This paper presents a quasi-static nonlinear buckling analysis of compressed single-layer graphene sheets (SLGSs) using the molecular mechanics method. Bonded interactions between carbon atoms are simulated using a modified parameter set of the DREIDING force field that leads to better agreement between simulated mechanical properties of graphene and reference literature data than the standard parameter set of this force field (see Mayo et al., J Phys Chem 1990; 94: 8897–8909). Identification of constraints of atoms of the SLGS edges with the boundary conditions of clamped and simply supported thin plates is made. The buckling loads and modes obtained by linear and nonlinear buckling analysis of a compressed quadratic SLGS with a side length of 6 nm are shown to be close to each other. In addition, it has been found by nonlinear buckling analysis that only equilibrium configurations with modes of initial post-buckling deformed configurations correlated with the one-half-wave column-like buckling mode have stable equilibrium configurations for clamped and simply supported SLGSs. As the edges of a simply supported SLGS approach each other, the geometry of this mode of post-buckling deformation with inclusion of the non-bonded van der Waals (vdW) interactions between carbon atoms becomes closer to the geometry of a single-walled carbon nanotube, and without inclusion of the vdW interactions, this mode has the geometry of a cylinder with a drop-shaped cross-section.
BerinskiiIEBorodichFM. Elastic in-plane properties of 2D linearized models of graphene. Mech Mater2013; 62: 60–68.
2.
PintoHPLeszczynskiJ. Fundamental properties of graphene. In: D’SouzaFKadishKM (eds.) Handbook of carbon nano materials (World Scientific Series on Carbon Nanoscience, vol. 5). Singapore: World Scientific Publishing, 2014, 1–37.
3.
KudinKNScuseriaGEYakobsonBI. C2F, BN, and C nanoshell elasticity from ab initio computations. Phys Rev B2001; 64: 235406.
4.
NovoselovKSGeimAKMorozovSV. Electric field effect in atomically thin carbon films. Sci2004; 306: 666–669.
5.
GeimAKNovoselovKS. The rise of graphenes. Nat Mater2007; 6: 183–191.
LeeCWeiXKysarJW. Measurement of the elastic properties and intrinsic strength of monolayer graphene. Sci2008; 321: 385–388.
8.
LiuFMingPLiJ. Ab initio calculation of ideal strength and phonon instability of graphene under tension. Phys Rev B2007; 76: 064120.
9.
SamadikhahKLarssonRBazooyarF. Continuum-molecular modelling of graphene. Comput Mater Sci2012; 53: 37–43.
10.
ArghavanSSinghAV. Atomic lattice structure and continuum plate theories for the vibrational characteristics of graphenes. J Appl Phys2011; 110: 084308.
11.
ArroyoMBelytschkoT. Finite crystal elasticity of carbon nanotubes based on the exponential Cauchy-Born rule. Phys Rev B2004; 69: 115415.
12.
BuHChenYZouM. Atomistic simulations of mechanical properties of graphene nanoribbons. Phys Lett A2009; 373: 3359–3362.
ChangTGaoH. Size-dependent elastic properties of a single-walled carbon nanotube via a molecular mechanics model. J Mech Phys Solids2003; 51: 1059–1074.
15.
DaviniC. Homogenization of a graphene sheet. Contin Mech Thermodyn2014; 26: 95–113.
GeorgantzinosSKGiannopoulosGIAnifantisNK. Numerical investigation of elastic mechanical properties of graphene structures. Mater Des2010; 31: 4646–4654.
18.
GeorgantzinosSKGiannopoulosGIKatsareasDE. Size-dependent non-linear mechanical properties of graphene nanoribbons. Comput Mater Sci2011; 50: 2057–2062.
19.
GiannopoulosGILiosatosIAMoukanidisAK. Parametric study of elastic mechanical properties of graphene nanoribbons by a new structural mechanics approach. Phys E2011; 44: 124–134.
20.
GiannopoulosGI. Elastic buckling and flexural rigidity of graphene nanoribbons by using a unique translational spring element per interatomic interaction. Comput Mater Sci2012; 53: 388–395.
21.
GuptaSSBatraRC. Elastic properties and frequencies of free vibrations of single-layer graphene sheets. J Comput Theor Nanosci2010; 7: 2151–2164.
22.
HaoPGaoYWZhouYH. The effect of electric charge on the mechanical properties of graphene. Sci China – Phys Mech Astron2013; 56(6): 1148–1156.
23.
