A Semi-Analytical Nonlinear Approach for Size-Dependent Analysis of Longitudinal Vibration in Terms of Axially Functionally Graded Nanorods

Reza Nazemnezhad ^*

School of Engineering, Damghan University, Damghan, Iran

1.     Introduction

The experimental and theoretical studies on the various behaviors of nano-sized structures have shown that they have superior properties in comparison with the structures at the macro-scale. These superior properties can be enhanced by using functionally graded materials (FGM) for nano-sized structures. FGMs have been used in the nano-technology field to attain the desired efficiency [1-4]. Since the behaviors of nano-sized FGM structures are different from those made of homogeneous materials, it is incumbent to be intimate with the behaviors of FGM nano-structures for NEMS/MEMS fabrication.

One of the nanoscale elements that attracted the attention of researchers is the nano-beam. There are many works regarding the analysis of its various mechanical behaviors, buckling [5, 6], post-buckling [7, 8], bending [9, 10], stability [11, 12], free and forced axial vibration [13, 14], free and forced torsional vibration [15, 16], free and forced transverse vibration [17-20], and wave propagation [21, 22]. Among the studies, the axial behavior of nanobeams (nanorods) has more recently been investigated. These investigations have been done based on three types of size-dependent theories, surface elasticity theory, strain gradient theory, and nonlocal elasticity theory. Free vibration, wave propagation, and instability of nanorods/nanotubes are investigated using the nonlocal elasticity theory. In these researches effects of various factors like elastic medium [23-27], magnetic fields [27-30], the shape of nanorod cross section (uniform [31, 32], non-uniform [33, 34], tapered [35, 36], cone-like [37]), crack [25, 38, 39], discontinuity [40], type of rod theory [26, 41-45], number of coupled nanorods [13, 30, 46-49], number of nanotube walls [23, 50-52], and type of nanorod material properties [53-56] have been considered. The other studies in the field of analysis of longitudinal behavior of nanorods/nanotubes are done based on the surface elasticity and strain gradient theories. Free vibration of homogeneous [57, 58], axially functionally graded [59-61], viscoelastic [62, 63], and embedded nanorods [64] are analyzed using the strain gradient theory. Finally using the surface elasticity theory, Nazemnezhad and Shokrollahi have considered free longitudinal oscillation of FGM nano-sized rods [65].

All of the mentioned works have only analyzed the linear axial behavior of nanorods/nanotubes based on different theories. There are only a few works studying the nonlinear axial vibration of nanotubes/nanorods. Nazemnezhad et al. [66] studied the nonlinear longitudinal oscillation and internal resonances of nanoscale rods incorporating the effects of the surface energy parameters. In another work, the CNTs had been modeled as a homogeneous nanorod, and their nonlinear free axial vibration was investigated [67]. Fatahi-Vajari and Azimzadeh [68] considered the nonlinear axial vibration of single-walled carbon nanotubes based on the Homotopy perturbation method. In this research, the nonlinear governing equation of motion was derived using doublet mechanics. The nonlinear coupled axial–torsional vibration of single-walled carbon nanotubes has also been analyzed [69]. In this study, the Galerkin method and homotopy perturbation method were used. In another different work, the internal resonances of nanorods were studied based on the nonlocal elasticity theory [70]. In this work, Nazemnezhad and Zare implemented the multi-mode Galerkin method to convert the partial differential equation to the ordinary differential type. Jamali Shakhlavi et al. [71] had studied similar research on internal resonances of Rayleigh nanorods. Finally, Jamali Shakhlavi et al. [72] considered the effects of thermal and magnetic fields and elastic medium on the nonlinear axial vibration of nanorods. The nonlocal elasticity was used and the von Kármán geometric nonlinearity was taken into account.

As mentioned in the first paragraph of this section, using functionally graded materials enhances the mechanical properties of structures. In addition, the literature survey showed that there is a lack of investigation of the nonlinear axial vibration of nanotubes/nanorods. The lack is what the effect of material on the nonlinear axial behavior of nanotubes/nanorods is. Therefore, this study presents an investigation of the nonlinear axial behavior of nanorods/nanotubes when they are made of FGMs. To this end, a comprehensive model is proposed to investigate the nonlocal parameter effect on the large amplitude oscillation of nano-size axially functionally graded (AFG) rods with von Kármán type nonlinearity. In this regard, the equation of motion and corresponding boundary conditions are obtained. Then, linear mode shapes and natural frequencies are extracted using the harmonic differential quadrature method. In the next step, using the multiple scales method, the nonlinear governing equation of motion is solved. Finally, the nonlocal parameter effects on the frequencies of AFG nanorods are investigated for various end conditions, nanorod lengths, nanorod diameter, and amplitude ratios.

