Document Type : Research Paper
Authors
University of Kashan, Kashan, Iran
Abstract
Keywords

Mechanics of Advanced Composite Structures 6 (2019) 105 – 115


Semnan University 
Mechanics of Advanced Composite Structures journal homepage: http://MACS.journals.semnan.ac.ir 
Discrepancies Between Free Vibration of FML and Composite Cylindrical Shells Reinforced by CNTs
M. Mohandes, A.R. Ghasemi^{*}
Department Solid Mechanics, University of Kashan, 8731753153, Kashan, Iran
Paper INFO 

ABSTRACT 
Paper history: Received 20180216 Received in revised form 20180528 Accepted 20190425 
In this study, discrepancies between the free vibration of fibermetal laminate (FML) and composite cylindrical shells reinforced by carbon nanotubes (CNTs) based on Love’s first approximation shell theory have been considered by beam modal function model. The representative volume elements consist of three and four phases for composite and FML structures, respectively, which include fiber, CNTs, polymer matrix and metal for FML cylindrical shells while the metal section is ignored for composite cylindrical shells. The modulus of carbon nanotubes reinforced composites cylindrical shell could be defined based on rule of mixture. In addition, the fiber phase can be reinforced by the obtained matrix using the extended rule of mixture. The frequencies of FML and composite cylindrical shells reinforced by CNTs have been compared to each other for different materials, layups, boundary conditions, axial and circumferential wave numbers. 



Keywords: Free vibration FML cylindrical shells Composite cylindrical shells CNTs Beam modal function model 


