown in Fig. 2, when the dimensionless moving velocity V = 0, the imaginary part of complex natural frequency Im(ω_{m, n}) is larger than zero, the real part Re(ω_{m, n}) is zero. With the increase of moving velocity in the range 0 < V < V_d1 = 0.0552, the imaginary part Im(ω_{1, 1}) of the complex eigenvalue of mode (1, 1) decreases, while the real part Re(ω_{1, 1}) remains zero, and the system is stable.

When the moving velocity increases to a critical value V_d1 = 0.0552, the natural frequency of mode (1, 1) disappears, i.e., the imaginary part Im(ω_{1, 1}) becomes zero. Subsequently, when V_d1 < V < V_de1 = 0.1042, the imaginary part Im(ω_{1, 1}) = 0, while the real part Re(ω_{1, 1}) > 0 and Re(ω_{1, 1}) < 0 occur, which means that the plate behaves divergence type of instability. V_d1 is the first-order critical divergence velocity, and V_de1 is called the boundary velocity of first-order divergent instability.

When V_de1 < V < V_f1 = 0.1145, Im(ω_{1, 1}) > 0 and Re(ω_{1, 1}) = 0, indicating that the system returns to be stable again. When the moving velocity increases from V_de1 to V_f1, the natural frequency ω_{1, 1} of mode (1, 1) increases form zero, while the natural frequency ω_{2, 1} of mode (2, 1) always decreases with the increase of moving velocity in the range 0 < V < V_f1.

When V = V_f1, the increasing natural frequency ω_{1, 1} and decreasing natural frequency ω_{2, 1} begin to couple together. In the case of V > V_f1, Im(ω_{1, 1}) = Im(ω_{2, 1}) > 0, Re(ω_{1, 1}) > 0 and Re(ω_{2, 1}) = − Re(ω_{1, 1}). This shows that the plate is in coupled-mode flutter type of instability, and V_f1 is the first-order critical flutter velocity.

For mode (1, 2) and mode (2, 2), the similar variation trend of the complex natural frequency with the moving velocity can also be seen in Fig. 2. When 0 < V < V_d2 = 0.0767, the system is stable. When V_d2 < V < V_de2 = 0.1102, the plate is in divergence type of instability, and V_d2 is the second-order critical divergence velocity, and V_de2 is called the boundary velocity of second-order divergent instability. When V_de2 < V < V_f2 = 0.1237, the system is stable again. When V > V_f2, the mode (1, 2) couples the mode (2, 2), and the plate behaves coupled-mode flutter type of instability, V_f2 is the second-order critical flutter velocity.

5.3 Effect of CNT distribution

Figure 3 depicts the effect of CNT distribution on the natural frequency and critical instability velocity of axially moving FG-CNTRC plates. It can be seen in Fig. 3a, in the subcritical range where the axial moving velocity is lower than the critical divergence velocity (0 < V < V_d1), for mode (1, 1) and mode (2, 1), the FGX-CNTRC plate has the highest natural frequencies ω_{1, 1} and ω_{2, 1}, followed by UD-, FGA-, FGO-CNTRC plates. At the same time, for axially moving FG-CNTRC plates with different CNT distributions, the first-order critical divergence velocity V_d1, the boundary velocity of first-order divergent instability V_de1, and the first-order critical flutter velocity V_f1 also follow the same ranking rule. In addition, the velocity range of divergent instability of FGX-CNTRC plate is the widest, while the FGO-CNTRC plate has the narrowest velocity range of divergent instability.

Fig. 3

Variations of the natural frequencies of axially moving FG-CNTRC plates with different CNT distributions ($V_{{{\text{cnt}}}}^{*}$ = 0.17, a/b = 1, b/h = 100, ΔT = 0): a modes (1, 1) and (2, 1); b modes (1, 2) and (2, 2)

For mode (1, 2) and mode (2, 2), the same variations of natural frequency and critical velocity can be seen in Fig. 3b.

5.4 Effect of CNT volume fraction

Figure 4 shows the variations of the natural frequencies of an axially moving FGA-CNTRC plates with moving velocity under different CNT volume fractions. With the increase of CNT volume fraction, the dimensionless natural frequencies ω_{m, n} of each mode increase in the subcritical range. Meanwhile, the critical divergence velocity (V_d1, V_d2), the boundary velocity of divergent instability (V_de1, V_de2), and the critical flutter velocity (V_f1, V_f2) also increase with the increase of CNT volume fraction. In addition, the higher the CNT volume fraction, the wider the velocity range of divergence instability.

