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

Authors: A. H. Meysami; H. Razavi; I. Fakhari Golpayegani; V. Bagheri
Journal: Advances in Materials Science and Engineering
 (2017)

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## 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.


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## Body

## 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 Figure1. In this figure, x is coordinate axis in the ridge direction of the cone, θ is the coordinate in environmental direction, and z denotes the centerline of the cone. Also (u,v,w) are displacements in (x,y,z) coordinate systems, respectively.Figure 1
Geometry and coordinate system of a conical shell.


### 2.2. Strain-Displacement Relations

Linear strain-displacement relations of the middle surface, according to Sanders’ kinematic relations, are as follows:(1)εxx0=∂u0∂x,εθθ0=1xsin⁡ϕ∂v0∂θ+u0x+w0xtan⁡ϕ,γxθ0=∂v0∂x+1xsin⁡ϕ∂u0∂θ-v0x.Also, curvatures are defined as follows:(2)kx=∂ψx∂x,kθ=1xsin⁡ϕ∂ψθ∂θ+ψxx,τ=∂ψθ∂x+1xsin⁡ϕ∂ψx∂θ-ψθx,(3)ψx=-∂w0∂x,ψθ=v0xtan⁡ϕ-1xsin⁡ϕ∂w0∂θ.Now, by substituting (3) in (2), the curves will be redefined as follows:(4)kx=∂∂x-∂w0∂x=-∂2w0∂x2,kθ=1xsin⁡ϕ∂∂θ+v0xtan⁡ϕ-1xsin⁡ϕ∂w0∂θ+1x-∂w0∂x=-1x2sin2⁡ϕ∂2w0∂θ2+cos⁡ϕx2sin2⁡ϕ∂v0∂θ-1x∂w0∂x,τ=∂∂xv0xtan⁡ϕ-1xsin⁡ϕ∂w0∂θ+1xsin⁡ϕ∂∂θ-∂w0∂x-v0x2tan⁡ϕ+1x2sin⁡ϕ∂w0∂θ=1xtan⁡ϕ∂v0∂x-1x2tan⁡ϕv0-1xsin⁡ϕ∂2w0∂x∂θ+1x2sin⁡ϕ∂w0∂θ-1xsin⁡ϕ∂2w0∂x∂θ-v0x2tan⁡ϕ+1x2sin⁡ϕ∂w0∂θ=-2xsin⁡ϕ∂2w0∂x∂θ+1xtan⁡ϕ∂v0∂x+2x2sin⁡ϕ∂w0∂θ-2v0x2tan⁡ϕ.Strain-displacement relations for an arbitrary point in the distance z from the middle surface are expressed as follows:(5)εxx=εxx0+zkx,εθθ=εθθ0+zkθ,γxθ=γxθ0+zτ.Derived equations can be expressed in matrix form as follows:(6)εxx0εθθ0γxθ0=∂u0∂x1xsin⁡ϕ∂v0∂θ+u0x+w0xtan⁡ϕ∂v0∂x+1xsin⁡ϕ∂u0∂θ-v0x,kxkθτ=-∂2w0∂x2-1x2sin2⁡ϕ∂2w0∂θ2+cos⁡ϕx2sin2⁡ϕ∂v0∂θ-1x∂w0∂x-2xsin⁡ϕ∂2w0∂x∂θ+1xtan⁡ϕ∂v0∂x+2x2sin⁡ϕ∂w0∂θ-2v0x2tan⁡ϕ,εxxεθθγxθ=εxx0εθθ0γxθ0+zkxkθτ.


### 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.(7)σxxσθθσxθ=Q11Q120Q12Q22000Q66εxxεθθγxθ,where(8)Q11=E11-ν12ν21,Q12=ν12E21-ν12ν21,Q22=E21-ν12ν21,Q66=G12.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:(9)Aij,Dij=∫-h/2h/2Qij1,z2dz,i,j=1,2,6,where Aij and Dij are extensional and bending stiffness, respectively.


