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http://dx.doi.org/10.12989/scs.2019.33.2.261

On axial buckling and post-buckling of geometrically imperfect single-layer graphene sheets  

Gao, Yang (State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University)
Xiao, Wan-shen (State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University)
Zhu, Haiping (School of Computing, Engineering and Mathematics, Western Sydney University)
Publication Information
Steel and Composite Structures / v.33, no.2, 2019 , pp. 261-275 More about this Journal
Abstract
The main objective of this paper is to study the axial buckling and post-buckling of geometrically imperfect single-layer graphene sheets (GSs) under in-plane loading in the theoretical framework of the nonlocal strain gradient theory. To begin with, a graphene sheet is modeled by a two-dimensional plate subjected to simply supported ends, and supposed to have a small initial curvature. Then according to the Hamilton's principle, the nonlinear governing equations are derived with the aid of the classical plate theory and the von-karman nonlinearity theory. Subsequently, for providing a more accurate physical assessment with respect to the influence of respective parameters on the mechanical performances, the approximate analytical solutions are acquired via using a two-step perturbation method. Finally, the authors perform a detailed parametric study based on the solutions, including geometric imperfection, nonlocal parameters, strain gradient parameters and wave mode numbers, and then reaching a significant conclusion that both the size-dependent effect and a geometrical imperfection can't be ignored in analyzing GSs.
Keywords
geometric imperfection; perturbation method; graphene sheets; nonlocal strain gradient theory;
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1 Allahyari, E. and Kiani, A. (2018), "Employing an analytical approach to study the thermo-mechanical vibration of a defective size-dependent graphene nanosheet", Eur. Phys. J. Plus., 133(6), 223. https://doi.org/10.1140/epjp/i2018-12058-2   DOI
2 Apuzzo, A., Barretta, R., Faghidian, S.A., Luciano, R. and de Sciarra, F.M. (2018), "Free vibrations of elastic beams by modified nonlocal strain gradient theory", Int. J. Eng. Sci., 133, 99-108. https://doi.org/10.1016/j.ijengsci.2018.09.002   DOI
3 Farokhi, H., Ghayesh, M.H. and Hussain, S. (2016), "Largeamplitude dynamical behaviour of microcantilevers", Int. J. Eng. Sci., 106, 29-41. https://doi.org/10.1016/j.ijengsci.2016.03.002   DOI
4 Fasolino, A., Los, J.H. and Katsnelson, M.I. (2007), "Intrinsic ripples in graphene", Nature Mater., 6, 858-861. https://doi.org/10.1038/nmat2011   DOI
5 Ghavanloo, E. (2017), "Axisymmetric deformation of geometrically imperfect circular graphene sheets", Acta Mech., 228(9), 3297-3305. https://doi.org/10.1007/s00707-017-1891-7   DOI
6 Ghayesh, M.H. (2013), "Subharmonic dynamics of an axially accelerating beam", Arch App. Mech., 82(9), 1169-1181. https://doi.org/10.1007/s00419-012-0609-5   DOI
7 Ghayesh, M.H. (2018a), "Dynamics of functionally graded viscoelastic microbeams", Int. J. Eng. Sci., 124, 115-131. https://doi.org/10.1016/j.ijengsci.2017.11.004   DOI
8 Ghayesh, M.H. (2018b), "Functionally graded microbeams: Simultaneous presence of imperfection and viscoelasticity", Int. J. Mech. Sci., 140, 339-350. https://doi.org/10.1016/j.ijmecsci.2018.02.037   DOI
