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Elastic buckling of single-layered graphene sheet
Computational Materials Science 45 (2009) 266–270
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Elastic buckling of single-layered graphene sheet A. Sakhaee-Pour Center of Excellence in Design, Robotics and Automation (CEDRA), Department of Mechanical Engineering, Sharif University of Technology, P.O. Box 11365-9567, Azadi Avenue, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 10 June 2008 Received in revised form 21 August 2008 Accepted 23 September 2008 Available online 8 November 2008 PACS: 62.23.Kn 62.25.g 62.20.mq 61.46.w 62.20.M-
a b s t r a c t The elastic buckling behavior of defect-free single-layered graphene sheet (SLGS) is investigated using an atomistic modeling approach. In this regard, the molecular structural mechanics that is comprised of equivalent structural beams is employed. The elastic buckling forces of the cantilever and bridge zigzag and armchair SLGSs with different side lengths and aspect ratios are calculated. It is discerned that the elastic buckling force per unit width of the SLGS changes nonlinearly with respect to the side length while it is insensitive to the aspect ratio. In addition, the elastic buckling force of the zigzag sheet is larger than that of the armchair sheet with equivalent geometrical parameters. The atomistic simulation results are also used to develop predictive equations via a statistical nonlinear regression model. The proposed equations can estimate the elastic buckling force of the SLGS within 5 percent difference with the molecular structural mechanics method. Ó 2008 Elsevier B.V. All rights reserved.
Keywords: Single-layered graphene sheet Elastic buckling
1. Introduction Stankovich et al. have recently achieved a valuable method of preparing graphene sheets for mass production usages [1]. The graphene sheets are envisaged to have most of the extraordinary properties of the carbon nanotubes (CNTs) while they can be made considerably cheaper by implementing the new method. Therefore, the probability of employing the nanostructures for ordinary applications has been increased. Consequently, this achievement can lead to a new era for nanoscience and nanotechnology in near future. A great deal of research has been conducted to explore the promising properties of the single-layered graphene sheets (SLGSs) after appearance of the new method of graphene sheet preparation. Stankovich et al. have proposed their findings related to the synthesis and exfoliation of isocyanate-treated graphene oxide nanoplatelets [2]. Implementing the chemical reduction, they have also been able to produce the graphene-based nanosheets [3]. In addition, Ferrari has reported the Raman spectroscopy of the SLGS [4]. Furthermore, Katsnelson and Novoselov have explored the unique electronic properties of the SLGSs [5]. They have stated that the graphene sheet is an unexpected bridge between condensed matter physics and quantum electrodynamics. Moreover, Meyer
E-mail address: [email protected] 0927-0256/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2008.09.024
et al. have achieved the ability of distinguishing between singleand multi-layered graphene sheets by analyzing electron diffraction [6]. On the other hand, Bunch et al. have reported the experimental results of using electromechanical resonators made from suspended single- and multi-layered graphene sheets [7]. In addition, the authors have studied the free vibrational behavior of the SLGS while considering the effects of chirality and aspect ratio as well as boundary conditions, and have developed predictive models for computing natural frequencies [8]. The potential applications of the SLGSs as mass sensors and atomistic dust detectors have further been investigated [9]. Also, the promising usage of the SLGS as strain sensor has been examined [10]. Since the analytical approaches are elaborate, an inclination toward numerical techniques was developed. Li and Chou presented a concept of modeling nanostructures with equivalent space-frame structures, the molecular structural mechanics [11]. In this regard, they proposed an equivalent structural beam with the capability of modeling interatomic forces of the covalent bonds. Then, they adopted the molecular structural mechanics method to compute the Young modulus of single-walled carbon nanotube (SWCNT). They further explored the radial deformation of the SWCNT under hydrostatic pressure [12]. Later, Tserpes and Papanikos introduced a three-dimensional finite element (FE) method, based on the Li and Chou concept, to investigate the influence of tube wall thickness on the SWCNT moduli of elasticity [13]. In addition, they examined the effect of Stone–Wales (SW) defect on the SWCNT
