Longitudinal Vibration Analysis of a Stepped Nonlocal Rod Embedded in Several Elastic Media

Mechanical properties of 1D nanostructures are of great importance in nanoelectromechanical systems (NEMS) applications. The free vibration analysis is a non-destructive technique for evaluating Young's modulus of nanorods and for detecting defects in nanorods. Therefore, this paper aims to study the longitudinal free vibration of a stepped nanorod embedded in several elastic media. The analysis is based on Eringen’s nonlocal theory of elasticity. The governing equation is obtained using Hamilton’s principle and then transformed into the nonlocal analysis. The dynamic stiffness matrix (DSM) method is used to assemble the rod segments equations. The case of a two-segment nanorod embedded in two elastic media is then deeply investigated. The effect of changing the elastic media stiffness, the segments stiffness ratio, boundary conditions and the nonlocal parameter are examined. The nano-rod spectrum and dispersion relations are also investigated. The results show that increasing the elastic media stiffness and the segment stiffness ratio increases the natural frequencies. Furthermore, increasing the nonlocal parameter reduces natural frequencies slightly at lower modes and significantly at higher modes.


Introduction
Recently, researchers have shown an increased interest in nanostructures due to their extraordinary properties. High strength, good thermal and electrical properties are some of the properties of nanostructures [1]. Nanostructures have many technological applications, such as, micro and nano electromechanical systems (MEMs and NEMs). In order to develop an accurate design of nanostructures, it is important to understand their behavior. Nanorods, nanobeams and nanowires are examples of 1D nanostructures. One of the popular nanostructures is carbon nanotubes (CNTs) were discovered by Iijima [2]. The vibrations of 1D nanostructures have recently received considerable attention due to their application in several nanotechnological instruments such as atomic force microscopy, nanomechanical cantilever sensors, etc. The natural frequency of 1D nanowires can be determined experimentally by driving these nanorods to their resonant frequency modes using mechanical or electrical methods. Then, by using vibration analysis, the nanowires Young's modulus can be evaluated. One of the advantages of this method is that it is nondestructive .
The vast research in nanostructures may be classified into two main categories [1]: the experimental methods and the analytical modeling. The experimental methods are a possible way to investigate nanostructures [3], although the experimental methods are quite challenging because controlling every parameter in nanoscale is a complex task.
The analytical modeling category can be divided into two sub-groups: quantum mechanics and continuum mechanics.
Quantum mechanics modelling may be grouped into the following categories: ab-initio method, density functional theory, Monte Carlo simulation, molecular mechanics and molecular dynamics. Molecular dynamics (MD) is one of the most popular quantum mechanics methods. Molecular dynamics is a simulation in the atomistic level [4]. Therefore, MD is time consuming, computationally expensive and not suitable for the analysis of nanostructures that consist of a large number of atoms and molecules [5].
Continuum mechanics, that is popular to engineers on its classical formulation, is used in nanostructures analysis [6]. In the classical continuum mechanics theory, the stresses at a point X depend on the strain at that point X only. The classical continuum theory requires some modifications at the nano scale because the size effect of the nanostructures has a great influence on its behavior [7].
In general, conventional continuum mechanics is a size effect-free theory. Therefore, much research is done to include the size effect into the classical continuum mechanics. This lead to many theories such as nonlocal elasticity theory, modified coupled stress theory, surface energy methods, multiscale models, and hybrid methods. In the current work, the analysis is based on nonlocal continuum elasticity theory [8].
In the nonlocal elasticity theory, the stress at a point X is a function of strain at that point X and all nearby points X ′ in the continuum [9]. Eringen theory is simple in deriving the equations and analysis of nanostructures. It is also worth noting that this theory can be divided into integral-based and differential-based models. A large number of studies investigated the static analysis [10], buckling [11]and vibration [12][13][14] of nanostructure using nonlocal continuum mechanics.
The longitudinal vibration of nanorod are studied using different analysis methods and parameters by many researchers. Aydogdu [15] studied the axial vibration of nanorod for clamped-clamped (C-C) and clamped-free (C-F) boundary conditions. Huang [16] examined the nonlocal long-range influence on the axial vibration of nanorod. Aydogdu [17] examined the effect of various parameters like C-C and C-F boundary conditions and the stiffness of the elastic medium. Adhikari et al. [18] investigated the free and forced axial vibration for nanorod using the dynamic finite element analysis. Demir and Civalek [19] examined the size effect on a continuous and discrete non-local models. They studied both the torsional and axial vibration analysis of microtubules. Güven [20] studied the longitudinal vibration of nano-rod using nonlocal strain gradient elasticity theory. The lateral deformation and shear strain effects are included by using the Love rod model and Bishop's correction. Adhikari et al. [21] used the dynamic finite element method to investigate a nanorod embedded in an elastic medium. They used conventional and dynamic finite elements methods for obtaining the dynamic response for the rod.

