Document Type : Research Paper
Authors
^{1} Mechanical Engineering Department, Shahrekord University, Shahrekord, Iran
^{2} Faculty of Engineering, Shahrekord University, Shahrekord, Iran
^{3} Nanotechnology Research Center, Shahrekord University, 8818634141, Shahrekord, Iran
^{4} Faculty of Mechanical Engineering, Semnan University, Semnan, Iran
Abstract
Keywords
Frequency Analysis of Functionally Graded Carbon Nanotubereinforced Cylindrical Panels by Meshfree Galerkin Method
^{a }Mechanical Engineering Department, Shahrekord University, Shahrekord, Iran
^{ b* }Faculty of Engineering, Shahrekord University, Shahrekord, Iran.
^{c }Nanotechnology Research Center, Shahrekord University, 8818634141, Shahrekord, Iran
^{d }Faculty of Mechanical Engineering, Semnan University, Semnan, Iran
KEYWORDS 

ABSTRACT 
Free vibration Carbon nanotubereinforced composite Meshfree method 
In this work, the threedimensional meshfree (3DMfree) method is used for free vibration analysis of functionally graded (FG) cylindrical panels reinforced by carbon nanotubes. The material properties of panels are considered to be changed linearly in the thickness direction, and the effective material properties of the panels are estimated by the rule of mixture. Five models of carbon nanotubes distribution, including a uniform distributed model and four FG distributed models, are considered. The weak form governing equations of motion are derived using Hamilton’s principle, and the moving least squares (MLS) approximation is used to construct the 3DMfree shape functions in cylindrical coordinates. Various boundary conditions are considered, and effects of boundary conditions, carbon nanotube distribution, the volume fraction of carbon nanotubes, and the panel geometry on the natural frequencies are studied. The results are compared with other results available in the literature, and a close agreement is observed. 
Carbon nanotubes (CNTs) have been widely accepted because of their many mechanical, electrical, and thermal properties and now attracted more attention in engineering applications. Usually, stiff fillers reinforce ordinary composites with macroscale dimensions embedded in a variety of matrixes to improve the mechanical property of the composite system. The discovery of CNTs changed the traditional macroscale filler reinforcements by newly discovered nanoscale reinforcements [1].
Recently, CNTs were considered for use as reinforcements in modern composite structures because of their high elastic properties and strength compared to conventional fibers. For example, Young’s modulus of a CNT is higher than 1 TPa, and its tensile strength is about 150 GPa; it means that they are a hundred times stronger than steel, whereas they are three to five times lighter than it. Some advantages make carbon nanotubereinforced composite used in many structures designed to undergo high mechanical and thermal loads with low weight [2].
Many researchers studied the vibration of conventional composites, but a few studies have utilized the vibration of carbon nanotube reinforced composites (CNTRCs); on the other hand, most of the research deals with the mechanical and thermal properties of CNTRCs.Some research about the vibration of CNTRCs is as follows:
The nonlinear bending analysis of plates made of carbon nanotubereinforced composite was studied by Shen [3]. In this work, the mechanical properties of carbon nanotubes were considered dependent on temperature, and the molecular dynamics simulation was used to compute the mechanical properties. Ke et al. [4] performed nonlinear free vibration analysis of functionally graded carbon nanotubereinforced composite beams. Their work is based on the Timoshenko beam theory, and the von Kármán geometric nonlinearity is considered. They assume that the mechanical property of carbon nanotubereinforced composite beams are varied in thickness direction based on the rule of the mixture, and finally, the Ritz method is used to extract the governing equations. Vibration behavior analysis of functionally graded carbon nanotubereinforced cylindrical panels was studied by Sobhani Aragh et al. [5]. The effective mechanical properties of the composite are derived with Eshelby–Mori–Tanaka's approach and the twodimensional generalized differential quadrature method (GDQM) was used to solve the governing equations and apply the corresponding boundary conditions. Wang and Shen [6] studied the nonlinear bending and vibration analysis of a sandwich plate; the sandwich plate was made of carbon nanotubereinforced composite face sheets, and their mechanical properties change along the thickness direction. Dynamic investigation of nanocomposite cylinders reinforced by singlewalled carbon nanotubes through the Mfree method conducted by MoradiDastjerdi et al.