Document Type : Research Paper
Authors
^{1} School of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran, Iran.  Department of Design, Fateh Sanat Kimia Company, Great Industrial Zone, Shiraz, Iran.
^{2} School of Engineering, Damghan University, Damghan, Iran
Abstract
Keywords
A Novel Method for Considering Interlayer Effects between Graphene Nanoribbons and Elastic Medium in Free Vibration Analysis
K. Kamali^{a,c}, R. Nazemnezhad^{b,*}
^{a} School of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran, Iran
^{b} School of Engineering, Damghan University, Damghan, Iran
^{c} Department of Design, Fateh Sanat Kimia Company, Great Industrial Zone, Shiraz, Iran
KEYWORDS 

ABSTRACT 
Elastic medium Sandwich theory Tensilecompressive effects Shear effects EulerBernoulli theory 

A complete inspectation on the free vibration of bilayer graphene nanoribbons (BLGNRs) modeled as sandwich beams considered tensilecompressive, and shear effects of van der Waals (vdWs) interactions between adjacent graphene nanoribbons (GNRs) as well as between GNRs and polymer matrix is performed in this research. In this modeling, nanoribbon layers play the role of sandwich beam layers and are modeled based upon the EulerBernoulli theory. In order to deliberate effects of vdWs interactions between adjacent GNRs as well as between GNRs and polymer matrix, their equivalent tensilecompressive and shear moduli are contemplated and applied in derivation of governing equations instead of employing conventional Winkler and Pasternak effects for elastic medium. The governing equations of motion are derived by considering the assumptions and employing sandwich beam theory, and natural frequencies are acquired by implementing harmonic differential quadrature method (HDQM). A detailed study is performed to examine the influences of the tensilecompressive and shear effects of vdWs interactions between adjacent GNRs as well as between GNRs and polymer matrix on the free vibration of BLGNRs. 
Graphenes can be synthesized either singlelayer or multilayer. Layers of multilayer graphenes are located next to each other by weak interactions known as vdWs interactions. These weak interactions change the mechanical and electrical properties of multilayer graphenes [1], where it can be attributed to the tensilecompressive and shear effects of vdWs interactions. Through an overview of the references studying the mechanical behavior of multilayer graphene sheets (MLGSs) and multilayer graphene nanoribbons (MLGNRs), it is found that they can be classified into two categories. In the first category, only tensilecompressive effects of van der Waals (vdWs) interactions between two graphene layers are contemplated on the mechanical behavior of MLGSs [211] or MLGNRs [1216]. For example, Ansari et al. [17] have explored effects of number of graphene layers and nonlocal parameters by employing ReissnerMindlin plate theory on the free vibration of MLGSs. In the second category, a handful of researchers have considered only shear effects of vdWs interactions between two graphene layers on the mechanical behavior of MLGNRs [18, 19]. In stance, Liu et al. [20] have explored the bending of cantilever bilayer and trilayer graphene nanoribbons incorporating the interlayer shear through employing Newmark’s composite beam theory. They indicate that considering of inplane displacement of nanoribbons has significant effect on the bending of multilayer graphene nanoribbons. This literature survey reveals that researchers have only considered one of the vdWs interactions effects between GNR layers, the tensilecompressive effect or the shear one. Consequently, as far as it’s reported there is no literature investigating the tensilecompressive and shear effects of vdWs interactions between adjacent graphene sheets (GSs) or GNRs, simultaneously.
It is known that in macro dimensions, the tensilecompressive and shear effects of the elastic medium are modeled by Winkler and Pasternak's terms defined in terms of transverse displacement. This approach is also applied in nano dimensions for embedded GSs or GNRs in elastic medium since vdWs interactions are observed not only between adjacent graphene layers but between elastic medium and graphene layers as well [10, 11, 15]. All of these studies have deliberetedonly transverse displacement of graphene layers. In other words, the inplane displacement of graphene layers has not been considered while it has been mentioned in previous paragraph that considering the inplane displacement of graphene layers has significant effect on the mechanical behavior of MLGSs and MLGNRs.