Fig. 4

Variations of the natural frequencies of axially moving FGA-CNTRC plates with different CNT volume fractions (a/b = 1, b/h = 100, ΔT = 0): a modes (1, 1) and (2, 1); b modes (1, 2) and (2, 2)

5.5 Effect of aspect ratio

The variations of natural frequencies of an axially moving FGA-CNTRC plate with various aspect ratios (a/b = 0.5, 1, 1.5, 2, 3, 4) are given in Fig. 5. For small aspect ratio (a/b = 0.5, 1, 1.5), in the subcritical range, the dimensionless natural frequencies ω_{m, n} of each mode decrease with the increase of moving velocity. When a/b = 0.5, ω_{1, 1} is very close to ω_{1, 2}, ω_{2, 1} is very close to ω_{2, 2}. As the aspect ratio increases from 0.5 to 1.5, the velocity range of divergent instability narrows gradually. When a/b = 1.5, the second-order critical divergence velocity V_d2 is almost equal to the boundary velocity of second-order divergent instability V_de2, and the velocity range of second-order divergent instability is close to disappearing (V_d2=V_de2). After that, as the aspect ratio continues to increase, the velocity range of second-order divergent instability reappears again.

Fig. 5

Variations of the natural frequencies of axially moving FGA-CNTRC plates with different aspect ratios ($V_{{{\text{cnt}}}}^{*}$ = 0.17, b/h = 100, ΔT = 0): a a/b = 0.5, b a/b = 1.0, c a/b = 1.5, d a/b = 2.0, e a/b = 3.0, (f) a/b = 4.0

For larger aspect ratio (a/b = 2, 3, 4), with the increase of moving velocity, the natural frequency ω_{2, 2} no longer monotonically decreases before the second-order critical flutter velocity, but increases first and then decreases. When a/b ≥ 4, the natural frequency ω_{2, 1} also increases first and then decreases with the increase of moving velocity. When a/b = 3, the velocity range of first-order divergent instability tends to disappears (V_d1=V_de1), while reappears as the aspect ratio continues to increase (a/b > 3). Besides, the boundary velocity of divergent instability, (V_de1, V_de2) is infinitely close to the critical flutter velocity (V_f1, V_f2).

In the case of aspect ratio changing continuously, Fig. 6 shows the stable and unstable regions which are constructed by plotting the critical divergence velocity, boundary velocity of divergent instability, and critical flutter velocity. It can be seen in Fig. 6a, with the increase of aspect ratio, that the first-order critical divergence velocity V_d1 decreases in the range of 0.5 < a/b < 2.1, increases in the range of 2.1 < a/b < 2.94, decreases in the range of 2.94 < a/b < 4.15, and increases in the range of a/b > 4.15. The boundary velocity of first-order divergent instability V_de1 first decreases and then increases, and V_d1=V_de1 when a/b = 2.94. The width of the region of first-order divergent instability first narrows, disappears until a/b = 2.95, and then begins to widen. With the aspect ratio increasing, the critical flutter velocity V_f1 first decreases to the minimum when a/b = 3.15, then begins to increase, and finally gradually approaches V_de1. The region of first-order flutter instability firstly grows larger with the increase of aspect ratio when a/b < 3.15 and then narrows when the aspect ratio continues to increase.

Fig. 6

Effects of aspect ratio on the critical divergence velocity and critical flutter velocity of axially moving FGA-CNTRC plates ($V_{{{\text{cnt}}}}^{*}$ = 0.17, b/h = 100, ΔT = 0): a modes (1, 1) and (2, 1); b modes (1, 2) and (2, 2)

From Fig. 6b, with the increase of aspect ratio, mode (1, 2) and mode (2, 2) have the similar phenomenon. When a/b = 1.46, the velocity range of second-order divergent instability disappears (V_d2=V_de2). The flutter instability region decreases to the narrowest when a/b = 1.6.

5.6 Effect of width-to-thickness ratio

Figures 7 and 8 describe the influence of width-to-thickness ratio b/h on the natural frequencies, critical divergence velocity, and critical flutter velocity of axially moving FGA-CNTRC plates.

