Document Type : Research Paper
Authors
^{1} Department of Civil Engineering, Madan Mohan Malaviya University of Technology, Gorakhpur273010, India
^{2} Department of Civil Engineering, Indian Institute of Technology, Roorkee247667, India
Abstract
Keywords
Main Subjects
Free Vibration Response of Agglomerated Carbon NanotubeReinforced Nanocomposite Plates
^{a }Department of Civil Engineering, Madan Mohan Malaviya University of Technology, Gorakhpur273010, India
^{b} Department of Civil Engineering, Indian Institute of Technology, Roorkee247667, India
KEYWORDS 

ABSTRACT 
FE method; HSDT; EshelbyMoriTanaka; Carbon nanotubes; Frequency. 
The current investigation deals with the effect of carbon nanotube (CNT) agglomeration on the free vibration behavior of nanocomposite plates created by inserting various graded distributions of carbon nanotube (CNT) in a polymeric matrix. In this study, affected material properties because of the CNT agglomeration effect were estimated first according to the twoparameter agglomeration model based on the EshelbyMoriTanaka approach for randomly oriented carbon nanotubes, and then a FEM code has been developed to model the FG plate using thirdorder shear deformation theory. In the used higherorder shear deformation theory, transverse shear stresses are represented by quadratic variation along the thickness direction, resulting in no need for a shear correction factor. Next, the present approach is implemented with the FEM by employing a C^{0}_{ }continuous isoparametric Lagrangian FE model with seven nodal unknowns per node. Finally, the effect of various levels of agglomeration by altering the agglomeration parameters, different CNT distribution patterns across the thickness direction, and various sidetothickness ratios along with various boundary conditions on the free vibration response of CNT reinforced composite plates explored parametrically. The generated result shows that the CNT agglomeration effect has a significant impact on the natural frequencies of the nanocomposite plate. 
The work of Iijima [1,2] enabled scientists to comprehend the enormous potential of Carbon Nanotubes (CNTs) and piqued the curiosity of many researchers who set out to find a practical use for CNTs that would benefit from them. Because of their superior mechanical and thermal qualities, carbon nanotubes (CNTs) have been recognized as the ideal candidate for reinforcing composite materials that may be used in a wide range of technological disciplines, including aerospace and mechanical engineering, since their discovery [3]. Functionally graded carbon nanotubereinforced composites (FGCNTRCs) have received a lot of interest in recent years due to their exceptional mechanical properties. FGCNTRC structures have numerous potential applications in aerospace, civil and ocean engineering, the automotive industry, and smart structures [41]. Nonetheless, as seen by many publications published on the subject, the characterization of the mechanical properties of CNTs is still an unresolved question [4]. Several ways to define the mechanical behavior of such composites can be discovered in the literature. They are commonly used for various structural purposes to improve the dynamic response or to provide a superior attitude in particular buckling issues. As a result of these considerations, it was decided to investigate the effect of CNT insertion into the polymeric matrix with various distributions and the agglomeration effect.
The simplified method for determining Young’s moduli, shear moduli, and Poisson's ratios of a CNTreinforced layer with orthotropic properties is to use an extended version of the Rule of Mixture, which can be found in [5]. The studies of Alibeigloo and Liew [6] and Alibeigloo [7], in which the elasticity is applied to explore the thermal and dynamic behavior of various CNTreinforced composite structures, respectively, provide some examples of this approach in use. The books of Alibeigloo and Liew [6] and Alibeigloo [7] contain examples of these uses.
In their most recent publications [8,9], Zhang, Lei, and Liew characterize the engineering constants of the material using the same micromechanical method to assess the mechanical properties of these composites. In these studies, the free vibration analysis is numerically solved using an upgraded version of the conventional Ritz approach.
Shi et. al. [10] suggest a strategy that is entirely distinct from any other one that has been used before to explore the CNTs agglomeration effect using a twoparameter theoretical model. The foundation of this concept is based on the idea that the spatial distribution of CNTs inside the matrix is not uniform, and as a consequence, certain regions of the composite material have a larger reinforcing particle concentration than others. After that, an evaluation of the effective mechanical characteristics of the composite, which in this instance possesses isotropic overall features, is carried out using a homogenization method that is based on the popular MoriTanaka method for granular composite materials [11]. The works [4,12], which present some parametric investigations to look into the impact that CNT agglomeration has on the vibrational behavior of various basic structures, take into account the current strategy. The purpose of these studies is to determine whether or not CNT agglomeration is beneficial.
Hedayati and Sobhani Aragh have especially considered the impact of graded agglomerated CNTs on annular sectorial plates' free vibration behavior sitting on the Pasternak foundation [12]. On the other hand, Sobhani Aragh et al. have evaluated the natural frequencies of CNT reinforced cylindrical shells [4,13]. In the research that was carried out and presented by Giovanni et. al. [14], the composite plates were assumed to be made of a purely isotropic elastic hosting matrix of one of three distinct types (epoxy, rubber, or concrete) and embedded singlewalled carbon nanotubes. The computations are done by discretizing the composite plates as finite elements using the finite element method (FE). It is determined how the modal characteristics are changed both locally and globally by the impacts of the CNT alignment and volume percentage, and it is then demonstrated that the lowest natural frequencies of CNTreinforced rubber composites can rise by up to 500 %. To generate and precisely solve the equations of motion, the analysis technique is based on the FSDT [15]. This is accomplished in order to record the fundamental frequencies of the rectangular functionally graded (FG) plates that are supported by an elastic base. Through the use of the finite element method, the purpose of this study is to ascertain the natural frequencies of an isotropic thin plate. The frequencies that were calculated have been contrasted with those that were determined using an exact Levytype solution. The Kirchhoff plate theory is used as the foundation for the calculation of the stiffness and mass matrices, which are done using the finite element method (FEM). The natural frequencies of the considered rectangular plate can be obtained with the help of this methodology, which is useful [16]. The impact of CNT agglomeration on the elastic characteristics of nanocomposites is assessed using a twoparameter micromechanics model of agglomeration. In this research, an analogous continuum model based on the EshelbyMoriTanaka method is used to determine the effective constitutive law of an elastic isotropic medium (matrix) with oriented straight CNTs. The results of this research work are presented in this article. The discretization of the equations of motion and the implementation of the various boundary conditions are both accomplished through the use of the generalized differential quadrature method (GDQM) for two dimensions [12]. With four unknowns, a new higherorder shear deformation theory has been devised, but it takes into consideration the transverse shear strains' parabolic fluctuation across the plate thickness. The flexure and free vibration analysis of FG plates is done using this theory. Therefore, a shear correction factor is not required to be used. The findings indicate that the present theory is capable of achieving the same level of precision as the existing higherorder theories of shear deformation, despite the latter's greater number of unanswered questions; however, the present theory's precision cannot be compared to that of 3D and quasi3D models that take into account the effect of thickness stretching [17]. Zhang et. al. [9] used both the elementfree IMLSRitz technique and firstorder shear deformation theory, also known as FSDT, to account for the impact of the plate’s transverse shear deformation. The study investigated how the vibration behavior of the plate was affected by various factors, including the volume fraction of carbon nanotubes, the plate's thicknesstowidth ratio, the plate's aspect ratio, and the boundary condition. Mareishi et. al. [5] studied the nonlinear free and forced vibration behavior of advanced nanocomposite beams resting on nonlinear elastic foundations. SWCNT volume fractions and dispersion patterns affect system behavior. Researchers studied the nonlinear forced and free vibration response of smart laminated nanocomposite beams and discovered this. Tornabene et. al. [18] looked at how agglomerated CNT affected the free vibration behavior of laminated composite plates and doublecurved shells. They used Carrera Unified Formulation (CUF), which is a method that permits the consideration of multiple Higherorder Shear Deformations Theories (HSDTs). Kiani [19] looked at how CNTbased nanocomposite plates that had layers of piezoelectric material on the top and bottom behaved when they were free to move. During the course of the research, the properties of the composite medium were determined with reference to a revised version of the rule of mixtures method that incorporates efficiency parameters. The electric potential was thought to be spread out in a straight line across the thickness of the piezoelectric material. The full set of motion and Maxwell equations for the system were found by using the Ritz formulation, which works for any inplane and outofplane boundary conditions. These equations describe the system's behavior. In this case, the researcher takes into consideration both open circuits and closed circuits as potential electrical boundary conditions for the free surfaces of the piezoelectric layers. According to the findings of the study, the resultant eigenvalue system was successfully solved in order to get the system's frequencies as well as the mode shapes. In the end, it was determined that the fundamental frequency of a plate with a closed circuit always had a higher value than the fundamental frequency of a plate with open circuit boundary conditions. A new eightunknown shear deformation theory was developed by Nguyen et. al. [20] for the bending and free vibration study of FG plates using the finiteelement method. The presented theory concurrently fulfills zeros for the transverse strains at the top and bottom surfaces of FG plates and is based on a full 12unknown higherorder shear deformation theory. The research utilized a rectangular element with four nodes and sixteen degrees of freedom at each node. In the end, the results were checked against the results that were already published in the relevant literature. Over the course of the research, parametric studies were carried out for a variety of powerlaw indices and sidetothickness ratios. Using both experimental measurements and an analytical method, Moghadam et. al. [21] investigated the effect of CNT agglomerates on the residual stresses in a fiberreinforced nanocomposite. In order to calculate the residual stress that was caused by thermal treatment, an analytical solution was utilized, which was founded on the traditional laminate theory. The observed residual stresses acquired using the incremental holedrilling method were found to be in good agreement with the theoretical residual stresses computed by each layer of the laminates. The study's findings demonstrated that poorly dispersed samples produced higher residual stresses when compared to perfectly dispersed samples, and this phenomenon was found to be more significant in the case of nanocomposites containing higher weight fractions of CNTs. Hamid et al. [22] studied the free vibration of sizedependent CNTRC nanoplates on a viscoPasternak foundation. Maleki et al. [23] solved the free vibration problem of threephase carbon fiber/nanotube/polymer nanocomposite conical shells using the GDQM. This was done in order to address the impact that the agglomeration of carbon nanotubes (CNTs) has on the dynamic responses of the shell. The final finding of the study was that the presence of agglomeration phenomena can significantly alter the dynamic behavior of the nanocomposite structure. Zhang et al. [34,35] studied the vibration analysis of carbon nanotube (CNT) reinforced functionally graded composite triangular plates subjected to inplane stresses and also investigated the effect of inplane forces on the vibration behavior of carbon nanotube (CNT) reinforced composite skew plates using firstorder shear deformation theory. Zhang and Salem [36] investigated the free vibration behavior of carbon nanotube (CNT) reinforced functionally graded thick laminated composite plates utilizing Reddy’s higherorder shear deformation theory (HSDT) in combination with elementfree IMLSRitz method with four types of CNT distributions. Finally, the influence of boundary conditions on the sequence of the first six mode shapes for various lamination arrangements is studied in detail. Mehar and Panda [37] studied in detail the vibration characteristics of carbonnanotubereinforced sandwich curved shell panels under the elevated thermal environment using the higherorder shear deformation theory. Mehar et. al. [3840] have done extensive theoretical and experimental investigations of vibration characteristics of carbonnanotube reinforced polymer composite structures.
In conclusion, from the abovedetailed literature review, the current manuscript is structured in a manner that helps to incorporate the CNTs agglomeration effect as can be seen that CNTs tend to agglomerate for lowvolume fraction distribution. Hence, if the abovementioned effect is not considered, will lead to erroneous results for the structures built from these CNT materials. The current work focuses on the analysis of functionally graded CNT reinforced plates, including the effect of agglomeration with different CNT distribution patterns, aspect ratio, and boundary conditions regarding its influence on the natural frequency of plate structure only. The analysis is based on the finite element method using Reddy’s HSDT model. The material properties of agglomerated CNTs are evaluated based on the EshelbyMoriTanaka method.
For any structural analysis, material modeling is very important. The application of CNTreinforced composite structures, demands to development of detailed modeling of the effective material properties of a such composite at the macroscopic level. Because molecular dynamics or other atomistic models are computationally intensive, micromechanical methods are used to describe the behavior of these materials in this work.
Material modeling of FGCNTRC is presented using the MoriTanaka method, considering the effect of agglomeration of CNT for various types of CNT distributions.
The FGCNTRC material is considered to be made up of an isotropic matrix (e.g., epoxy resin) and fiber (CNTs), with material qualities graded along the direction of thickness of the plate as per linear distribution (UD and FGV) of the fraction of volume of CNTs (fig.1).
The volume fractions (V_{cnt}) of CNTs in four types of functionally graded carbon nanotube plates are stated as follows:

