Flexural and Free Vibration Analysis of CNT-Reinforced Functionally Graded Plate

This paper examines the effect of uniaxially aligned carbon nanotube (CNT) on flexural and free vibration analysis of CNT-reinforced functionally graded plate. The mathematical model includes expansion of Taylor’s series up to the third degree in the thickness co-ordinate. Since there is a parabolic variation in transverse shear strain deformation across the thickness co-ordinate, the shear correction factor is not necessary. A nine-node two-dimensional (2D) C0 isoparametric element containing seven nodal unknowns per node was developed in the finite element code. The final material properties of CNT-reinforced functionally graded plate are estimated using the extended rule of mixture. The effect of CNT distribution, boundary condition, volume fraction and loading pattern are studied by developing a finite element code. An additional finite element code was developed for the study of the influence of concentrated mass on free vibration analysis of CNT-reinforced functionally graded plate.


Introduction
In the modern age, carbon nanotube (CNT)-reinforced composite plates have found considerable application in civil, mechanical, aeronautical and marine engineering due to their exceptional mechanical, thermal and electrical properties. The high tensile properties of CNT make CNT-reinforced composites preferable in tension-dominated applications such as pressure vessels. The concentrated mass is generally used to reduce the fundamental frequency to the desired value. The CNTs are allotropes of carbon having a length scale in the order of nanometres discovered by Iijima [1], having higher strength/weight ratio and lower density. Due to their superior properties, the CNTs are substantially preferable as a reinforcing choice for advanced composites. The Eshelby-Mori-Tanaka approach and a 2-D generalised differential quadrature method was used by Aragh et al. [2] for the frequency analysis of continuously graded CNT-reinforced cylindrical panel. The effect of singly walled carbon nanotubes (SWCNTs) on bending and vibration analysis of CNT-reinforced functionally graded (FG-CNT) plate was studied by Zhu et al. [3] with the help of the finite element method. Their mathematical model is based on the first order shear deformation theory (FSDT). Yas et al. [4] developed a three-dimensional model to study the vibration behaviour of functionally graded cylindrical panel reinforced with CNT. The element-free kp-Ritz method was used by Lei et al. [5] to study the free vibration analysis of CNT-reinforced composite (CNTRC) plate assuming an FSDT based displacement field. The deflection and stresses developed in CNT-reinforced composite cylinders have been studied by Dastjerdi et al. [6] using the mesh-free method. The FSDT-based displacement model

Effective Material of CNT-reinforced Functionally Graded Plates
In the present analysis, the geometry of CNT-reinforced plates is depicted in Figure 1 and is referred to the (φ 1 , φ 2 , ϕ) co-ordinates system. The FG-CNT-reinforced plate has a constant thickness h, with the length of the plate a, and width b. In this work, three types of functionally graded distribution (FG-O, FG-X and FG-V) and uniformly distributed (UD) of SWCNTs in polymer matrix across the thickness direction is considered. The extended rule of mixture [32,33], which contains the efficiency parameters, is incorporated for the calculation of effective material properties of the FG-CNT-reinforced composite plate.
Materials 2018, 11, x FOR PEER REVIEW 3 of 25 In the present analysis, the geometry of CNT-reinforced plates is depicted in Figure 1 and is referred to the ( , , ) co-ordinates system. The FG-CNT-reinforced plate has a constant thickness h, with the length of the plate a, and width b. In this work, three types of functionally graded distribution (FG-O, FG-X and FG-V) and uniformly distributed (UD) of SWCNTs in polymer matrix across the thickness direction is considered. The extended rule of mixture [32,33], which contains the efficiency parameters, is incorporated for the calculation of effective material properties of the FG-CNT-reinforced composite plate.
are Young's modulus and shear modulus of SWCNTs, respectively. The notations (E m , G m ) are known as Young's modulus and shear modulus of the polymer matrix. The CNT efficiency parameter (η 1 , η 2 , η 3 ) are the scale-dependent material properties. (ν m , ρ m ) and ν CNT 12 , ρ CNT represents the Poisson's ratio and mass density of matrix and SWCNT, respectively. The volume fractions of the SWCNT and matrix are denoted by V CNT and V m , respectively, and their additions are equal to unity.
The volume fraction of CNTs as a function of the thickness co-ordinate can be expressed as [32,33]: , w CNT denoted the mass fraction of the CNTs inside a CNT-reinforced plate. ρ CNT and ρ m are densities of the carbon nanotubes and matrix, respectively.