9 Ghayesh, M.H. (2018c), "Nonlinear vibration analysis of axially functionally graded shear-deformable tapered beams", Appl. Math. Model., 59, 583-596. https://doi.org/10.1016/j.apm.2018.02.017   DOI
10 Ghayesh, M.H. (2019a), "Viscoelastic dynamics of axially FG microbeams", Int. J. Eng. Sci., 135, 75-85. https://doi.org/10.1016/j.ijengsci.2018.10.005   DOI
11 Ghayesh, M.H. (2019b), "Viscoelastic mechanics of Timoshenko functionally graded imperfect microbeams", Compos. Struct., 225, 110974. https://doi.org/10.1016/j.compstruct.2019.110974   DOI
12 Ghayesh, M.H. (2019c), "Mechanics of viscoelastic functionally graded microcantilevers", Eur. J. Mech. A/Solid., 73, 492-499. https://doi.org/10.1016/j.euromechsol.2018.09.001   DOI
13 Ma, H.M., Gao, X.L. and Reddy, J.N. (2008), "A microstructure-dependent Timoshenko beam model based on a modified couple stress theory", J. Mech. Phys. Solid., 56(12), 3379-3391. https://doi.org/10.1016/j.jmps.2008.09.007   DOI
14 Liu, H., Lv, Z. and Wu, H. (2018), "Nonlinear free vibration of geometrically imperfect functionally graded sandwich nanobeams based on nonlocal strain gradient theory", Compos. Struct., 214, 47-61. https://doi.org/10.1016/j.compstruct.2019.01.090   DOI
15 Lu, L., Guo, X. and Zhao, J. (2017a), "A unified nonlocal strain gradient model for nanobeams and the importance of higher order terms", Int. J. Eng. Sci., 119, 265-277. https://doi.org/10.1016/j.ijengsci.2017.06.024   DOI
16 Lu, L., Guo, X. and Zhao, J. (2017b), "Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory", Int. J. Eng. Sci., 116, 12-24. https://doi.org/10.1016/j.ijengsci.2017.03.006   DOI
17 Malikan, M. and Nguyen, V.B. (2018), "Buckling analysis of piezo-magnetoelectric nanoplates in hygrothermal environment based on a novel one variable plate theory combining with higher-order nonlocal strain gradient theory", Physica E., 102, 8-28. https://doi.org/10.1016/j.physe.2018.04.018   DOI
18 Malikan, M., Nguyen, V.B. and Tornabene, F. (2018), "Electromagnetic forced vibrations of composite nanoplates using nonlocal strain gradient theory", Mater. Res. Express, 5(7), 075031. https://doi.org/10.1088/2053-1591/aad144   DOI
19 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. https://doi.org/10.1038/nature05545   DOI
20 Mcfarland, A.W. and Colton, J.S. (2005), "Role of material microstructure in plate stiffness with relevance to microcantilever sensors", J. Micromech. Microeng., 15(5), 1060-1067. https://doi.org/10.1088/0960-1317/15/5/024   DOI
21 Singh, S. and Patel, B.P. (2018), "A computationally efficient multiscale finite element formulation for dynamic and postbuckling analyses of carbon nanotubes", Comput. Struct., 195, 126-144. https://doi.org/10.1016/j.compstruc.2017.10.003   DOI
22 Shen, H.S. and Zhang, J.W. (1988), "Perturbation analyses for the postbuckling of simply supported rectangular plates under uniaxial compression", App. Math. Mech., 9(8), 793-804. https://doi.org/10.1007/BF02465403   DOI
23 Shen, H.S., Xiang, Y., Lin, F. and Hui, D. (2017), "Buckling and postbuckling of functionally graded graphene-reinforced composite laminated plates in thermal environments", Compos. Part B Eng., 119, 67-78. https://doi.org/10.1016/j.compositesb.2017.03.020   DOI
24 Shen, H.S., Xiang, Y., Fan, Y. and Hui, D. (2018), "Nonlinear bending analysis of FG-GRC laminated cylindrical panels on elastic foundations in thermal environments", Compos. Part B: Eng., 15, 148-157. https://doi.org/10.1016/j.compositesb.2017.12.048
