Abstract
Free vibration of a bimaterial circular nanotube is investigated. The tube is formed by bonding together a Si_{3}N_{4}/SUS304 functionally graded upper semi tube and a ZrO_{2}/Ti6Al4V functionally graded lower semi tube. The material properties of the tube are assumed to vary along the radius according to power law with the power index of upper semi tube differing from that of lower semi tube. Based on nonlocal elasticity theory and Hamilton’s principle, a refined beam model considering the effect of transverse shear deformation is used to derive the governing equations, then analytical solution is obtained by using a twosteps perturbation method. Our results were compared with the existing ones. The effects on tube’s linear and nonlinear frequency are analyzed of the factors, including small scale parameter, temperature, the double volume fraction indexes, slenderness ratio and different types of beam model. A new approach is suggested in this article to change the natural frequency of the tubes by adjusting constituent materials. In contrast to conventional approach, the new one can result in more accurate frequency control in the same dimensionless size of tubes.
Key words:
Functionally graded material; Nonlocal; Vibration; Bisemitubes; Perturbation method;
1. Introduction
Nonhomogeneous composite materials composed of functionally graded material (FGM) in which the effective material properties can be changed in a certain direction have captured extensive attention in a multitude of industries (Jha et al., 2013Jha, D. K., Kant, T., & Singh, R. K. (2013). A critical review of recent research on functionally graded plates. Composite Structures, 96(4), 833849.; Koizumi, 1997Koizumi, M. (1997). FGM activities in japan. Composites Part B Engineering, 28(12), 14.). Owing to superior performance, such material could satisfy the majority of design requirements of each component. In order to analyze its mechanical properties, classical formulas were modified and new theories were proposed (Huang and Li, 2010bHuang, Y., & Li, X. F. (2010b). Huang, y. and li,x.f. a new approach for free vibration of axially functionally graded beams with nonuniform crosssection. journal of sound and vibration, 329, 22912303. Journal of Sound & Vibration, 329(11), 22912303.; Shafiei et al., 2016aShafiei, N., Kazemi, M., & Ghadiri, M. (2016a). On sizedependent vibration of rotary axially functionally graded microbeam. International Journal of Engineering Science, 101, 2944.; Shafiei et al., 2016bShafiei, N., Kazemi, M., & Ghadiri, M. (2016b). Nonlinear vibration of axially functionally graded tapered microbeams. International Journal of Engineering Science, 102, 1226.; Zhang, 2013Zhang, D. G. (2013). Nonlinear bending analysis of fgm beams based on physical neutral surface and high order shear deformation theory. Composite Structures, 100(5), 121126.; Hosseini et al., 2016Hosseini, S. A. H., & Rahmani, O. (2016). Thermomechanical vibration of curved functionally graded nanobeam based on nonlocal elasticity. Journal of Thermal Stresses, 39(10), 12521267.; Thinh et al., 2016Thinh, T. I., Tu, T. M., Quoc, T. H., Long, N. V., Thinh, T. I., & Tu, T. M., et al. (2016). Vibration and buckling analysis of functionally graded plates using new eightunknown higher order shear deformation theory. Lat.am.j.solids Struct, 13(3), 456477.). However, with emerging new demands in manufacturing and engineering, most of the previous studies related to FGMs whose effective material properties only vary in the direction of thickness or length can’t meet those needs and challenges very well (NematAlla, 2003NematAlla, M. (2003). Reduction of thermal stresses by developing twodimensional functionally graded materials. International Journal of Solids & Structures, 40(26), 73397356.; Fan et al., 2013Fan, P., Fang, Z. Z., & Guo, J. (2013). A review of liquid phase migration and methods for fabrication of functionally graded cemented tungsten carbide. International Journal of Refractory Metals & Hard Materials, 36(1), 29.; Lü et al., 2009Lü, C.F., C.W. Lim, & W.Q. Chen. (2009). Sizedependent elastic behavior of FGM ultrathin films based on generalized refined theory. International Journal of Solids & Structures, 46(5), 11761185.; Wang et al., 2016Wang, J. F., Zhang, L. W., & Liew, K. M. (2016). A multiscale modeling of cntreinforced cement composites. Computer Methods in Applied Mechanics & Engineering, 309, 411433.; Lei et al., 2016Lei, Z. X., Zhang, L. W., & Liew, K. M. (2016). Buckling analysis of cntreinforced functionally graded laminated composite plates. Composite Structures, 152, 6273.; Gupta et al., 2015Gupta, A., & Talha, M. (2015). Recent development in modeling and analysis of functionally graded materials and structures. Progress in Aerospace Sciences, 79, 114.; Thang and NguyenThoi, 2016Thang, P. T., & NguyenThoi, T. (2016). Effect of stiffeners on nonlinear buckling of cylindrical shells with functionally graded coatings under torsional load. Composite Structures, 153, 654661.). Consequently, in recent years, more and more researchers have been studying structural