[7]. They used moving least squares (MLSs) shape functions to approximate the displacement field in an axisymmetric model. Lin and Xiang [8] studied free vibration analysis of carbon nanotubereinforced composite beams based on the first and thirdorder beam theories. They used both uniform and functionally graded distribution of carbon nanotube reinforcement. The Ritz method was employed to extract the natural frequencies, and they concluded that FGXCNT beams in all conditions exhibit the highest natural frequencies, and the beams with the FG_CNT model of reinforcements give the lowest natural frequencies. Free vibration analysis of carbon nanotubereinforced polymer composite structures with various geometries is carried out by Mehar et al. [9]. They used a generic higherorder shear deformation theory, and Hamilton’s principle derives the equations of motion of the structures, and finally, the solution of equations was performed by the finiteelement method. Karami et al. [10] presented an analytical study on functionally graded carbon nanotubereinforced composite plates. Their work is based on secondorder shear deformation theory, and also they included the size effects with nonlocal elasticity theory in their analysis. Vibration analysis of FGCNTRC in different geometries such as circular, annular, and sector plates is studied by Zhon et al. [11]. Firstorder shear deformation theory was used, and then the natural frequencies and corresponding mode shapes were extracted by the Ritzvariational method. The free vibration frequency of composite plates reinforced with singlewalled carbon nanotubes by using a refined simplified twovariable nthhigherorder theory is performed by Bouazza and Zenkour [12]. They used an exact closedform formulation based on this theory for solving the equations.
Bending analysis of nano/microbeams under the concentrated and distributed loads is investigated by Demir and Civalek [13]. They used Euler Bernoulli beam theory via the enhanced Eringen differential model. Also, the singularity function method is used to calculate the deflection of the concentrated and distributed loaded beams.
Dynamic instability of viscoelastic porous functionally graded nanobeam embedded on viscoPasternak medium subjected to an axially oscillating loading and the magnetic field is performed by Jalaeiand Civalek [14]. They employed Eringen’s differential law in conjunction with Timoshenko beam theory and Hamilton’s principle to derive equations of motion. Finally, Navier’s solution and Bolotin’s approach are utilized to obtain the dynamic instability region of viscoelastic porous FG nanobeam.
Civalek et al. [15] studied free vibration analysis of carbon nanotube‐reinforced composite microbeams. In their work, carbon nanotubes are distributed in a polymeric matrix with four different reinforcement patterns, and material properties of the CNTRC microbeams are predicted by using the rule of mixture. Hamilton's principle derives the microstructure‐dependent governing differential equations based on the couple stress theory. Finally, the obtained vibration equation is solved by using Navier's solution method.
Buckling and free vibration of rectangular polymeric laminate reinforced by graphene sheets are investigated by Karimi Zeverdejani and Tadi Beni [16]. In this paper, various patterns are considered for the augmentation of each laminate. Critical buckling load is evaluated for different parameters, including boundary conditions, reinforcement pattern, loading regime, and laminate geometric states.
Buckling and postbuckling of graphenereinforced laminated composite plates subjected to uniaxial and biaxial loadings are investigated by Karimi Zeverdejani et al. [17]. This paper uses polymethylmethacrylate for the matrix, and three patterns are considered for the plate crosssection. Graphene sheets are considered in both perfect and defective forms. The free vibration of polymer nanocomposite reinforced by graphene sheet is investigated by Karimi Zeverdejani and Tadi Beni [18]. In this work, the new sizedependent formulation is presented for nanocomposites based on the couple stress theory. For this purpose, the first shear deformation theory is applied. The effect of the scale parameter is investigated based on anisotropic couple stress theory. It is observed that graphene defects cause to diminish the lamina frequency.
Threedimensional (3D) vibration analysis of FG carbon nanotube reinforced cylindrical panel with the meshfree method has not been studied to the best of the authors’ knowledge. Also, there are two main shortcomings and limitations in many works performed before: twodimensional modeling using shell theories and limitations in modeling boundary conditions. So the major aim of the present work is a threedimensional free vibration study of FG carbon nanotube reinforced cylindrical panels under various boundary conditions using the Mfree method. The volume fraction of carbon nanotubes, the geometry of the panel, and boundary conditions are the parameters that their effects on natural frequencies are studied in this paper.
For validation of the 3DMfree approach, the results were compared with the available results in the literature for various material properties such as homogeneous, isotropic, and fiberreinforced cylindrical panels.
2.1. The material properties of CNTs
A carbon nanotube reinforced cylindrical panel of radius R_{a}, with dimension L × S, different boundary conditions at the edges = 0, and Also at the edges z= 0, z=L is shown in Fig. 1. The effective mechanical property of CNTRC cylindrical panels are obtained as [7,19]