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Gul et al. [22] used doublet mechanics to investigate C-C and C-F carbon nanotube embedded in an elastic media. Numanoğlu et al. [23] presented the effect of nonlocal parameter, boundary conditions, attachments and length on the axial vibration behavior of nanorod using Eringen's nonlocal theory. Loghmani et al. [24] analysed the free axial vibration of several cracked, stepped nanorod with attached mass at the tip of the rod. Ebrahimi et al. [25] investigated static stability and vibration analysis of a functionally graded nanobeams. The analysis is based on nonlocal theory of elasticity and modified couple stress theory to capture the size effects accurately. Civalek et al. [26] analyzed the lateral and axial vibration of embedded nanobeams based on Eringen's theory using finite element method. Yayli [27] used Eringen's theory to investigate the axial vibration of Rayleigh nanorod which is attached to axial elastic springs at both ends.
In an exact solution of multi-span rod or beams, the boundary conditions and the intermediate conditions of each segment are derived [28,29]. Then these equations are assembled. There are several methods to assemble the segments equations but the most common methods are numerical assembly technique (NAT) [30,31], transfer matrix method (TMM) [32,33] and DSM [34][35][36]. In the current manuscript, DSM method is adopted.
In the majority of studies, vibrations are described as a linear combination of a structure's modes. Another way to describe vibrational phenomena is to think of vibrations as moving waves along structures. Media with various material characteristics and cross-section areas usually have different impedances. Many researchers have investigated impedance mismatch in solid structures [37][38][39][40]. Both descriptions frequently provide complementary views for vibrational analysis and this was considered in the first verification example.
From the previous literature and to the best of author's knowledge, there is no single study which dealt with vibration of a stepped nonlocal rod embedded in several elastic media using DSM method. Therefore, the primary purpose of the current work is to fill this gap by solving such problem. The rest of the paper is organized as follows: in Sect. 2, a brief overview on the nonlocal Eringen's theory is explained. Then the equation of motion is derived in the local formulation using Hamilton's method. Also, the local formulation for equation of motion is transformed to the nonlocal formulation using nonlocal Eringen's theory. In Sect. 3, the DSM method is used for the analysis. The case of two-segment nanorod embedded in two elastic media is introduced and the verification results are presented and validated with the published results in Sect. 4. In addition, the results for a C-C and C-F boundary conditions are introduced and discussed. Finally, the conclusion of the study is given in Sect. 5.

A Brief Overview on the Nonlocal Elasticity Theory
The constitutive equations of linear, homogeneous, isotropic and nonlocal elastic solid with zero body forces are given by Eringen [41]. The stress tensor is defined by the integro-partial differential equation as: where ij and c ij are the stress tensor and classical stress tensor, respectively. X is a reference point,X ′ is any point in the body and (|X � − X|, ) denotes the nonlocal kernel function. The kernel function depends on the distance between the points X and X ′ and the nonlocal parameter = e 0 a . The parameters a and are the internal and external characteristic lengths, respectively. The parameter e 0 is used to calibrate the results of the nonlocal model to the atomic model. However, the identification of the small scaling parameter e 0 has not been fully understood. Several researchers reported several values for this parameter and these values are found to have a large scatter [9]. The integro-partial differential Eq. (1) is simplified to partial differential equation as: where C ijkl is elastic modulus tensor, kl is strain tensor and ∇ 2 denotes the second order spatial gradient. The nonlocal constitutive relation for 1D structures can be written as:

