The Effect of Volume Fraction of Single-Walled Carbon Nanotubes on Natural Frequencies of Polymer Composite Cone-Shaped Shell Made from Poly(Methyl Methacrylate)

Meysami, A. H.; Razavi, H.; Golpayegani, I. Fakhari; Bagheri, V.

doi:https://doi.org/10.1155/2017/7345438

Advances in Materials Science and Engineering

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Research Article | Open Access

Volume 2017 | Article ID 7345438 | https://doi.org/10.1155/2017/7345438

The Effect of Volume Fraction of Single-Walled Carbon Nanotubes on Natural Frequencies of Polymer Composite Cone-Shaped Shell Made from Poly(Methyl Methacrylate)

A. H. Meysami,¹H. Razavi,²I. Fakhari Golpayegani,²and V. Bagheri²

Academic Editor: Antonio Riveiro

Received27 Sept 2016

Revised31 Dec 2016

Accepted09 Jan 2017

Published14 Mar 2017

Abstract

In this paper, the effect of volume fraction of single-walled carbon nanotubes on natural frequencies of polymer composite cone-shaped shells made from Poly(Methyl Methacrylate) (PMMA) is studied. In order to determine the characterization of materials reinforced with nanoparticles, the molecular dynamics and mixture rule has been used. The motion equations of composite shell based on the classical thin shells theory using Hamilton’s principle are obtained. Then, using the Ritz method, approximate analytical solution of the natural frequency is presented. Results indicate that the nanotubes have a noticeable effect on the natural frequencies.

1. ‎Introduction

One of the most important phenomena in shells sciences is vibration due to dynamic loading, either one cyclic or impact loading. The shell and any other continuous structures have infinite natural frequencies and vibration mode shapes. If a structure vibrates with a frequency equal to its natural frequency, very little energy is needed to increase its amplitude rapidly and the structure becomes unstable. This can lead to uncomfortable consequences; for instance, the aircraft wing can be broken or the huge turbines may get out of their axis. In recent years, composite plates and shells reinforced with single-walled carbon nanotube (SWCNT) ‎are used widely in industry. Therefore, for stability of these structures, the natural frequency should be calculated and the difference between that and the frequency of time-dependent applied loads should be determined.

Single-walled carbon nanotubes (SWCNTs) and nanocones (SWCNCs) are made of one-‎atom-thick carbon sheets shaped into hallow cylinders and cones. The carbon sheets give ‎SWCNTs unique electric, chemical, and mechanical properties. In addition, extensive application of SWCNTs in nanoscale devices, NEMS, ‎sensors, and reinforcing particles in composite materials has brought these devices into ‎the focus of researchers’ attention [1]. ‎

The mechanical studies related to SWCNTs can be divided into two general categories: (a) analysis of ‎the static and dynamic behavior of SWCNT as a singular nanoscale part and (b) investigating ‎the mechanical behavior of composite shells reinforced by SWCNTs. It is important to ‎know that as dimensions are scaled down, many essential phenomena appear at the ‎nanoscale, which are not important at macroscale. Due to size dependency of ‎material properties‎ in SWCNTs, it is not possible to study the dynamic behavior and ‎mechanical properties of these components ‎through usual methods such as classical ‎continuum theory. Hence, the first kind can be studied through high order continuum ‎mechanics models for nanobeam [2, 3] and for ‎nanoshell including the nonlocal elasticity theory [4–8], the modified couple stress theory [9–13], the modified strain gradient theory [14–16], and the surface elasticity theory [17–20] and also molecular dynamic (MD) simulation [21–24] and ‎experimental approach [23, 25]. ‎

The size parameter and new developed material length scale parameter, which are ‎introduced in high order continuum theories to model the SWCNT as a singular nanoscale ‎device, result in more precise estimation of natural frequencies of the nanoscale parts. ‎Anyway, determination of Young’s modulus of SWCNCs and SWCNTs [26, 27], shear modulus [27], critical load in buckling [28], and mode ‎shapes [24] and in general determination of mechanical properties and ‎dynamic behaviors of nanoscale structures in various shapes, sizes, and supports are the ‎main goal of these types of analyses.‎