A. Sakhaee-Pour / Computational Materials Science 45 (2009) 266–270
fracture [14]. On the other hand, Cho reported the bending and shear moduli of the SWCNT using the FE method while considering the Poisson’s effect [15]. Also, Kalamkarov et al. applied the FE method to simulate the single- and multi-walled carbon nanotubes behavior under tension and torsion [16]. Moreover, the authors investigated the radial deformation of the SWCNT with different tube wall thicknesses [17]. In this study, the elastic buckling characteristic of the SLGS while considering the influences of chirality, boundary conditions and geometrical parameters are explored. The elastic buckling forces of the cantilever and bridge zigzag and armchair sheets with different side lengths and aspect ratios are presented. For this purpose, an atomistic modeling approach is used to simulate the twodimensional layer behavior. 2. Atomistic modeling of single-layered graphene sheets To analyze the elastic buckling characteristic of defect-free SLGS at a constant temperature, the molecular structural mechanics method is applied. In this atomistic modeling approach, the equivalent structural beams are employed to mimic interatomic forces of the carbon atoms which are connected via covalent bonds in the so-called honeycomb lattice. The elastic properties of the beam are computed in terms of the covalent bond stiffness to simulate the interatomic forces. The properties of the beam were developed by considering the equivalent potential energies of the molecular and structural mechanics [11]. In order to calculate the beam properties, force filed constants of the covalent bonds are used as:
EA ¼ kr ; L
EI ¼ kh ; L
GJ ¼ k/ L
ð1Þ
267
where the force field constants kr, kh and k/ represent stretching, angle bending and torsional stiffness of the covalent bonds, respectively, and E and G are moduli of elasticity and shear of the beam, respectively. In addition, A is the cross section area, I, the moment of inertia, J, the polar moment of inertia, and L the length of the beam. The beam length is assumed to be equal to the covalent bond distance of the carbon atoms in the hexagonal lattice. On the other hand, the equivalent structural beam is considered to have a circular cross section to explicitly derive the elastic properties. The elastic properties of the beam with circular cross section are obtained from the previous equation as [13]:
rffiffiffiffiffi kh d¼4 kr 2
E¼
kr L 4pkh
G¼
2 kr k/ L 2 8 kh
ð2Þ
p
Where d, E and G are, respectively, cross section diameter, moduli of elasticity and shear of the circular equivalent structural beam. Therefore, the molecular structural mechanics approach models the interatomic forces by adapting the beam properties in terms of the covalent bond characteristics. In the molecular structural mechanics approach, the SLGS is simulated as the equivalent space-frame structure; therefore, the equivalent model is employed to study the elastic buckling behavior. Consequently, the elastic buckling force of the model is computed by the structural stability method [18] that finds compressive forces by which the generated model does not have a unique equilibrium configuration. The compressive forces are
Fig. 1. Chirality and boundary conditions of the single-layered graphene sheets.
268
A. Sakhaee-Pour / Computational Materials Science 45 (2009) 266–270
Table 1 Geometrical parameters of the single-layered graphene sheet
Cantilever Zigzag SLGS
1
10
a/b=0.2165 a/b=0.4330 a/b=0.8660 a/b=1.7319 a/b=3.4638
Zigzag sheet
Armchair sheet 0.9845 nm 0.2887
6a6 6 ab 6
40.0722 nm 3.4638
6a6 6 ab 6
45.2869 nm 4.6188
calculated through the eigenvalue problem solution [19]. Then, the minimum force is considered as the buckling force of the SLGS. In order to study the elastic buckling behavior of the SLGS, the effects of chirality, geometrical parameters and boundary conditions are considered. In this regard, the zigzag and armchair SLGSs are adopted since chirality is envisaged to have a significant influence on the mechanical properties of the SLGS [20]. To define the SLGS geometry, the side length and width are assumed to be a and b with ab as an aspect ratio. a and b can also be found via the lattice translation vector [21]. The schematic illustrations of the SLGSs are shown in Fig. 1. On the other hand, the buckling force per unit width of the SLGS is calculated for different side lengths and aspect ratios to explore the geometrical parameters role. The ranges of the parameters are considered to be identical with the investigation done on the vibrational analysis of the SLGS [8] and presented in Table 1. In addition, the buckling forces are computed for cantilever and bridge configurations to examine the boundary conditions effects.