(1) 
where, , , , and are elasticity modules, shear module, Poisson's ratio and mass density of CNTs, respectively and , , and are those of matrix respectively. are the CNTs efficiency that determined by matching the elastic modulus of CNTRCs obtained by the MD simulation results with the numerical results are obtained from the rule of mixture.
Fig. 1. The design scheme of the CNTRC cylindrical panels
Furthermore, are CNTs and matrix volume fractions and defined as:

(2) 
Five profiles of linear variation of CNTs volume fraction in the thickness direction are considered (Fig. 2) and defined as:

(3) 

(4) 

(5) 

(6) 

(7) 
where

(8) 
where, is the mass fraction of CNTs.
The Poisson’s ratio is considered to change according to reference [3], i.e.

(9) 
2.2. Governing equations
Hamilton’s principle is used for extracting the weak forms of governing equations. In the absence of external works, Hamilton’s principle for free vibration analysis is indicated as below [20]:

(10) 
where T is kinetic energy, refers to elastic strain energy. More in detail, as follows:

(11) 

(12) 
In the above relations, , , , , and are stress vector, strain vector, displacement vector, velocity vector, and mass density, respectively. Also is the whole volume of the continuum body. Using Eqs. (11) and (12), in Eq. (10), the equation of motion can be expressed as weak form by the following relation:

(13) 
in which, ü is acceleration vector.
Fig. 2. Five profiles of linear variation of CNTs volume fraction in the thickness direction
For a linear elastic body, the general Hook’s law is defined as follow:

(14) 
where D for an orthotropic body [20] is as follows:

(15) 
In which:

(16) 
Stress and strain vectors in cylindrical coordinates are as follows:

(17) 
The meshfree Galerkin method based on moving least squares approximation (MLS) is used in this work. In this method, the field variable u(X) in the domain at an arbitrary point in the cylindrical coordinate is approximated as follows [21, 22, 23]:

(18) 
where , and m are the base vector, the variable coefficients vector, and the number of components of the base vector, respectively. In 3D space in cylindrical coordinate, a complete base vector basis of order s is given by:

(19) 
Furthermore, is a general function that, in MLS approximation, at an arbitrary point , is considered to minimize the weighted residual.
The weighted residual is defined as:

(20) 
in which, n, and are the number of nodes in the support domain of the point , the weight function, and virtual nodal value at the node , respectively. By minimization the weighted residual:

(21) 
is derived as follow:

(22) 
where is called the moment matrix given by:

(23) 
and

(24) 
is virtual nodal value vector and defined as follow:

(25) 
By substituting Eq. (22) in Eq. (18), we have:

(26) 
is MLS shape function of node located at and expressed as follow:

(27) 
Now by using the MLS approximation, the displacement vector in cylindrical coordinates can be approximated as follows:

(28) 
where:

(29) 
and



(30) 
The strain–displacement relationship is expressed in the matrix form as:

(31) 
in which , the strain matrix is given as:

(32) 
In the absence of body forces and surface tractions, using Eqs. (13), (28), and (31) in Eq. (14) leads to:



(33) 
Equation (33) should be satisfied for every arbitrary . Therefore, we have:

(34) 
or

(35) 
where:

(36) 
and

(37) 
N is the total number of nodes. Both of these integrals can be obtained simply by numerical integration. For numerical integration, the panel domain is first discretized by several cells, and the Gauss integration scheme is used for each cell; finally, the global stiffness matrix and the mass matrix are obtained numerically by sweeping all gauss integration points inside the domain . For performing the Gauss integration, each cell is mapped into a reference cell with preappreciated Gauss integration points and their corresponding weights. By mapping relations, the corresponding Gauss integration points in the cell of the real domain are derived, and integration is performed using the mapped Gauss integration points.
For imposing the essential boundary conditions, the global stiffness and mass matrixes are modified by transformation method as follow:

