1 Introduction

Micro/nano-tubes have been widely used in fabricate micro/nano-electromechanical systems (MEMSs/NEMSs) due to their superior properties,e.g.,transistors,semiconductors,sensor technologies,nano-oscillators,micro-electronic cooling systems,actuators,and composites^{[1, 2, 3]}.

The properties of micro/nano-tubes are closely related to their micro/nano-structures. Therefore,it is important to consider the micro/nano-scale effects. Two mainly theoretical approaches of atomistic and continuum mechanics have been used for the analyses of nano-structures. However,the atomistic method is computationally expensive,especially for large scale systems,and the classical continuum theory cannot capture the size effects which significantly affect the mechanical properties at nano-meter scales. Therefore,many non-classical continuum theories, e.g.,the couple stress theory,the nonlocal elasticity theory,and the strain gradient theory,are proposed.

Among the aforementioned theories,the classical couple stress theory is proposed and employed based on a higher-order continuum theory^{[4, 5, 6, 7]},where two material length-scale parameters are introduced to capture the size effects in the isotropic materials. Yang et al.^[8] modified the classical couple stress theory,in which only one constant was needed to take into account the size effects. The modified couple stress theory utilized an additional equilibrium relation to force the couple stress tensor to be symmetric. In recent years,many researchers have employed the modified couple stress theory to investigate the behavior of micro/nano-tubes^{[9, 10, 11]}.

A new class of composites is the functionally graded materials (FGMs),which have many applications in various industrials. These materials are inhomogeneous composites,and are commonly made from two materials. The composition and volume fraction of the materials vary with a smooth and continuous variation gradually from one surface to another. The gradient compositional variation of the constituents according to a predetermined profile causes corresponding changes in the mechanical properties of the material. One of the best advantages of FGMs is the minimization of the stress concentration that arises due to the mismatch in the elastic properties of the constituent phases. In macro-scale,many investigations have been concerned with the mechanical behavior of FGMs^{[12, 13, 14, 15]}.

Recently,FGMs have been used in micro/nano-structures and atomic force microscopes (AFMs)^{[16, 17, 18]}. It is necessary to enhance the knowledge about the mechanical response of the FGMTs for the next technological revolution,since these structures are emerging as the new generation of the micro/nano-tubes offering exciting physical and mechanical properties. Recently,there have been some attempts to understand and explain the mechanical response of FGMTs by use of different mathematical and numerical approaches^{[19, 20, 21]}.

The torsional deformation and vibration are very common for the micro/nano-tubes subjected to the external forces in some basic components of new nano-scale devices,e.g.,oscillators, actuators,and microelectronic cooling systems. Despite the importance of this issue,only few researchers have paid attention to the torsional vibration of micro/nano-tubes. Demir and Civalek^[22] investigated the size effects on the torsional and axial responses of micro-tubules by use of the nonlocal continuum rod model. Gheshlaghi and Hasheminejad^[23] studied the size dependent torsional linear vibration of nano-tubes by use of the modified couple stress theory. Islam et al.^[24] presented the torsional wave propagation and vibration of circular nano-structures based on Eringen’s nonlocal elasticity. Li^[25] developed two different nonlocal elasticity models for the torsional vibration analyses of carbon nano-tubes according to the weakened and enhanced models. Arda and Aydogdu^[26] investigated the torsional statics and dynamics of the carbon nano-tubes embedded in elastic media by use of the nonlocal elasticity theory. Lim et al.^[27] developed the analytical solutions for the free torsional vibration of the circular nano-rods/tubes based on a new elastic nonlocal stress model. However,the linear or nonlinear torsional vibration of functionally graded micro/nano-tubes has not been studied.

The main target of this paper is to present an analytical solution for the linear and nonlinear torsional vibration of FGMTs by use of the modified couple stress theory,which is a non-classical theory and can capture the size effects. The homotopy analysis method (HAM) is implemented to analytically solve the related nonlinear governing equation. The material properties of the FGMTs vary continuously across the radius according to the power law distribution.

2 Modified couple stress theory

According to the modified couple stress theory,the strain energy U for a deformable linear elastic material occupying the volume V can be written as follows^[28]:

where m_ij ,and are the stress tensor,the strain tensor,the deviatoric part of the couple stress tensor,and the symmetric curvature tensor,respectively,which are defined by In the above equations,i,j,k = x,y,z. is the Kronecker delta. u_i is the component of the displacement vector. l denotes the material length scale parameter,demonstrating the couple stress effect. and G(r) are Lame’s constants defined by where E(r) and stand for Young’s modulus and Poisson’s ratio,respectively. is the component of the rotation vector obtained by where e_ijk is the permutation symbol. 3 Motion equation of FGMTs

Consider an FGMT of the length L,the inner radius R_i,the outer radii R_o,the thickness t_r,the density ,and the cross-sectional area A. The inner and outer surfaces are made of different materials. A schematic of the geometry of the FGMT is represented in Fig. 1.