0
Buckling Force (N/nm)
0.8526 nm 0.2165
10
-1
10
-2
10
-3
10
0
5
10
15
20
25
30
35
Cantilever Armchair SLGS
1
10
a/b=0.2887 a/b=0.5774 a/b=1.1547 a/b=2.3094 a/b=4.6188
0
Table 2 Force field constants of the covalent bonds and graphene sheet thickness kr kh k/ t
6.52e7 N nm1 8.76e10 N nm rad2 2.78e10 N nm rad2 0.34 nm
Buckling Force (N/nm)
The elastic buckling analysis of the SLGS is performed using the force field constants [22] and graphene sheet thickness presented in Table 2. For the defect-free SLGS, the elastic buckling force is calculated by applying the molecular structural mechanics approach. The elastic buckling force of the cantilever SLGS is computed while considering the influences of the geometrical parameters, namely; side length and aspect ratio. The computational results for the buckling forces per unit widths of the cantilever zigzag and armchair SLGSs are demonstrated in Figs. 2 and 3, respectively. The atomistic simulation reveals the buckling force decreases nonlinearly with respect to the side length. In addition, the aspect ratio does not play an important role in the force variation. To shed light on the aspect ratio independency, it should be noted that the elastic buckling mode shape of the SLGS is hyperbolic cosine and it has no deflection in the width direction. Similar to the elastic buckling behavior, it has been predicted that the aspect ratio has a negligible effect on the SLGS fundamental frequency [8–10]. The elastic buckling behavior of the bridge SLGS is studied. In this regard, the elastic buckling forces of the zigzag and armchair sheets with different side lengths and aspect ratios are calculated. The elastic buckling forces of the zigzag and armchair sheets with bridge configuration are, respectively, shown in Figs. 4 and 5. It is discerned that the buckling force changes nonlinearly with side length while it is insensitive to the aspect ratio. The elastic buckling mode shape of the SLGS is the reason for the independency as explained for the cantilever boundary condition.
45
Fig. 2. Elastic buckling forces per unit width of the cantilever zigzag single-layered graphene sheets.
10
3. Results
40
Side Length (nm)
-1
10
-2
10
-3
10
-4
10
0
5
10
15
20
25
30
35
40
45
50
Side Length (nm) Fig. 3. Elastic buckling forces per unit width of the cantilever armchair singlelayered graphene sheets with different geometrical parameters.
Considering the results for the SLGSs with similar boundary conditions, it exhibits that the elastic buckling force is chirality dependent. It is found that the elastic buckling force of the zigzag sheet is slightly larger than the armchair sheet with equivalent geometrical parameters. For example, the buckling force per unit width of the bridge zigzag sheet with a = 2.1315 (nm) and a ¼ 0:4330 is 5.5801 (N/nm) while this value changes to 4.9941 b (N/nm) for the bridge armchair sheet with equivalent geometrical parameters. This is due to the covalent bonds orientations of the sheets; the ratio of the covalent bonds that are parallel to the loading direction is greater in the zigzag sheet. The elastic buckling forces of the cantilever and bridge configurations are compared with each other. For this purpose, SLGSs with identical geometrical parameters and chirality are considered. It is observed that the buckling forces of the clamped-free and clamped–clamped boundary conditions have similar trends. However, the buckling force of the bridge boundary condition is found to be 15–16 times more than that of the cantilever one. On the
A. Sakhaee-Pour / Computational Materials Science 45 (2009) 266–270
(b) Cantilever armchair sheet
Bridge Zigzag SLGS
2
269
10
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2:7523 a 0:5012 f ¼ a2:2152
a/b=0.2165 a/b=0.4330 a/b=0.8660 a/b=1.7319 a/b=3.4638
ð4Þ
1
Buckling Force (N/nm)
10
(c) Bridge zigzag sheet
f ¼
ð5Þ
0
10
(d) Bridge armchair sheet
f ¼
-1
10
-2
10
0
5
10
15
20
25
30
35
40
45
Side Length (nm) Fig. 4. Elastic buckling forces per unit width of the bridge zigzag single-layered graphene sheets.