Equation of Motion in the Classical Form
A schematic representation for stepped nanorod embedded in several elastic media is shown in Fig. 1. The total length of the rod is L, that has J segments and J + 1 nodes. The rod is constrained from the left and right sides by linear springs with stiffness K s1 , K sJ+1 respectively. The axial coordinate is x j and the axial displacement is u j . E j , j , L j , A j and K j are the Young's modulus, density, length, area and the surrounded elastic media stiffness of jth segment respectively. The stress concentration at each step is neglected during the present work.
The equation of motion of jth segment can be obtained using Hamilton's principle, which can be written as: where is the variational symbol, Π j is the potential or strain energy and Γ j is the kinetic energy of jth segment.
The axial displacement and strain fields of jth segment are given by: The kinetic and potential energies of jth segment embedded in elastic medium can be expressed as: where m j is the mass per unit length of jth segment. Substituting Eqs. (6,7) in Hamilton's Eq. (4) gives: Thus, the equation of motion of jth segment is obtained as: The axial force per unit length (N j ) of jth segment is expressed as: By differentiating Eq. (10) with respect to x j and substituting into Eq. (9), one can get:

Equation of Motion in the Nonlocal Form
The nonlocal constitutive equation of jth segment can be obtained from Eqs. (3, 10) as follows Equations (11,12) can be used to obtain the nonlocal equation of motion and the axial forces of jth segment The partial differential equation in Eq. (13) can be solved using the separation of variables method as: where T(t) = e i t , is the circular frequency and i 2 = −1. Substituting Eq. (15) into Eqs. (13) and (14) and transforming them into dimensionless form as: Fig. 1 Stepped nanorod embedded in several elastic media Equations (16) and (17) can be simplified by considering where j is the dimensionless wavenumber as shown below and where are the dimensionless frequency and stiffness of the 1st segment respectively and = e 0 a L is the dimensionless nonlocal parameter. The solution of Eq. (18) is presented in terms of C 1j and C 2 j as: The axial force may be expressed as:

Wave Characteristics (Spectrum and Dispersion Relations)
In this section, the wave propagation concepts is used to determine the wave parameters, namely the wavenumber ( j ) and the two different wave speeds (phase speed (C ph ) and group speed (C gr ) ). From Eq. (20) the wave frequency is a function of the wavenumber j , the nonlocal parameter , the material property ( j1 , E j1 ) , rod geometry (A j1 , l j ) , first elastic medium stiffness K 1 , and elastic media stiffness ratio (K j1 ) . If the wavenumber is real, the wave is said to be propagating. If the wavenumber is completely imaginary, the wave dampens out and is hence known to as evanescent mode. If wavenumber is complex, the wave will attenuate as it propagates. From U j j = C 1j cos j j + C 2j sin( j j ).
Eq. (20), the propagating mode in this case is dispersive (i.e., the wave shape changes with propagation). In case of = 0 and K 1 = 0 the wavenumber becomes linearly proportional to wave frequency, which results in nondispersive propagating mode (i.e., the wave retains its shape as it propagates). In our case, the waves will not start propagating except after certain frequency called cut-off frequency (Ω c ) . The wavenumber before this frequency will be imaginary. For the present case the cut off frequency is obtained by setting j = 0 as shown in Eq. (24). Moreover, the waves stop propagating after certain frequency called escape frequency (Ω es ) . The escape frequency is obtained by setting j tending to ∞ as shown in Eq. (25).
Then, the phase and group speeds should be determined. The phase speed is the speed of the individual particles that propagate in the structure. The transmission of any physical quantity in a waveguide is not connected with phase speed. It is related to the wavenumber through the relation During the propagation of waves, group of particles are travelled in bundles. The speed of each bundle is called the group speed. This is the velocity of energy transport, and it must be constrained. It is mathematically expressed as

DSM of jth Segment
The boundary conditions for dimensionless axial displacement and axial force of jth segment can be expressed as: At x j = 0(i.e., j = 0), U j = U 1 j and N j = −N 1 j At x j = L j (i.e., j = 1), U j = U 2 j and N j = N 2 j Substituting the boundary conditions into Eqs. (22,23) to obtain the displacements and forces vectors respectively as: The displacements and forces vectors may be expressed respectively as: where, and j is the DSM of jth segment without attachments.

