Modeling and simulation allow methodical variation of material properties beyond the capacity of
experimental methods. Due to the hexagonal structure of graphene, it is considered as frame-like
structure. In the frame, covalent C-C bonds are taken as beams joined together with carbon atoms
placed at the joints. Uniaxial beam elements, defined by their cross-sectional area, material properties,
and moment of inertia represent the covalent bonds. The parameters of the beam elements
are determined by establishing equivalence between structural and computational mechanics.
However, the bonds connecting the carbon atoms do not have physical existence as they are a
compromise between attractive and repulsive forces. Also, defects at nanoscale make graphene
different from frame-like structure. In addition, the topography of graphene makes it non-linear
structure and even the axial loading changes to eccentric loading. Here we show that, by using basic
statics principles, disparities between graphene and frame-likes structures can be highlighted.
References
[1]
Novoselov, K.S., Geim, A.K., Morozov, S.V., Jiang, D., Zhang, Y., Dubonos, S.V., et al. (2004) Electric Field Effect in Atomically Thin Carbon Films. Science, 306, 666-669. http://dx.doi.org/10.1126/science.1102896
Choi, W. and Lee, J.-W. (2012) Graphene Synthesis and Applications. CRC Press, Boca Raton.
[4]
Chan, Y. and Hill, J.M. (2010) Some Novel Plane Trajectories for Carbon Atoms and Fullerenes Captured by Two Fixed Parallel Carbon Nanotubes. The European Physical Journal D, 59, 367-374.
http://dx.doi.org/10.1140/epjd/e2010-00173-9
[5]
Sun, H. (1998) COMPASS: An Ab Initio Force-Field Optimized for Condensed-Phase Applications Overview with Details on Alkane and Benzene Compounds. The Journal of Physical Chemistry B, 102, 7338-7364.
http://dx.doi.org/10.1021/jp980939v
[6]
Rahman, R. (2013) The Role of Graphene in Enhancing the Stiffness of Polymeric Material: A Molecular Modeling Approach. Journal of Applied Physics, 113, Article ID: 243503. http://dx.doi.org/10.1063/1.4812275
[7]
Yanovsky, Y.G., Nikitina, E.A., Karnet, Y.N. and Nikitin, S.M. (2009) Quantum Mechanics Study of the Mechanism of Deformation and Fracture of Graphene. Physical Mesomechanics, 12, 254-262.
http://dx.doi.org/10.1016/j.physme.2009.12.007
[8]
Lu, Q., Gao, W. and Huang, R. (2011) Atomistic Simulation and Continuum Modeling of Graphene Nanoribbons under Uniaxial Tension. Modelling and Simulation in Materials Science and Engineering, 19, Article ID: 054006.
http://dx.doi.org/10.1088/0965-0393/19/5/054006
[9]
Theodosiou, T.C. and Saravanos, D.A. (2014) Numerical Simulation of Graphene Fracture Using Molecular Mechanics Based Nonlinear Finite Elements. Computational Materials Science, 82, 56-65.
http://dx.doi.org/10.1016/j.commatsci.2013.09.032
[10]
Ni, Z., Bu, H., Zou, M., Yi, H., Bi, K. and Chen, Y. (2010) Anisotropic Mechanical Properties of Graphene Sheets from Molecular Dynamics. Physica B: Condensed Matter, 405, 1301-1306.
http://dx.doi.org/10.1016/j.physb.2009.11.071
[11]
Liu, Y. and Xu, Z. (2014) Multimodal and Self-Healable Interfaces Enable Strong and Tough Graphene-Derived Materials. Journal of the Mechanics and Physics of Solids, 70, 30-41. http://dx.doi.org/10.1016/j.jmps.2014.05.006
[12]
Liu, F., Ming, P. and Li, J. (2007) Ab initio Calculation of Ideal Strength and Phonon Instability of Graphene under Tension. Physical Review B, 76, Article ID: 064120.