From the above consideration, two questions arise. First, what will the tensilecompressive and shear effects of vdWs interactions between adjacent graphene layers be on free vibration of GNRs when they are simultaneously considered? Second, what will the shear effect of elastic medium be on free vibration of GNRs if the inplane displacement of GNRs is considered? In order to cover the questions, a GNR is modeled based on the sandwich beam theory and tensilecompressive and shear effects of vdWs interactions between adjacent GNRs as well as elastic medium and GNRs are modeled as equivalent tensilecompressive and shear moduli in the equation of motion. The governing equations of motion are derived by applying Hamilton’s principle and solved numerically by HDQM. Consequently, natural frequencies are obtained for clampedclamped boundary conditions. Some comparison studies are performed to indicate the accuracy of formulation and solution procedure. The effects of equivalent tensilecompressive and shear moduli of vdWs interactions between elastic medium and GNRs on the first five natural frequencies of BLGNR are investigated. At last, the effects of equivalent tensilecompressive and shear moduli of vdWs interaction between elastic medium and GNRs on the natural frequencies of BLGNR are numerically compared with those of vdWs interactions between adjacent GNRs.
If we consider a BLGNR with the surrounding elastic medium in a continuum model as protrayed in Fig. 1, in order to have a better understanding of the study procedure, the flowchart diagram is presented for step by step understanding as Fig. 2. The model consists of five layers: two GNRs, a low density core connecting GNRs to each other (vdWs interactions), and two elastic mediums. Elastic mediums are bonded to the GNRs on the one side and connected to a fixed layer on the other side. All five layers are firmly bonded together, and vdWs interactions inertia is not notable. It is noteworthy to mention that considered elastic mediums are a type of vdWs interactions that connect GNRs to a polymer matrix such as Polyethylene [16]. The Cartesian coordinate system is applied, and the origin is located at the lefthand side of BLGNR in the middle of core thickness. The and coordinates of axes are in conformity with the length and thickness of BLGNR, respectively. Here , , , and denote length, width, thickness of GNR layers, thickness of core, and thickness of elastic mediums, respectively. The displacement components in accordance with and are displayed by and , respectively.
The vdWs interactions of elastic mediums are modeled in a way that they can withstand tensilecompressive and shear forces simultaneously. The vdWs interactions between GNR and polymer matrix can be modeled stronger or weaker than vdWs interactions between GNRs with each other. The GNRs are modeled based on EulerBernoulli beam theory. According to the theory the displacement field of the upper face ( and ) and the lower face ( and ) are defined as following [18].
in which , and , denote the displacements of an arbitrary point on midaxis of the top and bottom layers, respectively, and are the thickness of the nanoribbon layers and the core, respectively, and z is measured from the . The strain components of GNRs can be computed as [18]:
Fig. 1. Geometry and coordinate system of BLGNR embedded in an elastic medium. 
Fig. 2. The flowchart diagram for step by step understanding. 
weher and are the strain components of the top and bottom layers of graphene nanoribbon, respectively.
Pursuant to thesandwich beam theory, the longitudinal and transverse displacements ( and ) of the core (vdWs interaction between GNRs) are assumed linear through its thickness. On that account, the displacement components of the core can be obtained as following [18]:
As a general method, the elastic mediums are modeled as Winkler or Pasternak foundation [10, 11, 21, 22]. Winkler foundation considers only normal pressure, while the Pasternak foundation describes not only normal pressure but transverse shear stress as well. On the contrary to the conventional, the authors present a new model that contemplated the tensilecompressive and shear effects of the elastic medium for the first time. The modeling is based on the sandwich theory, and it is assumed that elastic medium has specific thickness, and longitudinal and transverse displacements of the elastic medium are varied linearly through the elastic medium thickness. Since the elastic medium is vdWs interactions between fixed matrix and GNRs, the displacement equations of the elastic medium can be acquired as below:
in which , and , denote the displacements of the top and bottom elastic mediums, is the elastic medium thickness, and and are measured from the . Since the core and elastic mediums are considered not to resist inplane loading, their longitudinal strains are insignificant. While bending and shear strains of the core ( and ) and elastic mediums ( and ) are significant and expressed as follow:
Now the stressstrain relations for the core, two faces, and two elastic mediums can be acquired as
where , , , and are the elastic modulus of a single layer of GNR, tensilecompressive modulus of the core, shear modulus of the core, tensilecompressive modulus of the elastic medium, and shear modulus of the elastic medium, respectively, and are the normal stresses in the x and z directions, respectively, and is the shear stress in the xz plane.
By employing Hamilton’s principle (equation (22)). Where U and T are potential energy and kinetic energy, and t is as time. and are defined as equation (2326).
(22) 
(23) 