HuangYWuJHwangKC. Thickness of graphene and single-wall carbon nanotubes. Phys Rev B2006; 74: 245413.
24.
KumarSHembramKPSSWaghmareUV. Intrinsic buckling strength of graphene: First-principles density functional theory calculations. Phys Rev B2010; 82: 115411.
25.
LarssonRSamadikhahK. Atomistic continuum modeling of graphene membranes. Comput Mater Sci2011; 50: 1744–1753.
26.
LiCChouTW. A structural mechanics approach for the analysis of carbon nanotubes. Int J Solids Struct2003; 40: 2487–2499.
27.
LuQHuangR. Nonlinear mechanics of single-atomic-layer graphene sheets. Int J Appl Mech2009; 1(3): 443–467.
ReddyCDRajendranSLiewKM. Equilibrium configuration and continuum elastic properties of finite sized graphene. Nanotech2006; 17: 864–870.
30.
RobertsMWClemonsCBWilberJP. Continuum plate theory and atomistic modeling to find the flexural rigidity of a graphene sheet interacting with a substrate. J Nanotech2010; 2010: 868492.
31.
SaitohKHayakawaH. Motion of a free-standing graphene sheet induced by the collision with an argon nanocluster: Analyses of the detection and heat-up of the graphene. Phys Rev B2010; 81: 115447.
32.
Sakhaee-PourA. Elastic properties of single-layered graphene sheet. Solid State Commun2009; 149: 91–95.
33.
ScarpaFAdhikariSSrikantha PhaniA. Effective elastic mechanical properties of single layer graphene sheets. Nanotech2009; 20: 065709.
34.
ScarpaFAdhikariSGilAJ. The bending of single layer graphene sheets: The lattice versus continuum approach. Nanotech2010; 21: 125702.
35.
ShenLShenHSZhangCL. Nonlocal plate model for nonlinear vibration of single layer graphene sheets in thermal environments. Comput Mater Sci2010; 48: 680–685.
36.
TserpesKIPapanikosP. Finite element modeling of the tensile behavior of carbon nanotubes, graphene and their composites. In: TserpesKISilvestreN (eds.) Modeling of carbon nanotubes, graphene and their composites (Springer Series in Materials Science, vol. 188). New York, NY: Springer International Publishing, 2014, 303–329.
37.
ZhangPHuangYGeubellePH. The elastic modulus of single-wall carbon nanotubes: A continuum analysis incorporating interatomic potentials. Int J Solids Struct2002; 39: 3893–3906.
38.
ZhouJHuangR. Internal lattice relaxation of single-layer graphene under in-plane deformation. J Mech Phys Solids2008; 56: 1609–1623.
39.
AnsariRSahmaniS. Prediction of biaxial buckling behavior of single-layered graphene sheets based on nonlocal plate models and molecular dynamics simulations. Appl Math Modell2013; 37: 7338–7351.
WilberJPClemonsCBYoungGW. Continuum and atomistic modeling of interacting graphene layers. Phys Rev B2007; 75: 045418.
42.
PradhanSC. Buckling of single layer graphene sheet based on nonlocal elasticity and higher order shear deformation theory. Phys Lett A2009; 373: 4182–4188.
43.
PradhanSCMurmuT. Small scale effect on the buckling of single-layered graphene sheets under biaxial compression via nonlocal continuum mechanics. Comput Mater Sci2009; 47: 268–274.
44.
RouhiSAnsariR. Atomistic finite element model for axial buckling and vibration analysis of single-layered graphene sheets. Phys E2012; 44: 764–772.
45.
Sakhaee-PourA. Elastic buckling of single-layered graphene sheet. Comput Mater Sci2009; 45: 266–270.
46.
KorobeynikovSNAlyokhinVVAnninBD. Buckling simulation of single layer graphene sheets by the molecular mechanics method. In: Proceedings of the 10th SSTA Conference, Gdansk, Poland, 16–18 October 2013, 207–210.
47.
AlekhinVVAnninBDBabichevAV. Natural vibrations and buckling of graphene sheets. Mech Solids2013; 48(5): 509–513.
48.
AlyokhinVVAnninBDBabichevAV. Free vibrations and buckling of graphene sheets. Dokl Phys2013; 58(11): 487–490.
49.
WilberJP. Buckling of graphene layers supported by rigid substrates. J Comput Theor Nanosci2010; 7: 2338–2348.
50.