© 2019 Published by Semnan University Press. All rights reserved. 
Due to the need for light structures with high strength and stiffness in the modern engineering, the composite structures are utilized more than the heavy metallic ones. Therefore, in the past decades, the applications of composite materials have increased in most advanced engineering fields such as aerospace and mechanical engineering and especially the automobile engineering structures. One of the most important subjects for the composite structures is the vibration analysis of various composite structures such as beams [1], plates [2] and specifically cylindrical shells [3 12]. Zhang [13] obtained the natural frequencies of laminated composite cylindrical shells subjected to different boundary conditions using wave propagation method. He found that the effect of boundary conditions in the small circumferential modes was more than the large ones. Lee et al. [14] studied vibration of laminated composite cylindrical shells with an interior plate. Malekzadeh et al. [15] investigated the free vibration of laminated cylindrical shell based on threedimensional elasticity theory. A mixed layerwise theory was used in their work to derive the equations of motion while the differential quadrature method (DQM) was utilized to solve them. Three dimensional state equations with DQM were used to analyze the static and free vibration of unsymmetric laminated composite cylindrical shell subjected to different boundary conditions by Alibeigloo [16].
Although when compared to metallic structures, composite structures have good characteristic, the composites have deficient behavior which could be modified by fiber metal laminates (FMLs). FMLs, which are socalled hybrid structures, have good characteristics of the metal such as ductility, impact and damage tolerances as well as the benefits of the fiber composite materials such as high strength and stiffness to weight ratios, excellent fatigue resistance and acceptable corrosion resistance. It was found that the fatigue crack growth rates in adhesive bonded sheet materials can be reduced, if they are built up by laminating and adhesively bonding thin sheets of them, instead of using one thick monolithic sheet [17]. Botelho et al. [18] investigated the damping behavior of the FMLs including different materials based on VoigtKelvin model. Shooshtari and Razavi [19] used multiple time scale method to study the linear and nonlinear vibration of crossply laminated composite and FML rectangular plates subjected to simply supported boundary condition. Rahimi et al. [20] studied the free vibration of FML annular plate with a central hole based on the threedimensional elasticity theory. They determined the natural frequencies of the plate with different boundary conditions using the combination of DQM, stateapace and Fourier series. The DQM was used to consider the dynamic response of geometrically nonlinear FML Timoshenko beam under unsteady temperature field by Fu et al. [21]. Fu and Tao [22] studied the nonlinear dynamic responses of FML Timoshenko beams subjected to thermal shock using the DQM. In their research, the influences of thermal shock, geometric nonlinearity and the conditions of viscoelasticity on the dynamic responses of the FML beams were considered. Zarei et al. [23] studied the dynamic response of glassreinforced aluminum laminate (GLARE) 53/2 plate based on higherorder shear deformation theory subjected to low velocity impact. With increasing the temperature, indentation value increased and contact force decreased. In addition, with dropping clamped edges, the FML plate became more flexible and also the energy absorption and indentation value grew. Tao et al. [24] investigated the nonlinear dynamic behavior of FML EulerBernoulli beams based on vonKarman assumption under moving loads in thermal environment. The effects of temperature, geometric nonlinearity, material parameters and velocity of the moving loads on the dynamic responses of the FML beam were considered. Moniri bidgoli and HeidariRarani [25] studied the buckling behavior of FML cylindrical shells based on firstorder shear deformation theory subjected to axial compression using analytical and numerical methods for simply supported boundary conditions. They found out that the effect of different layups on the buckling loads of FML cylindrical shells is considerable.
In the recent years, carbon nanotubes (CNTs)reinforced composite materials have attracted considerable attention for usage in different industries. Addition of CNTs to a matrix can significantly amend thermal, mechanical and electrical behaviors. Also, substituting the carbon fibers by CNTs can improve the composite properties such as tensile strength and elastic modulus. Khorshidi et al. [2627] considered the vibration and buckling of functionally graded (FG) rectangular nanoplate using nonlocal elasticity subjected to simply supported boundary condition based on exponential shear deformation theory. Emami et al. [28] considered free vibration of lattice cylindrical shells reinforced by CNTs based on firstorder shear deformation theory (FSDT). They found that with adding the CNTs, the natural frequencies increased. Thomas and Roy [29] investigated the vibration and damping of functionally graded carbon nanotubereinforced hybrid composite (FGCNTRHC) shells. In another research, Thomas and Roy [30] studied the vibration of uniform and functionally graded carbon nanotubereinforced composite (FGCNTRC) shells including , , and distributions. The results indicated that the volume fraction and distribution of CNTs influenced all the elastic properties of the composites. Dastjerdi et al. [31] studied static and free vibration of FG plates reinforced by wavy CNTs rested on Pasternak elastic foundation based on FSDT using developed meshfree method. The results showed that the developed meshfree method had an excellent convergence and accuracy to consider static and free vibration of the plate. Ansari et al. [32] applied variation differential quadrature method (VDQM) for vibration analysis of various FGCNTRC spherical shells rested on the elastic foundation subjected to different boundary conditions. They found out that the greatest value of the fundamental frequencies occurred for distribution. Furthermore, the results indicated that with increasing the thicknesstoradius ratio, the nondimensional frequencies decreases remarkably. Dastjerdi and MalekMohammadi [33] investigated the vibration and buckling of FG nanocomposite plate reinforced by CNTs having simply supported boundary condition. They demonstrated that both frequency and critical buckling loads increased by increasing (volume fraction of cluster with respect to total volume of representative volume element) and (length to thickness ratio) and decreasing (volume ratio of the CNTs inside the clusters over the total CNTs inside the representative volume element) and volume fraction of CNTs.
In this research, free vibration of CNT/fiber/polymer/metal laminate (CNTFPML) and CNT/fiber/polymer (CNTFP) cylindrical shells have been compared to eachother. Comparison between these two types of cylindrical shells would show the advantages of CNTFPML cylindrical shells compared to the CNTFP cylindrical shells. Consequently, the mixture of these four phases can obtain attractive results. The CNTs have been added to the matrix so as to reinforce the matrix in the first step; then the fiber phase was reinforced by the matrix (which has been reinforced by CNTs). The composite cylindrical shells reinforced by CNTs (which can be named as CNTFP cylindrical shells) have been prepared by this method. In order to make the FML cylindrical shells reinforced by CNTs (which can be named CNTFPML cylindrical shells), the adhesive fiber prepreg has been combined with the thin metal layers. The novelty of the present study consists of considering the free vibration of CNTFPML and CNTFP circular cylindrical shells having four and three phases including fiber, CNTs, polymer matrix, metal and fiber, CNTs, polymer matrix, respectively. The frequencies of CNTFPML and CNTFP have been compared to each other for different materials, layups, boundary conditions, axial and circumferential wave numbers.
The equations of motion for thin circular cylindrical shells are given by [34]:
(1a) 

(1b) 

(1c) 
where , and are the thickness, density and radius of the cylindrical shell. Forces and moments are defined as follow:
(2a) 