[13]
Mortazavi, B. and Rabczuk, T. (2015) Multiscale Modeling of Heat Conduction in Graphene Laminates. Carbon, 85, 1-7. http://dx.doi.org/10.1016/j.carbon.2014.12.046
[14]
Rafiee, M.A., Rafiee, J., Wang, Z., Song, H., Yu, Z. and Koratkar, N. (2009) Enhanced Mechanical Properties of Nanocomposites at Low Graphene Content. ACS Nano, 3, 3884-3890. http://dx.doi.org/10.1021/nn9010472
[15]
Ebrahimi, S., Ghafoori-Tabrizi, K. and Rafii-Tabar, H. (2012) Multi-Scale Computational Modelling of the Mechanical Behaviour of the Chitosan Biological Polymer Embedded with Graphene and Carbon Nanotube. Computational Materials Science, 53, 347-353. http://dx.doi.org/10.1016/j.commatsci.2011.08.034
[16]
Patterson, J. and Bailey, B. (2010) Solid-State Physics: Introduction to the Theory. 2nd Edition, Springer, Berlin.
http://dx.doi.org/10.1007/978-3-642-02589-1
[17]
Bianchini, F., Patera, L.L., Peressi, M., Africh, C. and Comelli, G. (2014) Atomic Scale Identification of Coexisting Graphene Structures on Ni(111). The Journal of Physical Chemistry Letters, 5, 467-473.
http://dx.doi.org/10.1021/jz402609d
[18]
Meyer, J.C., Kurasch, S., Park, H.J., Skakalova, V., Künzel, D., Gross, A., et al. (2011) Experimental Analysis of Charge Redistribution Due to Chemical Bonding by High-Resolution Transmission Electron Microscopy. Nature Materials, 10, 209-215. http://dx.doi.org/10.1038/nmat2941
[19]
Skowron, S.T., Lebedeva, I.V., Popov, A.M. and Bichoutskaia, E. (2015) Energetics of Atomic Scale Structure Changes in Graphene. Chemical Society Reviews, 44, 3143-3176. http://dx.doi.org/10.1039/C4CS00499J
[20]
Rutter, G.M., Crain, J.N., Guisinger, N.P., Li, T., First, P.N. and Stroscio, J.A. (2007) Scattering and Interference in Epitaxial Graphene. Science, 317, 219-222. http://dx.doi.org/10.1126/science.1142882
[21]
Ruffieux, P., Melle-Franco, M., Groning, O., Bielmann, M., Zerbetto, F. and Groning, P. (2005) Charge-Density Oscillation on Graphite Induced by the Interference of Electron Waves. Physical Review B, 71, Article ID: 153403.
http://dx.doi.org/10.1103/PhysRevB.71.153403
[22]
Osvath, Z., Vertesy, G., Tapaszto, L., Weber, F., Horvath, Z.E., Gyulai, J., et al. (2005) Atomically Resolved STM Images of Carbon Nanotube Defects Produced by Ar + Irradiation. Physical Review B, 72, Article ID: 045429.
http://dx.doi.org/10.1103/PhysRevB.72.045429
[23]
Chen, J.-H., Li, L., Cullen, W.G., Williams, E.D. and Fuhrer, M.S. (2010) Tunable Kondo Effect in Graphene with Defects. Nature Physics, 7, 535-538. http://dx.doi.org/10.1038/nphys1962
[24]
Gómez-Navarro, C., Meyer, J.C., Sundaram, R.S., Chuvilin, A., Kurasch, S., Burghard, M., et al. (2010) Atomic Structure of Reduced Graphene Oxide. Nano Letters, 10, 1144-1148. http://dx.doi.org/10.1021/nl9031617
[25]
Yan, H., Liu, C.C., Bai, K.K., Wang, X., Liu, M., Yan, W., et al. (2013) Electronic Structures of Graphene Layers on a Metal Foil: The Effect of Atomic-Scale Defects. Applied Physics Letters, 103, Article ID: 143120.
http://dx.doi.org/10.1063/1.4824206
[26]
Yan, H., Chu, Z.D., Yan, W., Liu, M., Meng, L., Yang, M., et al. (2013) Superlattice Dirac Points and Space-Depen- dent Fermi Velocity in a Corrugated Graphene Monolayer. Physical Review B, 87, Article ID: 075405.