### 2.4. Equations of Motion

In order to derive the equations of motion, Hamilton’s principle is applied:(10)∫t1t2δU+δV-δKdt=0,where the virtual strain energy, δU, is defined as(11)δU=∬∫-h/2h/2σxxδεxx+σθθδεθθ+σxθδγxθdzxsin⁡ϕdxdθ=∬∫-h/2h/2σxxδεxx°+zδkx+σθθδεθθ°+zδkθ+σxθδγxθ°+zδτdzxsin⁡ϕdxdθand the virtual work done by external loads acting upon the conical shell, δV, is defined as(12)δV=-∬qδw0xsin⁡ϕdxdθand virtual kinetic energy, δK, is defined as(13)δK=∭-h/2h/2ρu˙0δu˙0+v˙0δv˙0+w˙0δw˙0xsin⁡ϕdzdxdθ.Substituting δU in the Hamilton principle (10) and then integrating along the thickness, δV and δK yield(14)∫t1t2∬Nxxδεxx0+Mxxδkx+Nθθδεθθ0+Mθθδkθ+Nxθδγxθ0+Mxθδτ-qδw0-I0u˙0δu˙0+v˙0δv˙0+w˙0δw˙0xsin⁡ϕdxdθdt=0.By substituting (1) and (4) in (14), the following equation can be obtained:(15)∫t2t1∬Nxx∂δu0∂x-Mxx∂2δw0∂x2+Nθθ1xsin⁡ϕ∂δv0∂θ+δu0x+δw0xtan⁡ϕ+Mθθ-1x2sin2ϕ∂2δw0∂θ2+cos⁡ϕx2sin2ϕ∂δv0∂θ-1x∂δw0∂x+Nxθ∂δv0∂x+1xsin⁡ϕ∂δu0∂θ-δv0x+Mxθ-2xsin⁡ϕ∂2δw0∂x∂θ+1xtan⁡ϕ∂δv0∂x+2x2sin⁡ϕ∂δw0∂θ-2δv0x2tan⁡ϕ-qδw0-I0u˙0δu˙0+v˙0δv˙0+w˙0δw˙0∂δu0∂xxsin⁡ϕdxdθ∂δu0∂xdt=0,where stress resultants are defined as follows:(16)Nij=∫-h/2h/2σijdz,My¨=∫-h/2h/2σijzdz,ij=xx,θθ,xθ.Also,(17)I0=∫-h/2h/2ρdz.Now, applying integration by parts yields the following equations of motion:(18)∂Nxx∂x+Nxx-Nθθx+1xsin⁡ϕ∂Nxθ∂θ=I0u0¨,1xsin⁡ϕ∂Nθθ∂θ+∂Nxθ∂x+2Nxθx+Qθxtan⁡ϕ=I0v0¨,-Nθθxtan⁡ϕ+∂Qx∂x+Qxx+1xsin⁡ϕ∂Qθ∂θ+q=I0w0¨,where(19)Qx=Mxxx+∂Mxx∂x+1xsin⁡ϕ∂Mxθ∂θ-Mθθx,Qθ=1xsin⁡ϕ∂Mθθ∂θ+∂Mxθ∂x+2Mxθx.To express the equations of motion in terms of displacement, stress resultants versus displacement should be derived as follows:(20)Nxx=∫-h/2h/2σxxdz=∫-h/2h/2Q11εxx+Q12εθθdz=∫-h/2h/2Q11εxx0+Q12εθθ0+zQ11kx+Q12kθdz=∫-h/2h/2Q11εxx0+Q12εθθ0dz=A11εxx0+A12εθθ0.Similarly, the other stress resultants are obtained which can be represented in matrix form as follows:(21)NxxNθθNxθMxxMθθMxθ=A11A120000A12A22000000A66000000D11D120000D12D22000000D66εxx0εθθ0γxθ0kxkθτ,NxxNθθNxθ=A11A120A12A22000A66∂u0∂x1xsin⁡ϕ∂v0∂θ+u0x+w0xtan⁡ϕ∂v0∂x+1xsin⁡ϕ∂u0∂θ-v0x,MxxMθθMxθ=D11D120D12D22000D66-∂2w0∂x2-1x2sin2⁡ϕ∂2w0∂θ2+cos⁡ϕx2sin2⁡ϕ∂v0∂θ-1x∂w0∂x-2xsin⁡ϕ∂2w0∂x∂θ+1xtan⁡ϕ∂v0∂x+2x2sin⁡ϕ∂w0∂θ-2v0x2tan⁡ϕ.