components made of two or threedirectional functionally graded materials and subjected to different types of functionally graded distribution (Şimşek, 2015Şimşek, M. (2015). Bidirectional functionally graded materials (BDFGMS) for free and forced vibration of timoshenko beams with various boundary conditions. Composite Structures, 133, 968978.; Hao and Wei, 2016Hao, D., & Wei, C. (2016). Dynamic characteristics analysis of bidirectional functionally graded timoshenko beams. Composite Structures, 141, 253263; Lü et al., 2008Lü, C.F., W.Q. Chen, R.Q. Xu, & C.W. Lim. (2008). Semianalytical elasticity solutions for bidirectional functionally graded beams. International Journal of Solids & Structures, 45(1), 258275.; Nguyen et al., 2017Nguyen, D. K., Nguyen, Q. H., Tran, T. T., & Bui, V. T. (2017). Vibration of bidimensional functionally graded timoshenko beams excited by a moving load. Acta Mechanica, 228(1), 141155.; Pydah and Sabale, 2017Pydah, A., & Sabale, A. (2017). Static analysis of bidirectional functionally graded curved beams. Composite Structures, 160(15), 867876.; Nejad and Hadi, 2016aNejad, M. Z., & Hadi, A. (2016a). Nonlocal analysis of free vibration of bidirectional functionally graded eulerbernoulli nanobeams. International Journal of Engineering Science, 105, 111.; Nejad and Hadi, 2016bNejad, M. Z., & Hadi, A. (2016b). Eringen's nonlocal elasticity theory for bending analysis of bidirectional functionally graded eulerbernoulli nanobeams. International Journal of Engineering Science, 106, 19.; Nejad et al., 2016Nejad, M. Z., Hadi, A., & Rastgoo, A. (2016). Buckling analysis of arbitrary twodirectional functionally graded eulerbernoulli nanobeams based on nonlocal elasticity theory. International Journal of Engineering Science, 103, 110.).
In practical uses, FGMs may be fabricated into different structural components, e.g. plates, rods and tubes. Among these components, tube is one of the most important and frequently used structural components in many industries. Fig.1 shows some potential fields for functionally graded materials as well as tubes application(Hosseini et al., 2017Hosseini, M., Gorgani, H. H., Shishesaz, M., & Hadi, A. (2017). Sizedependent stress analysis of singlewall carbon nanotube based on strain gradient theory. International Journal of Applied Mechanics, 9(6), 1750087.; Jha et al., 2013Jha, D. K., Kant, T., & Singh, R. K. (2013). A critical review of recent research on functionally graded plates. Composite Structures, 96(4), 833849.; Zhang et al., 2014Zhang, X., Wen, Z., & Zhang, H. (2014). Axial crushing and optimal design of square tubes with graded thickness. ThinWalled Structures, 84(84), 263274.; Kiani and Eslami, 2013Kiani, Y., & Eslami, M. R. (2013). Thermomechanical buckling oftemperaturedependent fgm beams. Latin American Journal of Solids & Structures, 10(2), 223246.; Kakar, 2013Kakar, R. (2013). Magnetoelectroviscoelastic torsional waves in aeolotropic tube under initial compression stress. Latin American Journal of Solids & Structures, 11(4), 580597.; Djamaluddin et al. 2015Djamaluddin, F., Abdullah, S., Ariffin, A. K., Nopiah, Z. M., Djamaluddin, F., & Abdullah, S., et al. (2015). Multi objective optimization of foamfilled circular tubes for quasistatic and dynamic responses. Latin American Journal of Solids & Structures, 12(6), 11261143.; Huang et al. 2016Huang, B. W., Yu, P. P., Tseng, J. G., Huang, B. W., Yu, P. P., & Tseng, J. G. (2016). Dynamic properties of coupled tubearray structures with the axial loads. Lat.am.j.solids Struct, 13(12), 22202230.; Adeli et al. 2017Adeli, M. M., Hadi, A., Hosseini, M., & Gorgani, H. H. (2017). Torsional vibration of nanocone based on nonlocal strain gradient elasticity theory. European Physical Journal Plus, 132(9), 393.; Hadi et al. 2018Hadi, A., Nejad, M. Z., & Hosseini, M. (2018). Vibrations of threedimensionally graded nanobeams. International Journal of Engineering Science, 128, 1223.; Shishesaz et al. 2017Shishesaz, M., Hosseini, M., Tahan, K. N., & Hadi, A. (2017). Analysis of functionally graded nanodisks under thermoelastic loading based on the strain gradient theory. Acta Mechanica, 228, 128.). For efficient use of such components, it is necessary to study its mechanical behaviors like vibration, buckling and bending. Thereinto, the investigation on the vibration of FGM tubes and cylindrical shells was, is and remains to be a hot research subject. Yang et al. (2014Yang, T. Z., Ji, S., Yang, X. D., & Fang, B. (2014). Microfluidinduced nonlinear free vibration of microtubes. International Journal of Engineering Science, 76(4), 4755.) studied the effect of outer diameter, Poisson’s ratio and flow velocity on the vibration frequency. Zhong et al. (2016Zhong, J., Fu, Y., Wan, D., & Li, Y. (2016). Nonlinear bending and vibration of functionally graded tubes resting on elastic foundations in thermal environment based on a refined beam model. Applied Mathematical Modelling, 40(1718), 76017614.) utilized a twosteps perturbation method to analyze the influence of different dimension parameters on linear frequency and nonlinear frequency.