(24) 

(25) 

(26) 
where , and are the cross sectional area of the faces, the core, and the elastic mediums , respectively. Furthermore, , and are second moment of area of the faces, the core, and the elastic mediums, respectively, and is density of graphene nanoribbons. The developed coupled equations (equations (23)(26)) are the governing equations of motion in which their couplings are as results to the tensilecompressive and shear effects of vdWs interactions between carboncarbon atoms as well as vdWs interactions between nanoribbons and polymer matrix. Confirming to Hamilton’s principle the boundary conditions are also generated as follow:
(27) 

(28) 

(29) 

(30) 

(31) 

(32) 
In the present work, a BLGNR with clampclamp ends is inspected where equations of the boundary condition are presented as follow:
(33) 
2.1. Solution Procedure
Because of the coupling of the governing equations of motion (equations (23)(26)), the problem does not have an analytical or semianalytical solution. For this reason, HDQM [18] is employed. This method was initiated from the idea of conventional integral quadrature and is a numerical discretization technique for the approximation of derivatives [23, 24]. Following this idea, the nth order derivative of the function f(x) with N grid points, is approximated by a linear sum of all the functional values in the entire domain, that is,
(34) 
where represents the functional value at a grid point , and is the weighting coefficient of the nth order derivative. In order to generate a mesh in coordinate on the computational domain of BLGNR , Chebyshev distribution method is involved, which is described as follow:
(35) 
For free vibration analysis of BLGNRs, the dynamic displacement vectors are expressed as follow:
(36) 
where is the natural frequency of BLGNR. By substituting equation (37) into equations (23)(26) and (27)(32), using , and as dimensionless parameters, where is a carbon atom thickness, and implementing HDQM, the governing equations and boundary conditions are discretized. In order to avoid repetitive representations, only the discretized form of equation (25) is presented here, as follows:
(37) 
Writing boundary condition and governing equations in matrix form yields the following equations
(38) 

(39) 
where {W_{B}} and {W_{I}} are the functional values of the boundary and interior points, respectively. After mathematical simplifications on equations (38) and (39), the following final eigenvalue equation system can be acquired:
(40) 
Now, the natural frequencies and corresponding mode shapes of BLGNRs can be obtained by solving equation (40).
In order to study tensilecompressive and shear effects of the elastic medium on the natural frequencies of BLGNRs the bending rigidity, mass density, length, thickness of sandwich core, width and thickness of nanoribbon layers are considered to be, respectively: , , , , and . Moreover, the equivalent tensilecompressive and shear (in armchair direction) moduli of vdWs interactions between two GNR layers are taken 26.6 GPa, and 482 MPa, respectively [1], and the following parameters are defined
(41) 
The definitions of and say that vdWs interactions between polymer matrix and GNR will be stiffer than those between GNR layers if and ; and it is the other way round if and . It is important to note that and are constant in the above definitions.
3.1. Comparison studies
In this section, two comparison studies for natural frequencies are conducted in Tables 1 and 2 to validate the results of the present formulation and confirm its reliability. As the first comparison study, in Table 1 results of the present formulation are compared with the first and second natural frequencies of a CBLGNR with only contempalting the shear modulus effect of vdWs interactions between GNR layers (the tensilecompressive modulus is considered to be high enough, ) [18].In the second comparison study (see Table 2) the present results are compared with the outofplane/inphase (OPlIPh) and outofplane/antiphase (OPlAPh) natural frequencies of a simply supported double nanobeam [15] with deliberating only the tensilecompressive modulus effect of vdWs interactions of GNRs (the shear modulus of vdWs interactions between GNRs is considered to be zero, ) for three different values of the tensilecompressive modulus. As it can be observed from Tables 1 and 2, the results of the present study are in excellent agreement with those reported in literatures.
3.2. Benchmark Results
In order to inspect the tensilecompressive and shear effects of a surrounding elastic medium on the vibrational behavior of BLGNRs, numerical natural frequency results for different and values are given. To this end, the frequency ratio is defined as follow:
Plus, five natural frequencies of BLGNR without elastic medium are presented in Table 3. The natural frequencies of BLGNR with elastic medium can be computed by applying Table 3 and frequency ratio values.
In the following, results are presented in three sections. In the first section, only tensilecompressive effects and in the second section only shear effects of vdWs interactions of the elastic medium on the first five natural frequencies of BLGNR are investigated.
In the third section, the tensilecompressive and shear effects of vdWs interactions of the elastic medium on natural frequencies are simultaneously explored. Addintionally, in the last section, effects of aspect ratio of BLGNRs on the natural frequencies is contemplated for two different values of elastic medium moduli.
Firstly, in Fig. 3 variations of frequency ratio versus the mode number are plotted for various values of E^{*}( or ). It is noteworthy to mention that in Fig. 3 the value of is set to zero.
Table 1. Comparison of first and second natural frequencies of CBLGNR incorporating the shear modulus effect of vdWs interactions of GNR layers ( ) and . 

