Three-Dimensional Thermal Vibration of CFFF Functionally Graded Carbon Nanotube-Reinforced Composite Plates

Abstract

Purpose

Three-dimensional thermal vibration analysis of functionally graded carbon nanotube (FG-CNT) reinforcement composite plates is performed for uniform, linear and sinusoidally temperature distribution.

Methods

The reinforcement directions is considered through the thickness according to four reinforcement models as UD, FG-V, FG-O and FG-X are examined. As the components of the composite, the material properties of both the CNT forming the reinforcement phase and the polymer forming the matrix phase change depending on the temperature, and this is the focus of this study. The effective material properties of the FG-CNT reinforced composite are determined by the mixtures rule. The three displacements of the plates are expanded by a series of Chebyshev polynomials multiplied by appropriate functions to satisfy the essential boundary conditions. The natural frequencies are obtained by the Ritz method.

Results

It is shown that the numerical results of the current approach are compared with the results of other researchers for validation, the results appear to be in good agreement. The effects of the thickness-to-length ratio and different volume fraction distributions for cantilever (CFFF) boundary conditions in considered thermal environments are investigated. The effect of different boundary conditions such as clamped (CCCC), simply supported (SSSS), simply supported through the x-axis and clamped through the y-axis (SCSC) and simply supported through the x-axis and free through the y-axis (SFSF) is also examined. It is shown that the increase in the amount of temperature and the type of temperature distribution are effective on the decrease of frequencies.

Introduction

Composites, which consist of two or more separate materials combined in a macroscopic structural unit, are made from various combinations of the other three materials. Composites are generally used because they have desirable properties which could not be achieved by either of the constituent materials acting alone. The most common example is the fibrous composite consisting of reinforcing fibers embedded in a binder, or matrix material. Particle or flake reinforcement is also used, but it is not so effective as fibers [1]. Advanced composites made from graphite, silicon carbide, aramid polymer, boron or other higher modulus fibers are used mainly in more exotic applications such as aerospace structures where their higher cost can be justified based on improved performance [1]. Graphite or carbon fibers are the most widely used advanced fibers, and graphite/epoxy or carbon/epoxy composites are now used routinely in aerospace structures. Although carbon fibers were once prohibitively expensive, the cost has dropped significantly as production capacity and demand has increased recent years. Polymers are unquestionably the most widely used matrix materials in modern composites. The application where the superior potential of high specific strength and high specific stiffness composites was first realized was in military aircraft, where performance and maneuverability are highly dependent on weight. However, composite structural elements are used in various components of automotive, aerospace, marine and architectural structures, as well as skis, golf clubs and it is also used in consumer products such as tennis rackets [1].

A new and rapidly developing field in nanomaterials science has started with the first time obtaining of the C₆₀ molecule, which is formed by sixty carbon atoms taking the form of a soccer ball-shaped lattice structure, unlike crystal structures such as carbon, diamond and graphite [2]. Carbon nanotubes (CNTs) were first reported by Iijima [3], and it was demonstrated by Griebel and Hamaekers [4] that carbon can form a tube-shaped structure, with diameter ranges at nanoscale and lengths at micro scale.

Compared to carbon fiber-reinforced polymer composites, carbon nanotube-reinforced polymer composites have the potential to increase strength and stiffness and improve the structural, mechanical and electronic properties of the obtained composite due to CNTs low density and high aspect ratio [5,6,7,8]. Meguid and Sun [9] showed that the mechanical properties of the obtained composite would deteriorate if the CNT volume ratio exceeds a certain limit, and Han and Elliot [10] showed that the mechanical properties of the composite are also sensitive to the quantity and quality of the CNTs selected for the particular polymer. However, it was suggested to be used in the preferred direction in different gradients of CNT by Shen [11] to obtain nanocomposites desired performance. The fact that the mechanical properties of CNT and polymer change depending on temperature and that composite structures generally operate in high-temperature thermal environments have encouraged researchers to examine the behavior of CNT-reinforced polymer matrix composite structure components under mechanical and thermal loads for a high-quality design and production.

Three-dimensional thermoelastic analysis is researched by Alibeigloo and Liew [12] for FG-CNT/PmPV composite plates and by Alibeigloo [13] for FG-CNT/PmPV composite plates embedded in piezoelectric sensor and actuator layers. In both cases, the analyses are carried out using the Fourier series expansion and state-space method. Zhou and Song [14] studied three-dimensional nonlinear bending analysis of FG-CNT/PmPV composite plates using the element-free Galerkin method based on the S–R decomposition theorem. Zhu et al. [15] investigated bending and free vibration analyses of FG-CNT/PmPV composite plates by a finite element method based on the first-order shear deformation plate theory. For determination of the effective material properties of the considered plate, the rule of mixture is used. Static and free vibration analyses are carried out for FG-CNT/PmPV composite plates by Singh and Sahoo [16] using Navier’s solution technique based on trigonometric shear deformation theory. Garcia-Macias et al. [17] investigated static and free vibration analyses of FG-CNT/PMMA composite skew plates using an efficient finite element formulation based on the Hu–Washizu principle. The used shell theory is formulated in oblique coordinates and includes the effects of transverse shear strains by first-order shear deformation plate theory. Wattanasakulpong and Chaikittiratana [18] presented an exact solution based on generalized shear deformation plat theory for static and dynamic analyses of FG-CNT/PmPV composite plates with Pasternak elastic foundation including shear layer and Winklersprings. Static response and free vibration of FG-CNT/PMMA composite plates resting on Winkler–Pasternak elastic foundations using Navier solution based on the first-order shear deformation plate theory is investigated by Duc et al. [19].