(2b) 
where , and are the axial, circumferential and shear stresses at an arbitrary point of the cylindrical shell, respectively. The stress resultants are obtained by considering the axial , circumferential and shear strains at an arbitrary point of the cylindrical shell related to the middle surface strains , and and to the changes in the curvature and torsion of the middle surface , and . Therefore, stress resultants can be introduced based on the Love’s first approximation shell theory [35] as follows:
(3) 
where because orthotropic laminates as crossply and unidirectional layups are considered in this study. Stiffness coefficients can be expressed as the following equation [35]:
(4) 
where illustrates the transformed reduced stiffness coefficients. , and are extensional, coupling and bending stiffnesses, respectively. The stiffnesses for composite laminated cylindrical shells could be written as:
(5a) 

(5b) 

(5c) 
where and are the distances of the middle surface of the shell to outer and inner surfaces of the layer, respectively. In addition, denote the transformed reduced stiffness coefficients for the layer. Also, the stiffnesses for the FML cylindrical shells are defined as follow:
(6a) 

(6b) 

(6c) 
where and are the thickness and reduced stiffness of the metal layer, respectively. In the above equations, is related to the composite section and can be illustrated as follow:
(7) 

where , and are the longitudinal, transverse and shear modulus of the nanocomposite section, respectively. Also, and introduce the effective Poisson’s ratios of the nanocomposite cylindrical shell. The elastic modulus and Poisson’s ratios of the shell are presented as:
(8) 

where , and are the elastic moduli of fiber phase. In addition, and are the Poisson’s ratio and volume fraction of fiber, respectively. , and are the elastic moduli of matrix phase which has been reinforced by CNTs while and introduce the Poisson’s ratio and volume fraction of the matrix reinforced by CNTs, respectively. All of the parameters related to the new matrix which consists of a mixture of CNTs and the isotropic matrix are specified in the next section.
The effective mechanical properties of CNTRCs cylindrical shell are obtained based on the extended rule of mixture as follows [36]:
(9a) 

(9b) 

(9c) 
where , are Young's moduli and is shear moduli of CNTs, and are the CNTs efficiency parameters. Also, and introduce the corresponding properties for the isotropic matrix. In addition, the volume fraction of CNT and isotropic matrix are indicated by and , respectively, which are related by .
The effective density and Poisson's ratio of the CNTRCs shell can be defined as [36]:
(10) 

where the Poisson's ratio of the CNT and matrix phase are shown by and , respectively, and and are the density of the CNT and matrix phase, respectively.
The volume fraction of CNTs influences the free vibration of CNTRCs cylindrical shell remarkably. In this manuscript, the material properties of the CNTRCs vary continuously and smoothly through the thickness direction of the shell that their distribution are categorized by uniform distribution (UD) or functionally graded (FG) as shown in Fig. 1. Four types of CNT distributions including: , , and are considered in this study in which outer surface, the inner surface, the midplane and both outer and inner surfaces of the shell are CNTrich, respectively.
The volume fractions of the CNTs for each type of the distributions can be expressed as [37]:
(11) 

where
(12) 
where is the mass fraction of CNTs.
The governing equations of motion for the CNTFPML circular cylindrical shell are obtained by substituting the equation (3) into the equation (1) as follows:
(13a) 

(13b) 

(13c) 
The beam modal function model could be used to study the vibration of the cylindrical shell subjected to different boundary conditions. In the first step, the following harmonic solution should be used as [38]:
(14a) 

(14b) 

(14c) 
where , and are mode shapes in the longitudinal, torsional and flexural directions, respectively. denotes the number of circumferential waves in the mode shape and is the natural frequency of the vibration. The three modal displacements can be illustrated as [38]:
(15) 
where , and are constants to be determined. Although the exact value of for cylindrical shells is unknown, it is specified by the axial modal number or mode shapes of cylindrical shells in the axial direction for a given circumferential modal number . It is known that depends on the boundary conditions. In order to obtain the value of , it is assumed that the flexural mode shapes of the cylindrical shells in the axial direction are in the identical form with the flexural vibration of beam subjected to the same boundary condition. Therefore, the value of can be approximated using the beam modal function with appropriate boundary conditions [39]. In addition, there are many different studies indicating the use of the beam function method to obtain the approximate solution for cylindrical shells [3841]. Furthermore, the values of and (which are modal displacements) are related to the modal frequency and system parameters. By introducing equations (11) and (12) into equation (10), a displacement coefficient matrix in the nondimensional form can be expressed as:
(16) 
where
(17) 

where
(18) 