25 Soleimani, A., Dastani, K., Hadi, A. and Naei, M.H. (2019), "Effect of out-of-plane defects on the postbuckling behavior of graphene sheets based on nonlocal elasticity theory", Steel Compos. Struct., Int. J., 30(6), 517-534. https://doi.org/10.12989/scs.2019.30.6.517
26 Tahouneh, V., Naei, M.H. and Mashhadi, M.M. (2018), "The effects of temperature and vacancy defect on the severity of the SLGS becoming anisotropic", Steel Compos. Struct., Int. J., 29, 647-657. https://doi.org/10.12989/scs.2018.29.5.647
27 Wang, Q. (2005), "Wave propagation in carbon nanotubes via nonlocal continuum mechanics", J. Appl. Phys., 98, 124301. https://doi.org/10.1063/1.2141648   DOI
28 Barati, M.R. and Zenkour, A.M. (2017a), "Post-buckling analysis of refined shear deformable graphene platelet reinforced beams with porosities and geometrical imperfection", Compos. Struct., 181, 194-202. https://doi.org/10.1016/j.compstruct.2017.08.082   DOI
29 Bao, W., Miao, F., Chen, Z., Zhang, H., Jang, W., Dames, C. and Lau, C.N. (2009), "Controlled ripple texturing of suspended graphene and ultrathin graphite membranes", Nature Nanotechnol., 4, 562-566. https://doi.org/10.1038/nnano.2009.191   DOI
30 Barati, M.R. (2017), "A general nonlocal stress-strain gradient theory for forced vibration analysis of heterogeneous nanoporous plates", Eur. J. Mech. A-Solid., 67, 215-230. https://doi.org/10.1016/j.euromechsol.2017.09.001   DOI
31 Barati, M.R. and Zenkour, A.M. (2017b), "Investigating postbuckling of geometrically imperfect metal foam nanobeams with symmetric and asymmetric porosity distributions", Compos. Struct., 182, 91-98. https://doi.org/10.1016/j.compstruct.2017.09.008   DOI
32 Ghayesh, M.H. (2019g), "Viscoelastic nonlinear dynamic behaviour of Timoshenko FG beams", Eur. Phys. J. Plus, 134, 401. https://doi.org/10.1140/epjp/i2019-12472-x   DOI
33 Wang, J., Xie, H. and Guo, Z. (2017), "First-principles investigation on thermal properties and infrared spectra of imperfect graphene", Appl. Therm. Eng., 116, 456-462. https://doi.org/10.1016/j.applthermaleng.2016.12.087   DOI
34 Barati, M.R. and Zenkour, A.M. (2018a), "Vibration analysis of functionally graded graphene platelet reinforced cylindrical shells with different porosity distributions", Mech. Adv. Mater. Struct., 26(18), 1580-1588. https://doi.org/10.1080/15376494.2018.1444235   DOI
35 Barati, M.R. and Zenkour, A.M. (2018b), "Post-buckling analysis of imperfect multi-phase nanocrystalline nanobeams considering nanograins and nanopores surface effects", Compos. Struct., 184, 497-505. https://doi.org/10.1016/j.compstruct.2017.10.019   DOI
36 Barati, M.R. and Zenkour, A.M. (2018c), "Thermal post-buckling analysis of closed circuit flexoelectric nanobeams with surface effects and geometrical imperfection", Mech. Adv. Mater. Struct., 26(17), 1482-1490. https://doi.org/10.1080/15376494.2018.1432821   DOI
37 Barati, M.R. and Zenkour, A.M. (2018d), "Analysis of postbuckling of graded porous GPL-reinforced beams with geometrical imperfection", Mech. Adv. Mater. Struct., 26(6), 503-511. https://doi.org/10.1080/15376494.2017.1400622   DOI
38 Ghayesh, M.H. (2019e), "Nonlinear oscillations of FG cantilevers", Appl. Acoust., 145, 393-398. https://doi.org/10.1016/j.apacoust.2018.08.014   DOI
39 Ghayesh, M.H. (2019f), "Resonant vibrations of FG viscoelastic imperfect Timoshenko beams", J. Vib. Control, 25(12), 1823-1832. https://doi.org/10.1177/1077546318825167   DOI
40 Ghayesh, M.H. and Farokhi, H. (2015), "Chaotic motion of a parametrically excited microbeam", Int. J. Eng. Sci., 96, 34-45. https://doi.org/10.1016/j.ijengsci.2015.07.004   DOI
