Free Vibration Response of Agglomerated Carbon Nanotube-Reinforced Nanocomposite Plates

1. C. Maurya ^a* , S. M. A. Jawaid ^a , A. Chakrabarti ^b

^a Department of Civil Engineering, Madan Mohan Malaviya University of Technology, Gorakhpur-273010, India

^b Department of Civil Engineering, Indian Institute of Technology, Roorkee-247667, India

1.     Introduction

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 nanotube-reinforced composites (FG-CNTRCs) have received a lot of interest in recent years due to their exceptional mechanical properties. FG-CNTRC 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 CNT-reinforced 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 CNT-reinforced 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 two-parameter 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 Mori-Tanaka 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 single-walled 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 CNT-reinforced 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 Levy-type 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 two-parameter micromechanics model of agglomeration. In this research, an analogous continuum model based on the Eshelby-Mori-Tanaka 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 higher-order 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 higher-order 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 quasi-3D models that take into account the effect of thickness stretching [17]. Zhang et. al. [9] used both the element-free IMLS-Ritz technique and first-order 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 thickness-to-width ratio, the plate's aspect ratio, and the boundary condition. Mareishi et. al. [5] studied the nonlinear free and forced vibration behavior of advanced nano-composite 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 nano-composite beams and discovered this. Tornabene et. al. [18] looked at how agglomerated CNT affected the free vibration behavior of laminated composite plates and double-curved shells. They used Carrera Unified Formulation (CUF), which is a method that permits the consideration of multiple Higher-order Shear Deformations Theories (HSDTs). Kiani [19] looked at how CNT-based nano-composite 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 in-plane and out-of-plane 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 eight-unknown shear deformation theory was developed by Nguyen et. al. [20] for the bending and free vibration study of FG plates using the finite-element 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 12-unknown higher-order 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 power-law indices and side-to-thickness 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 fiber-reinforced 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 hole-drilling 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 size-dependent CNTRC nanoplates on a visco-Pasternak foundation. Maleki et al. [23] solved the free vibration problem of three-phase 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 in-plane stresses and also investigated the effect of in-plane forces on the vibration behavior of carbon nanotube (CNT) reinforced composite skew plates using first-order 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 higher-order shear deformation theory (HSDT) in combination with element-free IMLS-Ritz 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 carbon-nanotube-reinforced sandwich curved shell panels under the elevated thermal environment using the higher-order shear deformation theory. Mehar et. al. [38-40] have done extensive theoretical and experimental investigations of vibration characteristics of carbon-nanotube reinforced polymer composite structures.

In conclusion, from the above-detailed 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 low-volume fraction distribution. Hence, if the above-mentioned 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 Eshelby-Mori-Tanaka method.

2.     Material Modelling

For any structural analysis, material modeling is very important. The application of CNT-reinforced 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 FG-CNTRC is presented using the Mori-Tanaka method, considering the effect of agglomeration of CNT for various types of CNT distributions.

2.1. Material Modeling of FG-CNTRC

The FG-CNTRC 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 FG-V) 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:

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) V-Shape distributed CNT nanocomposite plate, (c) X-Shape distributed CNT nanocomposite plate, (d) O-Shape distributed CNT nanocomposite plate.

2.2. Modeling of Nanocomposite Material:

Several micromechanical models have been proposed to predict the properties of the material of CNT-reinforced 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 Mori-Tanaka’s [6] concept of average stress inside the matrix. Benveniste’s [9] revision of the effective modulus of elasticity tensor C of CNT-reinforced composites is as follows:

The symbol I is denoted as a fourth-order 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:

here symbol S represents the Eshelby tensor of the fourth order, as defined by Mura and Eshelby [8,10].

Here, a single-walled carbon nanotube having a solid cylinder of 1.424 nm diameter with (10,10) chirality index [11] is used for the analysis.

2.2.1.        Randomly Oriented CNT-Reinforced Composites:

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:

where g is given as:

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.

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

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]:

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:

The term K_m and G_m are used for bulk and shear moduli of the matrix, respectively.

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

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:

where  and represents the mass density of matrix and carbon nanotubes, respectively.

2.2.2.        Agglomeration of Carbon Nanotubes

A large proportion of carbon nanotubes in carbon nanotube-reinforced 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 FG-CNTRC without taking into account the CNT agglomeration effect, a new micromechanics model is developed and applied to a random oriented, agglomerated CNT-reinforced polymer composite to determine the effective properties of the material of a single-walled CNT reinforced polymer composite while taking into account the CNTs bundling effect. The influence of agglomeration on the elastic characteristics of CNT-reinforced composites having random orientation is investigated in the present study, which uses a two-parameter 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.

• Two Parameter Agglomeration Model

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 CNT-reinforced composites. Here, a micromechanical model has been built to check the CNTs agglomeration effect on the effectiveness of carbon nanotube-enhanced 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:

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.

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)_.