(38) 
in which

(39) 
After modification of stiffness and mass matrixes, the essential boundary conditions are enforced on the modified matrixes exactly like the finite element method. In the modified stiffness and mass matrixes, the row and columns corresponding to the degree of freedoms are removed that we want to be constrained.
In this section, the convergence results are obtained to find the sufficient node numbers and distribution in the problem domain and its boundaries for extracting the natural frequencies. So, the fundamental natural frequency parameter for a homogeneous isotropic cylindrical panel with (SSSS) boundary conditions and various geometrical parameters is computed and shown in Table 1.
Normalized fundamental frequencies ( ) for an isotropic FGM cylindrical panel (L/R_{a}=0.5, L/t=2) for various values of powerlaw index n are presented in Table 2 and compared with The semianalytical solution by Zahedinejad et al. [24] and a 2D higherorder deformation theory solution by Matsunaga [25].
The material properties in the thickness direction are as bellow:
For this simulation, dimensionless first natural frequencies for various thickness to radius ratios and span angles are extracted and listed in Table 3. The comparison is made in this table with the semianalytical results presented by Zahedinejad et al. [24].
Furthermore, the FG fiberreinforced cylindrical panel results were obtained and compared with semianalytical solutions through the differential quadrature method by Yas et al. [29] in Table 4. The fiber and matrix in this example are Tungsten ( ) and Cupper ( ), respectively, and the volume fractions of the constituent vary according to the powerlaw distribution from (%100 Cu) at inner to (%25 Cu & %75 W) at outer surfaces. Tables 1, 2, 3, and 4 show an excellent agreement between the results of the present model and other results in the literature. In addition, Table 1 shows that the proposed model has a good convergence rate.
Finally, in this section, the comparison is made for an FG carbon nanotube reinforced panel performed by Yas et al. [30].
In this example, the generalized differential quadrature method derived the normalized natural frequencies of an FG carbon nanotube reinforced panel with SSSS boundary conditions. The frequency parameter is defined as follows:
Table 1. Fundamental natural frequency parameter for various node distributions, (L/R_{a}=0.1, L/t=10 and ν=0.3, S=R_{a}α)
Node distribution 
S /L 


0.5 
1 
1.5 

30×3×25 
1.3175 
0.5560 
0.4105 
40×4×30 
1.3172 
0.5549 
0.4090 
45×5×35 
1.3171 
0.5546 
0.4085 
50×6×40 
1.3170 
0.5544 
0.4083 
[26] 
1.3360 
0.5563 
0.4044 
[27] 
1.31742 
0.55049 
0.39987 
[28] 
1.31597 
0.55136 
0.40266 
Table 2. Fundamental frequency parameters for an (SSSS) FG panel

n 






0 
0.5 
1 
4 
10 

[22] 
0.9334 
0.8213 
0.7483 
0.6011 
0.5460 
0.4752 
[21] 
0.9187 
0.8013 
0.7260 
0.5797 
0.5245 
0.4770 
Present method 
0.9205 
0.8017 
0.7272 
0.5841 
0.5322 
0.4685 
Table 3. Comparison fundamental frequency parameter for an (SSSS) FG panels, (R_{a}/L=1)

t/R_{a} 









0.3 
0.5 
0.3 
0.5 
[21] 
0.3113 
0.6731 
0.2279 
0.3768 
Present method 
0.3119 
0.6750 
0.2265 
0.3746 
Table 4. Comparison of fundamental frequency parameter for an FG fiberreinforced cylindrical panel (L/R_{a}=1, α=π/6, n=1)