41 Ghayesh, M.H. and Moradian, N. (2011), "Nonlinear dynamic response of axially moving, stretched viscoelastic strings", Arch. App. Mech., 81, 781-799. https://doi.org/10.1007/s00419-010-0446-3   DOI
42 Ghayesh, M.H., Yourdkhani, M., Balar, S. and Reid, T. (2010), "Vibrations and stability of axially traveling laminated beams", Appl. Math. Compos., 217(2), 545-556. https://doi.org/10.1016/j.amc.2010.05.088   DOI
43 Mirjavadi, S.S., Forsat, M., Hamouda, A.M.S. and Barati, M.R. (2019), "Dynamic response of functionally graded graphene nanoplatelet reinforced shells with porosity distributions under transverse dynamic loads", Mater. Res. Express, 6(7), 075045. https://doi.org/10.1088/2053-1591/ab1552   DOI
44 Ghayesh, M.H., Kazemirad, S. and Darabi, M.A. (2011), "A general solution procedure for vibrations of systems with cubic nonlinearities and nonlinear/time-dependent internal boundary conditions", J. Sound Vib., 330(22), 5382-5400. https://doi.org/10.1016/j.jsv.2011.06.001   DOI
45 Ghayesh, M.H., Kazemirad, S. and Reid, T. (2012), "Nonlinear vibrations and stability of parametrically exited systems with cubic nonlinearities and internal boundary conditions: A general solution procedure", Appl. Math. Model., 36(7), 3299-3311. https://doi.org/10.1016/j.apm.2011.09.084   DOI
46 Ghayesh, M.H. (2019d), "Asymmetric viscoelastic nonlinear vibrations of imperfect AFG beams", Appl. Acoust., 154, 121-128. https://doi.org/10.1016/j.apacoust.2019.03.022   DOI
47 Mirjavadi, S.S., Forsat, M., Barati, M.R., Abdella, G.M., Hamouda, A.M.S., Afshari, B.M. and Rabby, S. (2018a), "Postbuckling analysis of piezo-magnetic nanobeams with geometrical imperfection and different piezoelectric contents", Microsyst. Technol., 25(9), 3477-3488. https://doi.org/10.1007/s00542-018-4241-3
48 Mirjavadi, S.S., Afshari, B.M., Barati, M.R. and Hamouda, A.M.S. (2018b), "Transient response of porous fg nanoplates subjected to various pulse loads based on nonlocal stress-strain gradient theory", Eur. J. Mech. A-Solid., 74, 210-220. https://doi.org/10.1016/j.euromechsol.2018.11.004
49 Novoselov, K.S., Geim, A.K., Morozov, S.V., Jiang, D., Zhang, Y., Dubonos, S.V., Grigorieva, I.V. and Firsov, A.A. (2004), "Electric field effect in atomically thin carbon films", Science, 306(5696), 666-669. https://doi.org/10.1126/science.1102896   DOI
50 Nematollahi, M.S. and Mohammadi, H. (2019), "Geometrically nonlinear vibration analysis of sandwich nanoplates based on higher-order nonlocal strain gradient theory", Int. J. Mech. Sci., 156, 31-45. https://doi.org/10.1016/j.ijmecsci.2019.03.022   DOI
51 Pradhan, S.C. (2009), "Buckling of single layer graphene sheet based on nonlocal elasticity and higher order shear deformation theory", Phys. Lett. A, 373(45), 4182-4188. https://doi.org/10.1016/j.physleta.2009.09.021   DOI
52 Pradhan, S.C. and Murmu, T. (2009), "Small scale effect on the buckling of single-layered graphene sheets under biaxial compression via nonlocal continuum mechanics", Compos. Mater. Sci., 47(1), 268-274. https://doi.org/10.1016/j.commatsci.2009.08.001   DOI
53 Xu, X.J., Zheng, M.L. and Wang, X.C. (2017), "On vibrations of nonlocal rods: Boundary conditions, exact solutions and their asymptotics", Int. J. Eng. Sci., 119, 217-231. https://doi.org/10.1016/j.ijengsci.2017.06.025   DOI
54 Wang, Y., Feng, C., Zhao, Z. and Yang, J. (2018), "Eigenvalue buckling of functionally graded cylindrical shells reinforced with graphene platelets (GPL)", Compos. Struct., 202, 38-46. https://doi.org/10.1016/j.compstruct.2017.10.005   DOI