In this investigation,the material properties of the tube vary continuously across the radius r according to the power law distribution as follows^[20]:

The volume fraction of the materials can be written as follows:

where n is the power gradient index,and the subscripts i and o indicate the inner surface and the outer surface of the tube,respectively.

The displacement field of the micro/nano-tube along the x-,y-,and z-axes can be expressed as follows^[29]:

where is the angular rotation about the center of the twist. The von K´arm´an straindisplacement relation with the nonlinear terms is The strain components are obtained by substituting Eq. (10) into Eq. (11),i.e., The components of the rotation vector are found by expanding Eq. (7) in terms of the displacement field as follows: The components of the curvature tensor are obtained from Eqs. (4) and (13) as follows: Substituting Eq. (14) into Eq. (5) yields the components of the couple stress tensor as follows: By expanding Eq. (1),the strain energy U can be written as follows: The components of the stress field can easily be obtained by use of Eqs. (2) and (12),i.e, Now,substituting all of the derived components of the stress tensor,the strain tensor,the couple stress tensor,and the symmetric curvature tensor into Eq. (16) yields where The kinetic energy of the micro/nano-tube can be written as follows: With the assumed displacement field,we can obtain the kinetic energy as follows: The above equation can be simplified as follows: where The nonlinear vibrational governing equation of the micro/nano-tubes can be determined with the aid of Hamilton’s principle as follows:

Finally,substituting Eqs. (18) and (22) into Eq. (24) and using the fundamental lemma of the variation calculus yield the nonlinear governing differential equation as follows:

In this research,the following boundary conditions are considered for the FGMTs:

(i) Both ends are fixed,i.e.,= 0.

(ii) One end is fixed,and the another end is free,i.e.,= 0.

4 Method of solution

Equation (25) is a nonlinear partial differential equation. The Galerkin method together with the separation of variables is used to solve this partial differential equation. The rotation function can be expressed as follows:

where is the time dependent function that must be determined,and F(x) is defined based on the first mode shape of the micro/nano-tube vibration for the fixed-fixed and fixed-free boundary conditions as follows^[30]: Substituting Eq. (26) into Eq. (25),multiplying the obtained result by the function F(x),and then integrating the governing equation over the length of the nano-tube,we have where In the above equations, where the prime means differentiation with respect to x. 4.1 Analytical method

The resulted nonlinear time-dependent governing equation (29) is analytically solved by the HAM. This method does not require a small parameter in an equation,and has a significant advantage in providing an analytical approximate solution to a wide range of nonlinear problems in applied mechanics^[31]. To illustrate the basic idea of the method,consider the following nonlinear time-dependent differential equation:

where is a general differential operator,t denotes the time which is an independent variable, and f(t) is an unknown function. Liao^[31] constructed the so-called zeroth-order deformation equation by means of generalizing the concept of homotopy as follows: where p is the embedding parameter varying from 0 to 1,h is a nonzero auxiliary parameter, h(t) 6≠0 is an auxiliary function, is an auxiliary linear operator,f₀(t) denotes an initial guess of f(t),and is an unknown function. It is worth noting that one has great freedom to select the initial guess,the auxiliary linear operator,and the auxiliary function. When p changes from zero to unity,the solution varies from the initial approximation to the desired solution. In other words, By differentiating Eq. (33) with respect to p,the first-order deformation equation can be obtained as follows: Expanding in the form of the Taylor series with respect to the parameter p,we have where

The above series converges at p =1 by an appropriate selection of the auxiliary linear operator,the initial guess,the auxiliary parameter,and the auxiliary function,which reads