(1) 

(2) 
where, represents the CNTs mass fraction and and the densities of carbon nanotube and matrix, respectively. The material properties can be determined for this linear material property fluctuation by putting the value of into Eq. (1) for linear material property variation.
(a)
(b)
(c)
(d)
Fig. 1. (a) Uniformly distributed CNT nanocomposite plate, (b) VShape distributed CNT nanocomposite plate, (c) XShape distributed CNT nanocomposite plate, (d) OShape distributed CNT nanocomposite plate.
Several micromechanical models have been proposed to predict the properties of the material of CNTreinforced composites. In this research, the Mori–Tanaka technique is used to estimate the elastic properties of the equivalent fiber/polymer material. The equivalent inclusion average stress technique, commonly known as the Eshelby–Mori–Tanaka method, is based on Eshelby's [8] equivalent elastic inclusion notion and MoriTanaka’s [6] concept of average stress inside the matrix. Benveniste’s [9] revision of the effective modulus of elasticity tensor C of CNTreinforced composites is as follows:

(3) 
The symbol I is denoted as a fourthorder unit tensor. The matrix stiffness tensors are C_{m}, while the equivalent fiber stiffness tensors are C_{r} (CNT). The angle brackets in their overall configuration represent an average of all possible orientations for the inclusions. A_{r} is the tensor of the concentration of dilute mechanical strain, and it can be calculated as follows:

(4) 
here symbol S represents the Eshelby tensor of the fourth order, as defined by Mura and Eshelby [8,10].
Here, a singlewalled carbon nanotube having a solid cylinder of 1.424 nm diameter with (10,10) chirality index [11] is used for the analysis.
Two Euler angles show straight carbon nanotube orientation α and β, denoted by the arrows in Fig. 2. As a result, the base vectors of the global coordinate system and the base vectors of the local coordinate system are produced, which are related through the transformation matrix g, as follows:

(5) 
where g is given as:

(6) 
It is possible to characterize the orientation distribution of carbon nanotubes in composites by a function of probability density p(α, β) that meets the normalizing condition.

(7) 
Considering the random CNT orientation, the function of density for this case is,

(8) 
Calculation of Hill's elastic moduli for the reinforcing phase was accomplished by analyzing the equivalence of the two matrices that are presented below [13]:

(9) 
the terms k_{r}, l_{r}, m_{r}, n_{r}, and p_{r} in Eq. (9) represent Hill’s elastic moduli for the reinforcing phase (CNTs) of the composite calculated by the inverse of the compliance matrix of the equivalent fiber.
Fig. 2. Representative volume element (RVE) Composed of Randomly oriented straight CNT [12].
As for the composite's properties E_{L}, E_{T}, E_{Z}, G_{LT}, G_{TZ}, G_{TZ}, and ν_{LT}, which may be established using the rule of mixture technique, the first step is to determine the properties of the composite by performing a multiscale finite element analysis or molecular dynamics simulation analysis [14] on the composite.
Here, the composite is considered to be isotropic when the carbon nanotubes are orientated totally randomly in nature in the matrix. For this, the bulk modulus K and shear modulus G is calculated as follows:

(10) 

(11) 
The term K_{m} and G_{m }are used for bulk and shear moduli of the matrix, respectively.

(12) 


(13) 


(14) 


(15) 

Finally, the modulus of elasticity and Poison ratio of a CNTbased nanocomposite material are as follows:

(16) 

(17) 
Additionally, V_{cnt} and V_{m} represent the volume fractions of the carbon nanotubes and matrix, respectively, which fulfill the expression
V_{cnt} + V_{m}_{ }= 1. In a similar way, the mass density ρ is determined as follows:

(18) 
where and represents the mass density of matrix and carbon nanotubes, respectively.
A large proportion of carbon nanotubes in carbon nanotubereinforced composites has been found to be concentrated in agglomerates. Nanotubes agglomerate into bundles due to the van der Waals attractive interactions between them. After determining the material properties of FGCNTRC without taking into account the CNT agglomeration effect, a new micromechanics model is developed and applied to a random oriented, agglomerated CNTreinforced polymer composite to determine the effective properties of the material of a singlewalled CNT reinforced polymer composite while taking into account the CNTs bundling effect. The influence of agglomeration on the elastic characteristics of CNTreinforced composites having random orientation is investigated in the present study, which uses a twoparameter micromechanics agglomeration model to do this.
As per Fig. 3, it can be seen that the elastic characteristics of the surrounding material are distinct from the areas where inclusions have concentrated nanotubes.
In polymer matrix, the major cause of agglomeration of carbon nanotubes is the small diameter, due to which the elastic modulus gets reduced and the aspect ratio increases in the radial direction and hence producing low bending strength. It is crucial that carbon nanotubes are dispersed uniformly inside the matrix to achieve the desired features of CNTreinforced composites. Here, a micromechanical model has been built to check the CNTs agglomeration effect on the effectiveness of carbon nanotubeenhanced elastic modules.
Shi et al. [5] found that a substantial number of CNTs are concentrated in aggregates in the 7.5 % concentration sample. Carbon nanotubes are found to be unevenly distributed in the substrate, with a few areas having CNT concentrations larger than the average volume fraction. As illustrated in Fig. 3, these areas containing concentrated carbon nanotubes are considered spherical in this section and are referred to as 'inclusions' having a mix of varying elasticity characteristics from the surrounding material.
Fig. 3. Agglomeration of carbon nanotubes (CNTs) within the representative volume element (RVE)
The total volume V_{r} of CNTs in the RVE may be separated into two parts:

(19) 
where and are represented as the CNTs' volume dispersed in the matrix and the inclusions (concentrated regions), respectively.
To understand clearly the effect of carbon nanotube agglomeration, two parameters are introduced as ξ & ζ.

(20) 
where represents the volume of the RVE's sphere inclusions. In this case, represents the volume of the inclusion fraction in relation to the RVE's total volume V. Whenever is equal to one, CNTs are assumed to be distributed uniformly in the matrix, and as the value of decreases, the degree of agglomeration of carbon nanotubes becomes more severe (Fig. 5). The symbol denotes the nanotubes volume ratio distributed in the inclusions to the total volume of the CNTs. When the value is 1, all of the nanotubes are concentrated in the sphere regions. This is true if all nanotubes are dispersed evenly (i.e., = ) throughout the matrix. As the value of increases (i.e., ), the CNT’s spatial distribution becomes more. V_{cnt} denotes the average carbon nanotube volume fraction in the composite as per Eq. (21)_{.}

(21) 
The carbon nanotube volume fractions in the inclusions and the matrix are calculated using Eqs. (19)(21), and they are expressed as

(22) 

(23) 
Fig. 4. (Without agglomeration)
Fig. 5. (Complete agglomeration)
Fig. 6. (Partial agglomeration)
As a result, the Composite reinforced with carbon nanotubes is viewed as a system made up of sphereshaped inclusions embedded in a hybrid matrix. CNTs can be found in both the matrix as well as in the inclusions also. Hence to compute the composite system's overall property, first, we have to estimate the inclusion’s effective elastic stiffness and then the matrix.
Different micromechanics methods can be used to calculate the effective modulus of elasticity of the hybrid inclusions and matrix. Assuming that all CNT orientations are completely random and the nanotubes are transversely isotropic, the MoriTanaka scheme is used to estimate the hybrid matrix's elastic moduli, as described in the previous section. The carbon nanotubes are assumed to be oriented randomly within the inclusions, and thus the inclusions are isotropic. The term K_{in} and K_{out} represent the effective bulk moduli G_{in} and G_{out} represents the effective shear moduli of the inclusions and matrix, respectively given as:

(24) 

(25) 

(26) 

(27) 
Following that, the composite's effective bulk modulus K and effective shear modulus G are computed using the method of MoriTanaka as follows:

(28) 

(29) 
where,

(30) 
Finally, the CNTreinforced composite’s young modulus is calculated using Eq. (16).
The FGM plate's geometry used in this analysis is shown in Fig. 7. The plate's length and width are denoted by a and b, respectively, and its thickness is represented by h. The center of the FGCNT plate serves as the origin for material coordinates (x, y, and z). Plates are simply supported along their four edges, for the square plate. The aspect ratio considered is h/a = 0.1.
The inplane displacement variation of u, v, and displacement in transverse direction w across the plate thickness may be described as using Reddy's theory of higherorder shear deformation [15].

(31) 
where , and signify the displacement of a point along the (x, y, z) coordinates located at midplane, respectively. and denotes the bending rotations in the y and x directions, respectively, and , denotes the shear rotations assumed in the x, and y directions.
The relationship between the strain component and the strain displacement is defined as follows:

(32) 
The overall strain may be represented as mechanical strains for the purposes of plate analysis.