55 Wu, C.P. and Chen, Y.J. (2018), "Cylindrical Bending Vibration of Multiple Graphene Sheet Systems Embedded in an Elastic Medium", Int. J. Struct. Stab. Dyn., 19(4), 1950035. https://doi.org/10.1142/S0219455419500354   DOI
56 Wu, X., Mu, F., Wang, Y. and Zhao, H. (2018), "Application of atomic simulation methods on the study of graphene nanostructure fabrication by particle beam irradiation: A review", Compos. Mater. Sci., 149, 98-106. https://doi.org/10.1016/j.commatsci.2018.03.022   DOI
57 Yan, J.W. and Lai, S.K. (2018), "Superelasticity and wrinkles controlled by twisting circular graphene", Comput. Method. Appl. Mech. Eng., 338, 634-656. https://doi.org/10.1016/j.cma.2018.04.049   DOI
58 Yang, J., Wu, H. and Kitipornchai, S. (2016), "Buckling and postbuckling of functionally graded multilayer graphene plateletreinforced composite beams", Compos. Struct., 161, 111-118. https://doi.org/10.1016/j.compstruct.2016.11.048   DOI
59 Barretta, R. and Sciarra, F.M.D. (2018), "Constitutive boundary conditions for nonlocal strain gradient elastic nano-beams", Int. J. Eng. Sci., 130, 187-198. https://doi.org/10.1016/j.ijengsci.2018.05.009   DOI
60 Barati, M.R. and Zenkour, A.M. (2019), "Analysis of postbuckling behavior of general higher-order functionally graded nanoplates with geometrical imperfection considering porosity distributions", Mech. Adv. Mater. Struct., 26(12), 1081-1088. https://doi.org/10.1080/15376494.2018.1430280   DOI
61 Ebrahimi, F. and Barati, M.R. (2016), "Vibration analysis of piezoelectrically actuated curved nanosize fg beams via a nonlocal strain-electric field gradient theory", Mech. Adv. Mater. Struct., 25(4), 350-359. https://doi.org/10.1080/15376494.2016.1255830   DOI
62 Yang, J., Dong, J. and Kitipornchai, S. (2018a), "Unilateral and bilateral buckling of functionally graded corrugated thin plates reinforced with graphene nanoplatelets", Compos. Struct., 209, 789-801. https://doi.org/10.1016/j.compstruct.2018.11.025   DOI
63 Yang, Z., Liew, K.M. and Hui, D. (2018b), "Characterizing nonlinear vibration behavior of bilayer graphene thin films", Compos. Part B Eng., 145, 197-205. https://doi.org/10.1016/j.compositesb.2018.03.004   DOI
64 Belkorissat, I., Houari, M.S.A., Tounsi, A., Bedia, E.A. and Mahmoud, S.R. (2015), "On vibration properties of functionally graded nano-plate using a new nonlocal refined four variable mode", Steel Compos. Struct., Int. J., 18(4), 1063-1081. https://doi.org/10.12989/scs.2015.18.4.1063   DOI
65 Christy, P.A., Peter, A.J. and Lee, C.W. (2018), "Density functional theory on 13C NMR chemical shifts of fullerene", Solid State Commun., 283, 22-26. https://doi.org/10.1016/j.ssc.2018.08.001   DOI
66 Cong, P.H. and Duc, N.D. (2018), "New approach to investigate the nonlinear dynamic response and vibration of a functionally graded multilayer graphene nanocomposite plate on a viscoelastic Pasternak medium in a thermal environment", Acta Mech., 229(9), 3651-3670. https://doi.org/10.1007/s00707-018-2178-3   DOI
67 Ebrahimi, F. and Barati, M.R. (2017), "Hygrothermal effects on vibration characteristics of viscoelastic fg nanobeams based on nonlocal strain gradient theory", Compos. Struct., 159, 433-444. https://doi.org/10.1016/j.compstruct.2016.09.092   DOI
68 Ghayesh, M.H., Farokhi, H. and Alici, G. (2016), "Size-dependent performance of microgyroscopes", Int. J. Eng. Sci., 100, 99-111. https://doi.org/10.1016/j.ijengsci.2015.11.003   DOI