As can be shown in the open literature, most of the researchers change natural frequency of tubes through altering dimensions. However, they could not change the frequency of tubes to satisfy demands very well since dimension of tubes cannot be changed arbitrarily (Dohmann and Hartl, 1997Dohmann, F., & Hartl, C. (1997). Tube hydroformingresearch and practical application. Journal of Materials Processing Technology, 71(1), 174186.). To overcome this deficiency, we design double functionally graded distributions to change and control frequencies of tubes through adjusting respective material compositions. The design has an advantage to break the limits of dimensions.
It is a practicable technique to bond two materials together (Qiao and Chen, 2008Qiao, P., & Chen, F. (2008). An improved adhesively bonded bimaterial beam model for plated beams. Engineering Structures, 30(7), 19491957.; Suhir, 2016Suhir, E. (2016). Bimaterial assembly subjected to thermal stress: propensity to delamination assessed using interfacial compliance model. Journal of Materials Science Materials in Electronics, 27(7), 17.). The strengthening method can provide effective stress transfer from one to the other and secure the durability of components. Owing to good performance, bimaterial structures have been widely used in engineering practice, such as scarf joints (Le, 2011Le, J. L. (2011). General size effect on strength of bimaterial quasibrittle structures. International Journal of Fracture, 172(2), 151160.), pumping membrane (Hsu et al., 2000Hsu, C., Hsu, W., & Hsu, C. (2000). A twoway membranetype microactuator with continuous deflections. Journal of Micromechanics & Microengineering, 10(3), 387.), thermal sensor (Lim et al., 2005Lim, S. H., Choi, J., Horowitz, R., & Majumdar, A. (2005). Design and fabrication of a novel bimorph microoptomechanical sensor. Journal of Microelectromechanical Systems, 14(4), 683690.). Noël et al. (2016Noël, L., Miegroet, L. V., & Duysinx, P. (2016). Analytical sensitivity analysis using the extended finite element method in shape optimization of bimaterial structures. International Journal for Numerical Methods in Engineering, 107(8), 669695.) undertook an analytical sensitivity analysis for shape optimization of bimaterial structure. Huang et al. (2009Huang, C. S., Cheng, Y. T., Chung, J., & Hsu, W. (2009). Investigation of nibased thermal bimaterial structure for sensor and actuator application. Sensors & Actuators A Physical, 149(2), 298304.) studied deflection of two kinds of thermal bimaterial structure in which two materials present excellent interlaminar bonding ability. Zheng (2012Zheng, Y. (2012). Analysis of dualbeam asymmetrical torsional bimaterial cantilever for temperature sensing applications. Physics. DOI: arXiv:1208.0939) developed another analytical model to study torsional bimaterial cantilever, which can predict some performance.
Various methods have been developed to solve equations of nonlinear vibration, such as high dimensional harmonic balance method(Hall et al., 2002Hall, K. C., Thomas, J. P., & Clark, W. S. (2002). Computation of unsteady nonlinear flows in cascades using a harmonic balance technique. Aiaa Journal, 40(5), 879886.; Liu et al., 2007Liu, L., Dowell, E. H., & Hall, K. C. (2007). A novel harmonic balance analysis for the van der pol oscillator. International Journal of NonLinear Mechanics, 42(1), 212.), harmonic balance method(Dai et al., 2014Dai, H., Yue, X., Yuan, J., & Atluri, S. N. (2014). A time domain collocation method for studying the aeroelasticity of a two dimensional airfoil with a structural nonlinearity. Journal of Computational Physics, 270(3), 214237.), perturbation methods(Mook and Nayfeh,1979Mook, D. & Nayfeh, A. Nonlinear Oscillations, John Wiley & sons, New York, 1979.). Perturbation methods are frequently used. Nazemnezhad and HosseiniHashemi (2014Nazemnezhad, R., & HosseiniHashemi, S. (2014). Nonlocal nonlinear free vibration of functionally graded nanobeams. Composite Structures, 110(110), 192199.) studied nonlinear free vibration of FG beams under different boundary conditions with the aid of the multiplescale perturbation method. Ghadiri et al. (2017Ghadiri, M., Rajabpour, A., & Akbarshahi, A. (2017). Nonlinear forced vibration analysis of nanobeams subjected to moving concentrated load resting on a viscoelastic foundation considering thermal and surface effects. Applied Mathematical Modelling, 50, 676694.) used EulerBernoulli beam model to analyze forced vibration of beams subjected to moving concentrated load. To obtain analytical solution from nonlinear differential equations, they used the perturbation technique. Shen and Wang (2014Shen, H. S., & Wang, Z. X. (2014). Nonlinear analysis of shear deformable fgm beams resting on elastic foundations in thermal environments. International Journal of Mechanical Sciences, 81(4), 195206.) used a twosteps perturbation technique, to investigate vibration of beams exposed to different types of thermal environment.