(33) 
where represents the mechanical strain.
Again, in terms of total strain, the mechanical strain may be represented as

(34) 
while is the thickness coordinatesz function, and is the function of x and y.
Fig. 7. Geometry of the FGCNT Plate
This describes the overall strain as,

(35) 
The relation between stress and strain for FGM is as follows:

(36) 
where constitutive matrix

(37) 
In Eq. (37) the terms Q_{ij} are derived from the FG material properties, depending on the plate’s thickness (z) as shown below in Eq. (38).

(38) 
The FGM plate’s virtual work may be represented as

(39) 
With the help of Eq. (36), Eq. (39) can be rewritten as

(40) 
The following equation can be extended further using Eq. (35) as follows:

(41) 
In Eq. (41) the matrix [Q] represents the constitutive matrix with elasticity derived from the constituent’s elastic properties as given in Eq. (37). While [H] represents the 5 x 15 order matrix and includes the terms z and h as described below:

(42) 
Finally, we can rewrite Eq. (41) as

(43) 
where matrix [D] represents the rigidity matrix vector. For which the corresponding expression is given in Eq. (44) shown below.

(44) 
Figure 8 illustrates the isoparametric Lagrangian element’s geometry with nine nodes used in the analysis. In this element, there is a total of sixtythree degrees of freedom because each node has seven degrees of freedom (u, v, w, , , and ). In the xy plane coordinate system, this element has a rectangular geometry that is completely arbitrary. The element is transferred to plane in order to get a rectangular geometry.
Fig. 8. Ninenoded Isoparametric element with node numbering
For the present ninenode element the shape functions used are given below,

(45) 
The relationship between strain and displacement can be established using the nine shape functions mentioned above. The vector of a strain can be expressed in the following way:

(46) 
In Eq. (46) matrix [B] represents the straindisplacement matrix and matrix [X] represents the vector of nodal displacement for the element chosen and both matrices can be represented as follows:
,

(47) 
Midsurface displacement parameters (u_{o}, v_{o} and w_{o}) can be used to calculate acceleration at any location within the element, as

(48) 
In the above Eq. (48) the vector represents the nodal unknowns which is of order 7 x 1 and contains the terms of Eq. (33).
Again, the matrix is decoupled into matrix [C] which contains the shape functions (N_{i}) and global displacement vector .

(49) 
The mass matrix of an element can be expressed using Eq. (48) and (49),

(50) 
where the matrix [L] expression can be represented as

(51) 
while ρ is the estimated density of the composite material from Eq. (18). As a result, the governing equation for free vibration analysis is,

(52) 
In this section, many numerical examples were studied for the free vibration behavior of functionally graded nanocomposite plates with different distributions of carbon nanotube (Fig. 1) has been done by considering various agglomeration stages as shown in Figs. 46. This section is separated into two distinct sections. The first phase involves a convergence study and validation of the current formulation for isotropic plates [16] with varying aspect ratios, as no solution exists for the current problem. After confirming the effectiveness of the current formulation, the second step investigates the impacts of various agglomeration stages on the nondimensional frequency of the plate. In all the above phases, the influence of different boundary conditions (SSSS, CCCC, SCSC & SFSF) with different CNT distributions are investigated considering three stages of agglomeration (Fig.4) as (without agglomeration case), (complete agglomeration case) and (partial agglomeration case) are investigated. The properties of SWCNT (10,10) are listed in Table 1. The matrix substance employed in this situation has the following elastic characteristics: E_{m} = 2.1 GPa,
υ_{m} = 0.34, ρ_{m} = 1150 kg/m^{3}, and Table 1 lists the material characteristics of the reinforcement. The UD, FGV, FGX, and FGO type reinforcement distributions with various levels of agglomeration testing were taken into consideration. 7.5% of the value is taken into consideration, which is a significant number of carbon nanotubes [30].
Here, before the verification and convergence study the mechanical properties were verified with the experimental work done by Odegard et al. [33] and presented in Fig. 9. From Fig. 9 it can be observed that the EshelbyMoriTanaka scheme proposed by Shi et. al. [10] for the estimation of material properties and the results generated by Odegard et al. [33] are very close for the prediction of mechanical properties.
The result produced by the EMT approach for the agglomeration parameter ξ= 0.4 corresponding to ζ=1 (resembles the complete agglomeration behavior) is plotted in Fig. 9 with good agreement. The material for the matrix is used as E_{m} = 0.85 GPa and υ_{m} = 0.3, combined with the CNT properties given in Table 1 using the EMT approach to calculate overall mechanical properties for the analysis. The results generated here show, at the value of parameter ξ= 1 Young’s modulus has the higher increase in function of volume fraction, and as the value of ξ decreases, the increase in the CNT volume fraction does not correspond to the expected increase of mechanical properties because of the severity of the agglomeration effect.
Fig. 9 itself is selfexplanatory, and at the highest values of Young’s modulus, both agglomeration parameters are considered equal values. It is also possible to observe that the variation of the parameter of ξ has a higher impact on mechanical properties as compared to other parameters ζ. After a thorough study of the effect of two agglomeration parameters (ζ,ξ) on overall elastic properties, three different stages of agglomeration are generated in the next section to understand the free vibration behavior of square plate with four types of CNT distribution patterns along the thickness direction as shown in Figs. 46.
Fig. 9. Young’s modulus for different levels of agglomeration and CNT volume fraction
Table 1. Hill’s elastic moduli for SingleWalled Carbon Nanotubes (SWCNT) [31].
Carbon nanotubes 





SWCNT (10,10) 
271 
88 
17 
1089 
442 
Table 2. First six natural frequencies in Hz for isotropic plate (L = 0.6 m, B = 0.4 m) [16].

Plate thickness h = 0.00625 
Plate thickness h = 0.0125 
Plate thickness h = 0.025 
Plate thickness h = 0.05 

Mode No. 
Ref.[16] 
Present 
Ref.[16] 
Present 
Ref.[16] 
Present 
Ref.[16] 
Present 
1 
136.5 
136.60 
273.1 
272.48 
546.2 
540.74 
1092.5 
1050.91 
2 
262.6 
263.35 
525.2 
523.17 
1050.4 
1030.71 
2100.9 
1957.18 
3 
420.1 
419.76 
840.3 
834.01 
1680.7 
1630.81 
3361.5 
3359.37 
4 
472.7 
474.51 
945.4 
938.67 
1890.8 
1828.95 
3781.7 
3822.49 
5 
546.2 
547.49 
1092.5 
1082.78 
2185.0 
2102.56 
4370.1 
4443.28 
6 
756.35 
761.65 
1512.7 
1495.83 
3025.4 
3031.09 
6050.8 
6072.08 
A convergence study was carried out for free vibration analyses of agglomerated CNTreinforced functionally graded plates in order to determine the appropriate number of mesh sizes that should be used in order to achieve accurate results.
The convergence analysis for a simply supported FGCNTreinforced plate at the fundamental frequency is shown in Table 2. The results are computed for = 0.075 and a/h = 10 for different mesh sizes. Based on the results of these convergence studies, it has been determined that a mesh size of 16 x 16 is suitable for free vibration analysis of FGCNTreinforced plates. The outcomes of the free vibration analyses for an isotropic square plate are presented in Table 1 (E = 70 GPa, ρ = 2700 kg/m^{3}, and υ = 0.3). A comparison was made between the dimensionless frequency parameter of the isotropic plate and the HSDT results for a moderately thick plate [37].
A simply supported FGM plate consisting of aluminum (ceramic) and zirconium oxide (metal) is considered in the present problem. The properties of the constituents are: E_{c} = 151 GPa; E_{m} = 70 GPa; ϒ_{c} = ϒ_{m} = 0.3; ρ_{c} = 3000 kg/m^{3};
ρ_{m} = 2707 kg/m^{3}.The nondimensional natural frequency parameter used in the present study is .
In Table 3, the natural frequency obtained from the present study is compared with the results of Talha et. al. [32], which are also based on higherorder shear deformation theory. The thickness ratio (a/h) is taken as 20 and the volume fraction index (n) is varied from 0.5 to 10.
Table 3. Variation of the frequency parameter with the volume fraction index, n, for SSSS square (Al/ZrO2)
FGM plates (a/h = 20)
Mode 
n = 1 
n = 5 

Ref. [32] 
Present 
Ref. [32] 
Present 

1 
1.734 
1.668 
1.621 
1.568 
2 
4.332 
4.116 
4.046 
3.865 
3 
4.332 
4.116 
4.046 
3.865 
4 
6.869 
6.506 
6.405 
6.100 
5 
8.902 
8.067 
8.269 
7.556 
The free vibration behavior of a square plate, as shown in Fig. 1, was evaluated in the following subsections using the element Q9 with 16 x 16 elements (Table 4). In this section, various aspect ratio variations such as 5, 10, 20, 50, and 100 are taken for the purpose of analysis, along with a variety of boundary conditions. A complete parametric study is also done to find out more about the threeagglomeration stage. Different levels of agglomeration were tested on UD, FGV, FGX and FGO type of carbon nanotube distribution.
Table 4. Convergence study for the dimensional frequency of an agglomerated CNTreinforced plate with simply supported boundary conditions.
Mesh Size 
UD 
FGV 
FGX 
FGO 
8 x 8 
15.823 
13.822 
18.800 
12.157 
10 x 10 
15.819 
13.817 
18.796 
12.153 
12 x 12 
15.817 
13.815 
18.795 
12.151 
14 x 14 
15.817 
13.814 
18.794 
12.150 
16 x 16 
15.816 
13.814 
18.794 
12.150 
The dimensionless frequencies used in this study were obtained using the following expressions:

(53) 
After that, detailed parametric studies were carried out to investigate the effect of boundary conditions (SSSS, CCCC, SCSC, and SFSF), thickness ratio (a/h), agglomeration stage ( ), and CNT distribution pattern across the thickness direction on the free vibration behavior of an agglomerated CNTreinforced FG plate. These studies were carried out in order to determine how these factors influence the behavior of the plate during free vibration. Tables (5)(10) show the nondimensional frequencies of the first six modes for three distinct types of agglomeration stages for the FG–CNT reinforced plate. The results are computed for a/b =1 and different aspect ratios as a/h = 5, 10, 20, 50, and 100. The minimum and maximum nondimensional frequencies for the UD, FGV, FGX, and FGO types of CNT distribution over the thickness were noted for all boundary conditions taken into consideration. As a result, the maximum and minimum stiffness are produced by the UD, FGV, FGX, and FGO distributions, respectively. Additionally, it was discovered that the allsideclamped plate produces the maximum frequency parameter whereas the SFSF produces the least frequency parameters.
This is because the stiffer agglomerated CNTreinforced functionally graded plate results from the increased limitations at the boundary. Since the present study is based on the agglomeration effect of CNT, it can be seen through the result given in Table (5)(10) for three stages of CNT agglomeration by varying the twoagglomeration parameter and .
In this section agglomeration effect of CNT is not considered (ζ=ξ). The result presented in Table 5 is for without agglomeration effect of CNT with varying boundary conditions and aspect ratio. It can be seen that when compared to the other three distributions, the FGX provides the best vibrational characteristics since its natural frequencies assume higher values. This behavior is attained because the CNTs are in higher concentrations distributed to higher stress regions. It could also be noted from Table 5 that the third mode was omitted since it is symmetrical with the second mode in case all edges are simply supported and clamped. As the aspect ratio increases the nondimensional frequency also increases for all types of CNT distribution patterns considered in this study. But overall, one can observe that the FGX pattern has higher stiffness as compared to other types of CNT distribution patterns. This means CNTs are present in the matrix without forming clusters. Further, the result is generated for the other two stages of the agglomeration effect as complete agglomeration and partial agglomeration stage by varying the and parameters.
Table 5. The first six natural frequencies without the agglomeration effect for FGCNTreinforced plate
with different boundary conditions ( ).
CNT Distribution 
a/h 
Mode 
SSSS 
CCCC 
SCSC 
SFSF 
UD CNT 
5 
1 
14.548 
22.370 
18.233 
14.297 
2 
27.364 
39.433 
31.158 
21.791 