Lei et al. investigated free vibration [20] and buckling [21] of laminated FG-CNT/PmPV composite plates using kp-Ritz method based on first-order shear deformation theory. Malekzadeh and Zarei [22] investigated free vibration and Malekzadeh and Shojaee [23] investigated buckling behavior of quadrilateral laminated FG-CNT/PmPV composite plates by employing the differential quadrature method (DQM) based on the first-order shear deformation plate theory. Zhang et al. presented a free vibration analyses of FG-CNT/PmPV composite triangular plates [24] and buckling analysis of thick skew plates [25] using element-free IMSL-Ritz method based on the first-order shear deformation plate theory. For determination of the effective material properties of the considered plate, the rule of mixture is used. Zhang et al. [26] considered one more buckling analysis of FG-CNT/PmPV composite thick plates resting on Winkler foundations using element-free-based improved moving least squares-Ritz (IMLS-Ritz) method employed with first-order shear deformation theory. Farzam and Hassani [27] investigated thermal and mechanical buckling analysis of FG-CNT/PmPV composite plates using isogeometric analysis based on modified couple stress theory. Kiani [28] studied thermal shear buckling of FG-CNT/PMMA composite plates in uniform temperature distribution using the Ritz method with Chebyshev polynomials based on the first-order shear deformation theory. Kiani and Mirzaei [29] researched shear buckling of rectangular and skew FG-CNT/PMMA composite plates using the Ritz method whose shape functions are constructed according to the Gram–Schmidt process based on the first-order shear deformation theory. Mehrabadi et al. [30] investigated the mechanical buckling of a rectangular FG-CNT/PMMA composite plate reinforced by aligned and straight single-walled carbon nanotubes (SWCNTs) using the Mindlin plate theory considering the first-order shear deformation effect and variational approach. The material properties of SWCNT are determined according to molecular dynamics (MDs), and then the effective material properties at a point are estimated by either the Eshelby–Mori–Tanaka approach or the extended rule of mixture. Wu and Chang [31] considered a stability problem of FG-CNT/PmPV composite plates with surface-bonded piezoelectric actuator and sensor layers under bi-axial compression loads using a unified formulation of finite layer methods based on three-dimensional elasticity theory. Semi-analytical solutions to buckling and free vibration analysis of FG-CNT/PMMA composite thin plates applying the Galerkin technique with the classical plate theory presented by Wang et al. [32].

The free vibration analysis is considered for two directional FG-CNT/PmPV composite plates by Karamanlı and Aydoğdu [33] employing finite element method based on third-order shear deformation theory. A nonlinear vibration behaviors of FG-CNT/PMMA composite plates resting on an elastic foundation under uniform temperature distribution investigated by Wang and Shen [34] and the problem is solved by an improved perturbation technique based on a higher-order shear deformation plate theory. Guo and Zhang [35] studied the nonlinear oscillations and chaotic dynamics of a FG-CNT/PmPV composite plate subjected to the in-plane and the transverse excitations. The Galerkin method based on the Reddy’s third-order shear deformation plate theory and the geometric nonlinearity of Von Karman. For determination of the effective material properties of the considered plate, Mori–Tanaka theory and the Eshelby’s method are used. Quoc et al. [36] determined free vibration response of laminated piezoelectric FG-CNT/PMMA composite plates using the Navier technique based on a new four-variable refined plate theory. The free vibration of arbitrarily shaped FG-CNT/PMMA composite plates was considered by Fantuzzi et al. [37] by employing generalized Differential Quadrature Method based on the first-order shear deformation theory. The free vibration and bending analysis of graphene-reinforced composite circular and annular plates was considered by Bisheh et al. [38] by employing Differential Quadrature Method based on the three-dimensional elasticity theory. Shahrbabaki and Alibeigloo [39] applied the Ritz method to analyze free vibration based on three-dimensional elasticity theory. In this approach, orthogonal admissible functions were obtained from Jacobi polynomial. Effective material properties of the considered plate are estimated with the modified rule of mixture approach. Wang et al. [40] investigated free vibration and buckling behavior of FG-CNT-reinforced plates in the quadrilateral geometrical form by differential quadrature and finite element method based on the first-order shear deformation plate theory. Free vibration of regular and irregular plates with and without the temperature effect are considered for FG-CNT/PMMA composite plates and mechanical and thermal buckling considered for FG-CNT/PmPV composite plates under uniaxial and bi-axial in-plane load. Zhang et al. [41] considered free vibration of FG-CNT/PmPv composite plates subjected to in-plane loads using state-space Levy method based on Reddy’s third-order shear deformation theory. Zhang and Selim [42] studied vibration analysis of FG-CNT/PmPV laminated thick composite plates employing element-free IMLS-Ritz method based on Reddy’s higher-order shear deformation theory. The effective material properties of CNT-reinforced composite are estimated by a detailed and straightforward Mori–Tanaka approach.

Selim et al. [43] studied free vibration behavior of CNT-reinforced plates using element-free kp Ritz method based on Reddy’s higher-order shear deformation theory. Parametric studies performed for FG-CNT/PmPV composite plates to reveal the effects of CNT distribution, boundary conditions, side to thickness ratio on natural frequencies and performed for FG-CNT/PMMA composite plates to reveal the effects of uniform temperature distribution on natural frequencies. The unified formulation of RMVT-based FPMs is extended to the free vibration analysis of FG-CNT-reinforced composite plates and laminated fiber-reinforced composite by Wu and Li [44]. Lei et al. [45] analyzed free vibration of FG-CNT-reinforced composite plates using element-free kp-Ritz method based on first-order shear deformation theory. In this study the effect of the uniform temperature distribution for $V_{{{\text{CNT}}}}^{*}$ = 0.12 volume fraction on UD and FG-V distributed composite on natural frequencies are obtained. Effective material properties of the considered plate are estimated with the Eshelby–Mori–Tanaka approach.

As mentioned in the literature, there are many studies on the bending, buckling and vibration behavior of CNT-reinforced plates. However, although the properties of both CNT and polymer materials that form the components of the CNT-reinforced composite, and thus the properties of the composite, vary depending on temperature, there are limited studies in which the effect of temperature is taken into account. This conclusion has been the inspiration for this study. In the present study, vibration analysis of FG-CNT/PmPV-reinforced composite plates in different thermal environments such as uniform, linear and sinusoidally temperature distribution are considered using Ritz method based on three-dimensional elasticity. The effective material properties of the considered plate are estimated by the rule of mixture. Chebyshev polynomials are assumed admissible functions in the Ritz method. However, scope of the study on cantilever (CFFF) boundary condition, different boundary conditions such as CCCC, SSSS, SCSC and SFSF are also examined. Parametric studies are performed for the thickness-to-length ratio, aspect ratio, uniform, linear and sinusoidally temperature fields and different volume fraction distributions.