BunchJSvan der ZandeAMVerbridgeSS. Electromechanical resonators from graphene sheets. Sci2007; 315: 490–493.
51.
KhareRMielkeSLPaciJT. Coupled quantum mechanical/molecular mechanical modeling of the fracture of defective carbon nanotubes and graphene sheets. Phys Rev B2007; 75: 075412.
52.
Suarez-MartinezIGrobertNEwelsCP. Nomenclature of sp2 carbon nanoforms. Carb2012; 50: 741–747.
WernikJMMeguidSA. Atomistic-based continuum modeling of the nonlinear behavior of carbon nanotubes. Acta Mech2010; 212: 167–179.
55.
CornellWDCieplakPBaylyCI. A second generation force field for the simulation of proteins, nucleic acids, and organic molecules. J Am Chem Soc1995; 117: 5179–5197.
56.
AllingerNLYuhYHLiiJH. Molecular mechanics: The MM3 force field for hydrocarbons 1. J Am Chem Soc1989; 111: 8551–8566.
57.
MayoSLOlafsonBDGoddardWA III. DREIDING: A generic force field for molecular simulations. J Phys Chem1990; 94: 8897–8909.
58.
HolecDHartmannMAFischerFD. Curvature-induced excess surface energy of fullerenes: Density functional theory and Monte Carlo simulations. Phys Rev B2010; 81: 235403.
BrennerDW. Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of dimond films. Phys Rev B1990; 42: 9458–9471.
61.
StuartSJTuteinABHarrisonJA. A reactive potential for hydrocarbons with intermolecular interactions. J Chem Phys2000; 112(14): 6472–6486.
62.
BrennerDWShenderovaOAHarrisonJA. A second-generation reactive empirical bond order (REBO) potential energy expression for hydrocarbons. J Phys Condens Matt2002; 14: 783–802.
63.
BuehlerMJ. Atomistic modeling of materials failure. New York, NY: Springer, 2008.
64.
KrivtsovAM. Deformation and failure of solids with microstructure. (In Russian.) Moscow: Fizmatlit, 2007.
65.
AnsariRSahmaniSArashB. Nonlocal plate model for free vibrations of single-layered graphene sheets. Phys Lett A2010; 375: 53–62.
66.
AllingerNL. Molecular structure: Understanding steric and electronic effects from molecular mechanics. Hoboken, NJ: A John Wiley & Sons, 2010.
67.
BurkertUAllingerNL. Molecular mechanics. Washington, DC: American Chemical Society, 1982.
68.
KorobeynikovSNAlyokhinVVAnninBD. Using stability analysis of discrete elastic systems to study the buckling of nanostructures. Arch Mech2012; 64: 367–404.
69.
AnninBDAlekhinVVBabichevAV. Molecular mechanics method applied to problems of stability and natural vibrations of single-layer carbon nanotubes. Mech Solids2012; 47(5): 544–559.
70.
NasdalaLErnstG. Development of a 4-node finite element for the computation of nano-structured materials. Comput Mater Sci2005; 33: 443–458.
71.
NasdalaLKempeARolfesR. The molecular dynamic finite element method (MDFEM). CMC: Comput Mater Contin2010; 19(1): 57–104.
72.
NasdalaLKempeARolfesR. Are finite elements appropriate for use in molecular dynamic simulations?Compos Sci Tech2012; 72: 989–1000.
73.
WackerfußJ. Molecular mechanics in the context of the finite element method. Int J Numer Meth Eng2009; 77: 969–997.
74.
HollererSCeligojCC. Buckling analysis of carbon nanotubes by a mixed atomistic and continuum model. Comput Mech2013; 51: 765–789.
75.
HollererS. Buckling analysis of carbon nanotubes: A molecular mechanics approach using the finite element framework. Proc Appl Math Mech2011; 11: 221–222.
76.
GatesTSOdegardGMFranklandSJV. Computational materials: Multi-scale modeling and simulation of nanostructured materials. Compos Sci Tech2005; 65: 2416–2434.
GoldsteinRVChentsovAV. A discrete-continuous model of a nanotube. Mech Solids2005; 40(4): 45–59.
79.
GoldsteinRVChentsovAVKadushnikovRM. Methodology and metrology for mechanical testing of nano-and microdimensional objects, materials, and products of nanotechnology. Nanotech Russ2008; 3(1–2): 112–121.
80.
Sakhaee-PourAAhmadianMTVafaiA. Applications of single-layered graphene sheets as mass sensors and atomistic dust detectors. Solid State Commun2008; 145: 168–172.