3 
27.364 
39.433 
35.611 
27.364 

4 
32.117 
51.485 
35.932 
32.419 

5 
32.117 
51.485 
37.690 
33.499 

6 
38.698 
53.275 
49.969 
38.822 

FGV CNT 
1 
12.830 
20.386 
16.358 
13.028 

2 
27.488 
36.346 
31.242 
19.694 

3 
27.488 
36.346 
32.352 
27.540 

4 
28.688 
49.285 
32.628 
30.641 

5 
28.688 
51.120 
37.732 
32.451 

6 
38.648 
51.120 
45.652 
34.745 

FGX CNT 
1 
16.550 
23.590 
19.890 
15.272 

2 
28.102 
40.145 
31.999 
23.560 

3 
28.102 
40.145 
37.318 
28.102 

4 
34.760 
52.853 
37.620 
33.293 

5 
34.760 
52.853 
38.695 
34.794 

6 
39.742 
53.544 
51.284 
41.781 

FGO CNT 
1 
11.478 
18.754 
14.865 
11.966 

2 
26.203 
34.176 
30.005 
18.394 

3 
26.203 
34.176 
30.255 
27.027 

4 
27.027 
46.712 
30.775 
28.537 

5 
27.027 
50.851 
37.226 
32.021 

6 
38.223 
50.851 
42.760 
32.028 

UD CNT 
10 
1 
15.816 
27.091 
21.016 
17.315 
2 
37.828 
51.884 
44.532 
28.381 

3 
37.828 
51.884 
44.809 
42.565 

4 
54.728 
72.949 
62.317 
48.726 

5 
54.728 
86.138 
65.393 
54.728 

6 
58.200 
86.943 
75.380 
55.758 

FGV CNT 
1 
13.814 
23.986 
18.481 
15.365 

2 
33.216 
46.327 
39.440 
25.748 

3 
33.216 
46.327 
39.672 
37.801 

4 
51.330 
65.450 
58.184 
43.216 

5 
55.207 
77.587 
62.837 
50.294 

6 
55.207 
78.288 
70.228 
55.036 

FGX CNT 
1 
18.794 
30.923 
24.457 
19.793 

2 
43.863 
57.502 
50.420 
31.319 

3 
43.863 
57.502 
50.778 
47.960 

4 
56.205 
79.573 
63.998 
55.316 

5 
56.205 
92.781 
72.770 
56.205 

6 
66.210 
93.742 
77.390 
60.948 

FGO CNT 
1 
12.150 
21.361 
16.358 
13.693 

2 
29.489 
41.763 
35.294 
23.604 

3 
29.489 
41.763 
35.486 
33.877 

4 
45.922 
59.477 
52.479 
39.169 

5 
54.055 
70.892 
61.551 
45.835 

6 
54.055 
71.454 
63.636 
54.055 

UD CNT 
20 
1 
16.211 
29.024 
22.026 
18.691 
2 
40.028 
58.037 
48.492 
34.614 

3 
40.028 
58.037 
48.721 
44.660 

4 
63.292 
84.112 
73.294 
56.541 

5 
78.496 
101.299 
89.696 
65.314 

6 
78.496 
101.930 
89.844 
88.388 

FGV CNT 
1 
14.116 
25.380 
19.218 
16.377 

2 
34.911 
50.913 
42.414 
31.021 

3 
34.911 
50.913 
42.609 
40.852 

4 
55.288 
73.979 
64.242 
50.490 

5 
68.631 
89.216 
78.700 
57.637 

6 
68.631 
89.746 
78.824 
77.590 

FGX CNT 
1 
19.560 
34.529 
26.394 
22.184 

2 
47.912 
68.183 
57.500 
39.133 

3 
47.912 
68.183 
57.797 
55.223 

4 
75.200 
97.871 
86.169 
65.000 

5 
92.843 
117.163 
104.928 
75.659 

6 
92.843 
118.000 
105.127 
103.299 

FGO CNT 
1 
12.347 
22.287 
16.842 
14.402 

2 
30.623 
44.903 
37.306 
27.883 

3 
30.623 
44.903 
37.471 
35.956 

4 
48.631 
65.491 
56.689 
45.557 

5 
60.467 
79.135 
69.565 
51.118 

6 
60.467 
79.579 
69.669 
68.604 

UD CNT 
50 
1 
16.337 
29.706 
22.365 
19.315 
2 
40.774 
60.403 
49.926 
39.788 

3 
40.774 
60.403 
50.133 
48.212 

4 
65.193 
88.930 
76.480 
68.349 

5 
81.291 
107.793 
94.114 
69.924 

6 
81.291 
108.321 
94.238 
92.877 

FGV CNT 
1 
14.214 
25.870 
19.467 
16.839 

2 
35.490 
52.640 
43.482 
35.009 

3 
35.490 
52.640 
43.661 
42.001 

4 
56.786 
77.592 
66.673 
60.849 

5 
70.790 
94.014 
82.016 
61.271 

6 
70.790 
94.466 
82.123 
80.946 

FGX CNT 
1 
19.804 
35.912 
27.075 
23.295 

2 
49.353 
72.832 
60.317 
46.887 

3 
49.353 
72.832 
60.573 
58.202 

4 
78.772 
106.928 
92.189 
78.413 

5 
98.169 
129.526 
113.384 
83.830 

6 
98.169 
130.191 
113.541 
111.863 

FGO CNT 
1 
12.414 
22.613 
17.009 
14.733 

2 
31.014 
46.055 
38.021 
30.883 

3 
31.014 
46.055 
38.175 
36.734 

4 
49.672 
67.978 
58.365 
53.505 

5 
61.914 
82.345 
71.782 
54.726 

6 
61.914 
82.733 
71.874 
70.849 

UD CNT 
100 
1 
16.382 
29.902 
22.470 
19.511 
2 
40.992 
61.021 
50.319 
41.567 

3 
40.992 
61.021 
50.516 
48.618 

4 
65.908 
90.547 
77.596 
71.359 

5 
81.961 
109.303 
95.147 
76.095 

6 
81.961 
109.781 
95.261 
93.889 

FGV CNT 
1 
14.260 
26.0514 
19.568 
17.001 

2 
35.701 
53.196 
43.847 
36.385 

3 
35.701 
53.196 
44.016 
42.357 

4 
57.511 
79.140 
67.768 
62.357 

5 
71.400 
95.307 
82.924 
67.227 

6 
71.400 
95.709 
83.020 
81.809 

FGX CNT 
1 
19.863 
36.210 
27.227 
23.601 

2 
49.665 
73.818 
60.917 
49.761 

3 
49.665 
73.818 
61.160 
58.849 

4 
79.704 
109.246 
93.729 
86.005 

5 
99.234 
132.103 
115.102 
89.122 

6 
99.234 
132.705 
115.245 
113.585 

FGO CNT 
1 
12.459 
22.779 
17.104 
14.868 

2 
31.209 
46.545 
38.350 
31.949 

3 
31.209 
46.545 
38.494 
37.040 

4 
50.371 
69.425 
59.404 
54.688 

5 
62.436 
83.418 
72.546 
59.514 

6 
62.436 
83.756 
72.627 
71.554 
The present section deals with the complete agglomeration effect assuming that all the CNTs are aggregated in the spherically shaped inclusion. Here, in this section, three different combinations of ζ and ξ are considered for the analysis of this particular agglomeration stage. As, we can see from Table 6–8 as parameter ξ increases from 0.25 to 0.75 corresponding to ζ=1, the stage where ξ is equal to 0.25 means all CNTs are presented in the matrix as circular clusters have less stiffness as compared to ξ= 0.75 stage. The stage ζ=1 and ξ= 0.25 represents the worst case of the agglomeration stage. Next, as the value of ξ reaches towards ζ the CNTs which are present in stage 1 in a cluster will try to free from cluster effect by uniform mixing with the surrounding matrix. Overall, from Table 6 to Table 8 it can easily be understood that case 3 where ζ=1 and
ξ= 0.75 shows a higher value of nondimensional frequency as compared to the other two stages under the complete agglomeration effect.
According to the findings of the study, the elasticity of the material would be impacted more by the agglomeration of carbon nanotubes in proportion to the degree to which the values of the agglomeration parameters differed from one another. The same explanation can also be understood by glancing at the illustration that is labeled Fig. 9. The difference in the nondimensional frequency distributions is quite significant when contrasted with the frequency distributions of other cases of complete agglomeration. The difference between the two groups of findings is rather substantial when measured against the frequencies that were acquired in the section before this one without the agglomeration stage. The natural frequencies obtained for three different cases of complete agglomeration considering the UD, FGV, FGX, and FGO are listed in Table 6 – 8. From this table, one can conclude that for all cases of complete agglomeration observed, the FGO is the CNT distribution that has the worse dynamic behavior, when comparing it with the same states of agglomeration for the other CNT distributions.
When taken as a whole, it is possible to state that, for a stage that has been entirely agglomerated, the three CNT distributions that are being investigated will have lower natural frequencies if the distribution is more heterogeneous. It is possible to arrive at the conclusion that the FGX distribution demonstrates superior vibrational behavior in addition to the level of agglomeration because CNTs are distributed in regions with higher bending stress; despite this, the differences in natural frequencies between the distributions become smaller as the value of ξ decreases.
Table 6. The first six nondimensional natural frequencies for FGCNTreinforced plate with a full agglomeration effect
with different boundary conditions .
CNT Distribution 
a/h 
Mode 
SSSS 
CCCC 
SCSC 
SFSF 
UD CNT 
5 
1 
6.689 
10.212 
8.351 
6.498 
2 
12.263 
17.935 
13.973 
9.906 