Fig. 7

Variations of the natural frequencies of axially moving FGA-CNTRC plates with different width-to-thickness ratios ($V_{{{\text{cnt}}}}^{*} = 0.{17}$, a/b = 1, ΔT = 0): a modes (1, 1) and (2, 1); b modes (1, 2) and (2, 2)

Fig. 8

Effects of width-to-thickness ratio on the critical divergence velocity and critical flutter velocity of axially moving FGA-CNTRC plates ($V_{{{\text{cnt}}}}^{*}$ = 0.17, a/b = 1, ΔT = 0): a modes (1, 1) and (2, 1); b modes (1, 2) and (2, 2)

As can be seen in Fig. 7, at a constant moving velocity in the subcritical range, the natural frequencies ω_{m, n} of each mode decrease with the increase of b/h. The larger the width-to-thickness ratio, the lower the critical divergence velocity (V_d1, V_d2), the boundary velocity of divergent instability (V_de1, V_de2), and critical flutter velocity (V_f1, V_f2), which also can be seen in Fig. 8.

With the increase of width-to-thickness, the stable region and the region of divergent instability narrow, while the region of flutter instability widens. This indicates that the axially moving FGA-CNTRC plate with lower width-to-thickness ratio is more stable.

5.7 Effect of temperature rise

The axially moving FG-CNTRC plates considered in this paper are in the thermal environment. The natural frequency and stability of the static FGA-CNTRC plate in the thermal environment are analyzed firstly, as shown in Fig. 9. With the increase of temperature rise, the natural frequencies of each mode of the plate decrease. When ΔT = 10.7 K, the natural frequencies of mode (1, 1) and mode (1, 2) decrease to zero, i.e., Im(ω_1,1) = Im(ω_1,2) = 0, the plate is in divergence or buckling type of instability. The natural frequencies of mode (2, 1) and mode (2, 2) decrease to zero when ΔT = 41.2 K and ΔT = 46.9 K, respectively. Different from the stability characteristics of axially moving FG-CNTRC plates, with the further increase of temperature rise, the natural frequencies of the two modes of static FG-CNTRC plates are not coupled. This indicates that the static plates do not behave flutter type of instability.

Fig. 9

Variations of the complex natural frequencies of stationary FGA-CNTRC plate with respect to the temperature rise ($V_{{{\text{cnt}}}}^{*}$ = 0.17, a/b = 1, b/h = 100, V = 0): a imaginary parts of eigenvalues; b real parts of eigenvalues

Figure 10 shows the variations of the natural frequencies of axially moving FGA-CNTRC plates under different temperature rises. With the increase of temperature rise, the natural frequencies of each mode decrease, the critical divergence velocity (V_d1, V_d2), the boundary velocity of divergent instability (V_de1, V_de2), and critical flutter velocity (V_f1, V_f2) also decrease.

Fig. 10

Variations of the natural frequencies of axially moving FGA-CNTRC plates under different temperature rises ($V_{{{\text{cnt}}}}^{*}$ = 0.17, a/b = 1, b/h = 100): a modes (1, 1) and (2, 1); b modes (1, 2) and (2, 2)

When the temperature changes continuously, Fig. 11 demonstrates the effects of temperature rise on the critical divergence velocity and critical flutter velocity of axially moving FGA-CNTRC plates. From Fig. 11, with the increase of temperature rise, the stable region narrows while the unstable region widens.

Fig. 11

Effects of temperature rise on the critical divergence velocity and critical flutter velocity of axially moving FGA-CNTRC plates ($V_{{{\text{cnt}}}}^{*}$ = 0.17, a/b = 1, b/h = 100): a modes (1, 1) and (2, 1); b modes (1, 2) and (2, 2)

6 Conclusions

In this paper, the dynamic model of axially moving FG-CNTRC rectangular thin plates in thermal environment is established, and the stability of the gyroscope system is studied. In view of Reddy’s first-order shear deformation theory, the equations of motion of axially moving FG-CNTRC plats are derived by using Hamilton’s principle and Galerkin method. By the analysis of complex eigenvalue problems of the system, the variations of natural frequency of the axially moving FG-CNTRC plates with axial moving velocity are obtained, and the critical divergence velocity and critical flutter velocity are studied. The effects of CNT distribution, volume fraction, aspect ratio, width-to-thickness ratio, and temperature rise on the natural frequency and stability of axially moving FG-CNTRC plates are discussed. The following conclusions could be drawn:

• 1. In the subcritical range, the FGX-CNTRC plate has the highest natural frequencies of each mode, followed by UD-, FGA-, FGO-CNTRC plate. The critical divergence velocity, the width of velocity range of divergent instability, and the critical flutter velocity also follow the same ranking rule.

