Discrepancies Between Free Vibration of FML and Composite Cylindrical Shells Reinforced by CNTs

Discrepancies Between Free Vibration of FML and Composite Cylindrical Shells Reinforced by CNTs

M. Mohandes, A.R. Ghasemi^*

Department Solid Mechanics, University of Kashan, 87317-53153, Kashan, Iran

1. Introduction

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 three-dimensional 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 so-called 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 Voigt-Kelvin model. Shooshtari and Razavi [19] used multiple time scale method to study the linear and nonlinear vibration of cross-ply 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 three-dimensional elasticity theory. They determined the natural frequencies of the plate with different boundary conditions using the combination of DQM, state-apace 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 glass-reinforced aluminum laminate (GLARE) 5-3/2 plate based on higher-order 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 Euler-Bernoulli beams based on von-Karman 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 Heidari-Rarani [25] studied the buckling behavior of FML cylindrical shells based on first-order 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 lay-ups 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. [26-27] considered the vibration and buckling of functionally graded (FG) rectangular nanoplate using non-local 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 first-order 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 nanotube-reinforced hybrid composite (FG-CNTRHC) shells. In another research, Thomas and Roy [30] studied the vibration of uniform and functionally graded carbon nanotube-reinforced composite (FG-CNTRC) 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 mesh-free method. The results showed that the developed mesh-free 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 FG-CNTRC 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 thickness-to-radius ratio, the non-dimensional frequencies decreases remarkably. Dastjerdi and Malek-Mohammadi [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, lay-ups, boundary conditions, axial and circumferential wave numbers.

1. Fundumental Equations

The equations of motion for thin circular cylindrical shells are given by [34]:

where ,  and  are the thickness, density and radius of the cylindrical shell. Forces  and moments  are defined as follow:

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:

where  because orthotropic laminates as cross-ply and unidirectional lay-ups are considered in this study. Stiffness coefficients can be expressed as the following equation [35]:

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:

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:

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:

where ,  and  are the longitudinal, transverse and shear modulus of the nano-composite section, respectively. Also,  and  introduce the effective Poisson’s ratios of the nano-composite cylindrical shell. The elastic modulus and Poisson’s ratios of the shell are presented as:

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.

1. Material Properties of Cntrcs

The effective mechanical properties of CNTRCs cylindrical shell are obtained based on the extended rule of mixture as follows [36]:

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]:

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 mid-plane and both outer and inner surfaces of the shell are CNT-rich, respectively.

The volume fractions of the CNTs for each type of the distributions can be expressed as [37]:

where

where  is the mass fraction of CNTs.

1. Governing Equations of Motion

The governing equations of motion for the CNTFPML circular cylindrical shell are obtained by substituting the equation (3) into the equation (1) as follows:

1. Analytical Solution Procedure

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]:

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]:

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 [38-41]. 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 non-dimensional form can be expressed as:

where

where

The determinant of the coefficient matrix  set to zero for each value of  for a non-trivial solution of the equations of the motion. When the value of  is provided, the displacement coefficient matrix leads to a six-order polynomial in .

1. Numerical Results

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 non-existence 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 lay-ups of the cylindrical shell are considered cross-ply consisting of four-layered  for CNTFPML and three-layered  for CNTFP as shown in Fig. 1, the distribution of the CNTs is uni-directional (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 non-dimensional 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.

Table 1. Material properties of CNTFPML cylindrical shell

Fig. 2. Uniformaly distribution of CNTs

Table 2. Material properties of CNTFPML cylindrical shell

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.

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 .

Two different lay-ups of cross-ply and uni-directional 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 cross-ply and uni-directional 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 cross-ply and uni-directional lay-ups increases as growing the . It should be noted that the frequencies of cross-ply 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 cross-ply CNTFPML cylindrical shell are less and more than the uni-directional one for  and , respectively.

1. Conclusions

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 uni-directional lay-up of CNTFPML and CNTFP are more than the cross-ply 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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