### 2.5. Energy Functional

For the conical shell, the integral expressions of strain and kinetic energies are as follows:(22)U=12∬∫-h/2h/2σxxεxx+σθθδεθθ+σxθγxθdzxsin⁡ϕdxdθ,K=12∭-h/2h/2ρu˙02+v˙02+w˙02xsin⁡ϕdzdxdθ.The above equations can be rewritten as follows:(23)U=12∬∫-h/2h/2σxxεxx0+zkx+σθθεθθ0+zkθ+σxθγxθ0+zτdzxsin⁡ϕdxdθ,U=12∬Nxxεxx0+Mxxkx+Nθθεxx0+Mθθkθ+Nxθγxθ0+Mxθτxsin⁡ϕdxdθ,K=12∭ρu˙2+v˙2+w˙2xsin⁡ϕdzdxdθ=12I0∫02π∫0Lu˙02+v˙02+w˙02xsin⁡ϕdxdθ.Therefore, the energy functional, depending on the strain and kinetic energy, is expressed as follows:(24)Π=K-U.


### 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:(25)Π^=Kmax-Umax,where Umax and Kmax 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:(26)∂Π^∂ai=0;i=1,2,3⟹K11-ω2M11K12K13K21K22-ω2M22K23K31K32K33-ω2M33a1a2a3=000.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. (27)K11-ω2M11K12K13K21K22-ω2M22K23K31K32K33-ω2M33=0.


### 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:(28)u0=v0=w0=∂w0∂x=0,atx=0,u0=v0=w0=∂w0∂x=0,atx=L.According to the mentioned study, the authors will use the following displacement functions to provide an approximate analytical solution based on the Ritz method:(29)u0=a1x2x-L2cos⁡nθcos⁡ωt,v0=a2x2x-L2sin⁡nθcos⁡ωt,w0=a3x2x-L2cos⁡nθcos⁡ωt.


### 2.8. Mixtures Rule

Suppose that the composite shell that has been strengthened with carbon nanotubes uniformly along thex 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:(30)E11=η1VCNE11CN+VmEm,η2E22=VCNE22CN+VmEm,η3G12=VCNG12CN+VmGm,where E11CN, E22CN, and G12CN are Young’s and shear moduli of carbon nanotubes, respectively. Em and Gm are the corresponding properties, related to the background. ηj is the efficiency parameters of carbon nanotubes which is determined by adaptation of elastic moduli from molecular dynamics simulation results and the mixtures rules. VCN and Vm are volume fractions of carbon nanotubes and background, which are related as follows:(31)VCN+Vm=1.Suppose the volume fraction, VCN, as follows:(32)VCN=wzVCN∗,where(33)VCN∗=wCNwCN+ρCN/ρm-ρCN/ρmwCN,where wCN is the mass fraction of carbon nanotubes. For a uniform distribution of nanotubes, it is considered that VCN=VCN∗. Poisson’s ratio and mass density can be calculated as follows:(34)ν12=VCN∗ν12CN+Vmνm,ρ=VCN∗ρCN+Vmρm.Consequently, according to (9), Aij stiffness equations for a uniform distribution of nanotubes will be obtained as follows:(35)A11=11-ν12ν21VCN∗η1E11CN-Em+Emh,A12=ν121-ν12ν21η2E22CNEmE22CN+Em-E22CNVCN∗h,A22=11-ν12ν21η2E22CNEmE22CN+Em-E22CNVCN∗h,A66=η3G12CNGmG12CN+Gm-G12CNVCN∗h.According to the above equations, for a uniform distribution of nanotubes, there isDij=h2/12Aij; also(36)I0=VCN∗ρCN-ρm+ρmh.