• 2. The natural frequencies of each mode, critical divergence velocity, and critical flutter velocity increase with the increase of CNT volume fraction, while decrease with the width-to-thickness ratio. The higher CNT volume fraction, the wider the velocity range of divergence instability. The increase of width-to-thickness ratio leads to the narrowing of the stable region and the region of divergent instability, while the widening of the region of flutter instability. The axially moving FG-CNTRC plate with higher CNT volume fraction or lower width-to-thickness ratio is more stable.

• 3. With the increase of aspect ratio, the critical divergence velocity first decreases and then increases, and the region of divergent instability narrows until it disappears. After that, the critical divergence velocity first decreases and then increases, the region of divergent instability is broader. With increasing aspect ratio, the critical flutter velocity first decreases and then increases, and the region of flutter instability first widens and then narrows.

• 4. The natural frequencies of each mode, critical divergence velocity, and critical flutter velocity of axially moving FG-CNTRC plates decrease with the increase of temperature rise.

The present study can be instructive for the stability analysis of axially moving FG-CNTRC structures and other gyroscopic continuous systems.

Appendix

The elements of matrix M are

$$ \begin{aligned} {\mathbf{M}}_{{{11}}} & = \lambda_{2}^{2} I_{1} \int_{0}^{1} {\int_{0}^{1} {{\mathbf{UU}}^{{\text{T}}} } } {\text{d}}x{\text{d}}y,\quad {\mathbf{M}}_{{{14}}} = \lambda_{2} I_{2} \int_{0}^{1} {\int_{0}^{1} {{\mathbf{U\Phi }}_{x}^{{\text{T}}} } } {\text{d}}x{\text{d}}y, \\ {\mathbf{M}}_{{{22}}} & = \lambda_{3}^{2} I_{1} \int_{0}^{1} {\int_{0}^{1} {{\mathbf{VV}}^{{\text{T}}} } } {\text{d}}x{\text{d}}y,\quad {\mathbf{M}}_{{{25}}} = \lambda_{3} I_{2} \int_{0}^{1} {\int_{0}^{1} {{\mathbf{V\Phi }}_{y}^{{\text{T}}} } } {\text{d}}x{\text{d}}y, \\ {\mathbf{M}}_{{{33}}} & = I_{1} \int_{0}^{1} {\int_{0}^{1} {{\mathbf{WW}}^{{\text{T}}} } } {\text{d}}x{\text{d}}y, \\ {\mathbf{M}}_{{{41}}} & = \lambda_{2} I_{2} \int_{0}^{1} {\int_{0}^{1} {{{\varvec{\Phi}}}_{x} {\mathbf{U}}^{{\mathbf{T}}} } } {\text{d}}x{\text{d}}y,\quad {\mathbf{M}}_{{{44}}} = I_{3} \int_{0}^{1} {\int_{0}^{1} {{{\varvec{\Phi}}}_{x} {{\varvec{\Phi}}}_{x}^{{\text{T}}} } } {\text{d}}x{\text{d}}y, \\ {\mathbf{M}}_{{{52}}} & = \lambda_{3} I_{2} \int_{0}^{1} {\int_{0}^{1} {{{\varvec{\Phi}}}_{y} {\mathbf{V}}^{{\text{T}}} } } {\text{d}}x{\text{d}}y,\quad {\mathbf{M}}_{{{55}}} = I_{3} \int_{0}^{1} {\int_{0}^{1} {{{\varvec{\Phi}}}_{y} {{\varvec{\Phi}}}_{y}^{{\text{T}}} } } {\text{d}}x{\text{d}}y. \\ \end{aligned} $$

