Document Type : Research Paper
Authors
School of Engineering, Damghan University, Damghan, Iran
Abstract
Keywords
Main Subjects
Longitudinal Vibration of Nanobeams by TwoPhase Local/Nonlocal Elasticity, Rayleigh Theory, and Generalized Differential Quadrature Method
School of Engineering, Damghan University, Damghan, Iran
KEYWORDS 

ABSTRACT 
Longitudinal Vibration; Nanobeam; TwoPhase Elasticity. 
To solve a differential equation of motion via more reliable procedures, it is essential to realize their efficiency. Whether Rayleigh's theory can be a compatible platform with twophase local/nonlocal elasticity to render more reliable results compared to other theories or not is the main question that will be answered by this paper. Thus, nanobeam modeled by Rayleigh beam theory is analyzed by twophase local/nonlocal elasticity. Governing equation in presence of the axial and transverse displacements is derived by means of Hamilton’s principle and differential law of twophase elasticity. Next, fourthorder Generalized Differential Quadrature Method (GDQM) is utilized to attain the discretized twophase formulation. In order to confirm, the method and the results are compared with the exact solution prepared and presented in the literature. Moreover, the effects of various parameters such as geometrical properties like thickness, mode shape number, Local phase fraction coefficient, and nonlocal factor on the natural frequency are investigated to clarify that utilizing these theories with a common goal how ends with more accurate results, and how affects the natural frequencies. 
Focus on analyzing the natural behavior of nanoscale systems have been became one of the most crucial fields of study. The mechanical behavior of nanostructures enjoys vital importance due to their applications in nanodevices, such as nanomechanical resonators, nanoscale mass sensors, electromechanical nanoactuators, and nanogenerators. For instance, resonant frequencies are vastly vital in nanosensors to detect the type of viruses and bacteria. Theoretical results having relied on classical continuum theories have always been more convenient in contrast to other methods. Thus, new definitions or modifications of preliminary theories including strain gradient theory, nonlocal elastic theory, nonlocal strain gradient theory, and couple stress theory have been considered by numerous researchers. Some literature associated with the abovementioned theories is eventually cited [17]. Farajpour et al. [8] in a review paper discussed two modified continuumbased theories, the nonlocal elasticity, and the nonlocal strain gradient elasticity employed to estimate the mechanical behavior of nanostructures. In that review paper, first, these two modified elasticity theories are briefly explained. Then, the nonlocal motion equations for different nanostructures including nanorods, nanorings, nanobeams, nanoplates, and nanoshells are derived. Defined stress in nonlocal theories is a dependent variable on the whole strain domain. Eringen by an integral formula based on the definition presented a simplified and constructive theory [4].
The simplicity of the differential form of Eringen’s nonlocal elastic theory has attracted more interest to analyze mechanical problems. Furthermore, the hybrid of the differential form of this theory and strain gradient theory have both nonlocal and strain gradient effects. To enjoy further perception, more comprehensive studies are added [913]. The result in the mechanics of nanostructures studied by differential nonlocal shows that nonlocal parameter surging leads to a more softening effect. Although, the differential form of nonlocal elasticity enjoys some paradoxical influences. Challamel et al. [14] in a paper, investigated the selfadjointness of Eringen's nonlocal elasticity based on simple onedimensional beam models. It was shown that Eringen's model may be nonselfadjoint and that it can result in an unexpected stiffening effect for a cantilever's fundamental vibration frequency with respect to increasing Eringen's small length scale coefficient. FernandezSaez et al. [15] formulated the problem of the static bending of Euler–Bernoulli beams using the Eringen integral