## 2.1. Governing Equations of Motion

The conical shell model is shown in Figure1. In this figure, x is coordinate axis in the ridge direction of the cone, θ is the coordinate in environmental direction, and z denotes the centerline of the cone. Also (u,v,w) are displacements in (x,y,z) coordinate systems, respectively.Figure 1
Geometry and coordinate system of a conical shell.


## 2.2. Strain-Displacement Relations

Linear strain-displacement relations of the middle surface, according to Sanders’ kinematic relations, are as follows:(1)εxx0=∂u0∂x,εθθ0=1xsin⁡ϕ∂v0∂θ+u0x+w0xtan⁡ϕ,γxθ0=∂v0∂x+1xsin⁡ϕ∂u0∂θ-v0x.Also, curvatures are defined as follows:(2)kx=∂ψx∂x,kθ=1xsin⁡ϕ∂ψθ∂θ+ψxx,τ=∂ψθ∂x+1xsin⁡ϕ∂ψx∂θ-ψθx,(3)ψx=-∂w0∂x,ψθ=v0xtan⁡ϕ-1xsin⁡ϕ∂w0∂θ.Now, by substituting (3) in (2), the curves will be redefined as follows:(4)kx=∂∂x-∂w0∂x=-∂2w0∂x2,kθ=1xsin⁡ϕ∂∂θ+v0xtan⁡ϕ-1xsin⁡ϕ∂w0∂θ+1x-∂w0∂x=-1x2sin2⁡ϕ∂2w0∂θ2+cos⁡ϕx2sin2⁡ϕ∂v0∂θ-1x∂w0∂x,τ=∂∂xv0xtan⁡ϕ-1xsin⁡ϕ∂w0∂θ+1xsin⁡ϕ∂∂θ-∂w0∂x-v0x2tan⁡ϕ+1x2sin⁡ϕ∂w0∂θ=1xtan⁡ϕ∂v0∂x-1x2tan⁡ϕv0-1xsin⁡ϕ∂2w0∂x∂θ+1x2sin⁡ϕ∂w0∂θ-1xsin⁡ϕ∂2w0∂x∂θ-v0x2tan⁡ϕ+1x2sin⁡ϕ∂w0∂θ=-2xsin⁡ϕ∂2w0∂x∂θ+1xtan⁡ϕ∂v0∂x+2x2sin⁡ϕ∂w0∂θ-2v0x2tan⁡ϕ.Strain-displacement relations for an arbitrary point in the distance z from the middle surface are expressed as follows:(5)εxx=εxx0+zkx,εθθ=εθθ0+zkθ,γxθ=γxθ0+zτ.Derived equations can be expressed in matrix form as follows:(6)εxx0εθθ0γxθ0=∂u0∂x1xsin⁡ϕ∂v0∂θ+u0x+w0xtan⁡ϕ∂v0∂x+1xsin⁡ϕ∂u0∂θ-v0x,kxkθτ=-∂2w0∂x2-1x2sin2⁡ϕ∂2w0∂θ2+cos⁡ϕx2sin2⁡ϕ∂v0∂θ-1x∂w0∂x-2xsin⁡ϕ∂2w0∂x∂θ+1xtan⁡ϕ∂v0∂x+2x2sin⁡ϕ∂w0∂θ-2v0x2tan⁡ϕ,εxxεθθγxθ=εxx0εθθ0γxθ0+zkxkθτ.