Bridge Armchair SLGS
2
10
a/b=0.2887 a/b=0.5774 a/b=1.1547 a/b=2.3094 a/b=4.6188 1
10
Buckling Force (N/nm)
55:410 1 1:1825 a þ 2:3934 a2
0
10
-1
10
-2
10
0
5
10
15
20
25
30
35
40
45
50
Side Length (nm) Fig. 5. Elastic buckling forces per unit width of the bridge armchair single-layered graphene sheets.
other hand, it is noted that this ratio for the continuum plate is 16 and this indicates there is a good agreement between the continuum and atomistic-based modeling results. In order to avoid the time-consuming procedure of the implemented approach in finding the elastic buckling force of the SLGS, predictive equations are proposed. To this end, adjustable parameters of the equations are computed through a statistical nonlinear regression model [23]. The accuracy of the model is measured using a multiple coefficient of determination, R2, where 0 6 R2 6 1. In the present study, the value of R2 is found to be greater than 0.99 for all considered data, demonstrating a good fit of the predictive equations results to the molecular structural mechanics data. The predictive equations are: (a) Cantilever zigzag sheet
f ¼
1:6298 a1:9208
ð3Þ
1 2:0464 1:1713 a þ 3:9027 a2
ð6Þ
where a is the SLGS side length in (nm) and f represents the buckling force per unit width in (N/nm). These equations estimate the elastic buckling force within 5 percent difference with the atomistic simulation method in the ranges of the geometrical parameters (please see Table 1).
4. Conclusions The elastic buckling analysis of defect-free single-layered graphene sheet (SLGS) is performed through the molecular structural mechanics. In this atomistic simulation, equivalent structural beams are adopted to model interatomic forces of the covalently bonded carbon atoms in the so-called honey comb lattice. To explore the mechanical characteristic, the effects of side length, aspect ratio and chirality on the elastic buckling forces of the SLGSs with clamped-free and clamped–clamped configurations are studied. The atomistic modeling approach indicates that the buckling force decreases nonlinearly versus side length whereas this variation is not affected by the aspect ratio. In addition, the buckling force of the zigzag sheet is larger than the armchair sheet with equivalent geometrical parameters as a consequence of having more covalent bonds that are parallel to the loading direction. On the other hand, the buckling force of the cantilever and bridge SLGSs are compared with each other. It is observed that the bucking force of the bridge boundary condition is 15–16 times more than the cantilever one with identical chirality and geometrical parameters and this is in good agreement with the continuum theory results. Based on the molecular structural mechanics findings, equations for the SLGS buckling force prediction are developed using a nonlinear regression model. Since the aspect ratio does not play an important role in the buckling force variation, the equations are appeared as functions of the SLGS side length. These equations provide a quick tool for the buckling force calculation while their results remain within 5 percent difference with the atomistic modeling data. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
S. Stankovich et al., Nature 442 (2006) 282. S. Stankovich et al., Carbon 15 (2006) 3342–3347. S. Stankovich et al., Carbon 7 (2007) 1558–1565. A.C. Ferrari, Solid State Commun. 143 (2007) 47–57. M.I. Katsnelson, K.S. Novoselov, Solid State Commun. 143 (2007) 3–13. C.J. Meyer et al., Nature 446 (2006) 60. J. Bunch et al., Science 315 (2007) 490. A. Sakhaee-Pour, M.T. Ahmadian, R. Naghdabadi, Nanotechnology 19 (2008) 085702. A. Sakhaee-Pour, M.T. Ahmadian, A. Vafai, Solid State Commun. 4 (2008) 168– 172. A. Sakhaee-Pour, M.T. Ahmadian, Solid State Commun. 147 (2008) 336–340. C. Li, T.-W. Chou, Int. J. Solids Struct. 40 (2003) 2487–2499. C. Li, T.-W. Chou, Phys. Rev. B 69 (2004) 073401. K.I. Tserpes, P. Papanikos, Compos. B 36 (2005) 468–477.