The elements of matrix C_V are

$$ \begin{aligned} {\mathbf{C}}_{V11} & = 2\lambda_{2}^{2} I_{1} V_{0} \int_{0}^{1} {\int_{0}^{1} {{\mathbf{U}}\frac{{\partial {\mathbf{U}}^{{\text{T}}} }}{\partial x}} } {\text{d}}x{\text{d}}y,\quad {\mathbf{C}}_{V14} = 2\lambda_{2} I_{2} V_{0} \int_{0}^{1} {\int_{0}^{1} {{\mathbf{U}}\frac{{{{\varvec{\Phi}}}_{x}^{T} }}{\partial x}} } {\text{d}}x{\text{d}}y, \\ {\mathbf{C}}_{V22} & = 2\lambda_{3}^{2} I_{1} V_{0} \int_{0}^{1} {\int_{0}^{1} {{\mathbf{V}}\frac{{\partial {\mathbf{V}}^{{\text{T}}} }}{\partial x}} } {\text{dxdy}},\quad {\mathbf{C}}_{V25} = 2\lambda_{3} I_{2} V_{0} \int_{0}^{1} {\int_{0}^{1} {{\mathbf{V}}\frac{{\partial {{\varvec{\Phi}}}_{y}^{T} }}{\partial x}} } {\text{d}}x{\text{d}}y, \\ {\mathbf{C}}_{V33} & = 2I_{1} V_{0} \int_{0}^{1} {\int_{0}^{1} {{\mathbf{W}}\frac{{\partial {\mathbf{W}}^{{\text{T}}} }}{\partial x}} } {\text{dxdy}}, \\ {\mathbf{C}}_{V41} & = 2\lambda_{2} I_{2} V_{0} \int_{0}^{1} {\int_{0}^{1} {{{\varvec{\Phi}}}_{x} \frac{{\partial {\mathbf{U}}^{{\text{T}}} }}{\partial x}} } {\text{dxdy}},\quad {\mathbf{C}}_{V44} = 2I_{3} V_{0} \int_{0}^{1} {\int_{0}^{1} {{{\varvec{\Phi}}}_{x} \frac{{\partial {{\varvec{\Phi}}}_{x}^{T} }}{\partial x}} } {\text{d}}x{\text{d}}y, \\ {\mathbf{C}}_{V52} & = 2\lambda_{3} I_{2} V_{0} \int_{0}^{1} {\int_{0}^{1} {{{\varvec{\Phi}}}_{y} \frac{{\partial {\mathbf{V}}^{{\text{T}}} }}{\partial x}} } {\text{dxdy}},\quad {\mathbf{C}}_{V55} = 2I_{3} V_{0} \int_{0}^{1} {\int_{0}^{1} {{{\varvec{\Phi}}}_{y} \frac{{\partial {{\varvec{\Phi}}}_{y}^{T} }}{\partial x}} } {\text{d}}x{\text{d}}y. \\ \end{aligned} $$