Problem Formulation

Material properties of considered FG-CNT plate are assumed to be temperature dependent and reinforcement in thickness direction according to distribution as UD, FG-V, FG-O and FG-X models are examined. And also the reinforcement in thickness direction is examined of distribution as FG-V model in comparison study. In this study, thermal vibration in three different thermal environment with three-dimensional Ritz solution is performed.

Effective Material Properties of FG-CNT-Reinforced Composite Plates

The investigated FG-CNT reinforcement composite consists of CNT reinforcement phase that four type of distribution in the thickness direction and polymer matrix phase. The material properties both of two phases are assumed to be temperature dependent and reinforcement distribution models are assumed to be as uniform distribution (UD), V-type distribution (FG-V), O-type of distribution (FG-O) and X-type distribution (FG-X). The sum of volume fraction of reinforcement and matrix phases is

$$V_{{{\text{CNT}}}} + V_{{\text{m}}} = 1$$

(1)

and the CNT volume fraction distributions in the thickness direction examined in the study are as follows.

$$V_{{{\text{CNT}}}} \;(z) = V_{{{\text{CNT}}}}^{*} \quad {\text{(UD}})$$

(2)

$$V_{{{\text{CNT}}}} \;(z) = V_{{{\text{CNT}}}}^{*} \left( {1 + 2\frac{z}{h}} \right)\quad ({\text{FG - V}})$$

(3)

$$V_{{{\text{CNT}}}} \;(z) = 2V_{{{\text{CNT}}}}^{*} \left( {1 - 2\frac{z}{h}} \right)\quad ({\text{FG - O}})$$

(4)

$$V_{{{\text{CNT}}}} \;(z) = 4V_{{{\text{CNT}}}}^{*} \left( {\frac{\left| z \right|}{h}} \right)\quad ({\text{FG - X}}).$$

(5)

Here, the volume fraction of CNT ($V_{{{\text{CNT}}}}^{*}$) in the composite is defined as follows, based on the mass ratio of CNT.

$$V_{{{\text{CNT}}}}^{*} = \frac{{w_{{{\text{CNT}}}} }}{{w_{{{\text{CNT}}}} + \left( {\frac{{\rho^{{{\text{CNT}}}} }}{{\rho^{{\text{m}}} }}} \right) - \left( {\frac{{\rho^{{{\text{CNT}}}} }}{{\rho^{{\text{m}}} }}} \right)w_{{{\text{CNT}}}} }},$$

(6)

where w_CNT is the mass fraction of the CNT in the FG-CNT composite. In the study, the mass fraction of CNT is the same in all reinforcement distribution models. The effective material properties such as Young modulus (E_ii), shear modulus (G_ij) and Poisson ratio (υ_ij) are defined according to the mixtures rule are given as follows.

$$E_{11} \;(z,\;T) = \eta_{1} V_{{{\text{CNT}}}} \;(z)E_{11}^{{{\text{CNT}}}} \;(T) + V_{{\text{m}}} \;(z)\;E^{{\text{m}}} \;(T)$$

(7)

$$\frac{{\eta_{2} }}{{E_{22} \;(z,\;T)}} = \frac{{V_{{{\text{CNT}}}} \;(z)}}{{E_{22}^{{{\text{CNT}}}} \;(T)}} + \frac{{V_{{\text{m}}} \;(z)}}{{E^{{\text{m}}} \;(T)}}$$

(8)

$$\frac{{\eta_{3} }}{{G_{12} \;(z,\;T)}} = \frac{{V_{{{\text{CNT}}}} \;(z)}}{{G_{12}^{{{\text{CNT}}}} \;(T)}} + \frac{{V_{{\text{m}}} \;(z)}}{{G^{{\text{m}}} \;(T)}}$$

(9)

$$\nu_{12} = V_{{{\text{CNT}}}}^{*} \;\nu_{12}^{{{\text{CNT}}}} + V_{{\text{m}}} \nu^{{\text{m}}}$$

(10)

$$\nu_{21} = \nu_{12} \frac{{E_{22} \;(z,\;T)}}{{E_{11} \;(z,\;T)}}$$

(11)

$$\rho \;(z) = V_{{{\text{CNT}}}} \;(z)\rho^{{{\text{CNT}}}} + V_{{\text{m}}} \;(z)\rho^{{\text{m}}} ,$$

(12)

where $E_{ii}^{{{\text{CNT}}}}$ (i, j = 1, 2, 3) are the Young modulus, $G_{ij}^{{{\text{CNT}}}}$ (i, j = 1, 2, 3) are the shear modulus, $\upsilon_{ij}^{{{\text{CNT}}}}$ (i, j = 1, 2, 3) are the Poisson ratio of the CNTs in in-plane directions x and y and in thickness direction z, respectively. E^m is the Young modulus, G^m is the shear modulus, υ^m is the Poisson ratio of the isotropic matrix material. η_i (i = 1, 2, 3) are the size-dependent material properties. ρ^CNT, ρ^m and ρ are mass density per unit volume of CNT, matrix and FG-CNT composite, respectively. υ₁₂ is considered as constant over the thickness of plate. The thermal coefficients (α_ii) (i = 1, 2) of the FG-CNT reinforcement composite are defined as,

$$\alpha_{11} \;(z,\;T) = V_{{{\text{CNT}}}} \;(z)\alpha_{11}^{{{\text{CNT}}}} \;(T) + V_{{\text{m}}} \;(z)\alpha^{{\text{m}}} \;(T)$$

(13)

$$\alpha_{22} \;(z,\;T) = (1 + \nu_{12}^{{{\text{CNT}}}} )\;V_{{{\text{CNT}}}} \;(z)\alpha_{22}^{{{\text{CNT}}}} \;(T) + (1 + \nu^{{\text{m}}} )\;V_{{\text{m}}} \;(z)\alpha^{{\text{m}}} \;(T) - \nu_{12} \alpha_{11} \;(z,\;T)$$

(14)

where $\alpha_{ii}^{{{\text{CNT}}}}$ (i, j = 1, 2, 3) and α^m are the thermal coefficient of the CNTs in x and y directions, and isotropic matrix material, respectively. And the other effective material properties are assumed as, [13].

$$E_{33} \;(z,\;T) = E_{22} \;(z,\;T)$$

(15)

$$G_{23} \;(z,\;T) = G_{31} \;(z,\;T) = G_{12} \;(z,\;T)$$

(16)

$$\nu_{13} = \nu_{12}$$

(17)

$$\nu_{31} = \nu_{21}$$

(18)

$$\nu_{32} = \nu_{23} = \nu_{21}$$

(19)