The other kind of the problems related to SWCNTs is the macroscale composites, which ‎are reinforced by SWCNT particles. Therefore, the classical continuum theory developed ‎for plate and shell can be used to analyze the mechanical properties and dynamic ‎behaviors of these types of structures. One of the important issues is the effect of volume ‎fraction of SWCNTs on the dynamic behavior of composite shells and particularly cone ‎shaped shell of polymer composite of Poly(Methyl ‎Methacrylate) (PMMA)‎. PMMA is widely used in various applications for its many advantageous properties. Perhaps the most well-known of these properties is light transmission. Typical PMMA grades allow 92% of light to pass through it, which is more than glass or other plastics. This outstanding clarity enables the use of PMMA in many different optical and related applications. PMMA will not shatter, but, it breaks into large pieces. It dissolves in most organic solvents and has poor resistance to many chemical materials. However, its environmental stability is higher than most other plastics such as polyethylene and polystyrene; therefore it is often the first choice for outdoor applications [29]. Its durability and transparency have marked it as a versatile material for a wide range of fields and applications such as skylights, sanitary ware (bathtubs), bullet proof security barriers, signs and displays, LCD screens, and furniture. Methacrylate polymers are used extensively in dental and medical applications. PMMA is compatible with human tissue, and it can be implanted in the eye when the original lens has been removed due to cataracts disease [30]. In orthopedic surgery, PMMA bone cement is used to affix implants and to remodel lost bone. Bone cement acts like a grout and not so much like a glue in arthroplasty. Although sticky, it does not bond to either the bone or the implant; it primarily fills the spaces between the prosthesis and the bone preventing motion. A major consideration when using PMMA cement is the effect of stress shielding. Since PMMA has a Young’s modulus lower than that of natural bone [31], the stresses are loaded into the cement and so the bone no longer receives the mechanical signals to continue bone remodeling and so resorption will occur [32]. Nowadays, newly fabricated composites based on PMMA (whereas PMMA is used as matrix) are of high concern to the researchers. In this regard, Ke et al. [33] and Ansari et al. [34] have investigated, respectively, the nonlinear free vibration and nonlinear forced vibration behavior of nanocomposite beams reinforced with single-walled carbon nanotube (SWCNT) based on Timoshenko beam theory. In the problem of composite plates reinforced by CNTs, Ansari et al. [35, 36] have investigated numerically the nonlinear forced vibration behavior and the geometrically nonlinear primary resonance of third-order shear deformable functionally graded CNT rectangular plates with various edge supports subjected to a harmonic excitation transverse force. Zhang and Liew [37] have proposed an improved moving least-squares (IMLS) approximation for the field variables in the functionally graded CNT plates. They [38] also have studied that using the element-free kp-Ritz method based on first-order shear deformation theory. Ansari et al. [39] have proposed an analytical solution for the nonlinear postbuckling problem of piezoelectric FG-CNT reinforced composite cylindrical shells subjected to several loadings. They [40, 41] also have employed the variational formulation to study the buckling and vibration of axially compressed FG-CNTRC conical shells. Liew et al. [42] have studied free vibration of thin conical shells under various boundary conditions using Ritz method. Zhao and Liew [43] have studied the free vibration of conical shell panels made of functionally graded material using kp-Ritz method. Formica et al. [44] have studied the vibrational properties of composites reinforced with carbon nanotubes (CNTs), using a continuum model, based on the view of Mori et al.’s theory [45, 46]. Tornabene [47] has investigated the free vibration of FGM conical and cylindrical shells. In the mentioned paper, a kind of functional material has been used which is combination of metal and ceramic. In this study, the natural frequencies of conical shells reinforced with carbon nanotubes are obtained. In order to validate the model, the results are compared with the results of isotropic case in previous research. In addition, the effect of volume fraction of nanotubes and the effect of vertex angle of the cone on the natural frequencies are investigated.

2. Theoretical Approach

2.1. Governing Equations of Motion

The conical shell model is shown in Figure 1. In this figure, is coordinate axis in the ridge direction of the cone, is the coordinate in environmental direction, and denotes the centerline of the cone. Also () are displacements in () coordinate systems, respectively.