2.     Problem Formulation

2.1.  Governing Equation of Motion and Corresponding Boundary Conditions

In order to derive the nonlinear governing equation of motion and corresponding boundary conditions of AFG nano-size rods, an AFG nano-size rod with length L (0 ≤ x ≤ L), cross-sectional area A, and diameter D are desired in XYZ coordinates as shown in Figure 1.

Two different materials at the left and right ends constitute the AFG nano-size rod. The Poisson’s ratio is assumed to be constant but the shear modulus G(x), bulk elastic modulus E(x), and mass density ρ(x) of the AFG nanorod vary in the longitudinal direction according to the power low distribution as follow [73, 74]:

In Eq. 1, m is the gradient index and shows the profile of the mechanical properties of the AFG nano-size rod across its length, and the superscripts R and L imply the right (x=L) and left (x=0) ends of the AFG nanorod, respectively. In Figure 2 variations of the mass density and the Young’s modulus of elasticity across the AFG nanorod length are plotted for various values of the gradient index.

Based on the rod theory, the displacement field at any point of the AFG nano-size rod is given below:

where t is the time, and ,  and  are the displacement components along x, y, and z directions, respectively. To obtain the strain(s) in the AFG nanorod, the von-kármán strain-displacement relation with the nonlinear terms is used

According to Eqs. -(3), the non-zero strain in the cross-section of the AFG nanorod is obtained as

From Hooke’s law, the stress induced in the cross-section of the AFG nanorod is also obtained as

It is now possible to derive the governing equation of motion and the corresponding boundary conditions using Hamilton’s principle given as Eq. (6)

where T and U are the kinetic and potential energies of the AFG nano-size rod, respectively, and are given as

Now substituting Eqs. , (4)-(5) into Eqs. (7)-(8) results in

where  is the local force resultant and is described as

Equations (9) and (10) are the local governing equations of motion and corresponding boundary conditions, respectively.

In order to consider the nonlocal parameter effect on the governing equation of motion of the AFG nano-size rod, the nonlocal elasticity theory [75] is used. To this end, Eq. (9) can be stated in the nonlocal form as

where subscript nl denotes the nonlocal, and  is the force resultant in nonlocal form or it is the nonlocal force resultant. According to the nonlocal elasticity theory, the  can be obtained from Eq. (11) as follows

In Eq. (13),  is the one-dimensional Laplacian operator. Now,  is obtained from Eq. (13) as [70]

By obtaining the nonlocal force resultant, , the nonlocal governing equation of motion of the AFG nanorod can be given in terms of displacement by substituting Eq. (14) into Eq. (12) as

Eq. (15) can be simplified as

Eq. (16) can be expanded and rewritten as bellow

It is worth noting that by considering Eqs. (10) and (14), the local and nonlocal boundary conditions become the same and are given by

2.2. Free Vibration Analysis

2.2.1.          Linear Free Vibration Analysis

Ignoring the nonlinear components in Eqs. (17) and (18) make it possible to analyze the nonlocal linear free longitudinal vibration of AFG nano-size rods. The linear equations are as follows

Next, it is assumed that the AFG nano-size rod has a harmonic longitudinal displacement as

where  is the imaginary unit, and  is the nonlocal natural axial frequency. By substituting Eq. (21) into Eqs. (19) and (20), the governing equation of motion and the boundary conditions in terms of the displacement are obtained as

Since the coefficients of Eq. (22) are dependent on x, it is not possible to solve it analytically. To solve Eq. (22) and obtain the natural frequencies and corresponding mode shapes of AFG nanorods, the harmonic differential quadrature method (HDQM) [76], an accurate and useful method for analysis of structural components, is utilized in this study. Consider the domain of structure as  which is discretized by N points along
x-coordinate. If  represents the deformation function, then its derivative with respect to x at all points  can be expressed discretely as

in which  is the weighting coefficient in conjunction to the jth order derivative of , at the discrete point .

