Nonlinear Vibration Analysis of Functionally Graded Carbon Nanotube Reinforced Fluid-Conveying Tube in Thermal Environment


 In this paper, the nonlinear free vibration responses of functionally graded nanocomposite fluid-conveying tube reinforced by single-walled carbon nanotubes (SWNTs) in thermal environment is investigated. The SWCNTs gradient distributed in the thickness direction of the tube forms different reinforcement patterns. The materials properties of the functionally graded carbon nanotube-reinforced composites (FG-CNTRC) are estimated by rule of mixture. A higher-order shear deformation theory and Hamilton’s variational principle are employed to derive the motion equations incorporating the thermal and fluid effects. A two-step perturbation method is implemented to obtain the closed-form asymptotic solution for these nonlinear partial differential equations. The nonlinear frequency under several patterns of reinforcement are presented and discussed. We conducted a series of studies aimed at revealing the effects of the flow velocity, environment temperature, geometrical ratios and carbon nanotube volume fraction on the nature frequency.


Introduction
Due to carbon nanotubes have excellent mechanical properties, i.e. high specific strength, high specific modulus and low density [1][2][3][4][5] . Since they were discovered by Iijima [6] , they have been considered as excellent reinforcement of composite materials.
Studies by many scholars show that ， the mechanical [7] , electrical [8] and thermal properties [9] of the polymer can be significantly improved by mixing carbon nanotubes with the polymer matrix at a certain mass fraction. Functionally graded materials (FGM) was first proposed by Japanese researchers in the 1980s. The principle is that reinforced materials are distributed inhomogeneous in space, which can change the mechanical properties of beams, plates and shells. Shen [10] first applied the concept of functionally graded materials to SWNTS reinforced nanocomposite plates, allowing the SWNTS to be graded distributed along the desired direction in an isotropic matrix. Since then, many scholars including Shen have studied FG-CNTRC beam, tube, plate, shell and other structures [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] . Readers can refer to the comprehensive literature review prepared by Liew [27] for relevant researches.
Tube is the basic unit of many engineering structures, which is widely used in oil, chemical and nuclear industry. Therefore, many researchers have used different analytical and numerical methods to study the stability and vibration analysis of fluid conveying tube. Zhen [28] first studied the nonlinear vibration of the supercritical fluidconveying tube composed of FGM with initial curvature. Liang [29] investigate the stability and nonlinear parametric vibration of a spinning fluid-conveying tube using analytical and numerical methods. Rahim Abdollahi [30] studies the stability of a spinning fluid-conveying tube under simultaneous internal and external fluid loads using Euler-Bernoulli beam theory. Zhu [31] studies the free and forced vibration of a fluid-conveying tube composed of FGM under elastic support based on Euler-Bernoulli beam theory and considering von Karman assumption and damping effect. Zhou [32] explore the nonlinear vibrations of cantilever fluid-conveying tube under axial excitation. Tan [33] studies forced vibration of fluid-conveying tube under supercritical condition by using Timoshenko beam theory. Based on Euler-Bernoulli beam theory, Shahali [34] studies the nonlinear dynamic response of fluid-conveying tube under the action of uniform external cross flow. A.R. Askarian [35] studies the vibration response of fluid-conveying tube under general boundary conditions using Euler-Bernoulli beam theory. Lu [36] first studies the influence of vibration on fatigue performance of FGM fluid-conveying tube based on Euler-Bernoulli beam theory. Khodabakhsh [37] studies the post-buckling and nonlinear vibration of FGM fluid-conveying tube based on the Timoshenko beam theory. M. Heshmati [38] studies the stability and free vibration of FGM fluid-conveying tube with eccentric geometric defects.
As far as we know, there are no researches on nonlinear vibration of functionally graded carbon nanotube reinforced fluid-conveying tube in thermal environment by using higher-order shear deformation theory. In this paper, an investigation on the nonlinear free vibration responses of functionally graded nanocomposite fluidconveying tube reinforced by SWNTs in thermal environment is presented. A higherorder shear deformation theory and Hamilton's variational principle are employed to derive the governing equations incorporating the thermal and fluid effects. A two-step perturbation method is implemented to obtain the closed-form asymptotic solution for these nonlinear partial differential equations. The nonlinear frequency under several patterns of reinforcement are presented and discussed. We conducted a series of studies aimed at revealing the effects of the flow velocity, environment temperature, geometrical ratios /  The effective material properties of FG-CNTRC beams can be estimated by the rule of mixture [39] . ( ) 12 12 In which, 12 , CNT m ρ ρ are defined as the Poisson's ratios and mass density of the CNT and matrix, respectively.
In this paper, five types of the FG-CNTRC tube were considered, which have the form as: Where, equations (5)  to those from the MD simulations given by [40] . The mechanical properties of composite beams such as elastic modulus and thermal expansion coefficient) change significantly at high temperatures, so it is necessary to consider the temperature dependence of materials in order to accurately predict the behavior of FG-CNTRC tube. The Eqs.11-12 shows the relationship between the elastic modulus and thermal expansion coefficient of PMMA matrix and temperature.  m P T P P T P T P T P T = + ⋅ + ⋅ + ⋅ + ⋅ Table 1 Temperature-dependent material properties of (10,10) SWCNT ( [13] , [41] ). Tube length = 9.26 nm, tube mean radius = 0.68 nm, tube thickness = 0.067 nm Temperature (K) 11 CNT E (GPa) 22 CNT E (GPa) 12 CNT G (GPa) 11 CNT  Table 2 Temperature-dependent properties of PMMA ( [13] , [42] )