Displacement Fields and Strains
Based on the third-order shear deformation theory, the displacement field (u,v,w) can be determined as follows [34]: where (u 0 , v 0 , w 0 ) are the displacements along the (φ 1 , φ 2 , ϕ) directions, respectively, at the mid-plane (ϕ = 0). (θ 1 , θ 2 ) are the bending rotations about the φ 2 and φ 1 axes, respectively. (ξ 1 , ξ 2 , ζ 1 , ζ 2 ) are known as the higher order terms of Taylor's series expansion. The unknown terms (ξ 1 , ξ 2 , ζ 1 , ζ 2 ) are computed by applying zero shear stress at the lower and upper surfaces of a CNT-reinforced plate. Utilising the boundary conditions γ φ 1 φ 2 (φ 1 , φ 2 , ±h/2) = γ φ 1 φ 2 (φ 1 , φ 2 , ±h/2) = 0 at the top and bottom surfaces of the plate in Equation (7), we obtained Taylor's series expansion terms as Substituting Equation (8) into Equation (7), we obtain During the implementation of the displacement field represented in Equation (10), the problem of C 1 continuity is encountered due to the presence of first order derivatives of the transverse displacement component in the expression of in-plane fields. For applying efficient C 0 FE formulation, the derivatives are replaced by the appropriate substitution of an independent nodal unknowns as The higher order displacement field owning C 0 continuity can express as: Hence, the degree of freedom (basic field variables) according to present mathematical formulation for each node is where {δ} is named as the displacement vector. The strain vector from the above displacement field can be written as Further, the relations between the strain vector {ε} and the displacement vector {δ} can be expressed as {ε} = [B]{δ} (15) where the strain-displacement matrix [B] contains the derivatives of shape function. The in-plane and transverse shear strains are The strain relationships can be written as where, ε 0

Constitutive Relations
The linear stress-strain constitutive relationships for the CNT-reinforced plate are where the constitutive matrix The term Q ij can be obtained from the material properties which are the function of the depth of the plate.

Element Description
For the present C 0 finite element (FE) model, nine-node isoparametric Lagrangian elements with node-wise seven degrees of freedom are employed. The shape function (interpolation function) is used to express the generalised displacement vector and element geometry at any point within an element as: where N i is the shape function of nine-node isoparametric Lagrangian elements [35].

Flexural Analysis
The strain energy may be expressed as By using the Equation (18), the above expression can be represented as

Free Vibration Analysis
The governing equation of free vibration analysis of CNT-reinforced plates is expressed as [ where matrix [C] matrix contains shape function (N i ). The [L] matrix can be stated as: where the matrix [F] of order 3 × 7 contains ϕ and some constant quantities like that of [H] and ρ is known as the density which will be calculated from Equation (5).

Numerical Result and Discussion
In this section, many numerical examples were studied for the flexural and free vibration behaviour of CNT-reinforced functionally grade plates. PmPV [36] was for the matrix and for reinforcing the material armchair (10,10) SWCNTs were chosen. The material properties of SWCNT and the matrix at room temperature (300 K) are given as The CNT efficiency parameters for considered three types of volume fraction are given as: The quantities used in the present study are: For the flexural analysis For the free vibration analysis The loading patterns are used as: The details of end support conditions used in the present study are: 1. Clamped (CCCC):