Stress–Strain Relations Based on Three-Dimensional Elasticity

The considered moderately thick FG-CNT reinforcement plate in this paper is in the form of rectangular with length a, width b and thickness h. The origin of the coordinate system (x, y, z) is placed at the geometric center of the plate and the axes are parallel to the edges of the plate and the corresponding displacement components u, v and w along the x, y and z directions, respectively. For free vibration problem based on three-dimensional elasticity theory the displacement field is as follows

$$u\;(x,\;y,\;z;\;t) = U\;(x,\;y,\;z)\;e^{i\omega t}$$

(20)

$$v\;(x,\;y,\;z;\;t) = V\;(x,\;y,\;z)\;e^{i\omega t}$$

(21)

$$w\;(x,\;y,\;z;\;t) = W\;(x,\;y,\;z)\;e^{i\omega t}$$

(22)

where ω corresponds the natural frequency of the plate and $i = \sqrt { - 1}$. The strain components ε_ij (i, j = x, y, z) for small deformations are given as

$$\varepsilon_{x} = u_{,x}$$

(23)

$$\varepsilon_{y} = v_{,y}$$

(24)

$$\varepsilon_{z} = w_{,z}$$

(25)

$$\gamma_{yz} = v_{,z} + w_{,y}$$

(26)

$$\gamma_{xz} = u_{,z} + w_{,x}$$

(27)

$$\gamma_{xy} = u_{,y} + v_{,x}$$

(28)

where $_{,(x,\;y,\;z)} = \left( {\frac{\partial }{\partial x},\;\frac{\partial }{\partial y},\;\frac{\partial }{\partial z}} \right)$. The stress–strain relations for a linear elastic orthotropic material are given by the generalized Hooke’s law as follows

$$\sigma_{x} = C_{11} \varepsilon_{x} + C_{12} \varepsilon_{y} + C_{13} \varepsilon_{z}$$

(29)

$$\sigma_{y} = C_{12} \varepsilon_{x} + C_{22} \varepsilon_{y} + C_{23} \varepsilon_{z}$$

(30)

$$\sigma_{z} = C_{13} \varepsilon_{x} + C_{23} \varepsilon_{y} + C_{33} \varepsilon_{z}$$

(31)

$$\tau_{yz} = C_{44} \gamma_{yz}$$

(32)

$$\tau_{xz} = C_{55} \gamma_{xz}$$

(33)

$$\tau_{xy} = C_{66} \gamma_{xy}$$

(34)

where [C] is stiffness matrix and its components are defined as follows

$$C_{11} = \frac{{1 - \nu_{23} \nu_{32} }}{{E_{2} E_{3} \Delta }}$$

(35)

$$C_{12} = \frac{{\nu_{21} + \nu_{23} \nu_{31} }}{{E_{2} E_{3} \Delta }}$$

(36)

$$C_{13} = \frac{{\nu_{31} + \nu_{21} \nu_{32} }}{{E_{2} E_{3} \Delta }}$$

(37)

$$C_{22} = \frac{{1 - \nu_{13} \nu_{31} }}{{E_{1} E_{3} \Delta }}$$

(38)

$$C_{23} = \frac{{\nu_{32} + \nu_{12} \nu_{31} }}{{E_{1} E_{3} \Delta }}$$

(39)

$$C_{33} = \frac{{1 - \nu_{12} \nu_{21} }}{{E_{1} E_{2} \Delta }}$$

(40)

$$C_{44} = G_{23}$$

(41)

$$C_{55} = G_{31}$$

(42)

$$C_{66} = G_{12}$$

(43)

where Δ is defined as

$$\Delta = \left( {1 - \nu_{12} \nu_{21} - \nu_{23} \nu_{32} - \nu_{13} \nu_{31} - 2\nu_{21} \nu_{32} \nu_{13} } \right)/\left( {E_{1} E_{2} E_{3} } \right)$$

(44)

Thermal Analysis

In this study, thermal analysis is performed for FG-CNT reinforcement composite plates under uniform, linear and sinusoidal temperature rise through the thickness direction.

Uniform Temperature Rise

The temperature field under uniform temperature rise through the thickness is given as

$$T = T_{0} + \Delta T,$$

(45)

where T₀ is the temperature of free stress state that T₀ = 300 K and ΔT denotes the temperature change.

Linear Temperature Rise

The temperature field under linear temperature rise through the thickness is given as

$$T = T_{{\text{b}}} + \Delta T\left( {\frac{z}{h} + \frac{1}{2}} \right),$$

(46)

where ΔT = T_t − T_b is the temperature gradient, T_b and T_t at the bottom and top of the plate, T_b equals the initial temperature 300 K.

Sinusoidally Temperature Rise

The temperature field under sinusoidally temperature rise through the thickness is given as

$$T = T_{{\text{b}}} + \Delta T\left( {1 - \cos \left( {\frac{\pi }{2}\left( {\frac{z}{h} + \frac{1}{2}} \right)} \right)} \right),$$

(47)

where ΔT = T_t − T_b is the temperature gradient, T_b and T_t at the bottom and top of the plate, T_b equals the initial temperature 300 K.

Thermal Stresses Based on Three-Dimensional Elasticity

The plate is initially stress free at temperature T₀ and thermal stresses occur in the plate with temperature change. The initial stresses due to a temperature change of ΔT(z) are defined for an orthotropic plate as:

$$\sigma_{x}^{T} = - \left( {C_{11} \alpha_{1} (z,\;T) + C_{12} \alpha_{2} (z,\;T)} \right)\Delta T\;(z)$$

(48)

$$\sigma_{y}^{T} = - \left( {C_{12} \alpha_{1} (z,\;T) + C_{22} \alpha_{2} (z,\;T)} \right)\Delta T\;(z).$$

(49)

Three-Dimensional Ritz Solution in Thermal Environment

The linear elastic strain potential energy U_s of the plate can be given as,

$$U_{{\text{s}}} = \frac{1}{2}\int_{V} {\left[ {\sigma_{x} \varepsilon_{x} + \sigma_{y} \varepsilon_{x} + \sigma_{z} \varepsilon_{x} + \tau_{yz} \gamma_{yz} + \tau_{xz} \gamma_{xz} + \tau_{xy} \gamma_{xy} } \right]\;{\text{d}}V} .$$