The elements of matrix ${\mathbf{K}}$ are

$$ \begin{aligned} {\mathbf{K}}_{{{11}}} &= - \int_{0}^{1} {\int_{0}^{1} {\left( {a_{11} \lambda_{2}^{2} {\mathbf{U}}\frac{{\partial^{2} {\mathbf{U}}^{{\text{T}}} }}{{\partial x^{2} }} + a_{66} \lambda_{1}^{2} \lambda_{2}^{2} {\mathbf{U}}\frac{{\partial^{2} {\mathbf{U}}^{{\text{T}}} }}{{\partial y^{2} }}} \right)} } {\text{d}}x{\text{d}}y,\\ {\mathbf{K}}_{{{12}}} &= - \left( {a_{12} + a_{66} } \right)\lambda_{2}^{2} \int_{0}^{1} {\int_{0}^{1} {{\mathbf{U}}\frac{{\partial^{2} {\mathbf{V}}^{{\text{T}}} }}{\partial x\partial y}} } {\text{d}}x{\text{d}}y, \\ {\mathbf{K}}_{{{13}}} &= 0,K_{14} = - \int_{0}^{1} {\int_{0}^{1} {\left( {b_{11} \lambda_{2} {\mathbf{U}}\frac{{\partial^{2} {{\varvec{\Phi}}}_{x}^{T} }}{{\partial x^{2} }} + b_{66} \lambda_{1}^{2} \lambda_{2} {\mathbf{U}}\frac{{\partial^{2} {{\varvec{\Phi}}}_{x}^{T} }}{{\partial y^{2} }}} \right)} } {\text{d}}x{\text{d}}y,\\ {\mathbf{K}}_{{{15}}}& = - \left( {b_{12} + b_{66} } \right)\lambda_{1} \lambda_{2} \int_{0}^{1} {\int_{0}^{1} {\left( {{\mathbf{U}}\frac{{\partial^{2} {{\varvec{\Phi}}}_{y}^{T} }}{\partial x\partial y}} \right)} } {\text{d}}x{\text{d}}y, \\ {\mathbf{K}}_{{{21}}} &= - \left( {a_{12} + a_{66} } \right)\lambda_{2}^{2} \int_{0}^{1} {\int_{0}^{1} {{\mathbf{V}}\frac{{\partial^{2} {\mathbf{U}}^{{\text{T}}} }}{\partial x\partial y}} } {\text{d}}x{\text{d}}y,\\ {\mathbf{K}}_{{{22}}} &= - \int_{0}^{1} {\int_{0}^{1} {\left( {a_{66} \lambda_{3}^{2} {\mathbf{V}}\frac{{\partial^{2} {\mathbf{V}}^{{\text{T}}} }}{{\partial x^{2} }} + a_{22} \lambda_{2}^{2} {\mathbf{V}}\frac{{\partial^{2} {\mathbf{V}}^{{\text{T}}} }}{{\partial y^{2} }}} \right)} } {\text{d}}x{\text{d}}y, \\ {\mathbf{K}}_{{{23}}} &= 0,\quad {\mathbf{K}}_{{{24}}} = - \left( {b_{12} + b_{66} } \right)\lambda_{2} \int_{0}^{1} {\int_{0}^{1} {\left( {{\mathbf{V}}\frac{{\partial^{2} {{\varvec{\Phi}}}_{x}^{{\text{T}}} }}{\partial x\partial y}} \right)} } {\text{d}}x{\text{d}}y, \\ {\mathbf{K}}_{{{25}}} & = - \int_{0}^{1} {\int_{0}^{1} {\left( {b_{66} \lambda_{3} {\mathbf{V}}\frac{{\partial^{2} {{\varvec{\Phi}}}_{y}^{T} }}{{\partial x^{2} }} + b_{22} \lambda_{1} \lambda_{2} {\mathbf{V}}\frac{{\partial^{2} {{\varvec{\Phi}}}_{y}^{T} }}{{\partial y^{2} }}} \right)} } {\text{d}}x{\text{d}}y, \\ {\mathbf{K}}_{{{31}}} & = 0,\quad {\mathbf{K}}_{{{32}}} = 0,\quad {\mathbf{K}}_{{{33}}} = - \int_{0}^{1} {\int_{0}^{1} {\left( {a_{55} {\mathbf{W}}\frac{{\partial^{2} {\mathbf{W}}^{T} }}{{\partial x^{2} }} + a_{44} \lambda_{1}^{2} {\mathbf{W}}\frac{{\partial^{2} {\mathbf{W}}^{T} }}{{\partial y^{2} }}} \right)} } {\text{d}}x{\text{d}}y, \end{aligned}$$