http://dx.doi.org/10.1103/physrevb.87.075405
[27]
Matan, K., Williams, R.B., Witten, T.A. and Nagel, S.R. (2002) Crumpling a Thin Sheet. Physical Review Letters, 88, Article ID: 076101. http://dx.doi.org/10.1103/PhysRevLett.88.076101
[28]
Balankin, A.S. and Orlando, S.H. (2008) Entropic Rigidity of a Crumpling Network in a Randomly Folded Thin Sheet. Physical Review E, 77, Article ID: 051124. http://dx.doi.org/10.1103/PhysRevE.77.051124
[29]
Patel, M.U.M., Luong, N.D., Sepp?l?, J., Tchernychova, E. and Dominko, R. (2014) Low Surface Area Graphene/ Cellulose Composite as a Host Matrix for Lithium Sulphur Batteries. Journal of Power Sources, 254, 55-61.
http://dx.doi.org/10.1016/j.jpowsour.2013.12.081
[30]
Zhang, K., Duan, X., Zhu, X., Hu, D., Xu, J., Lu, L., et al. (2014) Nanostructured Graphene Oxide-MWCNTs Incorporated Poly(3,4-ethylenedioxythiophene) with a High Surface Area for Sensitive Determination of Diethylstilbestrol. Synthetic Metals, 195, 36-43. http://dx.doi.org/10.1016/j.synthmet.2014.05.005
[31]
Cranford, S.W. and Buehler, M.J. (2011) Packing Efficiency and Accessible Surface Area of Crumpled Graphene. Physical Review B, 84, Article ID: 205451. http://dx.doi.org/10.1103/PhysRevB.84.205451
[32]
Cranford, S. and Buehler, M.J. (2011) Twisted and Coiled Ultralong Multilayer Graphene Ribbons. Modelling and Simulation in Materials Science and Engineering, 19, Article ID: 054003.
http://dx.doi.org/10.1088/0965-0393/19/5/054003
[33]
Lee, D., Zou, X., Zhu, X., Seo, J.W., Cole, J.M., Bondino, F., et al. (2012) Ultrafast Carrier Phonon Dynamics in NaOH-Reacted Graphite Oxide Film. Applied Physics Letters, 101, Article ID: 021604.
http://dx.doi.org/10.1063/1.4736572
[34]
Becton, M., Zhang, L. and Wang, X. (2013) Effects of Surface Dopants on Graphene Folding by Molecular Simulations. Chemical Physics Letters, 584, 135-141. http://dx.doi.org/10.1016/j.cplett.2013.08.027
[35]
Zhao, J., Yang, B., Zheng, Z., Yang, J., Yang, Z., Zhang, P., et al. (2014) Facile Preparation of One-Dimensional Wrapping Structure: Graphene Nanoscroll-Wrapped of Fe3O4 Nanoparticles and Its Application for Lithium-Ion Battery. ACS Applied Materials & Interfaces, 6, 9890-9896. http://dx.doi.org/10.1021/am502574j
[36]
Bao, W., Miao, F., Chen, Z., Zhang, H., Jang, W., Dames, C., et al. (2009) Controlled Ripple Texturing of Suspended Graphene and Ultrathin Graphite Membranes. Nature Nanotechnology, 4, 562-566.
http://dx.doi.org/10.1038/nnano.2009.191
[37]
Cranford, S., Sen, D. and Buehler, M.J. (2009) Meso-Origami: Folding Multilayer Graphene Sheets. Applied Physics Letters, 95, 2013-2016. http://dx.doi.org/10.1063/1.3223783
[38]
Meyer, J.C., Geim, A.K., Katsnelson, M.I., Novoselov, K.S., Booth, T.J. and Roth, S. (2007) The Structure of Suspended Graphene Sheets. Nature, 446, 60-63. http://dx.doi.org/10.1038/nature05545
[39]
Parviz, D., Metzler, S.D., Das, S., Irin, F. and Green, M.J. (2015) Tailored Crumpling and Unfolding of Spray-Dried Pristine Graphene and Graphene Oxide Sheets. Small, 11, 2661-2668. http://dx.doi.org/10.1002/smll.201403466
[40]
Wang, X., Jin, J. and Song, M. (2013) An Investigation of the Mechanism of Graphene Toughening Epoxy. Carbon, 65, 324-333. http://dx.doi.org/10.1016/j.carbon.2013.08.032
[41]
Chen, Q., Liu, W., Guo, S., Zhu, S., Li, Q., Li, X., et al. (2015) Synthesis of Well-Aligned Millimeter-Sized Tetragon-Shaped Graphene Domains by Tuning the Copper Substrate Orientation. Carbon, 93, 945-952.