constitutive equation. Beams with different boundary and load conditions are analyzed and the results are compared with those derived from the differential approach showing that they are different in general. Appuzo et al. [16] investigated free vibrations of nanobeams by making recourse to the novel stressdriven nonlocal integral model (SDM). Natural frequencies evaluated according to the SDM are compared with those obtained by the Eringen differential law (EDM) and by the gradient elasticity theory (GradEla). SDM provides an effective methodology to describe nonlocal phenomena. To enjoy a more constructive method, Eringen’s combination of local and nonlocal has been used by researchers, for example, Wang et al. [17] analyzed static bending of nonlocal EulerBernoulli beams using Eringen’s twophase local/nonlocal model by an analytical study. Additionally, it was stated that the controversial nonlocal beam problem in the literature is well resolved by the reduction method, and Zhu et al. [18] adopted Eringen's twophase nonlocal integral model to carry out an analytical study on the buckling problem of EulerBernoulli beams by using a reduction method. Qing [19] combined strain and straindriven twophase nonlocal integral polar models to model the axisymmetric bending of circular microplates. The results showed that the purely straindriven nonlocal integral polar model turns to a traditional nonlocal differential polar model if the constitutive constraints are neglected. Khaniki [20] stated that the differential form of Eringen's nonlocal elastic theory has some inaccuracies in analyzing smallscale structures with different boundary conditions. Accordingly, in this work, a comprehensive study on the vibration behavior of doublelayered nanobeam systems (DNBS) was presented within the framework of Eringen's twophase local/nonlocal integral mode. Fakher, and HosseiniHashemi [21] through the exact solution corresponding to the vibrations of twophase Timoshenko nanobeams provided the shearlocking problem investigated in the case of the twophase finite element method (FEM). Moreover, since the FE model of local/nonlocal nanobeam is more complex than the classic one, due to the coupling of all elements together, they created an efficient lockingfree local/nonlocal FEM with a simple and efficient beam element. Employing nonlocal models with classical boundary conditions may lead to inconsistency. Thus, higherorder boundary conditions along with twophase differential equations can render the most accurate results compared with conventional patterns. Shaat et al. [22] formed consistent boundary conditions by iterative nonlocal residual approach. This approach solves the field equation in the local field with an imposed nonlocal residual. Rayleigh theory’s displacement field, which its simple form has been employed vastly thus far unlike this form utilized in this study for the first time, and twophase local/nonlocal elasticity are utilized to analyze the longitudinal vibration of nanobeams. Governing equations, in presence of the axial and transverse displacements, are derived by means of Hamilton’s principle and differential law of twophase elasticity. To solve numerically, the Generalized Differential Quadrature Method (GDQM) is utilized to attain the discretized twophase formulation. Eventually a verification, and the result are prepared. Also, the effects of various parameters such as, geometrical properties, nonlocal parameters, and local phase fraction coefficient on the natural frequencies are investigated. The unique feature of the present work is the effective use of twophase elasticity theory in modeling the longitudinal vibrations of nanobeams and the use of appropriate solution methods in the field of related differential equations, which has eliminated the contradictions compared to the common nonlocal differential theory. Also, the softening effect of nonlocal parameters appears more strongly in the twophase domain.
The equation of motion and boundary conditions of Rayleigh nanobeams are obtained as the first step. The geometry and coordinates of the nanobeam model, and displacement fields, according to the Rayleigh beam theory, are assumed as follows.