69 Ghayesh, M.H., Amabili, M. and Farokhi, H. (2013a), "Coupled global dynamics of an axially moving viscoelastic beam", Int. J. Non-linear Mech., 51, 54-74. https://doi.org/10.1016/j.ijnonlinmec.2012.12.008   DOI
70 Ghayesh, M.H., Amabili, M. and Farokhi, H. (2013b), "Threedimensional nonlinear size-dependent behaviour of Timoshenko microbeams", Int. J. Eng. Sci., 71, 1-14. https://doi.org/10.1016/j.ijengsci.2013.04.003   DOI
71 Hashemi, S.H. and Samaei, A.T. (2011), "Buckling analysis of micro/nanoscale plates via nonlocal elasticity theory", Physica E, 43(7), 1400-1404. https://doi.org/10.1016/j.physe.2011.03.012   DOI
72 Ghayesh, M.H., Farokhi, H. and Gholipour, A. (2017), "Vibration analysis of geometrically imperfect three-layered sheardeformable microbeams", Int. J. Mech. Sci., 122, 370-383. https://doi.org/10.1016/j.ijmecsci.2017.01.001   DOI
73 Gholipour, A., Farokhi, H. and Ghayesh, M.H. (2015), "In-plane and out-of-plane nonlinear size-dependent dynamics of microplates", Nonlinear Dyn., 79, 1771-1785. https://doi.org/10.1007/s11071-014-1773-7   DOI
74 Guo, H., Cao, S., Yang, T. and Chen, Y. (2018), "Vibration of laminated composite quadrilateral plates reinforced with graphene nanoplatelets using the element-free IMLS-Ritz method", Int. J. Mech. Sci., 142, 610-621. https://doi.org/10.1016/j.ijmecsci.2018.05.029   DOI
75 Hosseini, S.M. and Zhang, C. (2018), "Elastodynamic and wave propagation analysis in a FG graphene platelets-reinforced nanocomposite cylinder using a modified nonlinear micromechanical model", Steel Compos. Struct., Int. J., 27(3), 255-271. https://doi.org/10.12989/scs.2018.27.3.255
76 Sahmani, S., Bahrami, M. and Aghdam, M.M. (2016), "Surface stress effects on the nonlinear postbuckling characteristics of geometrically imperfect cylindrical nanoshells subjected to axial compression", Int. J. Eng. Sci., 99, 92-106. https://doi.org/10.1016/j.ijengsci.2015.10.010   DOI
77 Sadeghirad, A., Su, N. and Liu, F. (2015), "Mechanical modeling of graphene using the three-layer-mesh bridging domain method", Comput. Method. Appl. Mech. Eng., 294, 278-298. https://doi.org/10.1016/j.cma.2015.06.001   DOI
78 Sahmani, S. and Aghdam, M.M. (2017a), "Nonlinear instability of hydrostatic pressurized hybrid FGM exponential shear deformable nanoshells based on nonlocal continuum elasticity", Compos. Part B: Eng., 114, 404-417. https://doi.org/10.1016/j.compositesb.2017.01.038   DOI
79 Sahmani, S. and Aghdam, M.M. (2017b), "Nonlocal strain gradient shell model for axial buckling and postbuckling analysis of magneto-electro-elastic composite nanoshells", Compos. Part B Eng., 132, 258-274. https://doi.org/10.1016/j.compositesb.2017.09.004   DOI
80 Sahmani, S., Bahrami, M. and Aghdam, M.M. (2015), "Surface stress effects on the postbuckling behavior of geometrically imperfect cylindrical nanoshells subjected to combined axial and radial compressions", Int. J. Mech. Sci., 100, 1-22. https://doi.org/10.1016/j.ijmecsci.2015.06.004   DOI
81 Shafiei, N. and She, G.L. (2018), "On vibration of functionally graded nano-tubes in the thermal environment", Int. J. Eng. Sci., 133, 84-98. https://doi.org/10.1016/j.ijengsci.2018.08.004   DOI
82 Shahsavari, D., Karami, B. and Mansouri, S. (2017), "Shear buckling of single layer graphene sheets in hygrothermal environment resting on elastic foundation based on different nonlocal strain gradient theories", Eur. J. Mech. A-Solid., 67(C), 200-214. https://doi.org/10.1016/j.euromechsol.2017.09.004   DOI