Shear modulus (GPa) 
1^{st} frequency (GHz) 

2^{nd} frequency (GHz) 

Ref. [18] 
Present study 

Ref. [18] 
Present study 

0.25 
6.324 
6.324 

29.360 
29.360 
4.6 
10.246 
10.246 

62.591 
63.591 
Table 2. Comparison of OPlIPh and OPlAPh dimensionless natural frequencies of BLGNR ( ). 

Tensilecompressive modulus (GPa) 
First OPlIPl frequency 

First OPlAPh frequency 

Ref. [15] 
Present study 

Ref. [15] 
Present study 

10 
9.869 
9.869 

1220 
1220 
20 
9.869 
9.869 

1725.4 
1725.4 
30 
9.869 
9.869 

2113.1 
2113.1 
It is observed from Fig. 3 that all frequency ratio curves indicate a monotonically reducing trend as the mode number increases. This implies that the tensilecompressive effects of the elastic medium on natural frequencies decrease by increasing the mode number. Furthermore, Fig. 3 shows that the tensilecompressive effects of elastic medium are notably less influential at higher mode numbers. This is a result of this fact that as the mode number increases, the dominant displacement of BLGNR layers changes from out of plane to inplane. In general, it can also be observed in Fig. 3 that lower mode numbers are more dependent on the variations of the tensilecompressive modulus value than higher ones. Furthermore, Fig. 3 displays that the first five natural frequencies of BLGNRs are independent of the value of tensilecompressive modulus for . The reason of this is that increasing the value of causes the elastic medium to become stiffer, and accordingly, displacements of BLGNR layers become completely inplane. As a final point it is worth noting that since by increasing the tensilecompressive modulus of elastic medium displacements of BLGNR layers become completely inplane, it is expected that natural frequencies of BLGNRs become independent of the value of interlayer tensilecompressive modulus as well.
Continously, in order to consider the shear modulus effects of the elastic medium on the natural frequencies of BLGNRs, Fig. 4 is plotted. In Fig. 4 variations of frequency ratios versus mode number are inidacted for various values of ( or ) when . Fig. 4 shows that all frequency ratio curves have a monotonically decreasing trend as the mode number increases, like the one observed in Fig. 3. Consequently, an important result is that the shear effects of the elastic medium on the natural frequencies decrease by increasing the mode number.
Unlike Fig. 3 where variations of the frequency ratios versus the mode number are different for various values of the , it can be observed from Fig. 4 that variations of the frequency ratios versus the mode number are the same for various values of the . Furthermore, it can be observed from Fig. 4 that by increasing the mode number, variations of the frequency ratios become independent of values of the shear modulus of the elastic medium if . In another words, effects of the interlayer shear modulus on frequency ratios become more pronounced for . A final point to mention is that since by increasing the shear modulus of the elastic medium displacements of BLGNR layers become completely out of plane, it is expected that the natural frequencies of BLGNRs also become independent of the value of the interlayer shear modulus.
After deliberating tensilecompressive and shear moduli effects of the elastic medium on the frequency ratios of BLGNRs separately, now ispecting their effects are simultaneously desired. To this end, variations of the frequency ratios versus the mode number for various values of and are protrayed in Fig. 5. It is seen from Fig. 5 that the tensilecompressive and shear moduli effects of the elastic medium have a significant influence on low mode numbers, and their effects decrease by increasing the mode numbers. Also, as the stiffness of the elastic medium increases (increasing and ), the frequency ratio incrases. This implies that the natural frequencies of the embedded bilayer nanoribbons will increase by increasing and . The final point of Fig. 5 is that for low values of the and , variations of the frequency ratios are independent of the mode number.
Table 3. Natural frequencies of BLGNR without elastic medium. 