R_{a}/t 









20 
40 
60 
80 
100 
120 
140 
160 
180 
[26] 
0.7530 
0.4340 
0.3370 
0.2930 
0.2740 
0.2680 
0.2620 
0.2600 
0.2450 
Present 
0.7309 
0.4259 
0.3439 
0.3079 
0.2923 
0.2824 
0.2762 
0.2720 
0.2692 
Fig. 3. Comparison of frequency parameter for a CNT reinforced panel (R_{a}/t=100, , FGV)
Fig. 4. Comparison of frequency parameter for a CNT reinforced panel (R_{a}/t=100, , FGΛ)
Figures 3 and 4 show the variation of frequency parameters with length to mean radius ratio for an FG – CNT panel for different span angles. It should be noted that the results obtained by the present work are compared with the results of Yas et al. [30]. From figures 3 and 4, it is observed that the present work results are very close to those of Yas et al. [30].
After validation of numerical results, CNTRC panels are studied by the presented 3DMfree simulation. A CNTRC panel is made of Polymethylmethacrylate (PMMA) like matrix, with CNT as fibers aligned in the axial direction. PMMA is an isotropic material with GPa, kg/m^{3 }and . The (10,10) SWCNTs are selected as reinforcements. The material properties of SWCNT are TPa, TPa, TPa, kg/m^{3,} and [7]. In this work, the CNT efficiency parameters have been estimated by matching Young’s moduli E_{11} and E_{22} of CNTRCs obtained by the rule of the mixture to those from the MD simulations and are listed in Table 5. Also, the third efficiency parameter is [7].
The natural frequency parameter is defined as:
Fundamental frequencies of the panel for different span angles are listed in Table 6. This table shows that the maximum and minimum fundamental frequencies are obtained for FGX and FG◊ distribution of CNTs. In the former type, the volume fraction of CNT is zero at the mid surface of the panel and reaches its maximum value at the inner and outer surfaces on which maximum axial deformations occur, while for the latter, one maximum value of CNT volume fraction occurs at the mid surface of the panel, it may justify results obtained from this table. This table also shows that the fundamental frequency of the panel increases slightly with the increase in span angle. Table 7 depicts the effect of the geometry factor, t/R_{a,} on the first principal frequency of the panel. This table reveals that the fundamental frequency increases with the increase in t/R_{a}. An increase in this factor means getting thicker and, consequently, stiffer. It can justify the increase of fundamental natural frequency with the increase in t/R_{a}.
Table 5. Elasticity moduli of PMMA/CNT composites by the rule of the mixture and MD simulation
MD 

Rule of mixture 


E_{11} (GPa) 
E_{22} (GPa) 
E_{11} (GPa) 
E_{22} (GPa) 

0.12 
94.6 
2.9 
94.78 
0.137 
2.9 
1.022 
0.17 
138.9 
4.9 
138.68 
0.142 
4.9 
1.626 
0.28 
224.2 
5.5 
224.50 
0.141 
5.5 
1.585 
Table 6. First natural frequency parameter for various angle of CCFF panel, α. (t/R_{a}=0.1,L/R_{a}=1, )

UD 
FGX 
FGΛ 
FGV 
FG◊ 
α=π/6 
1.9021 
1.9576 
1.8004 
1.7922 
1.7330 
α=π/4 
1.9086 
1.9629 
1.8101 
1.8023 
1.7350 
α=π/3 
1.9088 
1.9637 
1.8118 
1.8022 
1.7345 
α=π/2 
1.9112 
1.9656 
1.8140 
1.8054 
1.7383 
Table 7. First natural frequency parameter for panel with various t/R_{a} ratios. (L/R_{a}=2, α=π/4, )

h/R_{m} 
UD 
FGX 
FGΛ 
FGV 
FG◊ 
CCFF 
0.01 
0.3451 
0.3822 
0.3215 
0.3201 
0.3057 

0.05 
1.0678 
1.2119 
0.9333 
0.9361 
0.8453 

0.1 
1.5717 
1.6851 
1.4234 
1.4297 
1.3269 

0.2 
1.9049 
1.9588 
1.7917 
1.8116 
1.7310 







CFFF 
0.01 
0.1206 
0.1249 
0.1194 
0.1195 
0.1177 

0.05 
0.2608 
0.2883 
0.2403 
0.2414 
0.2287 

0.1 
0.3894 
0.4454 
0.3412 
0.3445 
0.3151 

0.2 
0.6161 
0.6890 
0.5375 
0.5444 
0.4926 
The effects of boundary conditions on the natural frequency parameters are presented in Table 8. According to this table, panels with CCCC and CFFF boundary conditions have the highest and lowest natural frequencies, respectively, as expected. These boundary conditions cause maximum and minimum stiffness values among all boundary conditions listed in this table. Finally, the effects of the volume fraction of CNTs for two boundary conditions are presented in Table 9. This table indicates that allnatural frequencies rise with the increase in volume fraction of CNT. CNT is much stiffer than the matrix material. It justifies the increase of natural frequencies with an increase in volume fraction of CNT. Additionally, it is obvious that for both types of boundary conditions presented in this table, maximum and minimum values of fundamental frequencies deal with FGX and FG◊ distribution of CNT, but it is not the case for other frequencies.
Table 8. First natural frequency parameter for panel with various boundary conditions. (L/R_{a}=1,t/R_{a}=0.1,α=π/3, )