83 Zhang, J., Zhang, W., Ragab, T. and Basaran, C. (2018), "Mechanical and electronic properties of graphene nanomesh heterojunctions", Comp. Mater. Sci., 153, 64-72. https://doi.org/10.1016/j.commatsci.2018.06.026   DOI
84 Yengejeh, S.I., Kazemi, S.A., Ivasenko, O. and O chsner, A. (2017), "Simulations of Graphene Sheets Based on the Finite Element Method and Density Functional Theory: Comparison of the Geometry Modeling under the Influence of Defects", J. Nano. Res-sew., 47, 128-135. https://doi.org/10.4028/www.scientific.net/JNanoR.47.128   DOI
85 Zenkour, A.M. and Abouelregal, A.E. (2015), "Thermoelastic interaction in functionally graded nanobeams subjected to timedependent heat flux", Steel Compos. Struct., Int. J., 18(4), 909-924. https://doi.org/10.12989/scs.2015.18.4.909   DOI
86 Zhan, H.Z., Yang, F.P. and Wang, X. (2018), "Nonlinear dynamic characteristics of bi-graphene sheets/piezoelectric laminated films considering high order van der Walls force and scale effect", Appl. Math. Model., 56, 289-303. https://doi.org/10.1016/j.apm.2017.11.038   DOI
87 Zhu, X. and Li, L. (2017), "Closed form solution for a nonlocal strain gradient rod in tension", Int. J. Eng. Sci., 119, 16-28. https://doi.org/10.1016/j.ijengsci.2017.06.019   DOI
88 Ebrahimi, F., Barati, M.R. and Dabbagh, A. (2016), "A nonlocal strain gradient theory for wave propagation analysis in temperature-dependent inhomogeneous nanoplates", Int. J. Eng. Sci., 107, 169-182. https://doi.org/10.1016/j.ijengsci.2016.07.008   DOI
89 Ebrahimi, F. and Barati, M.R. (2018a), "Damping vibration analysis of graphene sheets on viscoelastic medium incorporating hygro-thermal effects employing nonlocal strain gradient theory", Compos. Struct., 185, 241-253. https://doi.org/10.1016/j.compstruct.2017.10.021   DOI
90 Ebrahimi, F. and Barati, M.R. (2018b), "Vibration analysis of biaxially compressed double-layered graphene sheets based on nonlocal strain gradient theory", Mech. Adv. Mater. Struct., 26(10), 854-865. https://doi.org/10.1080/15376494.2018.1430267   DOI
91 Faleh, N.M., Ahmed, R.A. and Fenjan, R.M. (2018), "On vibrations of porous FG nanoshells", Int. J. Eng. Sci., 133, 1-14. https://doi.org/10.1016/j.ijengsci.2018.08.007   DOI
92 Farokhi, H. and Ghayesh, M.H. (2015), "Nonlinear dynamical behaviour of geometrically imperfect microplates based on modified couple stress theory", Int. J. Mech. Sci., 90, 133-144. https://doi.org/10.1016/j.ijmecsci.2014.11.002   DOI
93 Farokhi, H. and Ghayesh, M.H. (2017), "Nonlinear resonant response of imperfect extensible Timoshenko microbeams", Int. J. Mech. Mater. Des., 13(1), 43-55. https://doi.org/10.1007/s10999-015-9316-z   DOI
94 Farokhi, H. and Ghayesh, M.H. (2018a), "Supercritical nonlinear parametric dynamics of Timoshenko microbeams", Commun. Nonlinear Sci., 59, 592-605. https://doi.org/10.1016/j.cnsns.2017.11.033   DOI
95 Farokhi, H. and Ghayesh, M.H. (2018b), "Nonlinear mechanics of electrically actuated microplates", Int. J. Eng. Sci., 123, 197-213. https://doi.org/10.1016/j.ijengsci.2017.08.017   DOI
96 Farokhi, H., Ghayesh, M.H. and Amabili, M. (2013), "Nonlinear resonant behavior of microbeams over the buckled state", Appl. Phys. A., 113, 297-307. https://doi.org/10.1007/s00339-013-7894-x   DOI
97 Kazemirad, S., Ghayesh, M.H. and Amabili, M. (2012), "Thermomechanical nonlinear dynamics of a buckled axially moving beam", Arch App. Mech., 83(1), 25-42. https://doi.org/10.1007/s00419-012-0630-8   DOI