3 
12.263 
17.935 
16.246 
12.263 

4 
14.706 
23.532 
16.394 
14.554 

5 
14.706 
23.532 
17.150 
15.226 

6 
17.343 
24.193 
22.750 
17.733 

FGV CNT 
1 
6.535 
10.021 
8.178 
6.372 

2 
12.105 
17.637 
13.793 
9.709 

3 
12.105 
17.637 
15.946 
12.105 

4 
14.404 
23.235 
16.092 
14.367 

5 
14.404 
23.235 
16.933 
14.955 

6 
17.119 
23.808 
22.355 
17.375 

FGX CNT 
1 
6.704 
10.210 
8.357 
6.498 

2 
12.225 
17.909 
13.930 
9.907 

3 
12.225 
17.909 
16.238 
12.225 

4 
14.715 
23.461 
16.386 
14.510 

5 
14.715 
23.461 
17.098 
15.213 

6 
17.289 
24.147 
22.724 
17.737 

FGO CNT 
1 
6.527 
10.025 
8.1755 
6.3744 

2 
12.134 
17.661 
13.826 
9.7129 

3 
12.134 
17.661 
15.957 
12.134 

4 
14.402 
23.291 
16.103 
14.402 

5 
14.402 
23.291 
16.973 
14.969 

6 
17.161 
23.848 
22.380 
17.38 

UD CNT 
10 
1 
7.298 
12.458 
9.681 
7.942 
2 
17.419 
23.800 
20.467 
12.908 

3 
17.419 
23.800 
20.596 
19.541 

4 
24.526 
33.410 
27.946 
22.345 

5 
24.526 
39.409 
30.007 
24.526 

6 
26.763 
39.785 
34.300 
25.431 

FGV CNT 
1 
7.114 
12.170 
9.447 
7.756 

2 
17.003 
23.286 
20.001 
12.631 

3 
17.003 
23.286 
20.126 
19.094 

4 
24.211 
32.718 
27.587 
21.824 

5 
24.211 
38.621 
29.352 
24.210 

6 
26.145 
38.986 
33.866 
24.887 

FGX CNT 
1 
7.3233 
12.487 
9.708 
7.960 

2 
17.469 
23.832 
20.507 
12.919 

3 
17.469 
23.832 
20.637 
19.576 

4 
24.451 
33.437 
27.860 
22.386 

5 
24.451 
39.425 
30.050 
24.451 

6 
26.819 
39.803 
34.196 
25.449 

FGO CNT 
1 
7.098 
12.153 
9.430 
7.7483 

2 
16.974 
23.269 
19.977 
12.637 

3 
16.974 
23.269 
20.102 
19.081 

4 
24.269 
32.708 
27.653 
21.817 

5 
24.269 
38.620 
29.329 
24.269 

6 
26.112 
38.983 
33.946 
24.903 

UD CNT 
20 
1 
7.489 
13.393 
10.170 
8.612 
2 
18.480 
26.755 
22.371 
15.794 

3 
18.480 
26.755 
22.477 
21.515 

4 
29.204 
38.744 
33.790 
25.869 

5 
36.208 
46.641 
41.337 
30.011 

6 
36.208 
46.936 
41.407 
40.725 

FGV CNT 
1 
7.294 
13.055 
9.909 
8.395 

2 
18.008 
26.096 
21.810 
15.440 

3 
18.008 
26.096 
21.913 
20.976 

4 
28.468 
37.809 
32.956 
25.269 

5 
35.303 
45.528 
40.327 
29.289 

6 
35.303 
45.813 
40.394 
39.729 

FGX CNT 
1 
7.518 
13.439 
10.207 
8.641 

2 
18.547 
26.839 
22.447 
15.821 

3 
18.547 
26.839 
22.554 
21.587 

4 
29.304 
38.853 
33.895 
25.925 

5 
36.327 
46.765 
41.460 
30.090 

6 
36.327 
47.061 
41.530 
40.845 

FGO CNT 
1 
7.276 
13.025 
9.885 
8.377 

2 
17.964 
26.043 
21.761 
15.427 

3 
17.964 
26.043 
21.864 
20.932 

4 
28.404 
37.740 
32.889 
25.242 

5 
35.227 
45.451 
40.249 
29.245 

6 
35.227 
45.735 
40.316 
39.656 

UD CNT 
50 
1 
7.549 
13.724 
10.333 
8.917 
2 
18.839 
27.900 
23.064 
18.291 

3 
18.839 
27.900 
23.160 
22.268 

4 
30.117 
41.065 
35.324 
31.236 

5 
37.554 
49.777 
43.469 
32.256 

6 
37.554 
50.022 
43.527 
42.895 

FGV CNT 
1 
7.352 
13.367 
10.064 
8.687 

2 
18.348 
27.178 
22.465 
17.840 

3 
18.348 
27.178 
22.559 
21.691 

4 
29.334 
40.009 
34.411 
30.517 

5 
36.579 
48.497 
42.346 
31.432 

6 
36.579 
48.735 
42.402 
41.787 

FGX CNT 
1 
7.579 
13.778 
10.374 
8.952 

2 
18.914 
28.008 
23.155 
18.348 

3 
18.914 
28.008 
23.251 
22.355 

4 
30.234 
41.220 
35.459 
31.303 

5 
37.700 
49.964 
43.636 
32.374 

6 
37.700 
50.211 
43.694 
43.059 

FGO CNT 
1 
7.332 
13.332 
10.037 
8.664 

2 
18.299 
27.108 
22.407 
17.804 

3 
18.299 
27.108 
22.500 
21.635 

4 
29.258 
39.909 
34.323 
30.478 

5 
36.484 
48.376 
42.238 
31.356 

6 
36.484 
48.613 
42.294 
41.680 

UD CNT 
100 
1 
7.570 
13.814 
10.382 
9.011 
2 
18.939 
28.188 
23.246 
19.161 

3 
18.939 
28.188 
23.337 
22.459 

4 
30.441 
41.808 
35.833 
32.938 

5 
37.866 
50.488 
43.954 
34.927 

6 
37.866 
50.710 
44.007 
43.372 

FGV CNT 
1 
7.372 
13.455 
10.111 
8.777 

2 
18.445 
27.456 
22.641 
18.673 

3 
18.445 
27.456 
22.730 
21.875 

4 
29.652 
40.730 
34.907 
32.090 

5 
36.880 
49.178 
42.811 
34.081 

6 
36.880 
49.394 
42.863 
42.245 

FGX CNT 
1 
7.600 
13.869 
10.423 
9.047 

2 
19.015 
28.299 
23.338 
19.230 

3 
19.015 
28.299 
23.430 
22.548 

4 
30.559 
41.967 
35.972 
33.063 

5 
38.015 
50.685 
44.126 
35.028 

6 
38.015 
50.909 
44.180 
43.543 

FGO CNT 
1 
7.352 
13.419 
10.084 
8.754 

2 
18.396 
27.384 
22.581 
18.629 

3 
18.396 
27.384 
22.670 
21.817 

4 
29.574 
40.627 
34.818 
32.009 

5 
36.782 
49.050 
42.699 
34.017 

6 
36.782 
49.266 
42.750 
42.134 
Table 7. The first six nondimensional natural frequencies for FGCNTreinforced plate with a full agglomeration effect
with different boundary conditions .
CNT Distribution 
a/h 
Mode 
SSSS 
CCCC 
SCSC 
SFSF 
UD CNT 
5 
1 
8.417 
12.883 
10.522 
8.209 
2 
15.567 
22.654 
17.734 
12.513 