Substituting Eq. (36) into Eq. (35),differentiating the resulting equation k times with respect to the parameter p,setting p = 0,and dividing all of the terms by k!,we can obtain the so-called kth-order deformation equation as follows:

where 4.2 Analytical results

By use of the transformation in Eq. (29),the new form of the governing equation can be obtained as follows:

where denotes the nonlinear frequency. The initial conditions are defined by where is the amplitude of the tube torsional vibration. The initial guess must be selected such that the initial conditions are satisfied. Therefore,we can write a convenient function satisfying the boundary conditions as follows: The linear and nonlinear operators can be expressed as follows: where One can adjust the convergence rate of the HAM solution by means of the auxiliary parameter . Liao^[32] studied the influence of this parameter for different solution expressions,and exhibited that the parameter could be selected in the range from -2 to 0 while the series converged the fastest to the exact solution in the whole region when = −1. Therefore,to achieve the accurate results by only few terms for the present problem,we set Starting from Eq. (35),we can write the first-order deformation equation as follows:

From Eqs. (45),(46),and (48),we have

subjected to

Substituting from Eq. (44) into Eq. (49) and solving the resulted equation,we have

The coefficient of the secular term sin must be identical to zero since the rotation of the micro/nano-tube is finite. Thus,

After solving Eq. (52),we can obtain the first-order approximation for the nonlinear frequency as follows:

Now,k should be set to be 2 in Eqs. (39) and (41) in order to obtain ,i.e.,

In a similar manner,we can obtain the second-order deformation equation and by substituting the appropriate equations into Eq. (54),applying Eq. (56),and solving the resulted equation,i.e., where Eliminating the secular term and solving the corresponding equation,we can obtain the secondorder approximation for the nonlinear frequency as follows:

Briefly,the explicit expressions for the linear and the nonlinear natural frequencies of the FGMT with the second-order approximation can be written as follows:

Similarly,we can obtain the third-order approximation as follows: where Let the coefficient of the secular term be zero. Then,we have

Eventually,the HAM solution to the third-order approximation can be obtained by collecting the resulted terms as follows:

The dimensionless linear and nonlinear natural frequencies can be defined,respectively,as follows: where and E_Ep are,respectively,the density and Young’s modulus of the epoxy,respectively. 5 Numerical examples

In this stage,some numerical results are presented to demonstrate the reliability and the efficiency of the developed analytical expressions. The obtained explicit relations in the previous section are used to study the linear and nonlinear torsional free vibration responses of FGMTs. The tubes with fixed-fixed and fixed-free boundary conditions are considered. Due to the lack of experimental data,the length scale parameter l is assumed to be 17.6 mm,which is reported for the epoxy. The non-dimensional form of the length scale parameter is used in the present work for the sake of generality. The material length scale parameter is mathematically the square root of the ratio of the modulus of the curvature to the modulus of the shear,and is physically a property of measuring the couple stress effect. This parameter can be found from the torsion tests of the slim cylinders with different diameters or the bending tests of the thin beams with different thicknesses. To determine the length scale parameter for a specific material,some classic experiments such as the micro-bending test,the micro-torsion test,and the specially micro/nano-indentation test have been be performed^[33]. The geometric properties of the tubes are

unless otherwise specified. 5.1 Validation study

Due to the lack of the similar solution concerning the problem of the torsional vibration of FGMTs in the open literature,the accuracy of the present solution is verified by comparing the results with those of Ref. [23] (see Tables 1 and 2). The geometrical parameters are considered to be identical to the values reported in Ref. [23] when L = 30 mm,R_o = 900 nm,R_i = 200 nm. Gheshlaghi and Hasheminejad^[23] investigated the torsional free vibration of fixed-fixed nanotubes,and reported the size dependency of the corresponding natural frequencies by use of the couple stress theory. They defined a dimensionless linear frequency parameter by the ratio of the non-classical linear natural frequency to the classical linear natural frequency,which we denote here by . To demonstrate the efficacy of the model,the natural frequencies corresponding to the higher modes of vibration are determined by changing the value of the parameter m in Eq. (27). Table 1 shows the variations of the parameter in terms of the dimensionless length scale parameter (l/R_o) for the first mode (m = 1) as well as the 100th mode (m = 100). It is seen that the results of the proposed method agree with those of Ref. [23]. The developed results exhibit that the size effect has a significant effect on the dimensionless frequency parameter, and this effect increases when the vibration modes increase. Moreover,the effects of the inner radius to the outer radius ratio (R_i/R_o) on the frequency parameter for the first mode and the 25th mode are illustrated in Table 2. It is seen that the numerical results agree well with each other.

5.2 Convergence and parametric studies

Without loss of generality,we assume here that the FGMT is made of aluminum (Al) and epoxy (Ep) with the properties as follows: E_Al = 70GPa, = 0.23, = 2 700 kg/m³, E_Ep = 1.44GPa, = 0.38,and = 1 220 kg/m³.