270 [14] [15] [16] [17]
A. Sakhaee-Pour / Computational Materials Science 45 (2009) 266–270
K.I. Tserpes, P. Papanikos, Compos. Struct. 79 (2007) 581–589. W.S. To Cho, Finite Elem. Anal. Des. 42 (2006) 404–413. A.L. Kalamkarov et al., Int. J. Solids Struct. 43 (2006) 6832–6854. A. Sakhaee-pour, A. Gerami, M.T. Ahmadian, Proc. IMechE Part C: J Mech. Eng. Sci. 6 (2008) 1109–1115. [18] W.F. Chen, E.M. Lui, Structural Stability Theory and Implementation, Elsevier Scientific Publication, New York, 1987.
[19] [20] [21] [22] [23]
C. Li, T.-W. Chou, Mech. Mater. 36 (2004) 1047–1055. K. Behfar, R. Naghdabadi, Compos. Sci. Technol. 65 (2005) 1159–1164. K.-T. Lau, D. Hui, Compos. Part B Eng. 33 (2002) 263–277. A.K. Rappe et al., J. Am. Chem. Soc. 114 (1992) 10024–10035. W. Mendenhall, J.E. Reinmuth, R. Beaver, Statistics for Management and Economics, PWS-Kent, Boston, MA, 1989.
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Elastic buckling of single-layered graphene sheet A. Sakhaee-Pour Center of Excellence in Design, Robotics and Automation (CEDRA), Department of Mechanical Engineering, Sharif University of Technology, P.O. Box 11365-9567, Azadi Avenue, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 10 June 2008 Received in revised form 21 August 2008 Accepted 23 September 2008 Available online 8 November 2008 PACS: 62.23.Kn 62.25.g 62.20.mq 61.46.w 62.20.M-
a b s t r a c t The elastic buckling behavior of defect-free single-layered graphene sheet (SLGS) is investigated using an atomistic modeling approach. In this regard, the molecular structural mechanics that is comprised of equivalent structural beams is employed. The elastic buckling forces of the cantilever and bridge zigzag and armchair SLGSs with different side lengths and aspect ratios are calculated. It is discerned that the elastic buckling force per unit width of the SLGS changes nonlinearly with respect to the side length while it is insensitive to the aspect ratio. In addition, the elastic buckling force of the zigzag sheet is larger than that of the armchair sheet with equivalent geometrical parameters. The atomistic simulation results are also used to develop predictive equations via a statistical nonlinear regression model. The proposed equations can estimate the elastic buckling force of the SLGS within 5 percent difference with the molecular structural mechanics method. Ó 2008 Elsevier B.V. All rights reserved.
Keywords: Single-layered graphene sheet Elastic buckling
1. Introduction Stankovich et al. have recently achieved a valuable method of preparing graphene sheets for mass production usages [1]. The graphene sheets are envisaged to have most of the extraordinary properties of the carbon nanotubes (CNTs) while they can be made considerably cheaper by implementing the new method. Therefore, the probability of employing the nanostructures for ordinary applications has been increased. Consequently, this achievement can lead to a new era for nanoscience and nanotechnology in near future. A great deal of research has been conducted to explore the promising properties of the single-layered graphene sheets (SLGSs) after appearance of the new method of graphene sheet preparation. Stankovich et al. have proposed their findings related to the synthesis and exfoliation of isocyanate-treated graphene oxide nanoplatelets [2]. Implementing the chemical reduction, they have also been able to produce the graphene-based nanosheets [3]. In addition, Ferrari has reported the Raman spectroscopy of the SLGS [4]. Furthermore, Katsnelson and Novoselov have explored the unique electronic properties of the SLGSs [5]. They have stated that the graphene sheet is an unexpected bridge between condensed matter physics and quantum electrodynamics. Moreover, Meyer
E-mail address: [email protected] 0927-0256/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2008.09.024