(1) 
shows Poisson’s ratio in Eq.1. In this theory, the inertia of the lateral motions by which the cross sections are extended or contracted in their planes is considered. But the contribution of shear stiffness to the strain energy is neglected. An element in the crosssection of the beam, located at the coordinates y and z, undergoes the lateral displacement and , respectively along the y and z directions. Nanobeam strain energy is [23]

(2) 
Axial force, , should be defined and by using Eq. (1), Eq. (2) is rewritten as

(3) 
The kinetic energy of the beam can be obtained as

(4) 
After simplification, the kinetic energy enjoys the shape below, thus

(5) 
where is the material density, and is defined by

(6) 
In the following, governing equations and boundary conditions will be obtained by employing Hamilton’s principle.

(7) 
According to Eqs. (3) and (5), variations of the potential and kinetic energies are presented.

(8) 
In this step, by substituting Eq. (8) into Eq. (7), the governing equation can be obtained.

(9) 
Also, corresponding boundary conditions of that are expressed as

(10) 
Note that Eq. (10) is satisfied if the beam is either fixed or free at the ends and . At a fixed end, and hence , while

(11) 
at a free end [23].
Twophase local/nonlocal elasticity is shown below comprising local and nonlocal parts.

(12) 

(13) 
The definition of the parameters employed in this study is listed in the nomenclature. Reference [24] presented the equal differential of the integral equation with two constitutive boundary conditions (CBC) as follows.

(14) 

(15) 

(16) 

(17) 
Eq. (15) is a differential form of Eq. (14), and two additional boundary conditions are essential to be satisfied.
By using the Eqs. (12), (13) and the primary form of axial force defined earlier, the twophase axial force is written as

(18) 
E is the elastic modulus, and A is the crosssection of the nanobeam. Now, by employing Eqs. (14), (15), and (18) as well as a governing equation, the differential form of axial force is derived as below.

(19) 
Eventually, by substituting Eq. (19) into governing equation, the expected form of governing equation is derived.

(20) 
And, according to Eqs. (16), (17), and (19) the CBCs of axial force will be obtained as

(21) 

(22) 
Amid other numerical means such as the Galerkin method and finite element method, the GDQM has numerous beneficial attributes, that is, simple mathematical principle, the high pace of convergence, high accuracy, small calculation amount, and less memory demand, etc. The generalized differential quadrature method (GDQM) has been proposed as a general numerical method to solve highorder differential equations. Based on the fourthorder derivatives GDQM, the rth order derivative of a function is considered as follows

(23) 
Hermite shape functions are as follows

(24) 

(25) 

(26) 

(27) 
where , , and are given in appendixI. Also, is equivalent to the . Also, the weighting coefficients, Lagrangian interpolation for the first derivative, and higherorder derivatives can be expressed as

(28) 

(29) 


(30) 
Both the equal space model and Chebyshev Gauss Lobatto model could be used, however owing to further reliable and quick convergence, the latter, that is eq. (32), is utilized. Also, the discretized form of the function could be defined as follows

(31) 

(32) 
To analyze the vibrational status of the system, harmonic vibration has been assumed, thus the displacement field can be shaped as shown below

(33) 

(34) 
To derive the discretized governing equations in the axial direction by utilizing Eqs. (33) and (34) into Eqs. (20), can be the final segment to achieve expected results.

(35) 
Furthermore, applying Eqs. (33) and (34) into Eqs. (21), and (22) finalize the discrete type of CBCs. The matrix, shown below, comprising a discrete governing equation and all of the boundary conditions shows that solving such an eigenvalue problem ends to the natural frequencies and corresponding mode shapes.

(36) 

(37) 

(38) 
To realize the influences of axial displacements on the vibration of a nanosensor, corresponding results are extracted and presented in three portions. The first is devoted to the verification of the employed method, and the other two portions presented the numerical achievements of this work. In all results, the material listed below in Table.1 is utilized.
Table 1. Material Properties
Material Properties 


1000 (kg/m^{3}) 

0.3 
/L 
0.1~0.5 
h/L 
0.05~0.1 
E 
1.44e9 (GPa) 
L 
2 (nm) 
In the first portion, the whole derivation process of formulas has been carried on by assuming . Thus, the employed theory became fully local, and the assumption made it possible to enjoy sufficient verification compared with the essential natural frequency formula obtained by Rao. [23]. The formula is indexed below for further assessment.

(39) 
Figures 1 and 2 comprising the first and second five natural frequencies, and their exact targets towards which curves converge show the path of progress of GDQM. For the first and second natural frequencies, there is a similar pattern, that is, on nearly 23 sample points, a sharp decline occurs, and they approach the exact solution. For the rest of the natural frequencies, various statuses should be considered. The rash decrease happened on almost 51 sample points. As expected, increasing the number of sample points cause further convergence and precise results. Overall, it can be obviously seen that it is possible to achieve the desired accuracy by utilizing 150 sample points. To enjoy the higher quality, the horizontal axis of the abovementioned figures is divided into 100.