81.
Sakhaee-PourAAhmadianMTVafaiA. Potential application of single-layered graphene sheet as strain sensor. Solid State Commun2008; 147: 336–340.
82.
AnsariRRouhiH. An explicit nonlocal frequency formula for monolayer graphene sheets. Int J Comput Meth Eng Sci Mech2013; 14: 40–44.
83.
BabaeiHShahidiAR. Free vibration analysis of quadrilateral nanoplates based on nonlocal continuum models using the Galerkin method: The effects of small scale. Mecc2013; 48: 971–982.
84.
YamakiN. Elastic stability of circular cylindrical shells. Amsterdam: North-Holland, 1984.
85.
KrivtsovAMMorozovNF. On mechanical characteristics of nanocrystals. Phys Solid State2002; 44(12): 2260–2265.
86.
EringenAC. Linear theory of nonlocal elasticity and dispersion of plane waves. Int J Eng Sci1972; 10: 425–435.
87.
EringenAC. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys1983; 54: 4703–4710.
88.
ArashBWangQ. A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes. In: TserpesKISilvestreN (eds.) Modeling of carbon nanotubes, graphene and their composites (Springer Series in Materials Science, vol. 188). New York, NY: Springer International Publishing, 2014, 57–82.
89.
LindsayLBroidoDA. Optimized Tersoff and Brenner empirical potential parameters for lattice dynamics and phonon thermal transport in carbon nanotubes and graphene. Phys Rev B2010; 81: 205441.
90.
LosJHGhiringhelliLMMeijerEJ. Improved long-range reactive bond-order potential for carbon. I. Construction. Phys Rev B2005; 72: 214102.
91.
KorobeynikovSNBabichevAV. Numerical simulation of dynamic deformation and buckling of nanostructures. In: GoldsteinRV (ed.) CD ICF Interquadrennial Conference Full Papers, 2007.
92.
AnninBDKorobeynikovSNBabichevAV. Computer simulation of a twisted nanotube buckling. J Appl Indus Math2009; 3(3): 318–333.
93.
KorobeynikovSNAnninBDBabichevAV. Buckling criteria for nanostructures and their applications in computer simulation of nanotube twisting. In: Fracture of Materials and Structures from Micro to Macro Scale: CD Proceedings of the 18th European Conference on Fracture, Dresden, Germany, 2010.
94.
KorobeinikovSNAgapovVPBondarenkoMI. The general purpose nonlinear finite element structural analysis program PIONER. In: Proceedings of the International Conference on Numerical Methods and Applications, 1989, 228–233.
95.
BatheK-J. Finite element procedures. Upper Saddle River, NJ: Prentice Hall, 1996.
96.
ChengHCLiuYLHsuYC. Atomistic-continuum modeling for mechanical properties of single-walled carbon nanotubes. Int J Solids Struct2009; 46: 1695–1704.
97.
AnninBDAlekhinVVBabichevAV. Computer simulation of nanotube contact. Mech Solids2010; 45(3): 352–369.
98.
KorobeynikovSN. Nonlinear equations of deformation of atomic lattices. Arch Mech2005; 57: 457–475.
99.
KorobeinikovSN. The numerical solution of nonlinear problems on deformation and buckling of atomic lattices. Int J Fract2004; 128(1): 315–323.
100.
BatheK-JDvorkinEN. On the automatic solution of nonlinear finite element equations. Comput Struct1983; 17: 871–879.
101.
SokolTWitkowskiM. The equilibrium path determination in nonlinear analysis of structures. In: Advances in Non-Linear Finite Element Methods: Proceedings of the 2nd International Conference on Computational Structures Technology, 1994, 35–45.
102.
WagnerWWriggersP. A simple method for the calculation of postcritical branches. Eng Comput1988; 5: 103–109.
103.
BabichevAV. Automating model construction and visualization of results of numerical simulation of deformation of nanostructures. (In Russian.) Comput Contin Mech2008; 1(4): 21–27.
104.
Solyanik-KrassaKV. Introduction to solid mechanics. (In Russian.) Leningrad: Leningrad University, 1976.
105.
GirifalcoLAHodakMLeeRS. Carbon nanotubes, buckyballs, ropes, and a universal graphitic potential. Phys Rev B2000; 62: 13,104–13,110.
106.
ShimaH. Buckling of carbon nanotubes: A state of the art review. Mater2012; 5: 47–84.