http://dx.doi.org/10.1016/j.carbon.2015.05.108
[42]
Bae, S., Kim, H., Lee, Y., Xu, X., Park, J.-S., Zheng, Y., et al. (2010) Roll-to-Roll Production of 30-Inch Graphene Films for Transparent Electrodes. Nature Nanotechnology, 5, 574-578. http://dx.doi.org/10.1038/nnano.2010.132
[43]
Ago, H., Kawahara, K., Ogawa, Y., Tanoue, S., Bissett, M.A., Tsuji, M., et al. (2013) PS-13-15. Epitaxial Growth and Electronic Properties of Large Hexagonal Graphene Domains on Cu(111) Thin Film. 2013 International Conference on Solid State Devices and Materials, Vol. 1, Fukuoka, 25-27 September 2013, 438-439.
[44]
Murdock, A.T., Koos, A., Britton, T.B., Houben, L., Batten, T., Zhang, T., et al. (2013) Controlling the Orientation, Edge Geometry, and Thickness of Chemical Vapor Deposition Graphene. ACS Nano, 7, 1351-1359.
http://dx.doi.org/10.1021/nn3049297
[45]
Hao, Y., Bharathi, M.S., Wang, L., Liu, Y., Chen, H., Nie, S., et al. (2013) The Role of Surface Oxygen in the Growth of Large Single-Crystal Graphene on Copper. Science, 342, 720-723. http://dx.doi.org/10.1126/science.1243879
[46]
Dai, G.-P., Wu, M.H., Taylor, D.K. and Vinodgopal, K. (2013) Square-Shaped, Single-Crystal, Monolayer Graphene Domains by Low-Pressure Chemical Vapor Deposition. Materials Research Letters, 1, 67-76.
http://dx.doi.org/10.1080/21663831.2013.772078
[47]
Warren, C. and Budynas, Y.R.G. (2002) Roark’s Formulas for Stress and Strain. 7th Edition, McGraw-Hill, New York.
[48]
Beer, F.P., Johnston, E.R.J., DeWolf, J.T. and Mazurek, D.F. (2012) Mechanics of Materials. 6th Edition, McGraw- Hill, New York.
[49]
Montazeri, A. and Rafii-Tabar, H. (2011) Multiscale Modeling of Graphene- and Nanotube-Based Reinforced Polymer Nanocomposites. Physics Letters A, 375, 4034-4040. http://dx.doi.org/10.1016/j.physleta.2011.08.073
Abedpour, N., Neek-Amal, M., Asgari, R., Shahbazi, F., Nafari, N. and Tabar, M.R.R. (2007) Roughness of Undoped Graphene and Its Short-Range Induced Gauge Field. Physical Review B, 76, Article ID: 195407.
http://dx.doi.org/10.1103/PhysRevB.76.195407
[52]
Fasolino, A., Los, J.H. and Katsnelson, M.I. (2007) Intrinsic Ripples in Graphene. Nature Materials, 6, 858-861.
http://dx.doi.org/10.1038/nmat2011
[53]
Dieter, G.E. (1988) Mechanical Metallurgy. SI Metric Edition, McGraw-Hill, New York.
[54]
Griffith, A.A. (1921) The Phenomena of Rupture and Flow in Solids. Philosophical Transactions of the Royal Society of London, Series A, Containing Papers of a Mathematical or Physical Character, 221, 163-198.
http://dx.doi.org/10.1098/rsta.1921.0006