Fig. 1. First five natural frequencies with





Fig. 2. Second five natural frequencies with
Figure 3 deals with generating an amplitude of the local phase fraction coefficient, and truly shows the 10 variants of natural frequencies, and how it changes along the horizontal axis or distinct . Figure 3 is presented the first ten natural frequencies. There, the first natural frequency experiences a sharp fluctuation which occurs almost in intervals 0~0.05. Then, it begins a stable climbing, however, the slope compared with other variants is considerable. What is essential to state here is that 0~0.05 interval is a region on which investigation does not result in a particular consequence, and with increasing local phase fraction coefficient, the order and natural frequencies raised in a hugely noticeable way. Thus, it can be stated that zeta parameter enjoys a vastly crucial role in final quantities.
Fig. 3. Natural Frequencies with Zeta amplitude
Table 2. demonstrates the trend of the first 5 natural frequencies with regard to distinct values of thickness and local phase fraction coefficient. Three and five independent values for the two abovementioned variables are defined. It is obvious that increasing h/L leads to an inconsiderable decline of natural frequencies. The decline for upper cases of natural frequency can be severe, and further. Additionally, this type of trend of natural frequency can be seen for the distinct value of the local phase fraction coefficient. What is mentionable in the next table, Table 3., is the demonstration of the relation of natural frequencies with nonlocal factors and local phase fraction coefficient that is not as clear as in the prior section. It can be stated that zeta 0.2, 0.4, and 0.6 for all quantities of /L enjoy similar effects on natural frequency in comparison with the previous table showed, but the two last cases of local phase fraction coefficient show a completely different approach. For 0.8 all differences are intensively low, and for 1, the behavior is predictable. In other words, the nonlocal factor is constructive in the process while the local phase fraction coefficient does not reach the ultimate value.
Table 2. The first five Natural Frequencies with various h and

h/L 
Mode Number 

1st 
2nd 
3^{rd} 
4th 
5^{th} 

0.2 
0.05 
18.9959 
29.8722 
41.1043 
51.5548 
63.0791 
0.075 
18.9854 
29.8237 
40.9603 
51.2477 
62.4990 

0.1 
18.9716 
29.7570 
40.7590 
50.8256 
61.7135 

0.4 
0.05 
22.5631 
38.0495 
54.3944 
69.8905 
86.3596 
0.075 
22.5376 
38.0009 
54.2133 
69.4787 
85.5764 

0.1 
22.5248 
37.9215 
53.9563 
68.9165 
84.5165 

0.6 
0.05 
25.3353 
44.5360 
64.7942 
84.2136 
104.4723 
0.075 
25.3189 
44.4761 
64.5753 
83.7217 
103.5302 

0.1 
25.3050 
44.3839 
64.2751 
83.0470 
102.2484 

0.8 
0.05 
27.7061 
50.1253 
73.6769 
96.4056 
119.8565 
0.075 
27.6992 
50.0518 
73.4260 
95.8436 
118.7762 

0.1 
27.6870 
49.9420 
73.0924 
95.0704 
117.3090 

1 
0.05 
29.8637 
55.0860 
81.5789 
107.2133 
133.4731 
0.075 
29.8328 
55.0240 
81.3025 
106.5870 
132.2703 

0.1 
29.8103 
54.9187 
80.9242 
105.7299 
130.6347 
Table 3. The first five Natural Frequencies with various K and

/L 
Mode Number 

1st 
2nd 
3rd 
4th 
5^{th} 

0.2 
0.1 
24.9743 
46.1374 
62.5482 
75.3673 
86.0071 
0.2 
22.8861 
38.6334 
50.0314 
60.1094 
69.7962 

0.3 
21.1401 
34.1179 
44.6697 
54.7428 
64.8907 

0.4 
19.8690 
31.4545 
42.1386 
52.2470 
62.8104 

0.5 
18.9716 
29.7570 
40.7590 
50.8256 
61.7135 

0.4 
0.1 
25.8425 
48.7280 
68.0464 
84.7580 
99.8715 
0.2 
24.6653 
43.8405 
59.9263 
75.0148 
89.6937 

0.3 
23.7058 
40.9163 
56.5348 
71.5662 
86.5978 

0.4 
23.0207 
39.1188 
54.8931 
69.8924 
85.2461 

0.5 
22.5248 
37.9215 
53.9563 
68.9165 
84.5165 

0.6 
0.1 
26.5934 
51.0556 
72.9800 
93.0233 
111.8100 
0.2 
26.2108 
48.2883 
68.1394 
87.1192 
105.6250 