(50)

The strain energy U_T from the initial stresses due to temperature rise can be given as,

$$U_{T} = \frac{1}{2}\int_{V} {\left[ {\sigma_{x}^{T} d_{xx} + 2\tau_{xy}^{T} d_{xy} + \sigma_{y}^{T} d_{yy} } \right]\;{\text{d}}V}$$

(51)

$$d_{ij} = u_{,i} u_{,j} + v_{,i} v_{,j} + w_{,i} w_{,j} \quad (i,\;j = x,\;y).$$

(52)

The kinetic energy T_p of the plate can be given as:

$$T_{{\text{p}}} = \frac{1}{2}\int_{V} {\rho \;(z,\;T)\left[ {\left( {\frac{\partial u}{{\partial t}}} \right)^{2} + \left( {\frac{\partial v}{{\partial t}}} \right)^{2} + \left( {\frac{\partial w}{{\partial t}}} \right)^{2} } \right]} \;{\text{d}}V.$$

(53)

According to thermal vibration problem the maximum energy functional Π of the elastic plate is defined as:

$$\Pi = \left( {U_{{{\text{smax}}}} + U_{{{\text{Tmax}}}} } \right) - T_{{{\text{max}}}} ,$$

(54)

where U_smax is the nondimensionalized maximum of linear elastic strain potential energy, U_Tmax is the nondimensionalized maximum of thermal strain potential energy and T_max is the nondimensionalized maximum of kinetic energy. The nondimensionalization process is performed using the following nondimensionalized parameters:

$$X = 2x/a$$

(55)

$$Y = 2y/b$$

(56)

$$Z = 2z/h$$

(57)

Thus, the nondimensionalized maximum values of the energy equations are written as follows.

$$U_{{{\text{smax}}}} = \frac{h}{4\lambda }\mathop \int \limits_{ - 1}^{1} \mathop \int \limits_{ - 1}^{1} \mathop \int \limits_{ - 1}^{1} \left[ {C_{11} \overline{\varepsilon }_{x}^{2} + C_{22} \overline{\varepsilon }_{y}^{2} + C_{33} \overline{\varepsilon }_{z}^{2} + 2\left( {C_{12} \overline{\varepsilon }_{x} \overline{\varepsilon }_{y} + C_{13} \overline{\varepsilon }_{x} \overline{\varepsilon }_{z} + C_{23} \overline{\varepsilon }_{y} \overline{\varepsilon }_{z} } \right) + C_{44} \overline{\gamma }_{yz}^{2} + C_{55} \overline{\gamma }_{xz}^{2} + C_{66} \overline{\gamma }_{xy}^{2} } \right]\;{\text{d}}Z\;{\text{d}}Y\;{\text{d}}X$$

(58)

$$U_{{{\text{Tmax}}}} = \frac{h}{4\lambda }\mathop \int \limits_{ - 1}^{1} \mathop \int \limits_{ - 1}^{1} \mathop \int \limits_{ - 1}^{1} - \left[ {\left( {C_{11} \alpha_{1} + C_{12} \alpha_{2} } \right)\Delta T\left\{ {\left( {\frac{\partial U}{{\partial X}}} \right)^{2} + \left( {\frac{\partial V}{{\partial X}}} \right)^{2} + \left( {\frac{\partial W}{{\partial X}}} \right)^{2} } \right\} + \left( {C_{12} \alpha_{1} + C_{22} \alpha_{2} } \right)\lambda^{2} \Delta T\left\{ {\left( {\frac{\partial U}{{\partial X}}} \right)^{2} + \left( {\frac{\partial V}{{\partial X}}} \right)^{2} + \left( {\frac{\partial W}{{\partial X}}} \right)^{2} } \right\}} \right]\;{\text{d}}Z\;{\text{d}}Y\;{\text{d}}X$$

(59)

$$T_{{{\text{max}}}} = \frac{abh}{{16}}\rho \omega^{2} \mathop \int \limits_{ - 1}^{1} \mathop \int \limits_{ - 1}^{1} \mathop \int \limits_{ - 1}^{1} \left[ {U^{2} + V^{2} + W^{2} } \right]\;{\text{d}}Z\;{\text{d}}Y\;{\text{d}}X$$

(60)

The above parameters are defined as follows:

$$\overline{\varepsilon }_{X} = \frac{\partial U}{{\partial X}}$$

(61)

$$\overline{\varepsilon }_{y} = \lambda \frac{\partial V}{{\partial Y}}$$

(62)

$$\overline{\varepsilon }_{z} = \frac{\lambda }{\gamma }\frac{\partial W}{{\partial Z}}$$

(63)

$$\overline{\gamma }_{xy} = \lambda \frac{\partial U}{{\partial Y}} + \frac{\partial V}{{\partial X}}$$

(64)

$$\overline{\gamma }_{yz} = \frac{\lambda }{\gamma }\frac{\partial V}{{\partial Z}} + \lambda \frac{\partial W}{{\partial Y}}$$

(65)

$$\overline{\gamma }_{zx} = \frac{\lambda }{\gamma }\frac{\partial U}{{\partial Z}} + \frac{\partial W}{{\partial X}}$$

(66)

and in here:

$$\lambda = a/b$$

(67)

$$\gamma = h/b.$$

(68)

In this thermal vibration problem of FG-CNT reinforcement plates, the Chebyshev polynomials are preferred which are the orthogonal polynomials reduced the computational effort [46]. In accordance with the Ritz method, each of the displacement amplitude functions is written as a triple series of Chebyshev polynomials, the displacement component of which is multiplied by a boundary function that satisfies the geometric boundary conditions of the plate. The displacement components are written in terms of nondimensionalized coordinates

$$U\;(X,\;Y,\;Z) = F_{u} \;(X,\;Y)\mathop \sum \limits_{i = 1}^{\infty } \mathop \sum \limits_{j = 1}^{\infty } \mathop \sum \limits_{k = 1}^{\infty } A_{ijk} P_{i} \;(X)\;P_{j} \;(Y)\;P_{k} \;(Z)$$