Now, applying Eq. (24) to Eqs. (22)-(23) results in the discretized form of the governing equation and boundary conditions as follows

Arranging the discretized form of Eq. (25) in matrix form, considering the discretized forms of Eq. (26), and separating the domain ( ) and boundary ( ) degrees of freedom (DOF) lead to the following assembled matrix

where

The nonlocal linear longitudinal frequencies,  and corresponding mode shapes,  can be obtained from Eq. (27) after doing some mathematical simplification as

2.2.2. Nonlinear Free Vibration Analysis Using the Method of Multiple Scales

For nonlinear free vibration analysis of nonlocal AFG nanorods, Galerkin’s method is used to convert Eq. (17) to an ordinary differential equation. To this end, the longitudinal displacement of the n^th mode of the AFG nanorod can be considered as

where U(x) is the nonlocal normalized linear mode shape which can be obtained from Eqs. (27)-(30) and T(t) is a time dependent function to be determined. Substituting Eq. (31) into Eq. (17) results in the following ordinary differential equation

Then, multiplying Eq. (32) by  and integrating from x=0 to x=L results in

where

Eq. (33) needs two initial conditions as follows

where T_max is a maximum amplitude corresponding to the time-dependent function T(t).

The method of multiple scales is implemented to solve Eq. (33). Before that, the following non-dimensional quantities are introduced

By substituting Eq. (37) into Eqs. (33), (35) and (36), the non-dimensional second order ordinary differential equation and initial conditions are obtained as

in which  and
.

The method of multiple scales considering a second-order uniform expansion of the solution of Eq. (38) is desired as

where ε is a small parameter measuring the amplitude of oscillation. In addition, we consider a fast time scale, q₀=t, denoting the main oscillatory behavior, and slow time scales  implying the amplitude and phase modulation.

Since the form of Eq. (38) is the same as the one obtained for nonlinear free transverse vibration of nanobeams [20], the procedure of extracting the nonlocal natural frequency equation of AFG nanorods is not presented here. If we follow the procedure presented in Ref. [20], the nonlocal natural frequency equation of AFG nanorods can be given as

where  is the nonlocal nonlinear natural frequency of the AFG nanorod, and the subscript and superscript nl denotes the nonlinear and nonlocal, respectively.

3.     Results and Discussion

Before the presentation of results, it is worth mentioning that for obtaining the nonlinear natural frequencies of AFG nanorods (Eq. (42)), the coefficients defined in Eq. (43) should be calculated. One of the parameters used in the coefficients is the linear normalized mode shapes (Eqs. (28) and (29)) which are obtained in discretized form. Therefore, we do a curve-fitting to obtain a function for each linear mode shape. This makes it possible to do mathematical operations on the functions to obtain nonlinear natural frequencies. In Fig. 3, linear mode shapes of AFG nano-size rods for various values of the gradient index are plotted for fixed-fixed (fi-fi) and fixed-free (fi-fr) end conditions.

3.1. Comparison Study

In this section, a comparison study is conducted to check the reliability of the present formulation and the analytical solution, and some numerical results are provided for the nonlinear vibration properties of AFG nano-size rods based on the nonlocal elasticity theory.

Table 1 gives a comparison between the natural frequencies of a fi-fi and a fi-fr boundary conditions presented by this work with those obtained by Nazemnezhad and Zare [70] for a homogeneous nanoscale rod. Both researchers have used the method of multiple scales to obtain the natural frequencies. In the present study, an AFG nanorod is investigated while in Ref [70] a homogeneous one is desired. Table 1 shows a good agreement in results.

3.2. Benchmark Results

The benchmark results presented here are obtained for nanorods with the mechanical properties given in Table 2.

We first examine the variations of linear and nonlinear nonlocal frequencies versus AFG nanorod diameters. In this regard, the length of the AFG nanorod is equal to 10 (nm) and the corresponding diameter changes are equal to 1-2 (nm) with a step of 0.25. Since the change in diameter is directly related to the thickness of the AFG nanorod, we have tried to select its values ​​so that the aspect ratio of the AFG nanorod is greater than 10, because in the calculation of nonlinear nonlocal frequencies, the Simple axial beam model has been used, which is suitable for this condition. The results show that with increasing AFG nanorod diameter, the linear nonlocal frequencies are constant, but the nonlinear nonlocal frequencies have increased, although this increase is not significant. Due to the fact that the AFG nanorod material leads to softness, with increasing the gradient index, linear and nonlinear nonlocal frequencies have decreased. Also, due to the fact that the system under study has a hardening effect, with increasing nonlinear amplitude, nonlinear nonlocal frequencies have increased.

The results of these changes are presented in Tables 3 and 4, respectively, for fi-fi and fi-fr boundary conditions in terms of the first frequency mode of the AFG nanorod. As the results show, the general trend of changes is the same for both boundary conditions, but with the difference that in the fi-fi boundary conditions in the first mode, the effect of hardening is maintained, while for the fi-fr boundary conditions, we see a softening effect in the first mode.