Governing equations
Based on the higher-order shear deformation beam model for tubes [43] , the displacement field of the tubes is expressed as: Here, ( ) ( ) It should be mentioned that, the position of neutral axis and centroid of the cross section is the same for present tubes. Based on the von Karman assumption, the nonlinear geometrical relationship of the tube can be expressed as: The constitutive relationships of the tube considering a uniform temperature field are as follows: 11 Where 11 12 13 , , E G G are Young's modulus and shear modulus, x α is the thermal expansion coefficient along x direction, and T ∆ is the temperature offset from the reference temperature at which the tube in a stress free state.
The Hamilton's variational principle is implemented to derive the partial differential equation of the motion as follows: The virtual strain energy of the tube s U δ is given by Substitute Eq.16 into Eq.19 results in the following relations: The virtual kinetic energy of the fluid-conveying tube k U δ is given by Substitute Eq.16 into Eq.22 results in the following relations: With [ ] 2 2 0 1 2 3 , , , , , , ρ is the density of the fluid, f v is the velocity of the flow and A is the cross sectional of the flow.
The virtual work of the external pressure w U δ is given by Where q is the load per unit length along z direction.
When the in-plane inertia is neglected, according to Hamiltonian variational principle, the governing equation represented by general force and moment is: From the first equation in Eq.27, we can find that the x N is a constant value. Due to the simply supported boundary conditions ( (0) ( ) 0 u u L = = ), the expressing of x N can be given as, Substituting Eq.17 back into Eq.21 results in the following relations: With [ ] Substituting Eq.30 into Eq.27, the governing equations represented by displacement components can be given as, For generality, the following dimensionless parameters are introduced: ; ; T T x x π ω ϕ π ω ϕ τ π ρ π π π π ρ ρ ρ ρ π π ρ π ρ π One thing to be noted that the present higher-order shear deformation model can

Solving method
A two-step perturbation technique [44] is used to solve the nonlinear partial differential equations. In this case, the displacements and transverse load are expended as the following from, Here, the ε is a small perturbation parameter with no physical meaning. It should be mentioned that, t ετ =  is introduced to delay the dynamic terms to higher-order perturbation equations.

A m A m A m A m B A A A m A A m A
The solution of Eq.36 is assumed as follows The solution of Eq.37 is assumed as follows   (2 ) sin( ) 4 Therefore, the final asymptotic solution is as follows Since the nonlinear free vibration has no transverse load, i.e.  0 0 q = . we can obtain the following duffing equation based on Galerkin method, Where the t  has been replaced byτ . 2  3  3  2  3  5  3  5  0  0  1  2  3  2  2  4  5  4  5   2  4  2  3  5  4  3  1  2  3  5  2  2  4  5  4  5   2 4 3 1 The closed-form solution of Eq can be expressed as,  Table 5 gives the comparisons of dimensionless fundamental frequencies of FGM tube composed of metal and ceramics, with that in Ref. [45] . In this example, the tube is not supported on an elastic foundation. For detailed material parameters, please refer to the original paper. It can be clearly found that the results are highly consistent with the literature. Ref. [45] 0     Fig.8 shows that the higher the volume fraction coefficient, the higher the critical buckling flow velocity of the tube. Fig.9 shows that the critical buckling flow velocity increases as the ratio of inner diameter to outer diameter ( / Ri Ro ) increases. As can be seen from Fig.10, the nonlinear fundamental frequency decreases as the temperature increases. When the fundamental frequency is zero, it corresponds to the critical buckling temperature. Same as Fig.7, V beam has the highest critical buckling temperature among several patterns of reinforcement, followed by X beam, UD beam, O beam and ∧ beam. Although the higher the volume fraction coefficient is, the higher the critical buckling flow velocity is, Fig.11 shows that the critical buckling temperature is the lowest when the volume fraction is 0.28. Fig.12 gives that as the geometrical ratio / L Ro increases, the critical buckling temperature decreases, but the rate of change decreases. Fig.13 shows that the critical buckling temperature increases with the increase of the ratio of inner to outer diameter ( / Ri Ro ), but the change decreases when the ratio is small.  2) V beam has the highest critical buckling flow velocity and temperature among several patterns of reinforcement, followed by X-beam, UD-beam, O-beam and ∧-beam. However, the nonlinearity of ∧-beam is the most sensitive to amplitude variation, followed by O-beam, UD-beam, X-beam, and V-beam.
3) Although the higher the volume fraction coefficient is, the higher the critical buckling flow velocity is, the critical buckling temperature is the lowest when the volume fraction is 0. 28