The Overall DSM
The overall DSM for all the system (stepped nanorod embedded in various elastic media with different boundary conditions) may be obtained as: where is the overall dimensionless attachments matrix in diagonal form as: are the left and right linear dimensionless stiffness respectively.
The overall DSM is function of the dimensionless natural frequency parameters Ω 1 . Newton-Raphson method is used to solve the DSM in Eq. (33). The solution is obtained by using very small initial guess Ω 0 = 10 −9 and using incremental step of Ω inc = 0.001 . The roots of this equation are the dimensionless natural frequency parameters Ω 1 .

Results and Discussions
This section is divided into three subsections. The first subsection describes a two-segment rod model. The second subsection includes some verification results which are selected from literature to verify the results of the present model. The third subsection includes some new results obtained using the present model. Figure 2 shows two-segment nanorod made from the same material (E 21 = 21 = 1) and embedded in two different elastic media. The dimensionless elastic media stiffness of the 1st and 2nd segment are K 1 and K 2 = K 21K1 respectively. The first and second segment dimensionless length are l 1 and l 2 = 1 − l 1 respectively. The dimensionless attached springs stiffness at the left and right are denoted by K s 1 and K s 3 , respectively. A 21 is the ratio of the second to the first cross sectional area. boundary conditions are investigated and compared with the results presented in [24]. Figures 3 and 4 show the first dimensionless natural frequency Ω 1 (1) for C-C and C-F rod respectively. Figure 3 shows that Ω 1 (1) decreases through the first quarter of step location and then increases through the second and third quarter then return to decreasing by going to the end. Figure 4 shows that Ω 1 (1) decreases through the first half of the step location ratio and return to increase by going to the end. It can be seen that the results are in good agreement with Loghmani, Yazdi [24] results.

Example 2
This example studies a uniform rod (A 21 = 1) that is fully embedded in one elastic medium (K 21 = 1) . Clamped-clamped (K � s 1 = K � s 3 = ∞ ) and C-F (K � s 1 = ∞ and K � s 3 = ∞ ) are investigated and compared with the solution presented in [22]. The analysis is carried out at nm. Also, the variation of nanorod length L = 1:5 nm is examined. Figures 5 and 6 show the first three dimensionless natural frequencies for C-C and C-F rod respectively for different lengths. The two figures indicate that with increasing the nanorod length the scale effect decreases, and the results close to the classical effect. From the figure results, It can be seen that the results are in good agreement with the results of Gul et al. [22].

Spectrum and Dispersion Relations for a Uniform Rod Partially Embedded in an Elastic Media
A uniform rod (A 21 = 1) partially embedded in an elastic media (K 21 = 0) is investigated. The total length of nanorod L = 25 nm and the two-segment length ratio are (l 1 = 0.25    The spectrum and dispersion curves obtained from local ( e 0 a = 0 ) and nonlocal rod model for the two segments of the rod are shown in Figs. 7 and 8. Figure 7 shows the variation of the nondimensional wavenumbers with nondimensional wave frequencies for both local and nonlocal theories. From this figure, the waves start propagating in the second segment at any excitation frequency and does not have a cut-off frequency, unlike the first segment, the waves start propagation at a certain frequency called cut-off frequency. The cut-off frequencies are independent of the nonlocal parameter and directly proportional to the elastic media stiffness. In the present study and for the first segment, the nondimensional cut-off frequency is at 3.1623 for both local and nonlocal theories.
For the case of local theory ( e 0 a = 0 ), the wavenumbers increase linearly with the nondimensional frequency for the second segment as shown in Fig. 7 . This linear variation denotes that the waves will propagate in nondispersive mode (i.e., the waves do not change their shapes as they propagate) as shown in Fig. 8, where the group speed is constant with the increase in frequency. Also, the wavenumbers increase monotonically for first segment with the increase in frequency and correspondingly, the group speed increase with increase in wave frequency. However, at higher frequencies, the group speeds attain a constant  value. On the other hand, with the introduction of nonlocal parameter, the wave behavior is completely changed. For both segments the wavenumbers have a nonlinear variation with frequency, which indicates that the waves are dispersive in nature. Also, the wavenumbers tend to infinity at a certain frequency called escape frequency and after this frequency there is no wave propagation. The escape frequencies are inversely proportional to the nonlocal parameter and directly proportional to the elastic media stiffness. In the present case for the first segment, the nondimensional escape frequency is at 50.0999 and 25.1992 and for second segment is at 50.0 and 25.0 for e 0 a = 0.5 and 1 nm, respectively.