2.2. Strain-Displacement Relations

Linear strain-displacement relations of the middle surface, according to Sanders’ kinematic relations, are as follows:Also, curvatures are defined as follows:Now, by substituting (3) in (2), the curves will be redefined as follows:Strain-displacement relations for an arbitrary point in the distance from the middle surface are expressed as follows:Derived equations can be expressed in matrix form as follows:

2.3. Stress-Strain Relationships

Since the shell made of composite is reinforced with carbon nanotubes, in general it is assumed that the shells material properties are orthotropic. On the other hand, because of the carbon nanotubes setting assumptions along the ridge of the cone-shaped shell, orthotropic stress-strain will be reducible to transversely isotropic case.whereYoung’s modulus, shear modulus, and Poisson’s ratio are obtained from the mixtures rule [50]. To simplify the equations, the following equations are considered:where and are extensional and bending stiffness, respectively.

2.4. Equations of Motion

In order to derive the equations of motion, Hamilton’s principle is applied:where the virtual strain energy, , is defined asand the virtual work done by external loads acting upon the conical shell, , is defined asand virtual kinetic energy, , is defined asSubstituting in the Hamilton principle (10) and then integrating along the thickness, and yieldBy substituting (1) and (4) in (14), the following equation can be obtained:where stress resultants are defined as follows:Also,Now, applying integration by parts yields the following equations of motion:whereTo express the equations of motion in terms of displacement, stress resultants versus displacement should be derived as follows:Similarly, the other stress resultants are obtained which can be represented in matrix form as follows:

2.5. Energy Functional

For the conical shell, the integral expressions of strain and kinetic energies are as follows:The above equations can be rewritten as follows:Therefore, the energy functional, depending on the strain and kinetic energy, is expressed as follows:

2.6. Formulation of the Eigen Value Problem

Removing harmonic component and insertion of the displacement function in (24), the maximum energy functional is expressed as follows:where and are the maximum strain energy and maximum kinetic energy, respectively. By minimizing the functional equation (25) with respect to the available coefficients, an eigen value equation is obtained as follows:To obtain the natural frequency of conical shape composite shell reinforced with nanotubes, it is sufficient to put the determinant of coefficients matrix equal to zero.

2.7. The Expression of Displacement Functions

To obtain the strain and kinetic energies, first the displacement functions should be determined. It should be noted that these functions must satisfy the boundary conditions. Conical shell that will be investigated here is a complete conical shell with a fixed base. Consequently, the boundary conditions can be expressed as follows:According to the mentioned study, the authors will use the following displacement functions to provide an approximate analytical solution based on the Ritz method:

2.8. Mixtures Rule

Suppose that the composite shell that has been strengthened with carbon nanotubes uniformly along the coordinate is made of a mixture of single-walled carbon nanotubes, which is assumed to be isotropic. Here, the main problem is determining the effective material properties of composites reinforced with carbon nanotubes. The effective physical properties can be estimated using the Mori-Tanaka model or mixtures rule [45, 50]. According to the rule of mixtures, Young’s moduli and shear moduli can be expressed as follows:where , , and are Young’s and shear moduli of carbon nanotubes, respectively. and are the corresponding properties, related to the background. is the efficiency parameters of carbon nanotubes which is determined by adaptation of elastic moduli from molecular dynamics simulation results and the mixtures rules. and are volume fractions of carbon nanotubes and background, which are related as follows:Suppose the volume fraction, , as follows:wherewhere is the mass fraction of carbon nanotubes. For a uniform distribution of nanotubes, it is considered that . Poisson’s ratio and mass density can be calculated as follows:Consequently, according to (9), stiffness equations for a uniform distribution of nanotubes will be obtained as follows:According to the above equations, for a uniform distribution of nanotubes, there is; also

3. Material Properties of Carbon Nanotubes

Based on the results of molecular dynamics simulations, size and temperature dependent material properties of single-walled carbon nanotubes [48] can be obtained numerically as in Table 1.

3.1. Material Properties of PMMA Polymer

The material properties of considered background in temperature equal to are assumed as follows:For short fiber composites is usually considered to be 0.2. However, there is no experimental method for determining the value of . Here, with adaption of Young’s moduli and obtained from the mixtures rule and molecular dynamics, efficiency parameters and are estimated. Comparison of the results shows that if the efficiency parameters are chosen properly, Young’s modulus, obtained from molecular dynamics, is well adapted to mixtures rule results, as shown in Table 2.