(69)

$$V\;(X,\;Y,\;Z) = F_{v} \;(X,\;Y)\mathop \sum \limits_{l = 1}^{\infty } \mathop \sum \limits_{m = 1}^{\infty } \mathop \sum \limits_{n = 1}^{\infty } B_{lmn} \;P_{l} \;(X)\;P_{m} \;(Y)\;P_{n} \;(Z)$$

(70)

$$W\;(X,\;Y,\;Z) = F_{w} \;(X,\;Y)\mathop \sum \limits_{p = 1}^{\infty } \mathop \sum \limits_{q = 1}^{\infty } \mathop \sum \limits_{r = 1}^{\infty } C_{pqr} P_{p} \;(X)\;P_{q} \;(Y)\;P_{r} \;(Z)$$

(71)

where $P_{s} \;(\zeta ) = \cos [(s - 1)\arccos (\zeta )]$ (s = 1, 2, 3,…; ζ = X, Y, Z) is the sth order one-dimensional Chebyshev polynomial and $F_{\alpha } (X,\;Y) = f_{\alpha }^{1} (X,\;Y)f_{\alpha }^{2} (X,\;Y)$ (α = U, V, W) is the boundary function satisfying the geometric boundary conditions, are as follows in terms of nondimensionalized coordinates and Chebyshev polynomials. The boundary functions used for boundary condition in this study that CC (f_u¹(X) = 1 − X², f_v¹(X) = 1 − X², f_w¹(X) = 1 − X², f_u²(Y) = 1 − Y², f_v²(Y) = 1 − Y², f_w²(Y) = 1 − Y²); SS(f_u¹(X) = 1, f_v¹(X) = 1 − X², f_w¹(X) = 1 − X²; f_u²(Y) = 1 − Y², f_v²(Y) = 1, f_w²(Y) = 1 − Y²); CF (f_u¹(X) = 1 + X, f_v¹(X) = 1 + X, f_w¹(X) = 1 + X; f_u²(Y) = 1 + Y, f_v²(Y) = 1 + Y, f_w²(Y) = 1 + Y) and FF (f_u¹(X) = 1, f_v¹(X) = 1, f_w¹(X) = 1; f_u²(Y) = 1, f_v²(Y) = 1, f_w²(Y) = 1).

In accordance with the Ritz method, by substituting the displacement components given by Eq. (72) at the maximum energy values given by Eq. (59) and substituting the maximum energy values in the maximum energy functional given by Eq. (55), the energy functional Π is obtained in terms of Chebyshev polynomials. Then the energy functional Π is minimized according to the unknown coefficients A_ijk, B_lmn and C_pqr.

$$\frac{\partial \Pi }{{\partial A_{ijk} }} = 0$$

(72)

$$\frac{\partial \Pi }{{\partial B_{lmn} }} = 0$$

(73)

$$\frac{\partial \Pi }{{\partial C_{pqr} }} = 0$$

(74)

As a result of the Ritz procedure, the eigenvalue problem given below is obtained, and the solution of the system of equations gives the natural frequencies of the free vibration problem occurring in the thermal environment under the influence of temperature.

$$\left( {\left[ {\begin{array}{*{20}c} {\left[ {K_{uu} } \right]} & {\quad \left[ {K_{uv} } \right]} & {\quad \left[ {K_{uw} } \right]} \\ { \left[ {K_{uv} } \right]^{T} } & {\quad \left[ {K_{vv} } \right]} & {\quad \left[ {K_{vw} } \right]} \\ { \left[ {K_{uw} } \right]^{T} } & {\quad \left[ {K_{vw} } \right]^{T} } & {\quad \left[ {K_{ww} } \right]} \\ \end{array} } \right] - \Omega^{2} \left[ {\begin{array}{*{20}c} {\left[ {M_{uu} } \right]} & {\quad 0} & {\quad 0} \\ {0 } & {\quad \left[ {M_{vv} } \right]} & {\quad 0} \\ 0 & {\quad 0} & {\quad \left[ {M_{ww} } \right]} \\ \end{array} } \right]} \right)\left\{ {\begin{array}{*{20}c} {\left\{ {A_{ijk} } \right\}} \\ {\left\{ {B_{lmn} } \right\}} \\ {\left\{ {C_{pqr} } \right\}} \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {\left\{ 0 \right\}} \\ {\left\{ 0 \right\}} \\ {\left\{ 0 \right\}} \\ \end{array} } \right\}$$

(75)

where [K_ij] and [M_ij] (i, j = u, v, w) are the stiffness matrix and diagonal mass matrix, respectively. The dimensionless coefficients {A_ijk}, {B_lmn} and {C_pqr} corresponding to the eigenvectors in the eigenvalue problem, represent the amplitude. Also, Ω is the non-dimensional frequency parameter and obtained as:

$$\Omega = \omega \;(a^{2} /h)\sqrt {\rho^{m0} /E^{m0} }$$

(76)

Here ω is the natural frequency and ρ^m0 and E^m0 are mass density per unit volume and Young modulus of matrix material at room temperature (T₀ = 300 K).

Numerical Results

In this study, a polymer matrix composite as defined Poly-co-vinylene (PmPV) reinforced by CNT in the thickness direction in three type of different form that UD, FG-V, FG-O and FG-X are examined. The mechanical properties of the matrix material PmPV, some of which are temperature dependent, are as follows [27]:

$$E^{{\text{m}}} = \left( {3.51 - 0.0047\;T} \right)\;{\text{GPa}}$$

(77)

$$\upsilon^{{\text{m}}} = 0.34$$

(78)

$$\rho^{{\text{m}}} = 1150\; {\text{kg/m}}^{3}$$

(79)

$$\alpha^{{\text{m}}} = 45\left( {1 + 0.0005\Delta T} \right) \times 10^{{{-}6}} {\text{/K}}$$

(80)

$$G^{{\text{m}}} = \frac{{E^{{\text{m}}} }}{{2(1 + \nu^{{\text{m}}} )}}\;{\text{GPa}}$$

(81)

The efficiency parameters of FG-CNT-reinforced composite are considered η₁ = 0.149 and η₂ = η₃ = 0.934 for VCNT* = 0.11; η₁ = 0.150 and η₂ = η₃ = 0.941 for VCNT* = 0.14; η₁ = 0.149 and η₂ = η₃ = 1.381 for VCNT* = 0.17. The temperature-dependent mechanical properties of the reinforcement material SWCNT which the type of armchair (10,10) are tabulated in Table 1 [27].