$$ \begin{aligned} {\mathbf{K}}_{{{34}}} &= - a_{55} \lambda_{2} \int_{0}^{1} {\int_{0}^{1} {{\mathbf{W}}\frac{{\partial {{\varvec{\Phi}}}_{x}^{T} }}{\partial x}} } {\text{d}}x{\text{d}}y,\quad {\mathbf{K}}_{{{35}}} = - a_{44} \lambda_{1} \lambda_{2} \int_{0}^{1} {\int_{0}^{1} {{\mathbf{W}}\frac{{\partial {{\varvec{\Phi}}}_{y}^{T} }}{\partial y}} } {\text{d}}x{\text{d}}y, \\ {\mathbf{K}}_{{{41}}} & = - \int_{0}^{1} {\int_{0}^{1} {\left( {b_{11} \lambda_{2} {{\varvec{\Phi}}}_{x} \frac{{\partial^{2} {\mathbf{U}}^{T} }}{{\partial x^{2} }} + b_{66} \lambda_{1}^{2} \lambda_{2} {{\varvec{\Phi}}}_{x} \frac{{\partial^{2} {\mathbf{U}}^{T} }}{{\partial y^{2} }}} \right)} } {\text{d}}x{\text{d}}y,\\ {\mathbf{K}}_{{{42}}} &= - \left( {b_{12} + b_{66} } \right)\lambda_{2} \int_{0}^{1} {\int_{0}^{1} {{{\varvec{\Phi}}}_{x} \frac{{\partial^{2} {\mathbf{V}}^{T} }}{\partial x\partial y}} } {\text{d}}x{\text{d}}y,\\ {\mathbf{K}}_{{{43}}} & = a_{55} \lambda_{2} \int_{0}^{1} {\int_{0}^{1} {{{\varvec{\Phi}}}_{x} \frac{{\partial {\mathbf{W}}^{T} }}{\partial x}} } {\text{d}}x{\text{d}}y, \\ {\mathbf{K}}_{{{44}}} & = - \int_{0}^{1} {\int_{0}^{1} {\left( {d_{11} {{\varvec{\Phi}}}_{x} \frac{{\partial^{2} {{\varvec{\Phi}}}_{x}^{{\text{T}}} }}{{\partial x^{2} }} + d_{66} \lambda_{1}^{2} {{\varvec{\Phi}}}_{x} \frac{{\partial^{2} {{\varvec{\Phi}}}_{x}^{{\text{T}}} }}{{\partial y^{2} }} - a_{55} \lambda_{2}^{2} {{\varvec{\Phi}}}_{x} {{\varvec{\Phi}}}_{x}^{T} } \right)} } {\text{d}}x{\text{d}}y, \\ {\mathbf{K}}_{{{45}}} & = - \left( {d_{12} + d_{66} } \right)\lambda_{1} \int_{0}^{1} {\int_{0}^{1} {{{\varvec{\Phi}}}_{x} \frac{{\partial^{2} {{\varvec{\Phi}}}_{y}^{{\text{T}}} }}{\partial x\partial y}} } {\text{d}}x{\text{d}}y, \\ {\mathbf{K}}_{{{51}}} & = - \;\left( {b_{12} + b_{66} } \right)\lambda_{1} \lambda_{2} \int_{0}^{1} {\int_{0}^{1} {{{\varvec{\Phi}}}_{y} \frac{{\partial^{2} {\mathbf{U}}^{T} }}{\partial x\partial y}} } {\text{d}}x{\text{d}}y, \\ {\mathbf{K}}_{{{52}}} & = - \int_{0}^{1} {\int_{0}^{1} {\left( {b_{66} \lambda_{1} \lambda_{2} {{\varvec{\Phi}}}_{y} \frac{{\partial^{2} {\mathbf{V}}^{T} }}{{\partial x^{2} }} + b_{22} \lambda_{1} \lambda_{2} {{\varvec{\Phi}}}_{y} \frac{{\partial^{2} {\mathbf{V}}^{T} }}{{\partial y^{2} }}} \right)} } {\text{d}}x{\text{d}}y,\\ {\mathbf{K}}_{{{53}}} &= a_{44} \lambda_{1} \lambda_{2} \int_{0}^{1} {\int_{0}^{1} {{{\varvec{\Phi}}}_{y} \frac{{\partial {\mathbf{W}}^{T} }}{\partial y}} } {\text{d}}x{\text{d}}y, \\ {\mathbf{K}}_{{{54}}} & = - \left( {d_{12} + d_{66} } \right)\lambda_{1} \int_{0}^{1} {\int_{0}^{1} {{{\varvec{\Phi}}}_{y} \frac{{\partial^{2} {{\varvec{\Phi}}}_{x}^{T} }}{\partial x\partial y}} } {\text{d}}x{\text{d}}y, \\ {\mathbf{K}}_{{{55}}}&= - \int_{0}^{1} {\int_{0}^{1} {\left( {d_{66} {{\varvec{\Phi}}}_{y} \frac{{\partial^{2} {{\varvec{\Phi}}}_{y}^{T} }}{{\partial x^{2} }} + d_{22} \lambda_{1}^{2} {{\varvec{\Phi}}}_{y} \frac{{\partial^{2} {{\varvec{\Phi}}}_{y}^{T} }}{{\partial y^{2} }} - a_{44} \lambda_{2}^{2} {{\varvec{\Phi}}}_{y} {{\varvec{\Phi}}}_{y}^{T} } \right)} } {\text{d}}x{\text{d}}y.\\ \end{aligned} $$