et al. have achieved the ability of distinguishing between singleand multi-layered graphene sheets by analyzing electron diffraction [6]. On the other hand, Bunch et al. have reported the experimental results of using electromechanical resonators made from suspended single- and multi-layered graphene sheets [7]. In addition, the authors have studied the free vibrational behavior of the SLGS while considering the effects of chirality and aspect ratio as well as boundary conditions, and have developed predictive models for computing natural frequencies [8]. The potential applications of the SLGSs as mass sensors and atomistic dust detectors have further been investigated [9]. Also, the promising usage of the SLGS as strain sensor has been examined [10]. Since the analytical approaches are elaborate, an inclination toward numerical techniques was developed. Li and Chou presented a concept of modeling nanostructures with equivalent space-frame structures, the molecular structural mechanics [11]. In this regard, they proposed an equivalent structural beam with the capability of modeling interatomic forces of the covalent bonds. Then, they adopted the molecular structural mechanics method to compute the Young modulus of single-walled carbon nanotube (SWCNT). They further explored the radial deformation of the SWCNT under hydrostatic pressure [12]. Later, Tserpes and Papanikos introduced a three-dimensional finite element (FE) method, based on the Li and Chou concept, to investigate the influence of tube wall thickness on the SWCNT moduli of elasticity [13]. In addition, they examined the effect of Stone–Wales (SW) defect on the SWCNT
A. Sakhaee-Pour / Computational Materials Science 45 (2009) 266–270
fracture [14]. On the other hand, Cho reported the bending and shear moduli of the SWCNT using the FE method while considering the Poisson’s effect [15]. Also, Kalamkarov et al. applied the FE method to simulate the single- and multi-walled carbon nanotubes behavior under tension and torsion [16]. Moreover, the authors investigated the radial deformation of the SWCNT with different tube wall thicknesses [17]. In this study, the elastic buckling characteristic of the SLGS while considering the influences of chirality, boundary conditions and geometrical parameters are explored. The elastic buckling forces of the cantilever and bridge zigzag and armchair sheets with different side lengths and aspect ratios are presented. For this purpose, an atomistic modeling approach is used to simulate the twodimensional layer behavior. 2. Atomistic modeling of single-layered graphene sheets To analyze the elastic buckling characteristic of defect-free SLGS at a constant temperature, the molecular structural mechanics method is applied. In this atomistic modeling approach, the equivalent structural beams are employed to mimic interatomic forces of the carbon atoms which are connected via covalent bonds in the so-called honeycomb lattice. The elastic properties of the beam are computed in terms of the covalent bond stiffness to simulate the interatomic forces. The properties of the beam were developed by considering the equivalent potential energies of the molecular and structural mechanics [11]. In order to calculate the beam properties, force filed constants of the covalent bonds are used as:
EA ¼ kr ; L
EI ¼ kh ; L
GJ ¼ k/ L
ð1Þ
267
where the force field constants kr, kh and k/ represent stretching, angle bending and torsional stiffness of the covalent bonds, respectively, and E and G are moduli of elasticity and shear of the beam, respectively. In addition, A is the cross section area, I, the moment of inertia, J, the polar moment of inertia, and L the length of the beam. The beam length is assumed to be equal to the covalent bond distance of the carbon atoms in the hexagonal lattice. On the other hand, the equivalent structural beam is considered to have a circular cross section to explicitly derive the elastic properties. The elastic properties of the beam with circular cross section are obtained from the previous equation as [13]:
rffiffiffiffiffi kh d¼4 kr 2
E¼
kr L 4pkh
G¼
2 kr k/ L 2 8 kh
ð2Þ
p
Where d, E and G are, respectively, cross section diameter, moduli of elasticity and shear of the circular equivalent structural beam. Therefore, the molecular structural mechanics approach models the interatomic forces by adapting the beam properties in terms of the covalent bond characteristics. In the molecular structural mechanics approach, the SLGS is simulated as the equivalent space-frame structure; therefore, the equivalent model is employed to study the elastic buckling behavior. Consequently, the elastic buckling force of the model is computed by the structural stability method [18] that finds compressive forces by which the generated model does not have a unique equilibrium configuration. The compressive forces are
Fig. 1. Chirality and boundary conditions of the single-layered graphene sheets.