Table 1 Temperature-dependent material properties of SWCNT at uniform temperature distribution in the thickness direction

Using the data given in Table 1, the material properties of SWCNTs whose material properties depend on temperature were defined as a third-order polynomial as follows, thus provided the estimation of material properties at temperatures other than these:

$$P\;(T) = P_{0} \left( {1 + P_{1} \Delta T + P_{2} \Delta T^{2} + P_{3} \Delta T^{3} } \right)$$

(82)

where P₀ is the material property of CNT at temperature T₀. P₀ and the coefficients of material properties depending on temperature, P_i (i = 0, 1, 2, 3) are given in Table 2 [27].

Table 2 Coefficients of the temperature-dependent material properties of SWCNT

The material properties of the CNT reinforcement polymer matrix composite plate examined in the study are temperature dependent, and it has been reinforced in four different forms as UD, FG-V, FG-O and FG-X in the thickness direction. The investigated problem is the thermal vibration problem and three different thermal environments are taken into account.

Convergence and Accuracy Studies

In this study, the natural frequencies are obtained by Ritz method and the Chebyshev polynomials which defined between the interval [− 1, 1] and also by a set of separable orthogonal polynomial functions are used as admissible functions. This ensures more rapid convergence and better stability in the numerical computation can be accomplished compared with other polynomial series [46]. As it is known, the natural frequencies obtained by the Ritz method converge to the exact values from the upper bound and more accurate results can be obtained by increasing the number of terms of the admissible functions. In the current study, 3D solutions are obtained using Chebyshev polynomials with 8 × 8 × 8 terms. The convergence study of SSSS square FG-CNT/PmPV composite with UD reinforcement in the stress-free temperature environment is performed for first six frequency parameters.

Throughout the study, including the convergence and comparison studies, the non-dimensional frequency parameter Ω is taken as given in Eq. (26). As can be seen in Table 3, where the convergence rate of the first six frequency parameters is given, the convergence rate is good for each h/b ratio examined, and the variation of the frequency parameters decreases as the number of terms increases, and the frequency parameters approach a definite value.

Table 3 Convergence and comparison of first six frequency parameters of SSSS square FG-CNT plates (V_CNT^* = 0.11, UD, ΔT = 0)

The material examined in this study is FG-CNT/PmPV. However, FG-CNT/PMMA was taken into account in the comparison of the accuracy of the results obtained from the thermal vibration problem as it is available in the literature. The results given in Table 4 for SSSS square FG-CNT/PMMA composite are compared at uniform temperature rise. For this case, the assumed efficiency parameters dependent on SWCNT volume fraction for PMMA/CNT composite are taken from Wang and Shen [34] and, hence, for V_CNT^* = 0.12 are η₁ = 0.137, η₂ = 1.022 and η₃ = 0.715, V_CNT^* = 0.17 are η₁ = 0.142, η₂ = 1.626 and η₃ = 1.138 and V_CNT^* = 0.28 are η₁ = 0.141, η₂ = 1.585 and η₃ = 1.109. From the comparison results given in Tables 3 and 4, it is seen that the results obtained in this study are consistent and stable.

Table 4 Comparison of natural frequency parameters of SSSS square FG-CNT/PMMA plates subjected to uniform temperature rise (h/b = 0.1, V_CNT^* = 0.12)

Parametric Studies

In the study is focused on to investigation effect of the temperature on frequency parameters in different thermal environment of Poly-co-vinylene (PmPV) matrix composite that reinforced by CNT in the thickness direction. The first six free vibration frequencies of the composite are obtained for the plate having CFFF, CCCC, SSSS, SCSC and SFSF boundary conditions at room temperature. The results are given in Tables 5, 6, 7. The first six free vibration frequencies of the composite are obtained for uniform temperature increase, linear temperature increase and sinusoidal temperature increase in the thickness direction when the lower surface of the plate at T_b = T₀ = 300 K and the upper surface of the plate at T_t = 500 K. The reinforcement modeling UD, FG-V, FG-O and FG-X of PmPV/FG-CNT plate at two side/side ratios 1 and 1.5, three different thickness/side ratios 0.05, 0.1 and 0.2 which corresponding, respectively, thin moderately thick and thick plates are considered for CFFF boundary condition and at side/side ratios 1, thickness/side ratios 0.05 are considered for CCC, SSSS, SCSC and SFSF boundary conditions. The results obtained for these conditions are given in Tables 8, 9, 10, 11, 12, 13, 14, 15, 16.

Table 5 Natural frequency parameters of FG-CNT-reinforced plate with CFFF boundary condition (a/b = 1, ΔT = 0)

Table 6 Natural frequency parameters of FG-CNT-reinforced plate with CFFF boundary condition (a/b = 1.5, ΔT = 0)

Table 7 Natural frequency parameters of FG-CNT-reinforced plate under different boundary conditions at room temperature (ΔT = 0, h/b = 0.05, a/b = 1)

Table 8 Natural frequency parameters of FG-CNT-reinforced plate with CFFF boundary condition subjected to uniform temperature rise with T₀ = 300 K, T_t = 500 K (a/b = 1, UD)

Table 9 Natural frequency parameters of FG-CNT-reinforced plate with CFFF boundary condition subjected to uniform temperature rise with T₀ = 300 K, T_t = 500 K (a/b = 1, FG-V)

Table 10 Natural frequency parameters of FG-CNT-reinforced plate with CFFF boundary condition subjected to uniform temperature rise with T₀ = 300 K, T_t = 500 K (a/b = 1, FG-O)

Table 11 Natural frequency parameters of FG-CNT-reinforced plate with CFFF boundary condition subjected to uniform temperature rise with T₀ = 300 K, T_t = 500 K (a/b = 1, FG-X)

Table 12 Natural frequency parameters of FG-CNT-reinforced plate with CFFF boundary condition subjected to uniform temperature rise with T₀ = 300 K, T_t = 500 K (a/b = 1.5, UD)

Table 13 Natural frequency parameters of FG-CNT-reinforced plate with CFFF boundary condition subjected to uniform temperature rise with T₀ = 300 K, T_t = 500 K (a/b = 1.5, FG-V)