In analysis of vibration problems involving nanostructures(Hosseini et al. 2018Hosseini, M., Hadi, A., Malekshahi, A., & Shishesaz, M. (2018). A review of sizedependent elasticity for nanostructures, Journal of Computational Applied Mechanics, 49, 197211.), the nonlocal elasticity theory proposed by Eringen (1983Eringen, A. C. (1983). on differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. 54, 47034710. Journal of Applied Physics, 54(9), 47034710.) is extensively used to study the smallscale effec ts. To analyze postbuckling of porous nanotubes, She et al. (2017aShe, G. L., Yuan, F. G., Ren, Y. R., & Xiao, W. S. (2017a). On buckling and postbuckling behavior of nanotubes. International Journal of Engineering Science, 121, 130142.) put forward a nonclassical model based on nonlocal elasticity theory. Rahmani and Pedram (2014Rahmani, O., & Pedram, O. (2014). Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal timoshenko beam theory. International Journal of Engineering Science, 77(7), 5570.) analyzed the sizedependent effect on the vibration of FGM nanobeams by utilizing nonlocal elasticity theory and obtained a closed form solution. Zenkour et al. (2015Zenkour, A. M., Abouelregal, A. E., K. A. Alnefaie, X. Zhang, & E. C. Aifantis. (2015). Nonlocal thermoelasticity theory for thermalshocknanobeams with temperaturedependent thermal conductivity. Journal of Thermal Stresses, 38(9), 205217.) used this theory to describe the smallscale effect on the behavior of respective field variables. Salehipour et al. (2015Salehipour, H., Shahidi, A. R., & Nahvi, H. (2015). Modified nonlocal elasticity theory for functionally graded materials. International Journal of Engineering Science, 90, 4457.). modified the nonlocal elasticity theory in analysis of FGM at nanoscale. The influence of nonlocal parameter was analyzed in detail based on an improved nonlocal elasticity theory. Hosseini and Rahmani (2016)Hosseini, M., Shishesaz, M., Tahan, K. N., & Hadi, A. (2016). Stress analysis of rotating nanodisks of variable thickness made of functionally graded materials. International Journal of Engineering Science, 109, 2953. investigated thermal buckling as well as nonlinear vibration of a curved FG beam based on Eringen’s different formulation. Narendar (2017Narendar, S. (2017). Nonlocal thermodynamic response of a rod. Journal of Thermal Stresses, 40(12), 111.) studied thermodynamic of a rod subjected to a moving heat source based on the Eringen’s nonlocal model. Differing from above references adopted Eringen’s equivalent differential formulation, Tuna and Kirca (2016Tuna, M., & Kirca, M. (2016). Exact solution of eringen's nonlocal integral model for bending of eulerbernoulli and timoshenko beams. International Journal of Engineering Science, 105, 8092.) analyzed buckling and vibration of beams by using the original integral constitutive equation. To obtain the exact solution of the original integral model, the Laplace transform method was adopted.
To date, there is no free vibration analysis of nanotubes consisted of functionally graded bisemitubes.From the perspective of bimaterial structure, we design a tube that is formed by bonding together a Si_{3}N_{4}/SUS304 functionally graded upper semi tube and a ZrO_{2}/Ti6Al4V functionally graded lower semi tube and used a twosteps perturbation method to obtain corresponding analytical solutions. Through adjusting respective material compositions, linear frequencies and nonlinear frequencies is studied in detail. The purpose of this work is to put forward alternative approach to change and control frequencies of tubes whose results can provide a route toward designing tubes and undertaking nananalysis of materials.
2. Basic theoretical formulation
2.1 Functionally graded bisemitubes
Proposed in this section, the basic equations of bifunctionally graded materials can serve as benchmark theoretical formulation for exactly analytical solutions of the equilibrium equations in later chapters.
An FGM tube with outer radius R _{0}, inner radius R _{i}, as well as length L shown in Fig. 2 is formed by bonding together a Si_{3}N_{4}/SUS304 functionally graded upper semitube with a ZrO_{2}/Ti6Al4V functionally graded lower semitube. The behavior of the FGM tube made of two types of functionally graded semitube must comply with general mechanics theorem; besides, both Si_{3}N_{4}/SUS304 functionally graded semitube material properties and ZrO_{2}/Ti6Al4V functionally graded semitube material properties should continuously vary along the radius directions, respectively. Thus, the effective material properties P _{f} including Young’s modulus, thermal expansion, Poisson’s ratio, as well as mass density, can be assumed to vary according to powerlaw and thus can be expressed as:
where p _{1}, p _{2}, p _{3}, p _{4} stand for the material constituents of SUS304, Si_{3}N_{4}, Ti6Al4V and ZrO_{2} respectively. The symbol of N _{1} aligned with N _{2} represent the volume fraction index in their respective angle intervals. The temperature factor is assumed to be a nonlinear function of temperature, which can be described as (Ghiasian et al., 2014Ghiasian, S. E., Kiani, Y., Sadighi, M., & Eslami, M. R. (2014). Thermal buckling of shear deformable temperature dependent circular/annular fgm plates. International Journal of Mechanical Sciences, 81(4), 137148)
Here P _{0}, P _{1,} P _{1}, P _{2}, and P _{3} denote the coefficients of Kelvin’s temperaturedependence which are tabulated in Table 1.