Clamped and simply supported (CCSS):
At

Convergence and Validation Study
To check the suitable number of mesh sizes to attain precise results, a convergence study was performed for both flexural and free vibration analyses of CNT-reinforced functionally graded plates. Tables 1 and 2 show the convergence study for the fundamental frequency and deflection of a clamped FG-CNT-reinforced plate. The results are computed for V * CNT = 0.11 and a/h = 10 for different mesh sizes. These convergence studies highlighted that for free vibration analysis and bending analysis of FG-CNT-reinforced plates, a 16 × 16 mesh size is satisfactory. Table 3 shows the results of the free vibration analyses for an isotropic square plate (ν = 0.3). The dimensionless frequency parameter of the isotropic plate was compared with HSDT results for a moderately thick plate [37] and an exact solution [38]. For more investigation, a detailed comparison has been done for free vibration and bending analyses considering three thickness ratios (a/h = 10, 20 and 50) and three volume fractions V * CNT = 0.11, 0.14 and 0.17 . The calculated frequency parameter shown in Tables 4 and 5 for simply supported boundary conditions are in line with previous result provided by Zhu et al. [3]. Table 6 shows the central deflection of the UD reinforced composite plate for CCCC, SSSS, SCSC and SFSF boundary conditions. Our numerical results confirm with previous result given by Zhu et al. [3].  Afterwards, the parametric studies have been conducted to examine the effect of boundary conditions (SSSS, CCCC, CCSS, CSCS, CCFF and CFCF), thickness ratios (a/h), concentrated mass, as well as, the volume fraction of CNT V * CNT on the flexural and free vibration behaviour of CNT-reinforced functionally graded plate. The non-dimensional frequency of the first six modes for FG-CNT-reinforced plate is presented in Tables 7-9 for the three-different types of V * CNT = 0.11, 0.14 and 0.17, respectively. The results are computed for a/b = 1 and a/h = 10. For the all considered boundary conditions, minimum and maximum non-dimensional frequency parameters were noted for FG-O and FG-X distribution among the other considered distribution. Rather than mid-section, the top and bottom section of the plate was chosen for the distribution of additional CNT to achieve maximum stiffness. Thus, the FG-O and FG-X distributions produce minimum and maximum stiffness, respectively. Further, it was also noticed that CFCF yields minimum frequency parameters while the all side-clamped plate yields the maximum frequency parameter. This is because the higher constraints at the boundary give a higher stiffness to the CNT-reinforced functionally graded plate. Here, approximately, a 6% increase in non-dimensional fundamental frequency was noticed when the volume fraction of CNT increases from 0.11 to 0.14, around a 25% increase was noticed when V * CNT changes from 0.11 to 0.17.  Figure 2 shows the effect of side-to-thickness ratio on the non-dimensional fundamental frequency of FG-CNT-reinforced plates. The results are calculated for V * CNT = 0.17 for CCSS, CSCS, CCFF and CFCF boundary conditions. Here it can be seen that the dimensionless frequency parameters increase along with the a/h ratio and it became insensitive from a/h = 60 onwards for all used boundary conditions. The effect of the concentrated mass on the free vibrations of FG-CNT-reinforced plates, having simply supported boundary conditions, is presented in Table 10. It can be noticed that increases in concentrated mass at the centre decreases the fundamental frequency parameter while no significant reduction is seen for any other mode of frequencies. Here, an approximate 28% decrease in the fundamental frequency is noticed when the value of the concentrated mass is increased by 0.5-1 and 1-2.      Figure 3 shows the effect of concentrated mass on the vibration behaviour of an FG-CNTRC plate having various types of boundary conditions. For all considered boundary conditions, the dimensionless frequency parameter decreases, with an increase in the concentrated mass; and the CFCF boundary conditions have the least effect of concentration among considered boundary conditions. The first mode shape of a UD-CNT-reinforced plate, with concentrated mass at the centre, is presented in Figure 4. Table 10. Dimensionless first six natural frequencies for an FG-CNT-reinforced plate with simply supported boundary conditions and concentrated mass at the centre ( * = 0.11, /ℎ = 10).