Uniform Rod Embedded in Two Different Elastic Media
A uniform rod (A 21 = 1) embedded in two different elastic media is investigated. The total length of nanorod L = 25 nm and the two-segment are equal (l 1 = l 2 = 0.5) . The analysis is carried out at e 0 a = 0.5:2 nm, K 21 = 0:5 and K 1 = 0:10.
Clamped-clamped rod In the case of C-C rod and for free vibration analysis, the stiffness of the boundary springs takes the following values: K s 1 =Ks 3 = ∞. Figure 9 shows the first and second dimensionless natural frequencies of C-C rod for different nonlocal parameters at (a, b, c and d). It shows that with increasing K 1 or K 21 the natural frequency increases. Figure 9a-d show Fig. 9 The first and second dimensionless natural frequency for C-C rod 1 3 that with increasing the nonlocal parameter the natural frequency in both 1st mode and 2nd mode decreases. Table 1 shows the first five dimensionless natural frequencies for C-C rod at different nonlocal parameters e 0 a,K 1 and K 21 . The table results show that increasing the values of K 1 , K 21 , results in an increase in the dimensionless natural frequencies. By increasinge 0 a , the dimensionless natural frequencies decrease in all modes however the decrease is a slightly bigger in the higher modes.
Clamped-free rod In the case of clamped-free boundary conditions the input values for the rod end conditions become as follows: K s 1 = ∞ and K s 3 = 0.  Figure 10 shows the first and second dimensionless natural frequencies of C-F rod for different nonlocal parameters at (a, b, c and d). Table 2 shows the first five dimensionless natural frequencies for clamped-free rod at different nonlocal parameters e 0 a,K 1 and K 21 . Table 2 and Figure 10 show that the dimensionless natural frequency increases with the increase in the of K 1 orK 12 . The subfigures (a, b, c and d) show that with increasing the nonlocal parameter, the natural frequency in both modes decreases. However, the trend of changing the natural frequency with K 1 , K 12 and e 0 a is the same for both C-C and C-F boundary conditions.

Conclusions
The natural frequencies of a stepped rod embedded in several elastic media and for different boundary conditions are introduced in the present work. Dynamics stiffness matrix method and Eringen's theory are used for this analysis. Two cases from the literature are used to validate the present model. Then,a selected case of two-segment nanorod embedded in two elastic media is deeply investigated. The effects of varying the nonlocal parameter and stiffness of elastic media and its ratio were considered. The spectrum and dispersion relations were obtained for partially embedded nanorod. The results show that the 1 3 start of wave propagation is independent of the nonlocal parameter but depends on the surrounded elastic media stiffness.
In local theory, the waves propagate in nondispersive mode with constant speed for the segment without surrounding elastic media. Unlike the other segment with surrounding elastic media, the group speed increase with frequency to attain a constant value. In the nonlocal theory, the waves propagate in dispersive mode from the cut-off frequency to the escape frequency where the wave speed reaches zero. The escape frequencies are inversely proportional to the nonlocal parameter and directly proportional to the elastic media stiffness.

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The dimensionless natural frequencies were obtained for C-C and C-F rod. The present results indicate that increasing the elastic media stiffness and its ratio, increase the dimensionless natural frequencies. Also, increasing nonlocal parameter decreases the dimensionless natural frequencies except for some points at the first mode of C-F boundary conditions. This decrease is more remarkable at the higher modes. The present results may be of great benefit in evaluating nanobeam elastic modulus based on the measurement of its resonance.
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