Temperaturedependent coefficients of material properties (Reddy and Chin, 1998Reddy, J. N., & Chin, C. D. (1998). Thermomechanical analysis of functionally graded cylinders and plates. Journal of Thermal Stresses, 21(6), 593626.).
The temperature can be separated into two parts: T=T _{0}+ΔT, where ΔT is the thermal increment from certain reference temperature T _{0}=300K at which a tube has no thermal strains. Notice from the expression of the effective material properties P _{f} that inner surface (r=R _{i}) of FGM tube is made of SUS304 and Ti6Al4V, whereas outer surface (r=R _{0}) of FGM tube is made of Si_{3}N_{4} and ZrO_{2}. Obviously, this differs from the conventional functionally graded tube or cylinder. When N _{1} and N _{2} are both equal to zero, the composite tube of the effective material properties P _{f} degenerates to the bimaterial tube consisted of two different materials.
2.2 nonlocal elasticity theory
When dealing with nanostructures, the effect of longrange interatomic forces can’t be ignored in the process of analysis. Eringen (1983Eringen, A. C. (1983). on differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. 54, 47034710. Journal of Applied Physics, 54(9), 47034710.) took the effect of longrange interatomic into account and put forward the nonlocal elasticity theory based on the experiment of phonon dispersion as well as atomic theory of the lattice dynamics. According to his theory, the stress at any point x in a body varies along with not only the strain at that point but also those at all other points x’ of the body. Therefore, the whole stress field σ _{ij}(x) can be arrived at
where the factor α (xx’, μ), small scale, is the nonlocal attenuation function, which incorporates into the constitutive equations the influences at the reference point x generated by the local strain at the source point x’, where xx’ represents the Euclidean distance. The factor α can be expressed as α=e _{0} a/L where e _{0} is a material constant determined by experiments or by reliable theoretical models. The parameters a and L are the internal (e.g. lattice parameter) and external characteristic lengths (e.g. crack length and wavelength) of the nanosolids, respectively.
However, it is difficult to obtain analytical results by applying the above expressions, therefore Eringen (1983Eringen, A. C. (1983). on differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. 54, 47034710. Journal of Applied Physics, 54(9), 47034710.) turned the integral constitutive relations to an equivalent differential form as
It is obviously observed that the nonlocal constitutive relation can degenerate to the classical elasticity theory when the result of μ is taken as zero.
2.3 Equilibrium equations
As shown in Fig. 2, the functionally graded nanotube of length L¸ inner radius R _{i} and outer radius R _{0}, is consisted of upper semitube made of functionally graded materials (Si_{3}N_{4}/SUS304) and lower semitube made of functionally graded materials (ZrO_{2}/Ti6Al4V). Besides, the nanotube is subjected to a uniform temperature field as well as a uniform transverse load.
Suppose that its middleaxis coincides with the Oxaxis of a Cartesian coordinate system Oxyz and the positive Zaxis is perpendicular to the Xaxis and directed upwards. The origin of a Cartesian coordinate system Oxyz is set at the middle surface of the tube
For circular tube, three displacement functions (u _{1}, u _{2}, u _{3}) can be listed as follows (Zhang and Fu, 2013Zhang, P., & Fu, Y. (2013). A higherorder beam model for tubes. European Journal of Mechanics A/solids, 38(3), 1219.)
where u(x, t) and w(x, t) stand for the Xdire and lateral displacements respectively, and θ(x, t) stands for the rotation of the normal relative to the Y axis. To easily deduce the relationship of the strain and displacement based on the von karman nonlinearity theory, the displacement function Eq. (4) can be rewritten in a simplification form
in which
The nonlinear straindisplacement expressions in line with the von karman nonlinearity theory can be induced as:
According to the Hooke’s law, for the case of a uniform thermal environment, the stresses associated with strain components from Eq. (6) can be determined as
in which
It should be mentioned that the Poisson’s ratio v of four materials in this tube are all set to be 0.3, the same value (She et al., 2017bShe, G. L., Yuan, F. G., & Ren, Y. R. (2017b). Thermal buckling and postbuckling analysis of functionally graded beams based on a general higherorder shear deformation theory. Applied Mathematical Modelling, 47.340357.). This assumption is made by referring the fact that, Cao and Evans (1989Cao, H. C., & Evans, A. G. (1989). An experimental study of the fracture resistance of bimaterial interfaces. Mechanics of Materials, 7(4), 295304.) in their experiment found the upper and lower beams possess similar Poisson’s ratio when the thickness ratio is equivalent to 0.1 and Other researchers (Yang et al., 2014Yang, T. Z., Ji, S., Yang, X. D., & Fang, B. (2014). Microfluidinduced nonlinear free vibration of microtubes. International Journal of Engineering Science, 76(4), 4755.) also demonstrated different Poisson’s ratio for FGM to have no obvious effect on the results.
The necessary boundary conditions for the stress of the tube at the places (r=R _{0}, R _{i}) must satisfy Eq. (8).