Mode number 

347.90 
252.01 
163.42 
92.94 
40.61 
Frequency (GHz) 
Fig. 3. Variations of frequency ratio with mode number for different values of 
Fig. 4. Variations of frequency ratio with mode number for different values of 
Fig. 5. Variations of frequency ratio versus mode number for various values of and 
Comparing Fig. 5 with Figs. 3 and 4 represent that when both the tensilecompressive and shear moduli of elastic medium are simultaneously considered, their effects on natural frequencies significantly increase in comparison with the case that their effects are separately investigated. For example, when and , the first frequency ratio becomes 8.57; and when and , the first frequency ratio becomes 3.38. But when and , the first frequency ratio reaches to 54.65. At last, in Tables 4 and 5, influences of the tensilecompressive and shear moduli of elastic medium are numerically compared with those of interlayer vdWs interactions. In Tables 4 and 5, natural frequencies and their value changes are categorized as a result of considering the elastic medium and interlayer vdWs moduli, respectively. The following findings can be highlighted by comparing the results shown in Tables 4 and 5:
Table 4. Frequency value and change in frequency value for three different cases of tensilecompressive and shear moduli of elastic medium (E_{c} = 26.6 GPa, G_{c} = 0.482 GPa). 

Mode number 
E^{*}=0.01 G^{*}=0 
E^{*}=0.05 G^{*}=0 
Change in frequency value (%) 
E^{*}=0.001 G^{*}=0.1 
E^{*}=0.001 G^{*}=0.5 
Change in frequency value (%) 
E^{*}=0.01 G^{*}=0.1 
E^{*}=0.05 G^{*}=0.5 
Change in frequency value (%) 

Frequency (GHz) 
Frequency (GHz) 
Frequency (GHz) 