268
A. Sakhaee-Pour / Computational Materials Science 45 (2009) 266–270
Table 1 Geometrical parameters of the single-layered graphene sheet
Cantilever Zigzag SLGS
1
10
a/b=0.2165 a/b=0.4330 a/b=0.8660 a/b=1.7319 a/b=3.4638
Zigzag sheet
Armchair sheet 0.9845 nm 0.2887
6a6 6 ab 6
40.0722 nm 3.4638
6a6 6 ab 6
45.2869 nm 4.6188
calculated through the eigenvalue problem solution [19]. Then, the minimum force is considered as the buckling force of the SLGS. In order to study the elastic buckling behavior of the SLGS, the effects of chirality, geometrical parameters and boundary conditions are considered. In this regard, the zigzag and armchair SLGSs are adopted since chirality is envisaged to have a significant influence on the mechanical properties of the SLGS [20]. To define the SLGS geometry, the side length and width are assumed to be a and b with ab as an aspect ratio. a and b can also be found via the lattice translation vector [21]. The schematic illustrations of the SLGSs are shown in Fig. 1. On the other hand, the buckling force per unit width of the SLGS is calculated for different side lengths and aspect ratios to explore the geometrical parameters role. The ranges of the parameters are considered to be identical with the investigation done on the vibrational analysis of the SLGS [8] and presented in Table 1. In addition, the buckling forces are computed for cantilever and bridge configurations to examine the boundary conditions effects.
0
Buckling Force (N/nm)
0.8526 nm 0.2165
10
-1
10
-2
10
-3
10
0
5
10
15
20
25
30
35
Cantilever Armchair SLGS
1
10
a/b=0.2887 a/b=0.5774 a/b=1.1547 a/b=2.3094 a/b=4.6188
0
Table 2 Force field constants of the covalent bonds and graphene sheet thickness kr kh k/ t
6.52e7 N nm1 8.76e10 N nm rad2 2.78e10 N nm rad2 0.34 nm
Buckling Force (N/nm)
The elastic buckling analysis of the SLGS is performed using the force field constants [22] and graphene sheet thickness presented in Table 2. For the defect-free SLGS, the elastic buckling force is calculated by applying the molecular structural mechanics approach. The elastic buckling force of the cantilever SLGS is computed while considering the influences of the geometrical parameters, namely; side length and aspect ratio. The computational results for the buckling forces per unit widths of the cantilever zigzag and armchair SLGSs are demonstrated in Figs. 2 and 3, respectively. The atomistic simulation reveals the buckling force decreases nonlinearly with respect to the side length. In addition, the aspect ratio does not play an important role in the force variation. To shed light on the aspect ratio independency, it should be noted that the elastic buckling mode shape of the SLGS is hyperbolic cosine and it has no deflection in the width direction. Similar to the elastic buckling behavior, it has been predicted that the aspect ratio has a negligible effect on the SLGS fundamental frequency [8–10]. The elastic buckling behavior of the bridge SLGS is studied. In this regard, the elastic buckling forces of the zigzag and armchair sheets with different side lengths and aspect ratios are calculated. The elastic buckling forces of the zigzag and armchair sheets with bridge configuration are, respectively, shown in Figs. 4 and 5. It is discerned that the buckling force changes nonlinearly with side length while it is insensitive to the aspect ratio. The elastic buckling mode shape of the SLGS is the reason for the independency as explained for the cantilever boundary condition.
45
Fig. 2. Elastic buckling forces per unit width of the cantilever zigzag single-layered graphene sheets.
10
3. Results
40
Side Length (nm)
-1
10
-2
10
-3
10
-4
10
0
5
10
15
20
25
30
35
40
45
50
Side Length (nm) Fig. 3. Elastic buckling forces per unit width of the cantilever armchair singlelayered graphene sheets with different geometrical parameters.