The virtual strain energy of the tube is assumed to be
in which Ω stands for the volume of the tube. By substituting Eq. (6) into Eq. (9), the virtual strain energy is reappraised as
Besides, the work performed by the external force is illustrated as
where
The kinetic energy of the studied tube is elevated as
So, Eq. (12) can be deduced to
In terms of the Hamilton principle, the corresponding governing equations can be induced as
Substituting Eq. (10), (11), and (13) into Eq. (14) and set these coefficients of δu, δθ, δw into zero, we obtain the nonlinear equilibrium equations can be expressed as
where N in Eq. (15) is a constant and can be written in the form.
Moreover, certain general forces and moments of inertia involved with Eq. (15) are presented as
Combined with the nonlocal constitutive relations, general forces of the tube are defined as
where Δ is the Laplace operator. Other coefficients in Eq. (18) and (19) are given by
Eq. (15) can be reformulated as
For simply supported ends, necessary boundary conditions of the tube are given by
For the calculation convenience, we introduce the nondimensional parameters.
where
General governing equations described in Eq. (20) can be rewritten in terms of dimensionless variables as
Meanwhile, the dimensionless boundary conditions can be described as
3. Solution of the model
In this section, we obtain corresponding analytical solutions of Eq. (22) and (23) by using a twosteps perturbation procedure. To begin with, to get a set of vibration equations, we suppose that the expanded form of dimensionless displacement, dimensionless rotation angle and dimensionless transverse load can be expressed as:
Where the small perturbation parameter(ε) has no physical meaning, which is introduced into Eq. (22) and (23). Then, we collect terms of the same order(ε) to arrive at
To solve the differential perturbation equations, asymptotic solutions of dimensionless displacement as well as dimensionless rotation angle, satisfying simply supported ends, are given by
Substituting Eq. (30) into Eq. (26), we could have
By substituting Eq. (30) and (31) into Eq. (27), we obtain the equation to determine${\lambda}_{q}^{1}$.
Later, the substitution of Eq. (30) and Eq. (31) into Eq. (28) yields:
${\lambda}_{q}^{3}$is determined by substituting Eq. (30), (31) and (33) into Eq. (29)
Eventually, the analytical solution of nondimension transverse load can be expressed as
Because the value of λ _{q} is equivalent to zero when solving free vibration problems, we apply the method of Galerkin in Eq. (35) to obtain the Duffing equation.
where
So, the analytical solution of Eq. (36) can be written as
where the symbol of${A}_{m}^{1}$ ($={W}_{m}/L$) is dimensionless amplitude of the vibration of the tube. ω _{L} and ω _{NL} stand for dimensionless linear frequency and dimensionless nonlinear frequency, respectively.
So far, most of the theories were established based on three classical models. They are Euler beam model, Timoshenko beam model as well as Reddy beam model. They could be taken to be the specific of our model.
When f=z, the displacement function Eq. (5) can be converted to that of EulerBernoulli model. Corresponding coefficients are
When f=0, the displacement function of Eq. (5) can be converted to that of Timoshenko model. The corresponding coefficients are
where the shear factor k _{s} is copied from Zhang and Fu (2013Zhang, D. G. (2013). Nonlinear bending analysis of fgm beams based on physical neutral surface and high order shear deformation theory. Composite Structures, 100(5), 121126.) and is given by
in which${\overline{R}}_{i}={R}_{i}/{R}_{0}$and the Poisson’s ratio v is assumed to be the average value of the four materials.
When$f=4{z}^{3}/(3{h}^{2})$,the displacement function Eq. (5) can be converted to that of Reddy model. Corresponding coefficients are the same as ones of the present model. More detail information can be found in Eq. (36).
4. Results and discussions
In this part, we fully utilize above analytical solution to discuss the vibration behavior of the nanotubes consisted of functionally graded bisemitubes. As is shown by the results, some novel behaviors of the bisemitubes are rather different from those of the conventional FGM nanotubes.
4.1 validation research
Owing to no existing data for the functionally graded bimaterial tube in the published literature, both dimensionless free vibration frequencies of functionally graded tube and nondimension frequencies of isotropic tube are calculated by present mathematical model. The results obtained in such a way can be conveniently used to verify the present deduced results directly. In Table. 2, the dimensionless frequency of a functionally graded nanotube with (R _{i}=0.5R _{0}, L=20R _{0,} R _{0}=1nm, T=300K, N=1) is used to verify the current solution. That this table demonstrates a small difference between the results reported by FuhGwo Yuan and the ones in this study clearly indicates that present solution is reliable and reasonable. To further validate the present model, the dimensionless natural frequency for an isotropic cylindrical shell with (R _{i}=0.998R _{0}, L=20R _{0}, v=0.3, T=300K) under simply supported ends are tabulated in Table 3. As is seen in Table 3, present results show a good agreement with published ones. Therefore, the two examples convince us the validation of the present model.
Comparisons of nondimension natural frequencies ($\omega \text{=}\Omega \sqrt{{\rho}_{0}/{E}_{0}}{L}^{2}/{R}_{0}$)for a simply supported functionally graded nanotube.
Comparisons of nondimension natural frequencies ($\omega =\Omega {R}_{0}\sqrt{(1{v}^{2}){\rho}_{0}/{E}_{0}}$) of an isotropic cylindrical shell with simply supported ends.