UD 
FGX 
FGΛ 
FGV 
FG◊ 
CCCC 
2.3390 
2.3987 
2.2745 
2.2596 
2.1998 
CCFF 
1.9077 
1.9637 
1.8022 
1.8118 
1.7345 
SSSS 
1.5852 
1.6958 
1.4460 
1.4775 
1.3724 
SSFF 
1.4814 
1.6039 
1.3284 
1.3374 
1.2308 
FFCC 
1.3458 
1.3740 
1.3609 
1.3554 
1.3410 
CFFF 
0.6431 
0.7132 
0.5720 
0.5771 
0.5270 
Table 9. First, five natural frequency parameters for panel with various ν^{*}. (L/R_{a}=1, t/R_{a}=0.1, α=π/3)








FFSS 

UD 
0.2334 
0.3238 
0.9982 
1.1283 
1.4188 


FGX 
0.2413 
0.3344 
1.0275 
1.1605 
1.4244 


FGΛ 
0.2353 
0.3251 
1.0055 
1.1349 
1.4290 


FGV 
0.2343 
0.3251 
1.0020 
1.1325 
1.4179 


FG◊ 
0.2277 
0.3156 
0.9771 
1.1036 
1.4226 










UD 
0.3016 
0.4184 
1.2895 
1.4577 
1.8830 


FGX 
0.3177 
0.4400 
1.3495 
1.5236 
1.8481 


FGΛ 
0.3056 
0.4216 
1.3059 
1.4726 
1.8574 


FGV 
0.3036 
0.4215 
1.2983 
1.4676 
1.8356 


FG◊ 
0.2907 
0.4030 
1.2497 
1.4129 
1.8449 










UD 
0.3160 
0.4383 
1.3510 
1.5271 
1.9209 


FGX 
0.3537 
0.4889 
1.4909 
1.6812 
1.9770 


FGΛ 
0.3262 
0.4487 
1.3927 
1.5688 
1.9960 


FGV 
0.3212 
0.4478 
1.3736 
1.5548 
1.9512 


FG◊ 
0.2955 
0.4103 
1.2757 
1.4434 
1.9706 








CCSS 

UD 
1.5336 
1.7017 
1.8165 
2.6248 
3.0612 


FGX 
1.5741 
1.7071 
1.8641 
2.6837 
3.1335 


FGΛ 
1.4603 
1.7022 
1.7627 
2.6038 
2.9348 


FGV 
1.4658 
1.7117 
1.7673 
2.6046 
2.9296 


FG◊ 
1.4093 
1.7068 
1.7070 
2.5409 
2.8381 










UD 
1.9605 
2.1985 
2.3298 
3.3806 
3.9188 


FGX 
2.0127 
2.2143 
2.4023 
3.4864 
4.0108 


FGΛ 
1.8676 
2.2045 
2.2664 
3.3669 
3.7617 


FGV 
1.8785 
2.2235 
2.2744 
3.3671 
3.7635 


FG◊ 
1.8070 
2.1883 
2.2139 
3.2617 
3.6519 










UD 
2.1056 
2.3041 
2.4821 
3.5656 
4.1968 


FGX 
2.1507 
2.3675 
2.5906 
3.7800 
4.2903 


FGΛ 
2.0530 
2.3460 
2.4637 
3.6121 
4.1161 


FGV 
2.0685 
2.3871 
2.4682 
3.5991 
4.1328 


FG◊ 
2.0225 
2.3665 
2.3776 
3.4352 
4.0644 
Treedimensional frequency analysis of FG carbon nanotubereinforced composite cylindrical panels was carried out by 3DMfree simulation. The Galerkin weak forms of governing equations with the platform of the Mfree method are constructed using the Hamilton principle.
The present results are compared with the same results in other works, and it is concluded following results:
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