3 
15.567 
22.654 
20.499 
15.567 

4 
18.532 
29.670 
20.685 
18.465 

5 
18.532 
29.670 
21.656 
19.237 

6 
22.016 
30.574 
28.725 
22.365 

FGV CNT 
1 
7.893 
12.226 
9.9318 
7.782 

2 
15.079 
21.620 
17.176 
11.842 

3 
17.493 
21.620 
19.464 
15.083 

4 
17.493 
28.738 
19.639 
17.882 

5 
17.493 
28.738 
20.985 
18.303 

6 
21.317 
29.235 
27.353 
21.129 

FGX CNT 
1 
8.518 
12.909 
10.590 
8.235 

2 
15.427 
22.589 
17.574 
12.564 

3 
15.427 
22.589 
20.523 
15.427 

4 
18.640 
29.407 
20.709 
18.299 

5 
18.640 
29.407 
21.463 
19.231 

6 
21.817 
30.436 
28.687 
22.471 

FGO CNT 
1 
7.756 
12.108 
9.802 
7.702 

2 
15.106 
21.510 
17.209 
11.729 

3 
15.106 
21.510 
19.298 
15.106 

4 
17.276 
28.808 
19.472 
17.919 

5 
17.276 
28.808 
21.024 
18.169 

6 
21.363 
29.128 
27.176 
20.898 

UD CNT 
10 
1 
9.172 
15.676 
12.174 
10.000 
2 
21.910 
29.973 
25.758 
16.295 

3 
21.910 
29.973 
25.919 
24.596 

4 
31.135 
42.098 
35.468 
28.129 

5 
31.135 
49.676 
37.785 
31.135 

6 
33.675 
50.146 
43.313 
32.072 

FGV CNT 
1 
8.554 
14.700 
11.385 
9.380 

2 
20.495 
28.223 
24.177 
15.394 

3 
20.495 
28.233 
24.326 
23.106 

4 
30.166 
39.734 
34.364 
26.415 

5 
30.166 
46.973 
35.552 
30.159 

6 
31.578 
47.408 
41.980 
30.297 

FGX CNT 
1 
9.329 
15.868 
12.352 
10.125 

2 
22.221 
30.229 
26.048 
16.411 

3 
22.221 
30.229 
26.214 
24.866 

4 
30.854 
42.371 
35.148 
28.453 

5 
30.854 
49.916 
38.128 
30.854 

6 
34.077 
50.398 
42.927 
32.292 

FGO CNT 
1 
8.370 
14.438 
11.162 
9.217 

2 
20.102 
27.804 
23.768 
15.217 

3 
20.102 
27.804 
23.911 
22.735 

4 
30.212 
39.216 
34.418 
26.015 

5 
30.212 
46.423 
35.015 
29.943 

6 
31.031 
46.843 
42.049 
30.212 

UD CNT 
20 
1 
9.408 
16.831 
12.778 
10.828 
2 
23.221 
33.636 
28.117 
19.925 

3 
23.221 
33.636 
28.251 
27.045 

4 
36.703 
48.722 
42.479 
32.601 

5 
45.510 
58.662 
51.974 
37.775 

6 
45.510 
59.030 
52.060 
51.210 

FGV CNT 
1 
8.758 
15.697 
11.905 
10.104 

2 
21.637 
31.418 
26.231 
18.740 

3 
21.637 
31.418 
26.354 
25.239 

4 
34.230 
45.568 
39.673 
30.607 

5 
42.466 
54.904 
48.569 
35.355 

6 
42.466 
55.242 
48.649 
47.859 

FGX CNT 
1 
9.586 
17.122 
13.009 
11.011 

2 
23.638 
34.167 
28.592 
20.128 

3 
23.638 
34.167 
28.729 
27.496 

4 
37.331 
49.433 
43.152 
32.992 

5 
46.265 
59.477 
52.767 
38.298 

6 
46.265 
59.858 
52.857 
51.985 

FGO CNT 
1 
8.5583 
15.357 
11.641 
9.889 

2 
21.158 
30.774 
25.672 
18.447 

3 
21.158 
30.774 
25.791 
24.707 

4 
33.496 
44.675 
38.859 
30.105 

5 
41.574 
53.858 
47.594 
34.689 

6 
41.574 
54.184 
47.671 
46.906 

UD CNT 
50 
1 
9.483 
17.240 
12.981 
11.205 
2 
23.665 
35.052 
28.975 
23.018 

3 
23.665 
35.052 
29.095 
27.977 

4 
37.834 
51.596 
44.379 
39.388 

5 
47.177 
62.541 
54.612 
40.542 

6 
47.177 
62.849 
54.684 
53.892 

FGV CNT 
1 
8.823 
16.046 
12.079 
10.433 

2 
22.023 
32.635 
26.970 
21.505 

3 
22.053 
32.635 
27.082 
26.044 

4 
35.218 
48.059 
41.323 
36.974 

5 
43.914 
58.253 
50.850 
37.783 

6 
43.914 
58.538 
50.918 
50.182 

FGX CNT 
1 
9.667 
17.570 
13.231 
11.415 

2 
24.121 
35.710 
29.526 
23.381 

3 
24.121 
35.710 
29.649 
28.506 

4 
38.553 
52.547 
45.210 
39.851 

5 
48.072 
63.692 
55.633 
41.271 

6 
48.072 
64.007 
55.707 
54.898 

FGO CNT 
1 
8.6184 
15.678 
11.800 
10.196 

2 
21.514 
31.892 
26.352 
21.066 

3 
21.514 
31.892 
26.461 
25.449 

4 
34.412 
46.980 
40.386 
36.342 

5 
42.910 
56.945 
49.697 
36.953 

6 
42.910 
57.222 
49.763 
49.045 

UD CNT 
100 
1 
9.509 
17.354 
13.041 
11.321 
2 
23.791 
35.412 
29.203 
24.089 

3 
23.791 
35.412 
29.317 
28.215 

4 
38.244 
52.530 
45.021 
41.390 

5 
47.567 
63.428 
55.217 
43.976 

6 
47.567 
63.707 
55.284 
54.487 

FGV CNT 
1 
8.848 
16.151 
12.136 
10.538 

2 
22.140 
32.964 
27.180 
22.458 

3 
22.140 
32.964 
27.286 
26.260 

4 
35.607 
48.930 
41.927 
38.556 

5 
44.271 
59.049 
51.398 
41.136 

6 
44.271 
59.307 
51.459 
50.716 

FGX CNT 
1 
9.693 
17.688 
13.293 
11.537 

2 
24.250 
36.087 
29.763 
24.516 

3 
24.250 
36.087 
29.880 
28.756 

4 
38.969 
53.508 
45.867 
42.157 

5 
48.481 
64.631 
56.271 
44.627 

6 
48.481 
64.917 
56.339 
55.528 

FGO CNT 
1 
8.642 
15.779 
11.856 
10.296 

2 
21.629 
32.209 
26.555 
21.967 

3 
21.629 
32.209 
26.659 
25.656 

4 
34.796 
47.830 
40.978 
37.691 

5 
43.252 
57.702 
50.219 
40.329 

6 
43.252 
57.952 
50.279 
49.552 
Table 8. The first six nondimensional natural frequencies for FGCNTreinforced plate with a full agglomeration effect with different boundary conditions .
CNT Distribution 
a/h 
Mode 
SSSS 
CCCC 
SCSC 
SFSF 
UD CNT 
5 
1 
10.760 
16.503 
13.467 
10.531 
2 
20.054 
29.052 
22.840 
16.050 

