A new approach of multi-scale finite element simulation is proposed to predict the fracture toughness and crack propagation in a single layer graphene sheet. In this simulation MD-based nonlinear beam element is developed for the atomistic model near the crack tip, whereas a plane element is employed for the continuum model in the far field of the cracked specimen. The material and section properties required in the nonlinear beam element are estimated through the equivalence between the potential energy of molecular dynamics and the elastic strain energy of continuum mechanics. With the estimated properties of beam element, the material properties of plane element are further estimated by applying loads on the specimen of graphene, which is formed by hexagonal lattice of carbon atoms. Coupling the nonlinear beam elements by plane elements with the local-global concept of multi-scaled modeling, a vast of computational time can be saved. The accuracy of near-tip stresses obtained in this simulation remedies the inaccuracy of linear elastic fracture mechanics, and can be used for the prediction of atomic bond-breaking, which leads to crack propagation. The associated critical load can then be applied in the continuum model for the cracked specimen to predict mode I and mode II fracture toughness of graphene. The obtained values were then verified by the published results measured or predicted by the other methods. By varying the size of cracks and orientation of applied loads, several interesting phenomena have been observed and discussed in this thesis.

[1] Geim, A. K., Novoselov, K. S., 2007, “The rise of graphene” Nat. Mater. 6, 183–191.
[2] Lee, C., Wei, X., Kysar, J.W., Hone, J., 2008, “Measurement of the elastic properties and intrinsic strength of monolayer graphene” Science 321, 385-388.
[3] Balandin, A.A., 2011, “Thermal Properties of Graphene, Carbon Nanotubes, and Nanostructured Carbon Materials” Nat. Mater. 10, 569-581.
[4] Nero, A.H.C., Guinea, F., Peres, N.M.R., Novoselov, K.S., Geim, A.K., 2009, “The electronic properties of graphene” Rev. Mod. Phys. 81,109.
[5] Fan, C.W., Liu, Y.Y., Hwu, C., 2009, “Finite element simulation for estimating the mechanical properties of multi-walled carbon nanotubes” Appl. Phys. A: Materials Science and Processing 95(3), 819-831.
[6] Lekawa‐Raus, A., Patmore, J., Kurzepa, L., Bulmer, J., Koziol, K., 2014, “Electrical Properties of Carbon Nanotube-Based Fibers and Their Future Use in Electrical Wiring” Adv. Funct. Mater. 24, 3661-3682. [7] Stankovich, S., Dikin, D. A., Dommett, G. H. B., Kohlhaas, K. M., Zimney, E. J., Stach, E. A., Piner, R. D., Nguyen, S. T. , Ruoff, R. S., 2006, “Graphene-based composite materials” Nature 442, 282–286
[8] Potts, J. R., Dreyer, D. R., Bielawski, C. W., Ruoff, R. S., 2011, “Graphene-based polymer nanocomposites” Polymer 52, 5–25.
[9] Monetta, T., Acquesta, A., Bellucci, F., 2015, “Graphene/Epoxy Coating as Multifunctional Material for Aircraft Structures” Aerospace 2(3), 423-434.
[10] Siochi, E.J., 2014, “Graphene in the sky and beyond” Nature Nanotechnology 9, 745-747.
[11] Geim, A.K., MacDonald, A.H., 2007, “Graphene: Exploring carbon flatland” Phys. Today 60,35.
[12] Chabot, V., Higgins, D., Yu, A.P., Xiao, X.C., Chen, Z.W., Zhang, J.J., 2014, “A review of graphene and graphene oxide sponge: material synthesis and applications to energy and the environment” Energy Environ. Sci. 7, 1564-1596.
[13] Shi, P.C., Guo, J.P., Liang, X., Cheng, S., Zhen, H., Wang, Y., Chen, C.H., Xiang, H.F., 2018, “Large-scale production of high-quality graphene sheets by a non-electrified electrochemical exfoliation method” Carbon, 126, 507-513.
[14] Anderson, T.L., 1991, “Fracture Mechanics: Fundamentals and Applications” CRC Press, Boca Raton.
[15] Omeltchenko, A., Yu, J., Kalia, R.K., Vashishta, P., 1997, “Crack front propagation and fracture in a graphite sheet: A molecular-dynamics study on a parallel computer” Phys. Rev. Lett. 78, 2148-2151.
[16] Zhang, P., Ma, L., Fan, F., Zeng, Z., Peng, C., Loya, P.E., 2014, “Fracture toughness of graphene” Nat. Commun. 4782,1-7.
[17] Khare, R., Mielke, S.L., Paci, J.T., Zhang, S., Ballarini, R., Schatz, G.C., Belytschko, T., 2007, “Coupled quantum mechanical/molecular mechanical modeling of the fracture of defective carbon nanotubes and graphene sheets” Phys. Rev. B 75, 075412.
[18] Xu, M., Tabarraei, A., Paci, J.T., Oswald, J., Belytschko, T., 2012, “A coupled quantum/continuum mechanics study of graphene fracture” Int. J. Fract. 173, 163-173.
[19] Baykasoglu, C., Mugan, A., 2012, “Coupled molecular/continuum mechanical modeling of graphene sheets” Phys. E 45: 151-161.
[20] Le, M.Q., Batra, R.C., 2016, “Mode-I stress intensity factor in single layer graphene sheets” Comp. mater. Sci. 118, 251-258.
[21] Zhang, B., Mei, L., Xiao, H., 2012, “Nanofracture in graphene under complex mechanical stresses” Appl. Phys. Lett. 101, 121915.1-5.
[22] Moura, M. J. B., Marder, M., 2013, “Tearing of free-standing graphene” Phys. Rev. E 88, 032405. [23] Le, M.Q., Batra, R.C., 2012, “Single-edge crack growth in graphene sheets under tension” Comp. mater. Sci. 69, 381-388.
[24] Yeh, Y.K., Hwu, C., 2017, “A modified molecular-continuum model for estimating the strength, and fracture toughness of graphene and carbon nanotube” Engineering Fracture Mechanics 176, 326-342.
[25] Li, C., Chou, T. W., 2003, “A structural mechanics approach for the analysis of carbon nanotubes” International Journal of Solids and Structures 40(10), 2487-2499.
[26] ANSYS User`s Manual, version16.1.
[27] The Weak Coupling Method for Coupling Continuum Mechanics with Molecular Dynamics, Dissertation (2009) by K. Fackeldey.
[28] Gao, H., Ji, B., Jäger, I.L., Arzt, E., Fratzl, P., 2003, “Materials become insensitive to flaws at nanoscale: Lessons from nature” Natl. Acad. Sci. U.S.A. 100(10), 5597-5600.
[29] Karihaloo, B.L., 1982, “On crack kinking and curving” Mechanics of Materials 1, 189-201.
[30] Cheng, Y.C., Wang, H.T., Zhu, Z.Y., Zhu, Y.H., Han, Y., Zhang, X.X., Schwingenschlogl, U., 2012, “Strain-activated edge reconstruction of graphene nanoribbons” Phys. Rev. B 85, 073406.