The determinant of the coefficient matrix set to zero for each value of for a nontrivial solution of the equations of the motion. When the value of is provided, the displacement coefficient matrix leads to a sixorder polynomial in .
In this section, the frequencies of CNTFPML circular cylindrical shells are compared to the frequencies of CNTFP circular cylindrical shells subjected to different boundary conditions by the beam modal function model. The difference between CNTFPML and CNTFP cylindrical shells is existence and nonexistence of metal in the their structurerespectively, so that the fiber phase is reinforced by the CNTs which have been combined with the polymer matrix for both of them. All properties and dimensions to study these circular cylindrical shells are presented as follows, except otherwise noted:
The boundary condition is simply supported, the materials of metal and composite layers are aluminum and carbon/epoxy reinforced by the CNTs, respectively, the layups of the cylindrical shell are considered crossply consisting of fourlayered for CNTFPML and threelayered for CNTFP as shown in Fig. 1, the distribution of the CNTs is unidirectional (UD) as shown in Fig. 2, . In addition, the considered material properties of the circular cylindrical shells are shown in Table 1.
6.1. Validation
The nondimensional frequency parameter of CNTFPML and CNTFP cylindrical shells without any reinforcement (CNTs) is considered to verify the accuracy of the presented model. Hence the frequencies of FML and composite laminated cylindrical shells are considered for validation in Table 2. The material properties of FML have been mentioned in the previous section and the material properties of composite laminated cylindrical shells are , , and . The results show excellent agreement between the used model in this manuscript and the other methods to obtain the frequencies.

z 
h 
Aluminium 
Aluminium 
w, zz 
u, x 
v, 
L 
(a)


(b)


Fig. 1. (a) FML cylindrical shell , (b) Composite cylindrical shell 
Table 1. Material properties of CNTFPML cylindrical shell
Material properties 

CNT 
Fiber 
Matrix 
Metal (Aluminum) 

Carbon 
Glass 





Fig. 2. Uniformaly distribution of CNTs
Table 2. Material properties of CNTFPML cylindrical shell
Present 
[42] 
Present 
[42] 