4.2 discussions of physical parameters
The formulation of the nondimension frequency is$\omega =\Omega \left({L}^{2}/{R}_{0}\right)\sqrt{{\rho}_{0}/{E}_{0}}$where$\Omega ={\omega}_{L}\left(\pi /L\right)\sqrt{{E}_{0}/{\rho}_{0}}$.The values of ρ _{0} and E _{0} are equivalent to 8166 kg/m^{3} and 201.04 GPa respectively. The effective properties of four types of material (SUS304, Si_{3}N_{4}, Ti6Al4V, ZrO_{2}) are tabulated in Table. 1.
Comparisons of different beam models on nondimension natural frequency for the tube. (R_{i}=0.8R_{0,} R_{0}=1nm, T=300K, μ=1nm, N_{1} = N_{2}=1)
Within the results of Table. 4, the effect of slenderness ratio and respective beam models on nondimension natural frequency for the tube can be illustrated as follows:

(i) The nondimension natural frequency of the present model is close to that of Timoshenko model, but smaller than that of Reddy model and that of Euler model.

(ii) A higher slenderness ratio tends to waken the stiffness of the tube. In another word, with the slenderness ratio becoming large, the frequency gradually ascends, regardless of which model is adopted.

(iii) The difference between the answers of four beam models becomes prominently with increasing the natural frequency number. That is due to the fact that increasing the natural frequency number needs more degrees of freedom. However, among those beam models, only the present model takes two directions of transverse shear into account. So, the results of the present model are the smallest and close to the actual. The results of EulerBernoulli are the biggest, because it dosen’t consider any transverse shear and has the biggest stiffness of the tube. The difference between Timoshenko model and Reddy model is that Reddy model takes highorder shear deformation into consideration so that the results of Reddy model are higher than those of Timoshenko model.
Effect of both volume indexes N_{1} and N_{2} on nondimension natural frequency for the tube. (R_{i}=0.99R_{0,}R_{0}=1nm,L=20R_{0,}T=300K, μ=1nm)
Results from Table. 5 distinctly reflects the influence of double material gradient indexes N _{1} and N _{2} on nondimension natural frequency for the tube. Fig. 3 shows the different types of functionally graded nanotube. When both the content of SUS304 and Ti6Al4V increase, dimensionless natural frequency of the tube decreases continuously. Table. 6 gives comparisons on nondimension natural frequency among various types of functionally graded tubes. It can be seen from this table that the nondimension natural frequency of the tube consisted of functionally graded bisemitubes is between that of conventional functionally graded tube(Si_{3}N_{4}/SUS304) and that of conventional functionally graded tube (ZrO_{2}/Ti6Al4V) when material indexes(N) of different types of functionally graded tube are equal. Therefore, some frequencies that cannot be obtained by Type A and Type B but can be obtained by present type so long as by adequately moderating Nvalue. Besides, compared with conventional functionally graded tube, the tube with double volume indexes has a smaller adjusting step size to produce more frequencies in the same conditions thus results in more accurate frequency control. Therefore, results show that this design can satisfy the requirement of the natural frequency very well when dimensionless sizes of the tube are kept unchanged.
Effect of inner radius R_{i} to outer radius R_{0} ratio on nondimension natural frequency for the tube.(T=300k,R_{0}=1nm,μ=1nm, L=20R_{0,} N_{1}=N_{2}=1)
The effect of the ratio of the inner radius R _{i} to the outer radius R _{0} on nondimension natural frequencies of the tube has been listed in Table. 7. It reveals that as the thickness of the tube continues to reduce, the dimensionless natural frequencies of the tube decrease at first, then increase remarkably. In terms of this trait, the effect of the thickness of the tube should be considered in design of vibrating tubes.
Effect of nonlocal parameter μ on nondimension natural frequency for the tube.(R_{i}=0.8R_{0,} R_{0}=1nm,T=300K,L=20R_{0,}N_{1}=N_{2}=1)
As for the effect of the nonlocal parameter μ on nondimension natural frequency for the tube, it can be observed from Table. 8 that the frequency is getting smaller when the value of parameter μ becomes larger and lager while other size parameters remain unchanged. Hence, we have demonstrated the nonlocal parameter μ has a tendency to reduce the nondimensionless natural frequencies of the tube.
Effect of dimensionless temperature λ_{T} on nondimension natural frequency for the tube.(R_{i}=0.8R_{0,}R_{0}=1nm,L=20R_{0,}N_{1}=N_{2}=1, μ=1nm)
It can be calculated from Eq. (36) and Eq. (37) that if the value of dimensionless temperature λ _{T} becomes large, the result of ω _{L} will descend. Table. 9 indicates that with the nondimension temperature elevating, the dimensionless frequencies of the tube subjected to uniform thermal environment diminish inversely.