## 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.(7)σxxσθθσxθ=Q11Q120Q12Q22000Q66εxxεθθγxθ,where(8)Q11=E11-ν12ν21,Q12=ν12E21-ν12ν21,Q22=E21-ν12ν21,Q66=G12.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:(9)Aij,Dij=∫-h/2h/2Qij1,z2dz,i,j=1,2,6,where Aij and Dij are extensional and bending stiffness, respectively.


## 2.4. Equations of Motion

In order to derive the equations of motion, Hamilton’s principle is applied:(10)∫t1t2δU+δV-δKdt=0,where the virtual strain energy, δU, is defined as(11)δU=∬∫-h/2h/2σxxδεxx+σθθδεθθ+σxθδγxθdzxsin⁡ϕdxdθ=∬∫-h/2h/2σxxδεxx°+zδkx+σθθδεθθ°+zδkθ+σxθδγxθ°+zδτdzxsin⁡ϕdxdθand the virtual work done by external loads acting upon the conical shell, δV, is defined as(12)δV=-∬qδw0xsin⁡ϕdxdθand virtual kinetic energy, δK, is defined as(13)δK=∭-h/2h/2ρu˙0δu˙0+v˙0δv˙0+w˙0δw˙0xsin⁡ϕdzdxdθ.Substituting δU in the Hamilton principle (10) and then integrating along the thickness, δV and δK yield(14)∫t1t2∬Nxxδεxx0+Mxxδkx+Nθθδεθθ0+Mθθδkθ+Nxθδγxθ0+Mxθδτ-qδw0-I0u˙0δu˙0+v˙0δv˙0+w˙0δw˙0xsin⁡ϕdxdθdt=0.By substituting (1) and (4) in (14), the following equation can be obtained:(15)∫t2t1∬Nxx∂δu0∂x-Mxx∂2δw0∂x2+Nθθ1xsin⁡ϕ∂δv0∂θ+δu0x+δw0xtan⁡ϕ+Mθθ-1x2sin2ϕ∂2δw0∂θ2+cos⁡ϕx2sin2ϕ∂δv0∂θ-1x∂δw0∂x+Nxθ∂δv0∂x+1xsin⁡ϕ∂δu0∂θ-δv0x+Mxθ-2xsin⁡ϕ∂2δw0∂x∂θ+1xtan⁡ϕ∂δv0∂x+2x2sin⁡ϕ∂δw0∂θ-2δv0x2tan⁡ϕ-qδw0-I0u˙0δu˙0+v˙0δv˙0+w˙0δw˙0∂δu0∂xxsin⁡ϕdxdθ∂δu0∂xdt=0,where stress resultants are defined as follows:(16)Nij=∫-h/2h/2σijdz,My¨=∫-h/2h/2σijzdz,ij=xx,θθ,xθ.Also,(17)I0=∫-h/2h/2ρdz.Now, applying integration by parts yields the following equations of motion:(18)∂Nxx∂x+Nxx-Nθθx+1xsin⁡ϕ∂Nxθ∂θ=I0u0¨,1xsin⁡ϕ∂Nθθ∂θ+∂Nxθ∂x+2Nxθx+Qθxtan⁡ϕ=I0v0¨,-Nθθxtan⁡ϕ+∂Qx∂x+Qxx+1xsin⁡ϕ∂Qθ∂θ+q=I0w0¨,where(19)Qx=Mxxx+∂Mxx∂x+1xsin⁡ϕ∂Mxθ∂θ-Mθθx,Qθ=1xsin⁡ϕ∂Mθθ∂θ+∂Mxθ∂x+2Mxθx.To express the equations of motion in terms of displacement, stress resultants versus displacement should be derived as follows:(20)Nxx=∫-h/2h/2σxxdz=∫-h/2h/2Q11εxx+Q12εθθdz=∫-h/2h/2Q11εxx0+Q12εθθ0+zQ11kx+Q12kθdz=∫-h/2h/2Q11εxx0+Q12εθθ0dz=A11εxx0+A12εθθ0.Similarly, the other stress resultants are obtained which can be represented in matrix form as follows:(21)NxxNθθNxθMxxMθθMxθ=A11A120000A12A22000000A66000000D11D120000D12D22000000D66εxx0εθθ0γxθ0kxkθτ,NxxNθθNxθ=A11A120A12A22000A66∂u0∂x1xsin⁡ϕ∂v0∂θ+u0x+w0xtan⁡ϕ∂v0∂x+1xsin⁡ϕ∂u0∂θ-v0x,MxxMθθMxθ=D11D120D12D22000D66-∂2w0∂x2-1x2sin2⁡ϕ∂2w0∂θ2+cos⁡ϕx2sin2⁡ϕ∂v0∂θ-1x∂w0∂x-2xsin⁡ϕ∂2w0∂x∂θ+1xtan⁡ϕ∂v0∂x+2x2sin⁡ϕ∂w0∂θ-2v0x2tan⁡ϕ.