3 
20.054 
29.052 
26.265 
20.054 

4 
23.719 
37.995 
26.503 
23.774 

5 
23.719 
37.995 
27.770 
24.675 

6 
28.360 
39.229 
36.828 
28.645 

FGV CNT 
1 
9.673 
15.147 
12.242 
9.653 

2 
19.247 
26.905 
21.912 
14.660 

3 
19.247 
26.905 
24.107 
19.266 

4 
21.546 
36.392 
24.319 
22.698 

5 
21.546 
36.392 
26.651 
22.841 

6 
27.179 
36.440 
33.956 
26.046 

FGX CNT 
1 
11.180 
16.697 
13.789 
10.689 

2 
19.798 
29.015 
22.549 
16.342 

3 
19.798 
29.015 
26.517 
19.798 

4 
24.243 
37.520 
26.754 
23.472 

5 
24.243 
37.520 
27.422 
24.815 

6 
27.999 
39.005 
36.925 
29.205 

FGO CNT 
1 
9.1787 
14.626 
11.729 
9.311 

2 
19.215 
26.271 
21.885 
14.202 

3 
19.215 
26.271 
23.348 
19.215 

4 
20.677 
35.711 
23.554 
22.081 

5 
20.677 
36.432 
26.624 
22.781 

6 
27.174 
36.432 
33.050 
25.106 

UD CNT 
10 
1 
11.712 
20.037 
15.553 
12.799 
2 
27.993 
38.341 
32.930 
20.918 

3 
27.993 
38.341 
33.136 
31.472 

4 
40.108 
53.877 
45.680 
36.017 

5 
40.108 
63.595 
48.330 
40.108 

6 
43.046 
64.194 
55.541 
41.144 

FGV CNT 
1 
10.442 
18.030 
13.931 
11.521 

2 
25.073 
34.724 
29.665 
19.070 

3 
25.073 
34.724 
29.843 
28.384 

4 
38.537 
48.978 
43.701 
32.453 

5 
38.537 
57.984 
43.888 
37.451 

6 
38.700 
58.512 
52.695 
38.503 

FGX CNT 
1 
12.341 
20.837 
16.278 
13.307 

2 
29.267 
39.477 
34.153 
21.430 

3 
29.267 
39.477 
34.378 
32.581 

4 
39.597 
55.166 
45.099 
37.331 

5 
39.597 
64.831 
49.832 
39.597 

6 
44.726 
65.472 
54.844 
42.087 

FGO CNT 
1 
9.8153 
17.082 
13.148 
10.923 

2 
23.690 
33.127 
28.170 
18.347 

3 
23.690 
33.127 
28.332 
26.988 

4 
36.721 
46.926 
41.676 
30.968 

5 
38.430 
55.726 
43.771 
35.964 

6 
38.430 
56.204 
50.384 
38.430 

UD CNT 
20 
1 
12.009 
21.492 
16.314 
13.834 
2 
29.647 
42.963 
35.906 
25.534 

3 
29.647 
42.963 
36.076 
34.545 

4 
46.867 
62.247 
54.257 
41.746 

5 
58.119 
74.955 
66.391 
48.300 

6 
58.119 
75.425 
66.502 
65.421 

FGV CNT 
1 
10.678 
19.166 
14.526 
12.349 

2 
26.398 
38.409 
32.035 
23.110 

3 
26.398 
38.409 
32.184 
30.837 

4 
41.789 
55.761 
48.491 
37.679 

5 
51.863 
67.220 
59.388 
43.336 

6 
51.863 
67.628 
59.484 
58.533 

FGX CNT 
1 
12.715 
22.655 
17.235 
14.566 

2 
31.311 
45.105 
37.808 
26.410 

3 
31.311 
45.105 
37.992 
36.350 

4 
49.382 
65.143 
56.974 
43.393 

5 
61.151 
78.295 
69.607 
50.450 

6 
61.151 
78.809 
69.728 
68.567 

FGO CNT 
1 
10.006 
18.005 
13.628 
11.612 

2 
24.775 
36.174 
30.118 
22.015 

3 
24.775 
36.174 
30.255 
29.006 

4 
39.280 
52.629 
45.673 
35.883 

5 
48.795 
63.518 
55.992 
40.958 

6 
48.795 
63.890 
56.080 
55.199 

UD CNT 
50 
1 
12.104 
22.007 
16.569 
14.305 
2 
30.207 
44.412 
36.986 
29.426 

3 
30.207 
44.412 
37.139 
35.714 

4 
48.295 
65.870 
56.653 
50.442 

5 
60.221 
79.842 
69.715 
51.774 

6 
60.221 
80.234 
69.808 
68.798 

FGV CNT 
1 
10.754 
19.565 
14.725 
12.726 

2 
26.847 
39.800 
32.885 
26.330 

3 
26.847 
39.800 
33.020 
31.759 

4 
42.942 
58.634 
50.401 
45.515 

5 
53.544 
71.064 
62.016 
46.141 

6 
53.544 
71.409 
62.098 
61.204 

FGX CNT 
1 
12.833 
23.313 
17.560 
15.141 

2 
32.013 
47.362 
39.173 
30.897 

3 
32.013 
47.362 
39.337 
37.816 

4 
51.151 
69.656 
59.957 
52.417 

5 
63.774 
84.422 
73.774 
54.687 

6 
63.774 
84.843 
73.873 
72.796 

FGO CNT 
1 
10.068 
18.326 
13.789 
11.928 

2 
25.142 
37.300 
30.808 
24.806 

3 
25.142 
37.300 
30.934 
29.758 

4 
40.235 
54.990 
47.246 
43.126 

5 
50.167 
66.644 
58.131 
43.425 

6 
50.167 
66.964 
58.207 
57.372 

UD CNT 
100 
1 
12.137 
22.152 
16.647 
14.452 
2 
30.368 
45.203 
37.277 
30.769 

3 
30.368 
45.203 
37.423 
36.016 

4 
48.821 
67.064 
57.476 
52.847 

5 
60.718 
80.968 
70.485 
56.245 

6 
60.718 
81.323 
70.569 
69.552 

FGV CNT 
1 
10.785 
19.694 
14.796 
12.851 

2 
26.994 
40.202 
33.144 
27.436 

3 
26.994 
40.202 
33.273 
32.021 

4 
43.437 
59.721 
51.161 
47.061 

5 
53.981 
72.022 
62.679 
50.443 

6 
53.981 
72.332 
62.753 
61.844 

FGX CNT 
1 
12.868 
23.476 
17.646 
15.311 

2 
32.189 
47.887 
39.500 
32.479 

3 
32.189 
47.887 
39.656 
38.164 

4 
51.707 
70.966 
60.846 
55.906 

5 
64.344 
85.750 
74.672 
58.911 

6 
64.344 
86.133 
74.763 
73.687 

FGO CNT 
1 
10.099 
18.447 
13.857 
12.040 

2 
25.282 
37.670 
31.050 
25.767 

3 
25.282 
37.670 
31.169 
29.997 

4 
40.718 
56.027 
47.978 
44.150 

5 
50.566 
67.498 
58.728 
47.617 

6 
50.566 
67.783 
58.796 
57.940 
Similarly, we can conclude the partial agglomeration effect in which some CNTs are present in cluster form and some CNTs are present in the matrix. In this partial agglomeration stage, two types of agglomeration stages were considered for the study in which ζ and ξ will have two different values to create a partial agglomeration stage. In the first stage where ζ= 0.25 and ξ= 0.5 by assuming the maximum number of CNTs are present in the cluster form, some CNTs will present in the matrix, while in the second stage ζ= 0.75 and
ξ= 0.5 is used to show that maximum number of CNTs are present in matrix and only little percentage of CNTs are forming clusters of CNT. Comparing both stages reveals that their behavior with respect to nondimensional frequency parameters is nearly identical. Although significant differences were found for the natural frequencies when comparing both cases of agglomeration under the partial agglomeration stage with the results obtained without the agglomeration stage discussed in section 5.3.1. Here, one can see the natural mode shapes do not suffer much difference from the agglomeration effect, for these two partially agglomerated states. In Table 12, the first six nondimensional frequencies for the UD, FGV, FGX, and FGO distributions are presented. The table shows highest nondimensional frequencies appear for ζ= 0.25, and once again the lesser the volume of CNTs inside the agglomerated inclusions, the better the dynamic free vibration behavior obtained in the CNTreinforced composite. Despite the agglomeration impact, the natural mode morphologies remained constant when compared to the findings obtained without the agglomeration stage for all forms of CNT dispersion.
Table 9. The first six nondimensional natural frequencies for FGCNTreinforced plate with a partial agglomeration effect under different boundary conditions. .
CNT Distribution 
a/h 
Mode 
SSSS 
CCCC 
SCSC 
SFSF 
UD CNT 
5 
1 
14.018 
21.561 
17.572 
13.782 
2 
26.390 
38.010 
30.049 
21.006 

3 
26.390 
38.010 
34.323 
26.390 

4 
30.952 
49.620 
34.633 
31.264 

5 
30.952 
49.620 
36.330 
32.291 

6 
37.322 
51.357 
48.165 
37.416 

FGV CNT 
1 
12.439 
19.744 
15.851 
12.618 

2 
26.542 
35.188 
30.167 
19.077 

3 
26.542 
35.188 
31.337 
26.593 

4 
27.803 
47.710 
31.604 
29.673 

5 
27.803 
49.339 
36.415 
31.339 

6 
37.322 
49.339 
44.213 
33.665 

FGX CNT 
1 
15.946 
22.757 
19.177 
14.734 

2 
27.124 
38.749 
30.884 
22.722 

3 
27.124 
38.749 
36.004 
27.124 

4 
33.521 
50.973 
36.297 
32.132 

5 
33.521 
50.973 
37.325 
33.578 

6 
38.359 
51.693 
49.497 
40.299 

FGO CNT 
1 
11.131 
18.173 
14.410 
11.595 

2 
25.400 
33.100 
29.073 
17.818 

3 
25.400 
33.100 
29.315 
26.090 

4 
26.090 
45.232 
29.708 
27.646 

5 
26.090 
49.052 
35.915 
30.908 

6 
36.897 
49.052 
41.421 
31.040 

UD CNT 
10 
1 
15.238 
26.104 
20.249 
16.692 
2 
36.448 
49.998 
42.911 
27.378 

3 
36.448 
49.998 
43.177 
41.033 

4 
52.781 
70.301 
60.099 
46.988 

5 
52.781 
83.015 
63.016 
52.781 

6 
56.080 
83.790 
72.661 
53.789 

FGV CNT 
1 
13.397 
23.252 
17.919 
14.895 

2 
32.208 
44.898 
38.233 
24.942 

3 
32.208 
44.898 
38.457 
36.641 

4 
49.765 
63.420 
56.395 
41.889 

5 
53.302 
75.172 
60.668 
48.729 

6 
53.302 
75.852 
68.062 
53.143 

FGX CNT 
1 
18.093 
29.792 
23.555 
19.072 

2 
42.247 
55.427 
48.582 
30.197 

3 
42.247 
55.427 
48.927 
46.220 

4 
54.249 
76.722 
61.769 
53.300 

5 
54.249 
89.475 
70.140 
54.249 

6 
63.793 
90.400 
74.651 
58.776 

FGO CNT 
1 
11.786 
20.715 
15.866 
13.278 

2 
28.601 
40.490 
34.225 
22.870 

3 
28.601 
40.490 
34.411 
32.850 

4 
44.534 
57.655 
50.882 
37.970 

5 
52.181 
68.712 
59.416 
44.425 

6 
52.181 
69.259 
61.693 
52.181 

UD CNT 
20 
1 
15.619 
27.963 
21.221 
18.010 
2 
38.565 
55.919 
46.721 
33.367 

3 
38.565 
55.919 
46.941 
44.960 

4 
60.979 
81.043 
70.618 
54.502 

5 
75.629 
97.605 
86.422 
62.948 

6 
75.629 
98.213 
86.565 
85.169 

FGV CNT 
1 
13.691 
24.612 
18.638 
15.881 

2 
33.858 
49.369 
41.132 
30.062 

3 
33.858 
49.369 
41.320 
39.615 

4 
53.618 
71.729 
62.295 
48.932 

5 
66.556 
86.499 
76.312 
55.880 

6 
66.556 
87.015 
76.433 
75.235 

FGX CNT 
1 
18.826 
33.241 
25.406 
21.359 

2 
46.120 
65.655 
55.359 
37.719 

3 
46.120 
65.655 
55.643 
53.169 

4 
72.396 
94.258 
82.973 
62.616 

5 
89.390 
112.851 
101.046 
72.884 

6 
89.390 
113.655 
101.237 
99.481 

FGO CNT 
1 
11.978 
21.620 
16.339 
13.971 

2 
29.708 
43.555 
36.189 
27.030 

3 
29.708 
43.555 
36.349 
34.878 

4 
47.176 
63.520 
54.988 
44.156 

5 
58.655 
76.752 
67.476 
49.577 

6 
58.655 
77.183 
67.577 
66.543 

UD CNT 
50 
1 
15.739 
28.620 
21.547 
18.609 
2 
39.282 
58.194 
48.100 
38.340 

3 
39.282 
58.194 
48.300 
46.449 

4 
62.809 
85.679 
73.684 
65.875 

5 
78.318 
103.852 
90.672 
67.371 

6 
78.318 
104.361 
90.792 
89.482 

FGV CNT 
1 
13.787 
25.091 
18.881 
16.331 

2 
34.421 
51.054 
42.173 
33.945 

3 
34.421 
51.054 
42.346 
40.736 

4 
55.076 
75.251 
64.663 
58.993 

5 
68.658 
91.178 
79.545 
59.402 

6 
68.658 
91.617 
79.648 
78.506 

FGX CNT 
1 
19.058 
34.561 
26.056 
22.421 

2 
47.496 
70.096 
58.050 
45.153 

3 
47.496 
70.096 
58.296 
56.015 

4 
75.810 
102.917 
88.727 
75.553 

5 
94.479 
124.669 
109.127 
80.693 

6 
94.479 
125.309 
109.278 
107.664 

FGO CNT 
1 
12.044 
21.938 
16.502 
14.292 

2 
30.088 
44.679 
36.886 
29.954 

3 
30.088 
44.679 
37.036 
35.637 

4 
48.188 
65.945 
56.621 
51.902 

5 
60.066 
79.884 
69.638 
53.056 

6 
60.066 
80.261 
69.727 
68.733 

UD CNT 
100 
1 
15.783 
28.808 
21.648 
18.797 
2 
39.493 
58.789 
48.479 
40.049 

3 
39.493 
58.789 
48.668 
46.840 

4 
63.498 
87.237 
74.760 
68.751 

5 
78.963 
105.306 
91.668 
73.328 

6 
78.963 
105.767 
91.778 
90.455 

FGV CNT 
1 
13.831 
25.266 
18.979 
16.489 

2 
34.626 
51.592 
42.526 
35.284 

3 
34.626 
51.592 
42.689 
41.081 

4 
55.775 
76.747 
65.720 
60.472 

5 
69.248 
92.433 
80.425 
65.177 

6 
69.248 
92.823 
80.518 
79.344 

FGX CNT 
1 
19.115 
34.847 
26.202 
22.713 

2 
47.795 
71.041 
58.624 
47.902 

3 
47.795 
71.041 
58.858 
56.634 

4 
76.706 
105.140 
90.205 
82.777 

5 
95.499 
127.135 
110.772 
85.839 

6 
95.499 
127.714 
110.909 
109.312 

FGO CNT 
1 
12.087 
22.098 
16.593 
14.423 

2 
30.276 
45.153 
37.203 
30.990 

3 
30.276 
45.153 
37.343 
35.933 

4 
48.862 
67.341 
57.622 
53.047 

5 
60.571 
80.922 
70.377 
57.7156 

6 
60.571 
81.251 
70.456 
69.416 
Table 10. The first six nondimensional natural frequencies for FGCNTreinforced plate with a partial agglomeration effect under different boundary conditions .
CNT Distribution 
a/h 
Mode 
SSSS 
CCCC 
SCSC 
SFSF 
UD CNT 
5 
1 
13.632 
20.962 
17.085 
13.397 
2 
25.640 
36.950 
29.196 
20.419 