The convergence study of the solution in predicting the dimensionless nonlinear torsional natural frequencies versus the amplitude of the vibration for fixed-fixed FGMTs are demonstrated in Table 3. The first-,second-,and third-order approximations for the non-dimensional nonlinear frequencies are obtained and shown,respectively,with . From the table, we can see that the convergence rate of the results is fast. To exhibit the discrepancy among the classical stress theory,the couple stress theory,and the linear and nonlinear vibration analyses for FGMTs,we depict the variations of different frequencies versus the dimensionless length scale parameter l/R_o in Fig. 2. According to the trend of the results,the importance of performing a nonlinear or even a linear non-classical analysis for the torsional vibration of FGMTs is evident. This verifies the size-dependency of the natural frequencies of FGMTs.

The dimensionless nonlinear torsional natural frequencies versus the vibration amplitude for fixed-fixed and fixed-free FGMTs are,respectively,shown in Figs. 3(a) and 3(b). Different values of the dimensionless length scale parameter are considered. From the figures,we can see that, when the vibration amplitude increases,the nonlinearity of the system increases,leading to the enhancement of the dimensionless nonlinear frequency. According to the developed results, the effect of the dimensionless material length scale parameter decreases when the vibration amplitude increases. Also,it is revealed that the dimensionless nonlinear natural frequency increases when the dimensionless material length scale parameter increases. Moreover,the results exhibit that the natural frequencies predicted for the fixed-fixed boundary condition are higher than those obtained for the micro/nano-tubes with fixed-free end conditions. In fact, the fixed-fixed nano-tubes have higher stiffness in comparison with that of the fixed-free tubes, and therefore the frequencies are enhanced.

In Fig. 4,the effects of the material gradient index on the dimensionless nonlinear frequency versus the vibration amplitude for the fixed-fixed and fixed-free boundary conditions are shown. The dimensionless material length scale parameter is considered to be l/R_o = 0.5. The developed results demonstrate that the dimensionless nonlinear natural frequency of the micro/nanotubes decreases by increasing the power law index. It means that the results shift from the natural frequency of the metallic micro/nano-tubes towards the epoxy one. The enhancement rate of the dimensionless nonlinear frequency increases with the decrease in the material gradient index for both the fixed-fixed condition and fixed-free boundary condition. However,this rate is magnified for the case of fixed-free FGMTs.

The effects of the material gradient index on the dimensionless nonlinear natural frequencies versus the vibration amplitude for two distinct values of the non-dimensional length scale parameter and the different boundary conditions are demonstrated in Fig. 5. It is observed that the size effect contribution on the dimensionless fundamental frequency significantly increases when the material gradient index decreases. The variations of the dimensionless response of FGMTs versus the time with two distinct values of the vibration amplitude are shown in Fig. 6, where l/R_o = 0.5,and n = 2.

6 Conclusions

The linear and nonlinear torsional free vibration analyses of functional gradient micro/nanotubes are analytically studied on the basis of the modified couple stress theory. Hamilton’s principle is used to obtain the nonlinear non-classical governing equation of motion. To examine the effect of the boundary conditions,two types of end conditions,i.e.,fixed-fixed and fixedfree,are considered. The present model accounts for the material length scale parameter and the two-constituent material variation through the radius of the micro/nano-tubes. The HAM is used to analytically solve the corresponding nonlinear differential equation. The developed explicit expressions for the nonlinear natural frequencies can be conveniently implemented to investigate the effects of different parameters such as the length scale,the material gradient index,and the vibration amplitude on the natural frequencies. To exhibit the accuracy,efficacy, and efficiency of the analytical solution,some numerical results are presented.

The main conclusions are summarized as follows:

(i) The results exhibit that the size effect has a significant effect on the dimensionless linear and nonlinear frequencies,and this effect increases when the vibration modes are higher.

(ii) According to the developed results,the importance of performing a non-classical analysis for the linear and nonlinear torsional vibrations of FGMTs is evident.

(iii)When the vibration amplitude increases,the nonlinearity of the system increases,leading to the enhancement of the dimensionless nonlinear frequency.

(iv) The dimensionless nonlinear natural frequency increases when the dimensionless length scale parameter increases.

(v) The natural frequencies predicted for the fixed-fixed boundary condition are higher than those obtained for the tubes with the fixed-free end conditions.

(vi) The enhancement rate of the dimensionless frequency increases when the material gradient index decreases for both the fixed-fixed boundary condition and the fixed-free boundary condition.