0.3 
25.8533 
46.4712 
66.0084 
84.8794 
103.6405 

0.4 
25.5592 
45.2477 
64.9206 
83.7323 
102.7439 

0.5 
25.3050 
44.3839 
64.2751 
83.0470 
102.2484 

0.8 
0.1 
27.2848 
53.2243 
77.5309 
100.5206 
122.4807 
0.2 
27.6184 
52.2618 
75.3441 
97.6204 
119.3487 

0.3 
27.7528 
51.3164 
74.1748 
96.2959 
118.1916 

0.4 
27.7567 
50.5421 
73.5118 
95.5462 
117.6309 

0.5 
27.6870 
49.9420 
73.0924 
95.0704 
117.3090 

1 
0.1 
27.9381 
55.2773 
81.7886 
107.4436 
132.2327 
0.2 
28.9256 
55.8997 
81.8592 
107.0441 
131.6063 

0.3 
29.4801 
55.6877 
81.4785 
106.4668 
131.1149 

0.4 
29.7353 
55.2888 
81.1658 
106.0355 
130.8222 

0.5 
29.8103 
54.9187 
80.9242 
105.7299 
130.6347 
Twophase local/nonlocal elasticity as the essential theory is utilized to investigate the longitudinal vibration of Rayleigh nanobeams. The generalized differential quadrature method solves the differential equation of motion. Rayleigh's theory and twophase local/nonlocal elasticity demonstrate a reliable linkage that ends with accessible results. In this study, results truly show that employing the main theories result in acceptable consequences. Nonlocal factors and local phase fraction coefficients may significantly decrease or increase the frequencies. The details were described in the prior section, but it can be mentioned that the geometrical parameters can enjoy a more significant effect on natural frequency than other mentioned parameters.
Moreover, the displayed discrepancy between distinct mode shapes and natural frequencies, in Fig. 3 with regard to the local phase fraction coefficient illustrates the importance of the employed independent variable. The compatibility of GDQM with vibration analysis of twophase Rayleigh nanobeam is confirmed by comparison with an analytical procedure, even though considerable amounts of sample points were needed to attain. One of the main novelties of the current work is employing twophase elasticity theory without contradiction in modelling longitudinal vibrations of nanobeams as well as the development of proper solution procedures in the domain of corresponding differential equations that in comparison with common nonlocal differential theory eliminated contradictions and consequently provides Integral form of results in accordance with differential one.
Moreover, by comparing the results of twophase theory and nonlocal, it can be said that in addition to the elimination of contradictions softening effect of nonlocal parameter consideration appears with more intensity in the twophase domain (with a quantity close to zero) that this enjoys more discrepancy in higher modes. Eventually, this is essentially considering that according to the complexity related to the manner of creation of atomic linkage between the nanostructure and supportive surfaces, utilizing classical boundary conditions in modelling nanostructures can make errors, thus employing higher order boundary conditions along with twophase differential equations enjoys vital importance. Thus, it can be mentioned here that the development of models concerning boundary conditions is constructive to enhance the accuracy of prediction of mechanics of nanomachines, and more research in this field is needed.
L 
Length 
b 
Width 
h 
Thickness 
U, V, W 
Displacement fields 

Strain energy 
Vo 
Volume of the nanobeam 

Normal stress 

Normal strain 

Axial force 
T 
Kinetic energy 

Polar moment of the inertia of the crosssection 

Stress in twophase state 

Local phase fraction coefficient 

Kernel function 

Strain tensor 

Elasticity tensor 

Volume of the domain 

Nonlocal factor 
& 
Constant coefficients. 

Number of sample points, 

Location of sample points, 

Hermite shape functions for all sample points, 

Hermite shape functions for Rth order derivative at the first sample point 

Hermite shape functions for Rth order derivative at the last sample point 

Constant coefficients 

Lagrangian interpolation test functions 

Kronecker delta 

Poisson’s ratio 

Density 
Appendix I
Appendix II
Transformation of Integral order, the first part
The Second part
The last part needs to have another integration by parts, on the x variable.
Thus:
References
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