## 2.5. Energy Functional

For the conical shell, the integral expressions of strain and kinetic energies are as follows:(22)U=12∬∫-h/2h/2σxxεxx+σθθδεθθ+σxθγxθdzxsin⁡ϕdxdθ,K=12∭-h/2h/2ρu˙02+v˙02+w˙02xsin⁡ϕdzdxdθ.The above equations can be rewritten as follows:(23)U=12∬∫-h/2h/2σxxεxx0+zkx+σθθεθθ0+zkθ+σxθγxθ0+zτdzxsin⁡ϕdxdθ,U=12∬Nxxεxx0+Mxxkx+Nθθεxx0+Mθθkθ+Nxθγxθ0+Mxθτxsin⁡ϕdxdθ,K=12∭ρu˙2+v˙2+w˙2xsin⁡ϕdzdxdθ=12I0∫02π∫0Lu˙02+v˙02+w˙02xsin⁡ϕdxdθ.Therefore, the energy functional, depending on the strain and kinetic energy, is expressed as follows:(24)Π=K-U.


## 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:(25)Π^=Kmax-Umax,where Umax and Kmax 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:(26)∂Π^∂ai=0;i=1,2,3⟹K11-ω2M11K12K13K21K22-ω2M22K23K31K32K33-ω2M33a1a2a3=000.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. (27)K11-ω2M11K12K13K21K22-ω2M22K23K31K32K33-ω2M33=0.


## 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:(28)u0=v0=w0=∂w0∂x=0,atx=0,u0=v0=w0=∂w0∂x=0,atx=L.According to the mentioned study, the authors will use the following displacement functions to provide an approximate analytical solution based on the Ritz method:(29)u0=a1x2x-L2cos⁡nθcos⁡ωt,v0=a2x2x-L2sin⁡nθcos⁡ωt,w0=a3x2x-L2cos⁡nθcos⁡ωt.


## 2.8. Mixtures Rule

Suppose that the composite shell that has been strengthened with carbon nanotubes uniformly along thex 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:(30)E11=η1VCNE11CN+VmEm,η2E22=VCNE22CN+VmEm,η3G12=VCNG12CN+VmGm,where E11CN, E22CN, and G12CN are Young’s and shear moduli of carbon nanotubes, respectively. Em and Gm are the corresponding properties, related to the background. ηj is the efficiency parameters of carbon nanotubes which is determined by adaptation of elastic moduli from molecular dynamics simulation results and the mixtures rules. VCN and Vm are volume fractions of carbon nanotubes and background, which are related as follows:(31)VCN+Vm=1.Suppose the volume fraction, VCN, as follows:(32)VCN=wzVCN∗,where(33)VCN∗=wCNwCN+ρCN/ρm-ρCN/ρmwCN,where wCN is the mass fraction of carbon nanotubes. For a uniform distribution of nanotubes, it is considered that VCN=VCN∗. Poisson’s ratio and mass density can be calculated as follows:(34)ν12=VCN∗ν12CN+Vmνm,ρ=VCN∗ρCN+Vmρm.Consequently, according to (9), Aij stiffness equations for a uniform distribution of nanotubes will be obtained as follows:(35)A11=11-ν12ν21VCN∗η1E11CN-Em+Emh,A12=ν121-ν12ν21η2E22CNEmE22CN+Em-E22CNVCN∗h,A22=11-ν12ν21η2E22CNEmE22CN+Em-E22CNVCN∗h,A66=η3G12CNGmG12CN+Gm-G12CNVCN∗h.According to the above equations, for a uniform distribution of nanotubes, there isDij=h2/12Aij; also(36)I0=VCN∗ρCN-ρm+ρmh.