Comparisons of different beam models on the amplitudefrequency of the studied tube made of SuS303, Si_{3}N_{4,} ZrO_{2} and Ti6Al4V. (R_{i}=0.8R_{0,} R_{0}=1nm, T=300K, μ=1nm, N_{1}=N_{2}=1)
Fig. 4 presents comparisons of different beam models on the amplitudefrequency of the tube made of SuS303, Si_{3}N_{4}, ZrO_{2} and Ti6Al4V. As could be seen in this figure, the results of present model are higher than those of other beam models and the results of Euler beam model is the lowest when L=10R _{0}. However, for large slenderness ratio(L/R _{0}≥30), all the resultsω _{NL}/ω _{L} become nearly the same value. In conclusion, transverse shear plays a crucial role in the vibration behavior of short tubes, whereas its effect on the vibration behavior of long enough tubes is neglectable.
The influence of nonlocal parameter μ with the amplitudefrequency of the tube. (R _{0}=1nm, T=300K, R _{i}=0.8R _{0}, L=20R _{0}, N _{1}=N _{2}=1)
The influence of dimensionless temperature λ_{T} with respective to respective to the amplitudefrequency of the tube. (R _{0}=1nm, R _{i}=0.8R _{0}, L=20R _{0}, N _{1}=N _{2}=1, μ=1nm)
Fig. 5 depicts the influence of nonlocal parameter μ on the relation of amplitudefrequency of the tube. These curves reveal that the ratio of the nonlinear frequency to linear frequency(RNFLF) of the tube can be remarkably improved by means of increasing the value of nonlocal parameter μ. Thus, the smallscale parameter μ plays an indispensable role in the nonlinear vibration problem.
Fig. 6 present the effect of nondimension temperature on the relation of amplitudefrequency of the tube. It can be found that the RNFLF increases when the tube is exposed to a rising thermal environment. The results of Fig. 6 can be predicted from Eq. (36) and Eq. (37). The reason is that a higher dimensionless temperature λ _{T} can result in lower denominator of the expression of ω _{NL} /ω _{L}.
The influence of material index N_{2} and N_{1} with to the amplitudefrequency of the tube. (R_{0}=1nm, T=300K, R_{i}=0.8R_{0}, L=20R_{0}, N_{1}=1)
The influence of material index from different types of respective functionally graded tube on the amplitudefrequency. (R _{0}=1nm, T=300K, R _{i}=0.8R _{0}, L=20R _{0})
Fig. 7 shows the influence of material index N _{2} and N _{1} on the variation law of amplitudefrequency of the tube. From this figure, a conclusion can be drawn that with both material indexes N _{1} and N _{2} increasing, corresponding amplitude frequency curves become lower. In other words, as the content of SUS304 and Ti6Al4V continues to ascend, the nonlinear vibration frequencies reduce gradually.
Fig. 8 exhibits the difference between the tube consisted of functionally graded bisemitubes and the tube made of functionally graded materials (Si_{3}N_{4} and SuS304). Shown in this figure, the RNFLF of the former tube is larger than that of the later tube for the high value of the phase of SUS304.
The influence of inner radius R_{i} above the amplitude frequency of the tube. (R_{0}=1nm, T=300K, μ=1nm, R_{i}=0.8R_{0}, N_{1}=N_{2}=1)
The influence of slenderness ratio above the amplitude frequency of the tube.(R_{0}=1nm, T=300K, μ=1nm, L=20R_{0}, N_{1}=N_{2}=1)
Fig. 9 describes the influence of inner radius R _{i} above the amplitude frequency of the tube. It can be seen that when taking larger inner radius under the same outer radius, the nonlinear to linear frequencies ratio goes down.
Fig. 10 shows the influence of slenderness ratio above the amplitude frequency of the tube. The figure clearly plots that a bigger slenderness ratio leads to a smaller ω _{NL} /ω _{L} curve. That is due to the fact that a large slenderness is able to waken the stiffness of the studied tube.
5. Conclusions
This paper focuses on free vibration of nanotubes formed by bonding together a Si_{3}N_{4}/SUS304 functionally graded upper semitube and a ZrO_{2}/Ti6Al4V functionally graded lower semitube. Firstly, four types of material distribution of the FGM bisemitubes were assumed. By using a twosteps perturbation method, the analytical solution was obtained to carry out a vibration analysis in detail. Finally, some important conclusions are outlined.

(1) The ascending of slenderness ratio can increase the natural frequencies, whereas the ascending of dimensionless temperature and nonlocal parameter μ can decrease the natural frequencies.

(2) The natural frequencies reduce with the increase of the content of SUS304 and Ti6Al4V.

(3) Compared with conventional functionally graded tube, obviously, the tube with two volume indexes results in more accurate frequency control in the same dimensionless size of tubes.

(4) The decrease of the thickness of the tube makes the natural frequencies firstly decline and then rise.

(5) The increment of dimensionless temperature λ _{T} and nonlocal parameter μ can improve the RNFLF while the increase of slenderness ratio and the decrease of the thickness of tubes can make the RNFLF decline.

(6) The influence of double volume indexes on the relation of amplitudefrequency of bisemitubes is more obvious than the influence of one volume index on the relation of amplitudefrequency of conventional functionally graded tubes.
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Available online November 07, 2018.
Publication Dates

Publication in this collection
2019
History

Received
27 June 2018 
Reviewed
09 Sept 2018 
Accepted
13 Sept 2018