The elements of matrix ${\mathbf{K}}_{V}$ are

$$ \begin{aligned} {\mathbf{K}}_{V11} &= \lambda_{2}^{2} I_{1} V_{0}^{2} \int_{0}^{1} {\int_{0}^{1} {{\mathbf{U}}\frac{{\partial^{2} {\mathbf{U}}^{{\text{T}}} }}{{\partial x^{2} }}} } {\text{d}}x{\text{d}}y,\quad {\mathbf{K}}_{V14} = \lambda_{2} I_{2} V_{0}^{2} \int_{0}^{1} {\int_{0}^{1} {{\mathbf{U}}\frac{{\partial^{2} {{\varvec{\Phi}}}_{x}^{T} }}{{\partial x^{2} }}} } {\text{d}}x{\text{d}}y, \\ {\mathbf{K}}_{V22} &= \lambda_{3}^{2} I_{1} V_{0}^{2} \int_{0}^{1} {\int_{0}^{1} {{\mathbf{V}}\frac{{\partial^{2} {\mathbf{V}}^{{\text{T}}} }}{{\partial x^{2} }}} } {\text{d}}x{\text{d}}y,\quad {\mathbf{K}}_{V25} = \lambda_{3} I_{2} V_{0}^{2} \int_{0}^{1} {\int_{0}^{1} {{\mathbf{V}}\frac{{\partial^{2} {{\varvec{\Phi}}}_{y}^{T} }}{{\partial x^{2} }}} } {\text{d}}x{\text{d}}y, \\ {\mathbf{K}}_{V33} & = I_{1} V_{0}^{2} \int_{0}^{1} {\int_{0}^{1} {{\mathbf{W}}\frac{{\partial^{2} {\mathbf{W}}^{{\text{T}}} }}{{\partial x^{2} }}} } {\text{d}}x{\text{d}}y, \\ {\mathbf{K}}_{V41} & = \lambda_{2} I_{2} V_{0}^{2} \int_{0}^{1} {\int_{0}^{1} {{{\varvec{\Phi}}}_{x} \frac{{\partial^{2} {\mathbf{U}}^{{\text{T}}} }}{{\partial x^{2} }}} } {\text{d}}x{\text{d}}y,\quad {\mathbf{K}}_{V44} = I_{3} V_{0}^{2} \int_{0}^{1} {\int_{0}^{1} {{{\varvec{\Phi}}}_{x} \frac{{\partial^{2} {{\varvec{\Phi}}}_{x}^{T} }}{{\partial x^{2} }}} } {\text{d}}x{\text{d}}y, \\ {\mathbf{K}}_{V52} & = \lambda_{3} I_{2} V_{0}^{2} \int_{0}^{1} {\int_{0}^{1} {{{\varvec{\Phi}}}_{y} \frac{{\partial^{2} {\mathbf{V}}^{{\text{T}}} }}{{\partial x^{2} }}} } {\text{d}}x{\text{d}}y,\quad {\mathbf{K}}_{V55} = I_{3} V_{0}^{2} \int_{0}^{1} {\int_{0}^{1} {{{\varvec{\Phi}}}_{y} \frac{{\partial^{2} {{\varvec{\Phi}}}_{y}^{{\text{T}}} }}{{\partial x^{2} }}} } {\text{d}}x{\text{d}}y. \end{aligned} $$

The elements of matrix ${\mathbf{K}}_{T}$ are

$$ {\mathbf{K}}_{T33} = N_{x}^{T} \int_{0}^{1} {\int_{0}^{1} {{\mathbf{W}}\frac{{\partial^{2} {\mathbf{W}}^{{\text{T}}} }}{{\partial x^{2} }}} } {\text{d}}x{\text{d}}y + \lambda_{1}^{2} N_{y}^{T} \int_{0}^{1} {\int_{0}^{1} {{\mathbf{W}}\frac{{\partial^{2} {\mathbf{W}}^{{\text{T}}} }}{{\partial y^{2} }}} } {\text{d}}x{\text{d}}y. $$