## 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.Table 1
Temperature-dependent material properties of single-walled carbon nanotubes [48].
Temperature(K)
E
11
C
N
T
P
a
E
22
C
N
T
P
a
G
12
C
N
T
P
a
α
11
C
N
×
1
0
-
6
/
K
α
22
C
N
×
1
0
-
6
/
K
300
5.6466
7.0800
1.9445
3.4584
5.1682


### 3.1. Material Properties of PMMA Polymer

The material properties of considered background in temperature equal to300°K are assumed as follows:(37)νm=0.34,ρm=1.15g/cm3,Em=2.5GPa,Gm=Em21+υm=0.933Gpa.For short fiber composites η1 is usually considered to be 0.2. However, there is no experimental method for determining the value of ηj. Here, with adaption of Young’s moduli E11 and E22 obtained from the mixtures rule and molecular dynamics, efficiency parameters η1 and η2 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.Table 2
Young’s moduli for the composite PMMA/CNT: mixtures rule in comparison with molecular dynamics.
V
C
N
∗
Molecular dynamics [49]
Mixtures rule [49, 50]
E
11
E
22
E
11
η
1
E
22
0.12
94.6
2.9
94.78
0.137
2.9
0.17
138.9
4.9
138.68
0.142
4.9
0.28
224.2
5.5
224.50
0.141
5.5


## 3.1. Material Properties of PMMA Polymer

The material properties of considered background in temperature equal to300°K are assumed as follows:(37)νm=0.34,ρm=1.15g/cm3,Em=2.5GPa,Gm=Em21+υm=0.933Gpa.For short fiber composites η1 is usually considered to be 0.2. However, there is no experimental method for determining the value of ηj. Here, with adaption of Young’s moduli E11 and E22 obtained from the mixtures rule and molecular dynamics, efficiency parameters η1 and η2 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.Table 2
Young’s moduli for the composite PMMA/CNT: mixtures rule in comparison with molecular dynamics.
V
C
N
∗
Molecular dynamics [49]
Mixtures rule [49, 50]
E
11
E
22
E
11
η
1
E
22
0.12
94.6
2.9
94.78
0.137
2.9
0.17
138.9
4.9
138.68
0.142
4.9
0.28
224.2
5.5
224.50
0.141
5.5


## 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 h2/12R22 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.Table 3
Nondimensional frequency of a fixed conical shell with a fixed vertex.
β
n
0
1
2
Leisa (presented results)
Leisa (presented results)
Leisa (presented results)
15% diff
0.223180.22310.01165
0.15410.15410.00
0.8010.8010.00
30% diff
0.80030.80040.00975
0.49380.49380.00
0.21590.21570.0764
45% diff
1.50141.50140.00533
0.72560.72550.12277
0.27660.27640.0767
60% diff
1.76591.76600.00113
0.54500.54490.01
0.19270.19260.0745