4. Results and Validation

It should be noted that there are various shell models according to the thickness of the shell ‎and various models can be used for the variable density of filling the polymer with nanotubes. ‎In general the vibration solution for polymer composite cone-shaped shell made from PMMA ‎is hard. However, in this section, thin shells model with uniform distribution of nanotubes in ‎composite cone-shaped shell has been used; therefore results are extracted for special case. Validation of current study has been undertaken with [44, 51]. In this case, the parameter is considered to be equal to for the conical shell. It is considered in Table 3 that the errors of the present results in comparison with the solution presented in [51] are less than 1% for every vertex angle.

4.1. Numerical Results of the PMMA/CNT Composite

In this section, by considering the fixed PMMA/CNT composite shell, the numerical results of the nondimensional frequencies with respect to the different volume fractions of carbon nanotubes, different values of , and various are presented in Tables 4, 5 and 6. In addition, frequency variations are plotted with respect to the number of modes “n” in Figures 2, 3, and 4 for specific volume fractions of PMMA/CNT composite shell. The figures show that the most important effects of using nanotubes are in elementary modes of PMMA/CNT composite shell. Also, for “n” greater than 3, the frequencies obtained for different tend to converge. It can be concluded that values of nondimensional frequencies in aforementioned range do not depend on the angle of cone apexes and the volume fraction.

5. Conclusion

In this paper, the effect of volume fraction of single-walled carbon nanotubes on natural frequencies of cone-shaped polymer composite shells made of PMMA is studied. There are various shell models according to the thickness of the shell ‎and various models can be used for the variable density of filling the polymer with nanotubes. ‎In general the vibration solution for polymer composite cone-shaped shell made from PMMA ‎is hard. However, a complete conical shell with a fixed base is investigated. Equations of motion were developed based on Hamilton’s principle. Thin shells model with uniform distribution of nanotubes in ‎composite cone-shaped shell has been used; therefore results are extracted for special case. Regardless of the volume fraction amount of nanotubes, it is observed that the greatest decrease in nondimensional frequency parameter with increasing “n” parameter occurs in the first modes. In some cases, this reduction has increased up to 4 times. In addition, for number of modes “n” greater than 3, the frequencies obtained for different angle of cone apexes tend to converge. It is considered that the errors of the present results in comparison with the solution presented by Leisa are less than 1% for every vertex angle. It is concluded that values of nondimensional frequencies in aforementioned range do not depend on the angle of the cone apexes and the volume fraction.

Nomenclature

	Ritz approximation coefficients
:	Extensional stiffness for a uniform distribution of nanotubes
:	Bending stiffness for a uniform distribution of nanotubes
:	Young’s moduli
:	Young’s moduli of carbon nanotubes
:	Young’s moduli related to the background
:	Shear moduli
:	Shear moduli of carbon nanotubes
:	Shear moduli related to the background
	Shell thickness
:	Density resultant
	Virtual kinetic energy
:	Matrix components
:	Curvature parameters
:	Maximum kinetic energy
	Length of the cone
:	Moment resultant
:	Moment resultant components
:	Mode number
:	Force resultant
:	Force resultant components
:	Orthotropic elastic constants
:	Orthotropic elastic constants in axial and circumferential direction
:	Displacement in directions
:	Velocity in directions
	Virtual strain energy
:	Maximum strain energy
:	Volume fractions of carbon nanotubes
:	Volume fractions related to the background
:	Mass fraction of carbon nanotubes
:	Coordinate axis in the ridge direction of the cone
:	Centerline of the cone
	Cone vertex angle
	Energy functional
:	Maximum energy functional
:	Strain-displacement relations for an arbitrary point in the distance from the middle surface
:	Linear strain-displacement relations of the middle surface
	Coordinate in environmental direction
	Half cone angle
:	Stress components
:	Poisson’s ratio
:	Poisson’s ratio of carbon nanotubes
:	Poisson’s ratio related to the background
:	Efficiency parameter of carbon nanotubes
	Natural frequency
	Mass density
:	Mass density of carbon nanotubes
:	Mass density related to the background.

Competing Interests

The authors declare that they have no competing interests.

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Copyright © 2017 A. H. Meysami et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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