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

In this paper, the effect of volume fraction of single-walled carbon nanotubes on natural frequencies of polymer composite coneshaped 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 usingHamilton’s principle are obtained.Then, using theRitzmethod, approximate analytical solution of the natural frequency is presented. Results indicate that the nanotubes have a noticeable effect on the natural frequencies.


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][5][6][7][8], the modified couple stress theory [9][10][11][12][13], the modified strain gradient theory [14][15][16], and the surface elasticity theory [17][18][19][20] and also molecular dynamic (MD) simulation [21][22][23][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 singlewalled 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.

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 (, V, ) are displacements in (, , ) coordinate systems, respectively.

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:

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. where Young'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.

Equations of Motion.
In order to derive the equations of motion, Hamilton's principle is applied: where the virtual strain energy, , is defined as and the virtual work done by external loads acting upon the conical shell, , is defined as and virtual kinetic energy, , is defined as Substituting  in the Hamilton principle (10) and then integrating along the thickness,  and  yield By 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: where To 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: Advances in Materials Science and Engineering 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  max and  max 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.

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  CN  11 ,  CN  22 , and  CN 12 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. CN and   are volume fractions of carbon nanotubes and background, which are related as follows: Suppose the volume fraction,  CN , as follows: where where  CN is the mass fraction of carbon nanotubes.For a uniform distribution of nanotubes, it is considered that  CN =  * CN .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  = (ℎ 2 /12)  ; also

Material Properties of Carbon Nanotubes
Based on the results of molecular dynamics simulations, size and temperature dependent material properties of singlewalled carbon nanotubes [48] can be obtained numerically as in Table 1.

Material Properties of PMMA Polymer.
The material properties of considered background in temperature equal to 300 ∘ K are assumed as follows: ]  = 0.34, For short fiber composites  1 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  11 and  22 obtained from the mixtures rule and molecular dynamics, efficiency parameters  1 and  2 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.

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 ℎ 2 /12 2 2 is considered to be equal to 10 −5 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.

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      : Young's moduli related to the background   : S h e a r m o d u l i  CN  : Shear moduli of carbon nanotubes   : Shear moduli related to the background ℎ: S h e l l t h i c k n e s s  0 : D e n s i t y r e s u l t a n t : Virtual

Figure 1 :
Figure 1: Geometry and coordinate system of a conical shell.

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

Table 2 :
Young's moduli for the composite PMMA/CNT: mixtures rule in comparison with molecular dynamics.

Table 3 :
Nondimensional frequency of a fixed conical shell with a fixed vertex.
kinetic energy   : M a t r i x c o m p o n e n t s   ,   , : Curvature parameters  max : Maximum kinetic energy : L e n g t h o f t h e c o n e   : Moment resultant   ,   ,   : Moment resultant components : M o d e n u m b e r   : F o r c er e s u l t a n t   ,   ,   : Force resultant components   : Orthotropic elastic constants   ,   : Orthotropic elastic constants in axial and circumferential direction  0 , V 0 ,  0 : Displacement in , ,  directions u 0 , V 0 , ẇ 0 : V elocityin, ,  directions : Maximum energy functional   ,   ,   : Strain-displacement relations for an arbitrary point in the distance  from the middle surface  0  ,  0  ,  0  : Linear strain-displacement relations of the middle surface : Coordinate in environmental direction : H a l f c o n e a n g l e   ,   ,   : Stress components ] : Mass density related to the background.