FML 
Composite 

1 
0.1731 
0.1731 
0.083908 
0.083908 
2 
0.0645 
0.0645 
0.030009 
0.030009 
3 
0.0329 
0.0329 
0.015193 
0.015193 
4 
0.0260 
0.0260 
0.012176 
0.012176 
5 
0.0320 
0.0320 
0.015231 
0.015231 
6.2. Comparison Between Free Vibration Analysis of CNTFPML and CNTFP Cylindrical Shells
The frequencies of CNTFPML and CNTFP cylindrical shells for two carbon and glass fibers are compared to each other for different values of and is presented in Fig. 3. As shown in the figure, the frequencies of CNTFPML are greater than CNTFP for because CNTFPML is stiffer than the CNTFP due to the existence of metal. The thin metal layers have been combined to composite laminated layers; consequently, the elastic modulus of their combine has become more than thao of the composite cylindrical shell. When the elastic modulus of a structure increases, the frequencies of that structure would increase. Similar to this illustration, the frequencies of CNTFPML and CNTFP with carbon fiber are greater than the glass one because of greater elastic modulus. It can be noted that the frequencies of CNTFPML are less than CNTFP for .
The effects of length to radius ratio on the frequencies of both CNTFPML and CNTFP circular cylindrical shells are considered in Figs. 4(a) and 4(b). As depicted in the figures, with increasing the , the frequencies of both structures decrease for short cylinders. As growing the length of cylindrical shell, the change slope of frequencies of the shells declines. Although the frequencies of the CNTFPML long cylindrical shell ( ) for and are maximum and minimum, respectively, the frequencies of CNTFP cylindrical shell do not have a constant procedure for different values of . Admittedly, the frequencies of CNTFP cylindrical shell for in are maximum and in are minimum, but this procedure changes with raising the . It can be seen that for , the difference between the frequencies of CNTFPML cylindrical shell subjected to various values of is greater than the CNTFP one.
In the Fig. 5, the frequencies of CNTFPML and CNTFP circular cylindrical shells for different values of and are compared to each other. It can be seen that with increasing the , the frequencies of CNTFPML would change more than those of the CNTFP. Although with growing the , in the first step the frequencies of both structures decrease and then increase, the decline procedure of CNTFPML is more than the CNTFP. In addition, as growing the , the frequencies of both structures are converged.
In the Fig. 5, the frequencies of CNTFPML and CNTFP circular cylindrical shells for different values of and are compared to each other. It can be seen that with increasing the , the frequencies of CNTFPML would change more than those of the CNTFP. Although with growing the , in the first step the frequencies of both structures decrease and then increase, the decline procedure of CNTFPML is more than the CNTFP. In addition, as growing the , the frequencies of both structures are converged.
A comparison between the frequencies of CNTFPML and CNTFP circular cylindrical shells for different boundary conditions is indicated in Fig. 6. As predicted, not only are the frequencies of CNTFPML greater than the CNTFP, but also the frequencies of clamped boundary condition are more than the other ones since the clamp boundary condition is fixed in all directions and there is no displacement in all directions. So, the stiffness of the structure in this boundary condition is more than the others such as simply boundary condition which can displace in one direction. Therefore, the frequency of clamped boundary condition is more than the others. Also, with increasing the value of , the frequencies of CNTFPML and CNTFP cylindrical shells increase rapidly and moderately, respectively. In addition, the difference between the frequencies of CNTFPML subjected to various boundary conditions is greater than those of the CNTFP.
Fig. 3. The effect of material properties on the frequencies of CNTFPML and CNTFP cylindrical shells with respect to n 
(a) 
(b) Fig. 4. The effect of on the frequencies of (a) CNTFP, (b) CNTFPML, cylindrical shell with respect to 
The effect of weight fraction of CNTs on the frequencies of the cylindrical shells is considered in Fig. 7. The figure illustrates that the frequencies of CNTFPML for all of the weight fractions of CNTs in are greater than those of the CNTFP. Also, the frequencies are approximately identical for CNTFP, but they are different for CNTFPML in . Moreover, although the frequencies of CNTFP increase with growing the weight fraction of CNTs for all values of the , they decrease and increase with growing the weight fraction of CNTs for and , respectively. It can be understood from the figure that the frequencies of CNTFPML and CNTFP drop dramatically and gradually, respectively, and then they raise by increasing the .
Fig. 5. The effect of on the frequencies of CNTFPML and CNTFP cylindrical shells with respect to 
Fig. 6. The effect of boundary condition on the frequencies of CNTFPML and CNTFP cylindrical shells with respect to 
Fig. 7. The effect of CNTs weight fraction on the frequencies of CNTFPML and CNTFP cylindrical shells with respect to 
Fig. 8. The effect of layups on the frequencies of CNTFPML and CNTFP cylindrical shells with respect to 
Two different layups of crossply and unidirectional for composite section of CNTFPML and CNTFP cylindrical shells are considered in Fig. 8 in order to study the discrepancies of free vibration of these two structures. The figure demonstrates that the frequencies of two structures for crossply and unidirectional are approximately identical in . With growing the , initially the frequencies decrease and then increase but the decline for the CNTFPML is more remarkable than the CNTFP. Also, the difference between crossply and unidirectional layups increases as growing the . It should be noted that the frequencies of crossply CNTFP are greater than the unidirectional one for all values of the , but this procedure does not govern for CNTFPML cylindrical shell. The frequencies of crossply CNTFPML cylindrical shell are less and more than the unidirectional one for and , respectively.
In this manuscript, the frequencies of CNTFPML and CNTFP cylindrical shells based on Love’s first approximation shell theory using beam modal function model subjected to different boundary conditions for different material properties are compared to each other. The representative volume element consists of three and four phases for the composite and FML structures, respectively, which include fiber, CNTs, polymer matrix and metal for the FML cylindrical shells and the metal section is ignored for the composite cylindrical shells. In order to generate the CNTFPMLs cylindrical shell, in the first step the CNTs have been added to the matrix and then the reinforced matrix has been utilized to reinforce the fiber phase. Finally, the adhesive fiber prepreg has been combined with the thin metal layers. The results show that the frequencies of CNTFPML are more than the CNTFP for , but with increasing the , this procedure is converted. Also, the frequencies of CNTFPML and CNTFP made of carbon fiber are more than the glass one. Moreover, the frequencies of CNTFPML cylindrical shell decrease with increasing the in , while the frequencies of CNTFP cylindrical shell do not have this regular process. Furthermore, the difference between the frequencies CNTFPML for various values of is more than the CNTFP in . In addition, frequencies comparison between the CNTFPML and CNTFP for different weight fractions of CNTs indicates that although the frequencies of CNTFP for different weight fractions of CNTs in are approximately identical, they are different for CNTFPML. Admittedly, the frequencies of unidirectional layup of CNTFPML and CNTFP are more than the crossply in but with increasing the , this procedure is converted.
Acknowledgements
The authors are grateful to the University of Kashan for supporting this work by Grant No.463853/06.
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