3 
25.640 
36.950 
33.369 
25.640 

4 
30.095 
48.243 
33.670 
30.377 

5 
30.095 
48.243 
35.317 
31.389 

6 
36.261 
49.921 
46.822 
36.377 

FGV CNT 
1 
12.134 
19.231 
15.450 
12.287 

2 
25.705 
34.259 
29.220 
18.581 

3 
25.705 
34.259 
30.530 
25.754 

4 
27.109 
46.445 
30.790 
28.895 

5 
27.109 
47.854 
35.295 
30.366 

6 
36.158 
47.854 
43.067 
32.810 

FGX CNT 
1 
15.422 
22.081 
18.581 
14.282 

2 
26.270 
37.650 
29.913 
22.011 

3 
26.270 
37.650 
34.945 
26.270 

4 
32.496 
49.404 
35.231 
31.123 

5 
32.496 
49.404 
36.171 
32.595 

6 
37.152 
50.255 
48.087 
39.072 

FGO CNT 
1 
10.891 
17.752 
14.087 
11.323 

2 
24.831 
32.301 
28.395 
17.386 

3 
24.831 
32.301 
28.632 
25.299 

4 
25.299 
44.122 
28.808 
26.990 

5 
25.299 
47.599 
34.846 
29.974 

6 
35.779 
47.599 
40.435 
30.327 

UD CNT 
10 
1 
14.821 
25.385 
19.693 
16.231 
2 
35.447 
48.617 
41.729 
26.613 

3 
35.447 
48.617 
41.988 
39.901 

4 
51.281 
68.356 
58.392 
45.688 

5 
51.281 
80.715 
61.276 
51.281 

6 
54.536 
81.469 
70.634 
52.291 

FGV CNT 
1 
13.072 
22.675 
17.480 
14.522 

2 
31.421 
43.770 
37.286 
24.286 

3 
31.421 
43.770 
37.506 
35.728 

4 
48.544 
61.817 
54.989 
40.841 

5 
51.607 
73.261 
58.740 
47.475 

6 
51.607 
73.925 
66.358 
51.463 

FGX CNT 
1 
17.462 
28.807 
22.755 
18.437 

2 
40.821 
53.667 
46.995 
29.221 

3 
40.821 
53.667 
47.327 
44.718 

4 
52.541 
74.337 
59.827 
51.538 

5 
52.541 
86.740 
67.900 
52.541 

6 
61.697 
87.636 
72.342 
56.923 

FGO CNT 
1 
11.539 
20.269 
15.529 
12.989 

2 
27.993 
39.597 
33.483 
22.328 

3 
27.993 
39.597 
33.666 
32.133 

4 
43.573 
56.362 
49.763 
37.115 

5 
50.599 
67.156 
57.616 
43.409 

6 
50.599 
67.693 
60.324 
50.599 

UD CNT 
20 
1 
15.191 
27.196 
20.635 
17.515 
2 
37.508 
54.384 
45.439 
32.440 

3 
37.508 
54.384 
45.653 
43.726 

4 
59.307 
78.816 
68.680 
52.991 

5 
73.555 
94.922 
84.049 
61.212 

6 
73.555 
95.513 
84.188 
82.829 

FGV CNT 
1 
13.360 
24.014 
18.187 
15.493 

2 
33.039 
48.163 
40.132 
29.291 

3 
33.039 
48.163 
40.316 
38.650 

4 
52.318 
69.970 
60.776 
47.681 

5 
64.940 
84.374 
74.449 
54.497 

6 
64.940 
84.878 
74.567 
73.396 

FGX CNT 
1 
18.154 
32.079 
24.508 
20.613 

2 
44.493 
63.398 
53.432 
36.467 

3 
44.493 
63.398 
53.705 
51.322 

4 
69.869 
91.060 
80.118 
60.489 

5 
86.290 
109.055 
97.594 
70.417 

6 
86.290 
109.828 
97.777 
96.087 

FGO CNT 
1 
11.730 
21.167 
15.998 
13.676 

2 
29.088 
42.636 
35.429 
26.421 

3 
29.088 
42.636 
35.587 
34.145 

4 
46.187 
62.170 
53.827 
43.147 

5 
57.423 
75.115 
66.048 
48.512 

6 
57.423 
75.538 
66.147 
65.133 

UD CNT 
50 
1 
15.309 
27.836 
20.957 
18.099 
2 
38.207 
56.601 
46.783 
37.284 

3 
38.207 
56.601 
46.977 
45.177 

4 
61.089 
83.332 
71.666 
64.047 

5 
76.174 
101.007 
88.189 
65.523 

6 
76.174 
101.502 
88.306 
87.031 

FGV CNT 
1 
13.454 
24.484 
18.425 
15.936 

2 
33.590 
49.818 
41.154 
33.107 

3 
33.590 
49.818 
41.323 
39.751 

4 
53.744 
73.426 
63.097 
57.518 

5 
67.000 
88.970 
77.621 
57.926 

6 
67.000 
89.398 
77.722 
76.607 

FGX CNT 
1 
18.374 
33.325 
25.123 
21.621 

2 
45.794 
67.599 
55.975 
43.585 

3 
45.794 
67.599 
56.213 
54.016 

4 
73.100 
99.263 
85.566 
72.993 

5 
91.105 
120.248 
105.243 
77.835 

6 
91.105 
120.864 
105.389 
103.834 

FGO CNT 
1 
11.794 
21.483 
16.160 
13.995 

2 
29.465 
43.750 
36.120 
29.314 

3 
29.465 
43.750 
36.267 
34.897 

4 
47.186 
64.568 
55.441 
50.813 

5 
58.819 
78.220 
68.190 
51.869 

6 
58.819 
78.589 
68.277 
67.304 

UD CNT 
100 
1 
15.351 
28.019 
21.055 
18.282 
2 
38.411 
57.180 
47.151 
38.950 

3 
38.411 
57.180 
47.336 
45.557 

4 
61.759 
84.847 
72.712 
66.867 

5 
76.801 
102.422 
89.158 
71.304 

6 
76.801 
102.871 
89.265 
87.978 

FGV CNT 
1 
13.497 
24.654 
18.520 
16.090 

2 
33.788 
50.341 
41.496 
34.421 

3 
33.788 
50.341 
41.655 
40.087 

4 
54.419 
74.875 
64.120 
58.998 

5 
67.573 
90.191 
78.476 
63.552 

6 
67.573 
90.573 
78.568 
77.423 

FGX CNT 
1 
18.429 
33.597 
25.261 
21.900 

2 
46.080 
68.496 
56.522 
46.206 

3 
46.080 
68.496 
56.747 
54.604 

4 
73.957 
101.381 
86.976 
79.822 

5 
92.074 
122.586 
106.803 
82.871 

6 
92.074 
123.144 
106.936 
105.396 

FGO CNT 
1 
11.836 
21.638 
16.248 
14.123 

2 
29.647 
44.211 
36.428 
30.336 

3 
29.647 
44.211 
36.565 
35.185 

4 
47.837 
65.920 
56.410 
51.930 

5 
59.310 
79.232 
68.909 
56.466 

6 
59.310 
79.555 
68.987 
67.970 
The influence that the sidetothickness ratio has on the nondimensional fundamental frequency of FGCNT reinforced plate is illustrated in Fig. (10)(15). For the different boundary conditions of SSSS, CCCC, SCSC, and SFSF, the results are calculated for =0.075. Here, it can be seen that the a/h ratio increases with increasing dimensionless frequency parameters, and it becomes insensitive after
a/h = 50 for all applied boundary conditions.
Overall, from the three stages of the agglomeration effect, without agglomeration stage led to give higher nondimensional frequency as compared to the other two stages because nonuniform CNT distribution in the matrix affects the overall material properties of the nanocomposite. The effect if not considered will propagate the erroneous overall result.
Fig. 10. Variation of dimensionless frequency vs. a/h ratio for different types of CNTreinforced plate with various boundary conditions, including without
agglomeration effect


Fig. 11. Variation of dimensionless frequency vs. a/h ratio for different types of CNTreinforced plate with various boundary conditions for complete agglomeration 



Fig. 12. Variation of dimensionless frequency vs. a/h ratio for different types of CNTreinforced plate with various boundary conditions for complete agglomeration 



Fig. 13. Variation of dimensionless frequency vs. a/h ratio for different types of CNTreinforced plate with various boundary conditions for complete agglomeration 



Fig. 14. Variation of dimensionless frequency vs. a/h ratio for different types of CNTreinforced plate with various boundary conditions for partial agglomeration 



Fig. 15. Variation of dimensionless frequency vs. a/h ratio for different types of CNTreinforced plate with various boundary conditions for partial agglomeration 



Fig. 16. Variation of dimensionless frequency vs. agglomeration parameter ξ for different types of CNT reinforced plate with different boundary conditions for complete agglomeration effect for a/h =5 (Case1) 



Fig. 17. Variation of dimensionless frequency vs. agglomeration parameter ξ for different types of CNT reinforced plate with different boundary conditions for complete agglomeration effect for a/h =10 (Case2) 



Fig. 18. Variation of dimensionless frequency vs. agglomeration parameter ξ for different types of CNTreinforced plate with different boundary conditions for complete agglomeration effect for a/h =20 (Case3) 



Fig. 19. Variation of dimensionless frequency vs. agglomeration parameter ξ for different types of CNT reinforced plate with different boundary conditions for complete agglomeration effect for a/h =50 (Case3). 



Fig. 20. Variation of dimensionless frequency vs. agglomeration parameter ξ for different types of CNT reinforced plate with different boundary conditions for complete agglomeration effect for a/h =100 (Case5) 
In the current work, an investigation into the free vibration behavior of CNTreinforced functionally graded plates, including the effect of agglomeration, was carried out using a C0 FE model that was developed using Reddy's HSDT. It is presumed that the CNT distribution will be uniform or functionally graded all the way through the thickness of the plate. The EshelbyMoriTanaka approach, which is based on a twoparameter model ζ and ξ, is utilized in order to compute the properties of an agglomerated CNTreinforced composite plate at any point. By adjusting these two parameters, it was possible to capture all three stages of the agglomeration effect. Several parametric studies were conducted to determine the effect of reinforcing phase features such as agglomeration and volume fraction distribution along the thickness. These studies examine how these factors affect the dynamic behavior of these structures.
The most important contribution of this work was the introduction of the carbon nanotube agglomeration model into the constitutive rules that define mechanical behavior. In addition, Reddy's wellknown HSDT model is utilized in order to perform an analysis of the free vibrations of plates with varying parameters such as aspect ratio, CNT distribution across the thickness, and three distinct stages of agglomeration. The overall concise outcomes of the present study are as follows:
Nomenclature
CNT 
Carbon nanotube 
FG 
Functionally Graded Materials 
h 
Thickness 
UD 
Uniformly Distributed 

Carbon nanotube volume fraction 
SSSS 
All four edges simply supported 
CCCC 
All four edges clamped 
SCSC 
Two adjacent edges simply supported and the remaining two adjacent edges clamped 
SFSF 
Two adjacent edges simply supported and the remaining two adjacent edges Free 
FGV 
VType CNT distribution pattern along the thickness direction 
FGX 
XType CNT distribution pattern along the thickness direction 
FGO 
OType CNT distribution pattern along the thickness direction 
ζ, ξ 
Agglomeration parameter 
N_{i} 
Shape function 
Acknowledgments
The authors would like to acknowledge Madan Mohan Malaviya University of Technology, Gorakhpur U.P273010 India, for the financial support of this work.
Conflicts of Interest
The corresponding author declares that there are no competing interests on behalf of the other authors.
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