Table 1. Comparison study of the present work with Ref. [70] in terms of the first third natural frequencies (GHz)
 for AFG nanorod (µ=2 nm, R=2 nm, T_max = 2, m=0)

Table 2. Mechanical properties of AFG nanorod

Next, the linear and nonlinear frequency changes of the AFG nanorod are studied based on the variations of the non-local parameter and the gradient index for the first and second AFG nanorod modes in terms of both fi-fi and fi-fr boundary conditions. These results are derived for AFG nanorods with a length of 10 (nm) and a diameter of 1 (nm) in terms of three nonlinear amplitudes of 0, 1 and 2. The results show that the decreasing effects of nonlocal parameters on linear and nonlinear frequencies for the first mode are negligible. Also, the decreasing slope of the graphs for the lower gradient index is higher, which indicates that the stricter the AFG nanorod, the greater the decreasing effects of the nonlocal parameter on its linear and nonlinear frequencies. Here, too, the trend of general results changes for both fi-fi and fi-fr boundary conditions is similar, except that the frequency-reducing effects of the non-local parameter for
fi-fr boundary conditions are less than fi-fi. The detailed results of the study of these changes can be clearly seen in Figs 4a to 4f.

In this section, the longitudinal frequency changes of AFG nanorods based on nonlinear amplitudes for different non-local parameter values and gradient indices are evaluated. In this evaluation, the length and diameter of the AFG nanorod are considered equal to 10 and 1 (nm), respectively. As shown in Fig 5, the variations of the longitudinal frequencies of the AFG nanorod versus the nonlinear amplitude changes are greater for the fi-fi than fi-fr boundary conditions. Also, the incremental effect of nonlinear nonlocal frequencies due to the increase of nonlinear amplitude for the second mode of AFG nanorod frequency is higher than the first mode, but this incremental effect will be more in higher modes (third mode onwards). In addition, the results show that the incremental effects of increasing the nonlinear amplitude are the same for different gradient index nanorods. In other words, the nonlinear effects of material changes along the AFG nanorod are neutral. The results of these studies are clearly shown in the various sections of Fig 5a, 5b, 5c, and 5d for both boundary conditions.

Finally, the frequency changes of the AFG nanorod versus its length for different non-local parameter values and gradient index are presented in Fig 6. These results are derived for both fi-fi (Figs. 6a, 6b, and 6c) and fi-fr (Figs. 6d, 6e, and 6f) boundary conditions for an AFG nanorod with nonlinear amplitudes of 0, 1, and 2. Figs 6a, 6b, and 6c show that the decreasing effects of linear and nonlinear frequencies due to the non-local parameter in the fi-fi boundary conditions are greater than the fi-fr, which has been observed in previous results. Another important point is that with increasing AFG nanorod length, linear and nonlinear nonlocal frequencies coincide, which shows that the effects of changes in material, nonlocal parameter, and nonlinear amplitude on long AFG nanorods on its frequencies are negligible. The bottom line is that the larger the AFG nanorod length, the higher the nonlinear and nonlocal coefficients must be for the results to be significant. Also on thick nanorods, the effects of nonlinear factors are less than on thin nanorods, see Fig. 6.

4.     Conclusions

In the present study, large amplitude free longitudinal vibration behavior of axially functionally graded nanorods with material gradation is investigated. Different boundary conditions are considered namely fi-fi and fi-fr. However, the present methodology can be employed for other types of classical and non-classical boundaries as well. Hamilton’s principle is utilized for the mathematical formulation and the boundary conditions, the problem can be solved in two parts, linear and nonlinear with the help of Harmonic Differential Quadrature and Multiple Scale methods, respectively. The methodology is general in nature as it can be applied to another type of material gradation and rod patterns. The obtained results are confirmed from previously published results and were found to be in good agreement. Results pertaining to various AFG nanorod parameters for fi-fi and fi-fr boundary conditions are furnished as highlight curves for the fundamental modes. For all combinations of the system parameters, the hardening type of nonlinearity is observed in terms of fi-fi boundary conditions while the softening type is demonstrated for initial modes in terms of fi-fr boundary conditions. Nonlinear nonlocal analysis of rod-type structural components of NEMS (Nano-Electro-Mechanical-Systems) is a topic of major interest in Engineering and Medical sciences.

Acknowledgments

The financial support of the research council of Damghan University with grant number 97/Engi/133/307 is acknowledged.

Conflicts of Interest

The author declares that there is no conflict of interest regarding the publication of this article.

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