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

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 sphere-shaped 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 Mori-Tanaka 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:

Following that, the composite's effective bulk modulus K and effective shear modulus G are computed using the method of Mori-Tanaka as follows:

where,

Finally, the CNT-reinforced composite’s young modulus is calculated using Eq. (16).

3.     Formulation

3.1. In-Plane Displacement Fields and Strains

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 in-plane displacement variation of u, v, and displacement in transverse direction w across the plate thickness may be described as using Reddy's theory of higher-order shear deformation [15].

where ,  and  signify the displacement of a point along the (x, y, z) coordinates located at mid-plane, 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:

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

where  represents the mechanical strain.

Again, in terms of total strain, the mechanical strain may be represented as

while  is the thickness coordinates-z function, and  is the function of x and y.

Fig. 7. Geometry of the FGCNT Plate

This describes the overall strain as,

3.2. Constitutive Relationship

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

where constitutive matrix

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).

3.3. Virtual work in FGCNT plate

The FGM plate’s virtual work may be represented as

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

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

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:

Finally, we can rewrite Eq. (41) as

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

4.     Finite Element Formulation

4.1. Element Description

Figure 8 illustrates the isoparametric Lagrangian element’s geometry with nine nodes used in the analysis. In this element, there is a total of sixty-three degrees of freedom because each node has seven degrees of freedom (u, v, w, , , and ). In the x-y 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. Nine-noded Iso-parametric element with node numbering

For the present nine-node element the shape functions used are given below,

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:

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

4.2. Governing Equation for The Analysis of Free Vibrations

Mid-surface displacement parameters (u_o, v_o and w_o) can be used to calculate acceleration at any location within the element, as

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 .

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

where the matrix [L] expression can be represented as

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

5.     Numerical Results & Discussion

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. 4-6. 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/m³, and Table 1 lists the material characteristics of the reinforcement. The UD, FG-V, FG-X, and FG-O 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 Eshelby-Mori-Tanaka 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 self-explanatory, 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. 4-6.

Fig. 9. Young’s modulus for different levels of agglomeration and CNT volume fraction

Table 1. Hill’s elastic moduli for Single-Walled Carbon Nanotubes (SWCNT) [31].

Table 2. First six natural frequencies in Hz for isotropic plate (L = 0.6 m, B = 0.4 m) [16].

5.1. Validation Study-Isotropic Plate Under Free Vibration Case

A convergence study was carried out for free vibration analyses of agglomerated CNT-reinforced 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 FG-CNT-reinforced 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 FG-CNT-reinforced plates. The outcomes of the free vibration analyses for an isotropic square plate are presented in Table 1 (E = 70 GPa, ρ = 2700 kg/m³, 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].

5.2.  Validation Study- FGM Plate Under Free Vibration Effect

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/m³;
ρ_m = 2707 kg/m³.The non-dimensional 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 higher-order 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)

5.3. New Results for the Effect of The Agglomeration on FG-CNTRs Square Plates Free Vibrations Behavior

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 three-agglomeration stage. Different levels of agglomeration were tested on UD, FG-V, FG-X and FG-O type of carbon nanotube distribution.

Table 4. Convergence study for the dimensional frequency of an agglomerated CNT-reinforced plate with simply supported boundary conditions.

The dimensionless frequencies used in this study were obtained using the following expressions:

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 CNT-reinforced 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 non-dimensional 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 non-dimensional frequencies for the UD, FG-V, FG-X, and FG-O 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, FG-V, FG-X, and FG-O distributions, respectively. Additionally, it was discovered that the all-side-clamped plate produces the maximum frequency parameter whereas the SFSF produces the least frequency parameters.

This is because the stiffer agglomerated CNT-reinforced 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 two-agglomeration parameter and .

5.3.1.          Free Vibration Analysis Without Agglomeration Effect

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 FG-X 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 non-dimensional frequency also increases for all types of CNT distribution patterns considered in this study. But overall, one can observe that the FG-X 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 FG-CNT-reinforced plate
with different boundary conditions ( ).

5.3.2.          Free Vibration Analysis Under Complete Agglomeration Effect

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, FG-V, FG-X, and FG-O are listed in Table 6 – 8. From this table, one can conclude that for all cases of complete agglomeration observed, the FG-O 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 FG-X 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 non-dimensional natural frequencies for FG-CNT-reinforced plate with a full agglomeration effect
with different boundary conditions .

Table 7. The first six non-dimensional natural frequencies for FG-CNT-reinforced plate with a full agglomeration effect
with different boundary conditions .

Table 8. The first six non-dimensional natural frequencies for FG-CNT-reinforced plate with a full agglomeration effect with different boundary conditions .

5.3.3.          Free Vibration Analysis Under Partial Agglomeration Effect

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, FG-V, FG-X, and FG-O 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 CNT-reinforced 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 non-dimensional natural frequencies for FG-CNT-reinforced plate with a partial agglomeration effect under different boundary conditions. .