### 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 ofβ, and various VCN∗ 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 β 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.Table 4
Nondimensional frequency variation of fixed PMMA/CNT composite shell forVCN∗=0.
β
n
0
1
2
3
4
5
15
0.223141
0.126679
0.05998
0.060032
0.121503
0.269507
30
0.800276
0.38597
0.15421
0.100899
0.140962
0.277957
45
1.50132
0.526649
0.193304
0.115026
0.145165
0.27607
60
1.7659
0.374958
0.133334
0.085712
0.126232
0.260175
75
0.739415
0.105912
0.04092
0.042762
0.100341
0.240933Table 5
Nondimensional frequency variation of fixed PMMA/CNT composite shell forVCN∗=0.12.
β
n
0
1
2
3
4
5
15
0.444175
0.281997
0.203815
0.203427
0.264869
0.41323
30
1.21041
0.517539
0.258945
0.203185
0.241179
0.376852
45
2.27495
0.61614
0.260705
0.175626
0.200352
0.327925
60
3.3595
0.433397
0.174726
0.115444
0.149108
0.279531
75
4.15994
0.136299
0.055915
0.050495
0.105159
0.244842Table 6
Nondimensional frequency variation of fixed PMMA/CNT composite shell forVCN∗=0.28.
β
n
0
1
2
3
4
5
15
0.880079
0.580785
0.43786
0.437286
0.549704
0.821152
30
2.26357
0.986998
0.514793
0.413378
0.483145
0.731539
45
4.19439
1.14393
0.494933
0.339869
0.385371
0.618985
60
6.17178
0.797481
0.324699
0.21642
0.278094
0.51685
75
7.64184
0.250454
0.103402
0.093363
0.193304
0.448963Figure 2
Nondimensional frequency variation of fixed PMMA/CNT composite shell forVCN∗=0.Figure 3
Nondimensional frequency variation of fixed PMMA/CNT composite shell forVCN∗=0.12.Figure 4
Nondimensional frequency variation of fixed PMMA/CNT composite shell forVCN∗=0.28.


## 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 ofβ, and various VCN∗ 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 β 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.Table 4
Nondimensional frequency variation of fixed PMMA/CNT composite shell forVCN∗=0.
β
n
0
1
2
3
4
5
15
0.223141
0.126679
0.05998
0.060032
0.121503
0.269507
30
0.800276
0.38597
0.15421
0.100899
0.140962
0.277957
45
1.50132
0.526649
0.193304
0.115026
0.145165
0.27607
60
1.7659
0.374958
0.133334
0.085712
0.126232
0.260175
75
0.739415
0.105912
0.04092
0.042762
0.100341
0.240933Table 5
Nondimensional frequency variation of fixed PMMA/CNT composite shell forVCN∗=0.12.
β
n
0
1
2
3
4
5
15
0.444175
0.281997
0.203815
0.203427
0.264869
0.41323
30
1.21041
0.517539
0.258945
0.203185
0.241179
0.376852
45
2.27495
0.61614
0.260705
0.175626
0.200352
0.327925
60
3.3595
0.433397
0.174726
0.115444
0.149108
0.279531
75
4.15994
0.136299
0.055915
0.050495
0.105159
0.244842Table 6
Nondimensional frequency variation of fixed PMMA/CNT composite shell forVCN∗=0.28.
β
n
0
1
2
3
4
5
15
0.880079
0.580785
0.43786
0.437286
0.549704
0.821152
30
2.26357
0.986998
0.514793
0.413378
0.483145
0.731539
45
4.19439
1.14393
0.494933
0.339869
0.385371
0.618985
60
6.17178
0.797481
0.324699
0.21642
0.278094
0.51685
75
7.64184
0.250454
0.103402
0.093363
0.193304
0.448963Figure 2
Nondimensional frequency variation of fixed PMMA/CNT composite shell forVCN∗=0.Figure 3
Nondimensional frequency variation of fixed PMMA/CNT composite shell forVCN∗=0.12.Figure 4
Nondimensional frequency variation of fixed PMMA/CNT composite shell forVCN∗=0.28.


## 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.



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