98 Hosseini, M., Gorgani, H.H., Shishesaz, M. and Hadi, A. (2017), "Size-dependent stress analysis of single-wall carbon nanotube based on strain gradient theory", Int. J. App. Mech., 9(6), 1750087. https://doi.org/10.1142/S1758825117500879   DOI
99 Hussein, A. and Kim, B. (2018), "Graphene/polymer nanocomposites: The active role of the matrix in stiffening mechanics", Compos. Struct., 202, 170-181. https://doi.org/10.1016/j.compstruct.2018.01.023   DOI
100 Karami, B., Janghorban, M. and Li, L. (2018), "On guided wave propagation in fully clamped porous functionally graded nanoplates", Acta Astronaut., 143, 380-390. https://doi.org/10.1016/j.actaastro.2017.12.011   DOI
101 Kiani, Y. (2017), "Isogeometric large amplitude free vibration of graphene reinforced laminated plates in thermal environment using NURBS formulation", Comput. Method. Appl. M., 332, 86-101. https://doi.org/10.1016/j.cma.2017.12.015   DOI
102 Korobeynikov, S.N., Alyokhin, V.V. and Babichev, A.V. (2018), "On the molecular mechanics of single layer graphene sheets", Int. J. Eng. Sci., 133, 109-131. https://doi.org/10.1016/j.ijengsci.2018.09.001   DOI
103 Kumar, D. and Srivastava, A. (2016), "Elastic properties of CNTand graphene-reinforced nanocomposites using RVE", Steel Compos. Struct., Int. J., 21(5), 1085-1103. https://doi.org/10.12989/scs.2016.21.5.1085   DOI
104 Li, L., Tang, H. and Hu, Y. (2018), "The effect of thickness on the mechanics of nanobeams", Int. J. Eng. Sci., 123, 81-91. https://doi.org/10.1016/j.ijengsci.2017.11.021   DOI
105 Lim, C.W., Zhang, G. and Reddy, J.N. (2015), "A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation", J. Mech. Phys. Solids, 78, 298-313. https://doi.org/10.1016/j.jmps.2015.02.001   DOI
106 She, G.L., Yuan, F.G. and Ren, Y.R. (2017d), "Thermal buckling and post-buckling analysis of functionally graded beams based on a general higher-order shear deformation theory", Appl. Math. Model., 47, 340-357. https://doi.org/10.1016/j.apm.2017.03.014   DOI
107 Shahsavari, D., Karami, B., Fahham, H.R. and Li, L. (2018), "On the shear buckling of porous nanoplates using a new sizedependent quasi-3d shear deformation theory", Acta Mech., 229(11), 4549-4573. https://doi.org/10.1007/s00707-018-2247-7   DOI
108 She, G.L., Yuan, F.G. and Ren, Y.R. (2017a), "Nonlinear analysis of bending, thermal buckling and post-buckling for functionally graded tubes by using a refined beam theory", Compos. Struct., 165, 74-82. https://doi.org/10.1016/j.compstruct.2017.01.013   DOI
109 She, G.L., Yuan, F.G., Ren, Y.R. and Xiao, W.S. (2017b), "On buckling and postbuckling behavior of nanotubes", Int. J. Eng. Sci., 121, 130-142. https://doi.org/10.1016/j.ijengsci.2017.09.005   DOI
110 She, G.L., Yuan, F.G. and Ren, Y.R. (2017c), "Research on nonlinear bending behaviors of FGM infinite cylindrical shallow shells resting on elastic foundations in thermal environments", Compos. Struct., 170, 111-121. https://doi.org/10.1016/j.compstruct.2017.03.010   DOI
111 She, G.L., Yuan, F.G., Karami, B., Ren, Y.R. and Xiao, W.S. (2019), "On nonlinear bending behavior of FG porous curved nanotubes", Int. J. Eng. Sci., 135, 58-74. https://doi.org/10.1016/j.ijengsci.2018.11.005   DOI
112 Shen, H.S. (2007), "Thermal postbuckling behavior of shear deformable FGM plates with temperature-dependent properties", Int. J. Mech. Sci., 49(4), 466-478. https://doi.org/10.1016/j.ijmecsci.2006.09.011   DOI
113 Shen, H.S. (2013), A two-step perturbation method in nonlinear analysis of beams, plates and shells, Higher Education Press.