1 
167.94 
347.90 
107.15 
66.29 
68.81 
3.80 
168.21 
367.26 
118.33 

2 
187.50 
366.63 
95.54 
107.79 
113.52 
5.31 
188.39 
377.95 
100.62 

3 
230.76 
375.42 
62.68 
173.22 
180.38 
4.13 
232.16 
380.93 
64.08 

4 
300.07 
399.27 
33.06 
259.32 
267.44 
3.13 
301.88 
403.30 
33.60 

5 
347.90 
442.39 
27.16 
354.75 
375.02 
5.71 
354.75 
448.90 
26.54 

6 
395.55 
467.65 
18.23 
366.32 
380.93 
3.99 
397.60 
491.83 
23.70 

7 
466.46 
512.51 
9.87 
471.33 
491.15 
4.21 
471.55 
520.36 
10.35 

8 
515.70 
609.98 
18.28 
494.44 
503.57 
1.85 
517.90 
619.21 
19.56 

9 
658.75 
695.81 
5.63 
642.75 
652.18 
1.47 
661.07 
712.89 
7.84 

10 
695.81 
734.71 
5.59 
699.26 
712.89 
1.95 
699.26 
745.20 
6.57 

11 
762.32 
762.69 
0.05 
765.40 
777.72 
1.61 
765.43 
777.91 
1.63 

12 
824.68 
886.75 
7.53 
812.49 
822.13 
1.19 
827.07 
897.80 
8.55 

13 
1012.63 
1043.71 
3.07 
1003.23 
1013.02 
0.98 
1015.07 
1055.17 
3.95 

14 
1043.71 
1063.80 
1.92 
1046.02 
1055.17 
0.88 
1046.02 
1075.35 
2.80 

15 
1089.15 
1089.20 
0.00 
1091.32 
1100.01 
0.80 
1091.33 
1100.04 
0.80 

Table 5. Frequency value and change in frequency value for three different cases of tensilecompressive and shear moduli of core (E^{*} = 0, G^{*} = 0). 

Mode number 
E_{c}=26.6 G_{c}=0 
E_{c}=42 G_{c}=0 
Change in frequency value (%) 
E_{c}=4000 G_{c}=0.482 
E_{c}=4000 G_{c}=4.8 
Change in frequency value (%) 
E_{c}=26.6 G_{c}=0.482 
E_{c}=42 G_{c}=4.8 
Change in frequency value (%) 

Frequency (GHz) 
Frequency (GHz) 
Frequency (GHz) 

1 
23.96 
23.96 
0.00 
40.61 
69.95 
72.25 
40.61 
69.95 
72.25 

2 
66.05 
66.05 
0.00 
92.94 
164.77 
77.28 
92.94 
164.77 
77.28 

3 
129.48 
129.48 
0.00 
163.42 
282.98 
73.16 
163.42 
282.98 
73.16 

4 
214.04 
214.04 
0.00 
252.01 
347.90 
38.05 
252.01 
347.90 
38.05 

5 
319.73 
319.73 
0.00 
347.90 
417.09 
19.89 
347.90 
417.09 
19.89 

6 
347.90 
347.90 
0.00 
360.43 
565.43 
56.88 
360.43 
565.43 
56.88 

7 
347.90 
347.90 
0.00 
466.20 
695.81 
49.25 
466.20 
695.81 
49.25 

8 
446.57 
446.57 
0.00 
489.46 
727.67 
48.67 
489.46 
727.67 
48.67 

9 
594.55 
594.55 
0.00 
638.29 
904.78 
41.75 
638.29 
904.78 
41.75 

10 
695.81 
695.81 
0.00 
695.81 
1041.59 
49.69 
695.81 
1041.59 
49.69 

11 
695.81 
695.81 
0.00 
762.29 
1043.71 
36.92 
762.29 
1043.71 
36.92 

12 
763.66 
763.66 
0.00 
808.42 
1099.87 
36.05 
808.42 
1099.87 
36.05 

13 
953.92 
953.92 
0.00 
999.43 
1209.79 
21.05 
999.43 
1209.79 
21.05 

14 
1043.71 
1043.71 
0.00 
1043.71 
1311.14 
25.62 
1043.71 
1311.14 
25.62 

15 
1043.71 
1043.71 
0.00 
1089.14 
1391.62 
27.77 
1089.14 
1391.62 
27.77 

The tensilecompressive and shear effects of vdWs interactions between adjacent GNRs as well as elastic medium and GNRs on free vibration of BGNRs are investigated. In order to explore inplane displacements of GNRs and both tensilecompressive and shear effects of vdWs interactions, sandwich beam theory is utilized. Governing equations of motion are acquired and solved numerically by HDQM. Results show that lower mode numbers are more dependent on the tensilecompressive effects of the elastic medium than higher ones. Furthermore, the tensilecompressive effects of elastic medium on natural frequencies are more than the shear effects of elastic medium, especially at low mode numbers. It is also observed that the effects of interlayer shear are pronounced at low mode numbers while it’s the other way round for the effects of interlayer tensilecompressive. This study implies that for an accurate analysis of multilayer graphene nanoribbons embedded in an elastic medium the tensilecompressive and shear effects of vdWs interactions between adjacent GNRs as well as elastic medium and GNRs must be contemplated simultaneously.
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