CNT Distribution
First Six Minimum Frequencies  Figure 3 shows the effect of concentrated mass on the vibration behaviour of an FG-CNTRC plate having various types of boundary conditions. For all considered boundary conditions, the dimensionless frequency parameter decreases, with an increase in the concentrated mass; and the CFCF boundary conditions have the least effect of concentration among considered boundary conditions. The first mode shape of a UD-CNT-reinforced plate, with concentrated mass at the centre, is presented in Figure 4. The maximum deflection of an FG-CNT-reinforced plate having various side-to-thickness ratios for * = 0.11, 0.14 and 0.17 subjected to sin-sin loading are presented in Tables 11-13, respectively. The results are calculated for UD, FG-V, FG-O and FG-X distribution of CNT across the transverse The maximum deflection of an FG-CNT-reinforced plate having various side-to-thickness ratios for V * CNT = 0.11, 0.14 and 0.17 subjected to sin-sin loading are presented in Tables 11-13, respectively. The results are calculated for UD, FG-V, FG-O and FG-X distribution of CNT across the transverse direction, having an aspect ratio a/b = 1. A decrease in deflection is noted when the V * CNT increases because of the higher value of V * CNT , imparts a higher stiffness in CNT-reinforced plate, thus the deflection is reduced. The maximum deflection decreases with an increase in the a/h ratio irrespective of boundary conditions and types of distribution. Our finding confirms that there is approximately a 7% reduction in the maximum deflection for all considered end support as the value of V * CNT increased from 0.11 to 0.14 and approximately a 36% decrease is found when V * CNT increases from 0.11 to 0.17. FG-X and FG-O distribution yields minimum and maximum deflection, respectively. Table 11. Central transverse deflection of an FG-CNT-reinforced square plate subjected to sin-sin loading V * CNT = 0.11.  Table 12. Central transverse deflection of an FG-CNT-reinforced square plate subjected to sin-sin loading V * CNT = 0.14. direction, having an aspect ratio a/b = 1. A decrease in deflection is noted when the increases because of the higher value of * , imparts a higher stiffness in CNT-reinforced plate, thus the deflection is reduced. The maximum deflection decreases with an increase in the a/h ratio irrespective of boundary conditions and types of distribution. Our finding confirms that there is approximately a 7% reduction in the maximum deflection for all considered end support as the value of * increased from 0.11 to 0.14 and approximately a 36% decrease is found when * increases from 0.11 to 0.17. FG-X and FG-O distribution yields minimum and maximum deflection, respectively.     Figure 5 shows the variation of deflection of UD, FG-V, FG-O and FG-X type CNT-reinforced plates along the centre line subject to the various types of mechanical load. The results are obtained for V * CNT = 0.11. It can be seen that, for all types of CNT distribution in the thickness direction, the graph of deflection along the length is of the same nature. The minimum and maximum deflections were noticed for cos-cos type of loading and uniform loading, respectively. The axial stress developed in a CNT-reinforced functionally graded plate under sin-sin loading is plotted in Figure 6 against the thickness co-ordinate for CCSS, CSCS, CCFF and CFCF support conditions. The non-dimensional axial stress decreases with an increase in constraints at end support. It is interesting to note that for all types of boundary conditions, except CFCF, the nature of the graph along thickness co-ordinate is the same, for CFCF type boundary conditions, the nature of the graph is opposite to other taken boundary conditions. The minimum and maximum deflections were noticed for cos-cos type of loading and uniform loading, respectively. The axial stress developed in a CNT-reinforced functionally graded plate under sin-sin loading is plotted in Figure 6 against the thickness co-ordinate for CCSS, CSCS, CCFF and CFCF support conditions. The non-dimensional axial stress decreases with an increase in constraints at end support. It is interesting to note that for all types of boundary conditions, except CFCF, the nature of the graph along thickness co-ordinate is the same, for CFCF type boundary conditions, the nature of the graph is opposite to other taken boundary conditions.

Conclusion
In the present work, a C 0 FE model based on Reddy's TSDT was developed to investigate the flexural and free vibration behaviour of CNT-reinforced functionally graded plates. The CNT distribution through the thickness of plate is assumed to be uniform or functionally graded. The properties of CNT-reinforced plates at any point are calculated using the modified rule of mixture in which efficiency parameters are introduced into the rule of mixtures approach. The influence of the concentrated mass, volume fraction, side-to-thickness ratios, loading pattern and end support condition on the dimensionless bending and frequency parameter were also studied. Based on the present results, it can be concluded that:  Among the considered distribution pattern of CNT, FG-X pattern results in higher dimensionless frequency parameter and lower deflection, while FG-O pattern yields lower dimensionless frequency parameters and higher dimensionless deflections.
 An increase in the dimensionless frequency parameters and decrease in the deflection of FG-CNT-reinforced plate is found when the volume fraction of CNT is increased.
 With the increase in side-to-thickness ratio, an increase in dimensionless frequency and a decrease in deflection is noticed.
 The greater constraints on boundaries results in lower values of deflection and higher values of dimensionless frequency parameters.
 The concentrated mass at the centre decreases the fundamental frequency parameter.

Conclusions
In the present work, a C 0 FE model based on Reddy's TSDT was developed to investigate the flexural and free vibration behaviour of CNT-reinforced functionally graded plates. The CNT distribution through the thickness of plate is assumed to be uniform or functionally graded. The properties of CNT-reinforced plates at any point are calculated using the modified rule of mixture in which efficiency parameters are introduced into the rule of mixtures approach. The influence of the concentrated mass, volume fraction, side-to-thickness ratios, loading pattern and end support condition on the dimensionless bending and frequency parameter were also studied. Based on the present results, it can be concluded that:

•
Among the considered distribution pattern of CNT, FG-X pattern results in higher dimensionless frequency parameter and lower deflection, while FG-O pattern yields lower dimensionless frequency parameters and higher dimensionless deflections. • An increase in the dimensionless frequency parameters and decrease in the deflection of FG-CNT-reinforced plate is found when the volume fraction of CNT is increased.

•
With the increase in side-to-thickness ratio, an increase in dimensionless frequency and a decrease in deflection is noticed.

•
The greater constraints on boundaries results in lower values of deflection and higher values of dimensionless frequency parameters.

•
The concentrated mass at the centre decreases the fundamental frequency parameter.