Table 14 Natural frequency parameters of FG-CNT-reinforced plate with CFFF boundary condition subjected to uniform temperature rise with T₀ = 300 K, T_t = 500 K (a/b = 1.5, FG-O)

Table 15 Natural frequency parameters of FG-CNT-reinforced plate with CFFF boundary condition subjected to uniform temperature rise with T₀ = 300 K, T_t = 500 K (a/b = 1.5, FG-X)

Table 16 Natural frequency parameters of FG-CNT-reinforced plate under different boundary conditions with T₀ = 300 K, T_t = 500 K (V_CNT^* = 0.11, h/b = 0.05, a/b = 1)

It is seen that the frequency values for the FG-X model are greater than the other models and the frequency values for the FG-O model are smaller than in all boundary conditions and the frequency values obtained at a/b = 1.5 ratio are greater than a/b = 1 ratio. However, the frequency values increase as the volume ratio increases and decrease as the thickness increases at the same volume ratio. At the same volume ratio and thickness values, with the increase of the a/b ratio, the amount of increase in the second and higher frequencies is greater than the amount of increase in the fundamental frequencies. For all reference parameters, the highest frequency values were obtained in the thermal environment where the temperature increased sinusoidally, and the lowest frequency values were obtained in the uniform temperature increase. The frequency values obtained in linear temperature variation were always obtained between the frequency values obtained from the uniform and sinusoidal temperature change.

When it is investigated according to boundary conditions, it is seen that the frequency values increase with increasing constraints on boundaries. And accordingly, the frequencies for each volume ratio are sorted from the highest to the lowest value according to boundary conditions CCCC, SCSC, SSSS, SFSF and CFFF, respectively.

The variation of the fundamental frequency parameters with temperature was investigated graphically for the 0.11, 0.14 and 0.17 values of the V_CNT^* while h/b = 0.1 and a/b = 1 in considered three thermal environments. Graphs are given in Fig. 1a–d for the reinforcement type that UD, FG-V, FG-O and FG-X, respectively. The curves show the effect of three different temperature distributions on the fundamental frequency parameters at each volume ratio. The fundamental frequency values for each volume ratio are higher for the reinforcement type that FG-X and the fundamental frequency values for each volume ratio are smaller for the reinforcement type that FG-O. The fundamental frequency values decrease as the temperature increases and show the greatest change in the uniform temperature distribution. In general, it is seen that the behavior of the material is more stable in the sinusoidal temperature distribution compared to other temperature distributions.

Fig. 1

The fundamental frequency parameters of FG-CNT-reinforced plate with CFFF boundary condition (h/b = 0.1)

The greatest decrease in frequency values is observed in the uniform temperature distribution and the least decrease in the sinusoidal temperature distribution. When compared in terms of volume ratios, the rate of decrease in frequency value is higher at V_CNT^* = 0.11 for all boundary conditions. When compared in terms of fundamental frequency and high frequencies, the frequency that is least affected by temperature changes in all volume ratios is the fundamental frequency.

Figures 2, 3, 4, 5 show the mode shapes for the first 6 frequencies of the FG-CNT plate with CFFF boundary condition for all reinforcement models, at room temperature and uniform, linear and sinusoidal temperature increase. The mode shapes describe the deformation of the vertical component (W) of the FG-CNT composite plate when vibrating at natural frequency at room temperature and at different thermal environments. In the uniform temperature condition, fluctuations are observed in the nodal lines starting from the fourth frequency. In linear and sinusoidal temperature distribution, there is a tendency to increase in the number of waves starting from the fourth frequency.

Fig. 2

First six mode shapes of square FG-CNT-reinforced plate with CFFF boundary condition (UD, V_CNT^* = 0.14, h/b = 0.05, T_b = 300 K, T_t = 500 K)

Fig. 3

First six mode shapes of square FG-CNT-reinforced plate with CFFF boundary condition (FG-V, V_CNT^* = 0.14, h/b = 0.05, T_b = 300 K, T_t = 500 K)

Fig. 4

First six mode shapes of square FG-CNT-reinforced plate with CFFF boundary condition (FG-O, V_CNT^* = 0.14, h/b = 0.05, T_b = 300 K, T_t = 500 K)

Fig. 5

First six mode shapes of square FG-CNT-reinforced plate with CFFF boundary condition (FG-X, V_CNT^* = 0.14, h/b = 0.05, T_b = 300 K, T_t = 500 K)

Conclusions

The free vibration of FG-CNT/PmPV composite plate constructed by embedding a single wall carbon nanotube with chiral index (10,10) in different volume fractions into PmPV polymer matrices is investigated using Ritz method based on three-dimensional elasticity for three different thermal condition. Assumed thermal conditions are uniform, linear and sinusoidal temperature distribution in the study. The following conclusions can be carried out from the obtained results for the considered problem.

• The Chebyshev polynomials chosen as admissible functions in Ritz method provide high accuracy, consistent in the computation and rapid convergence.

• An increase in the volume ratio of CNT results in an increase in the natural frequency, and an increase in thickness at the same volume ratio results in a decrease in the natural frequency.

• The greatest and smallest frequency values for the reinforcement models were obtained for the FG-X model and for the FG-O model, respectively.

• The frequencies for each volume ratio are sorted from the highest to the lowest value according to boundary conditions CCCC, SCSC, SSSS, SFSF and CFFF, respectively.

• An increase in temperature causes the natural frequency to decrease.

• The greatest and smallest frequency values for the three thermal conditions examined were obtained in sinusoidal and uniform temperature distribution, respectively.

• It is observed that in the uniform temperature distribution, frequencies sharply decrease and in the sinusoidal temperature distribution, frequencies monotonically decrease.

• The frequencies obtained in the linear temperature distribution were greater than the values obtained in the uniform temperature distribution and smaller than the values obtained in the sinusoidal temperature distribution for each condition. However, the variation of frequencies are more similar to the behavior in the sinusoidal temperature condition.

• As the temperature increases, the frequency decreases, and decrease ratio of frequency always higher at the lowest volume ratio of the CNT.

• For given boundary condition and at a certain temperature, in the change of temperature distributions, changes are also observed in the order of the mode shapes.

• Mode shapes are not greatly affected by the reinforcement model and temperature, but the greatest irregularity is seen in the FG-X model and uniform temperature distribution.