Considering the results for the SLGSs with similar boundary conditions, it exhibits that the elastic buckling force is chirality dependent. It is found that the elastic buckling force of the zigzag sheet is slightly larger than the armchair sheet with equivalent geometrical parameters. For example, the buckling force per unit width of the bridge zigzag sheet with a = 2.1315 (nm) and a ¼ 0:4330 is 5.5801 (N/nm) while this value changes to 4.9941 b (N/nm) for the bridge armchair sheet with equivalent geometrical parameters. This is due to the covalent bonds orientations of the sheets; the ratio of the covalent bonds that are parallel to the loading direction is greater in the zigzag sheet. The elastic buckling forces of the cantilever and bridge configurations are compared with each other. For this purpose, SLGSs with identical geometrical parameters and chirality are considered. It is observed that the buckling forces of the clamped-free and clamped–clamped boundary conditions have similar trends. However, the buckling force of the bridge boundary condition is found to be 15–16 times more than that of the cantilever one. On the
A. Sakhaee-Pour / Computational Materials Science 45 (2009) 266–270
(b) Cantilever armchair sheet
Bridge Zigzag SLGS
2
269
10
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2:7523 a 0:5012 f ¼ a2:2152
a/b=0.2165 a/b=0.4330 a/b=0.8660 a/b=1.7319 a/b=3.4638
ð4Þ
1
Buckling Force (N/nm)
10
(c) Bridge zigzag sheet
f ¼
ð5Þ
0
10
(d) Bridge armchair sheet
f ¼
-1
10
-2
10
0
5
10
15
20
25
30
35
40
45
Side Length (nm) Fig. 4. Elastic buckling forces per unit width of the bridge zigzag single-layered graphene sheets.
Bridge Armchair SLGS
2
10
a/b=0.2887 a/b=0.5774 a/b=1.1547 a/b=2.3094 a/b=4.6188 1
10
Buckling Force (N/nm)
55:410 1 1:1825 a þ 2:3934 a2
0
10
-1
10
-2
10
0
5
10
15
20
25
30
35
40
45
50
Side Length (nm) Fig. 5. Elastic buckling forces per unit width of the bridge armchair single-layered graphene sheets.
other hand, it is noted that this ratio for the continuum plate is 16 and this indicates there is a good agreement between the continuum and atomistic-based modeling results. In order to avoid the time-consuming procedure of the implemented approach in finding the elastic buckling force of the SLGS, predictive equations are proposed. To this end, adjustable parameters of the equations are computed through a statistical nonlinear regression model [23]. The accuracy of the model is measured using a multiple coefficient of determination, R2, where 0 6 R2 6 1. In the present study, the value of R2 is found to be greater than 0.99 for all considered data, demonstrating a good fit of the predictive equations results to the molecular structural mechanics data. The predictive equations are: (a) Cantilever zigzag sheet
f ¼
1:6298 a1:9208
ð3Þ
1 2:0464 1:1713 a þ 3:9027 a2
ð6Þ
where a is the SLGS side length in (nm) and f represents the buckling force per unit width in (N/nm). These equations estimate the elastic buckling force within 5 percent difference with the atomistic simulation method in the ranges of the geometrical parameters (please see Table 1).
4. Conclusions The elastic buckling analysis of defect-free single-layered graphene sheet (SLGS) is performed through the molecular structural mechanics. In this atomistic simulation, equivalent structural beams are adopted to model interatomic forces of the covalently bonded carbon atoms in the so-called honey comb lattice. To explore the mechanical characteristic, the effects of side length, aspect ratio and chirality on the elastic buckling forces of the SLGSs with clamped-free and clamped–clamped configurations are studied. The atomistic modeling approach indicates that the buckling force decreases nonlinearly versus side length whereas this variation is not affected by the aspect ratio. In addition, the buckling force of the zigzag sheet is larger than the armchair sheet with equivalent geometrical parameters as a consequence of having more covalent bonds that are parallel to the loading direction. On the other hand, the buckling force of the cantilever and bridge SLGSs are compared with each other. It is observed that the bucking force of the bridge boundary condition is 15–16 times more than the cantilever one with identical chirality and geometrical parameters and this is in good agreement with the continuum theory results. Based on the molecular structural mechanics findings, equations for the SLGS buckling force prediction are developed using a nonlinear regression model. Since the aspect ratio does not play an important role in the buckling force variation, the equations are appeared as functions of the SLGS side length. These equations provide a quick tool for the buckling force calculation while their results remain within 5 percent difference with the atomistic modeling data. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
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