Steel and Composite Structures

  • 제34권2호
  • /
  • Pages.261-277
  • /
  • 2020
  • /
  • 1229-9367(pISSN)
  • /
  • 1598-6233(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Influence of vacancy defects on vibration analysis of graphene sheets applying isogeometric method: Molecular and continuum approaches

  • Tahouneh, Vahid (School of Mechanical Engineering, College of Engineering, University of Tehran) ;
  • Naei, Mohammad Hasan (School of Mechanical Engineering, College of Engineering, University of Tehran) ;
  • Mashhadi, Mahmoud Mosavi (School of Mechanical Engineering, College of Engineering, University of Tehran)
  • 투고 : 2019.08.02
  • 심사 : 2019.12.02
  • 발행 : 2020.01.25
⟨ 이전 논문 다음 논문 ⟩

초록

The main objective of this research paper is to consider vibration analysis of vacancy defected graphene sheet as a nonisotropic structure via molecular dynamic and continuum approaches. The influence of structural defects on the vibration of graphene sheets is considered by applying the mechanical properties of defected graphene sheets. Molecular dynamic simulations have been performed to estimate the mechanical properties of graphene as a nonisotropic structure with single- and double- vacancy defects using open source well-known software i.e., large-scale atomic/molecular massively parallel simulator (LAMMPS). The interactions between the carbon atoms are modelled using Adaptive Intermolecular Reactive Empirical Bond Order (AIREBO) potential. An isogeometric analysis (IGA) based upon non-uniform rational B-spline (NURBS) is employed for approximation of single-layered graphene sheets deflection field and the governing equations are derived using nonlocal elasticity theory. The dependence of small-scale effects, chirality and different defect types on vibrational characteristic of graphene sheets is investigated in this comprehensive research work. In addition, numerical results are validated and compared with those achieved using other analysis, where an excellent agreement is found. The interesting results indicate that increasing the number of missing atoms can lead to decrease the natural frequencies of graphene sheets. It is seen that the degree of the detrimental effects differ with defect type. The Young's and shear modulus of the graphene with SV defects are much smaller than graphene with DV defects. It is also observed that Single Vacancy (SV) clusters cause more reduction in the natural frequencies of SLGS than Double Vacancy (DV) clusters. The effectiveness and the accuracy of the present IGA approach have been demonstrated and it is shown that the IGA is efficient, robust and accurate in terms of nanoplate problems.

키워드

참고문헌

  1. Abbassian, F., Dawswell, D.J. and Knowles, N.C. (1987), "Free Vibration Benchmarks. Atkins Engineering Sciences": Glasgow.
  2. Aghababaei, R. and Reddy, J.N. (2009), "Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates", J. Sound Vib., 326(1-2), 277-289. https://doi.org/10.1016/j.jsv.2009.04.044.
  3. Ahmed Houari, M.S., Bessaim, A., Bernard, F., Tounsi, A. and Mahmoud, S.R. (2018), "Buckling analysis of new quasi-3D FG nanobeams based on nonlocal strain gradient elasticity theory and variable length scale parameter", Steel Compos. Struct., 28(1), 13-24. https://doi.org/10.12989/scs.2018.28.1.013.
  4. Ali, G.A.M., Makhlouf, S.A., Yusoff, M.M. and Chong, K.F. (2015), "Structural and electrochemical characteristics of graphene nanosheets as supercapacitor electrodes", Rev. Adv. Mater. Sci., 41(1), 35-43.
  5. Ali, M.A., Mondal, K., Wang, Y., Jiang, H., Mahal, N.K., Castellano, M.J., Sharma, A. and Dong, L. (2017), "In situ integration of graphene foam-titanium nitride based bio-scaffolds and microfluidic structures for soil nutrient sensors", Lab. Chip., 17(2),274-285. https://doi.org/10.1039/C6LC01266C
  6. Al-Sherbini, A.S., Bakr, M., Ghoneim, I. and Saad, M. (2017), "Exfoliation of graphene sheets via high energy wet milling of graphite in 2-ethylhexanol and kerosene", J. Adv. Res., 8(3), 209-215. https://doi.org/10.1016/j.jare.2017.01.004.
  7. Ansari, R., Ajori, S. and Motevalli, B. (2012), "Mechanical properties of defective single-layered graphene sheets via molecular dynamics simulation", Superlattices Microstruct., 51(2), 274-289. https://doi.org/10.1016/j.spmi.2011.11.019.
  8. Ansari, R., Sahmani, S. and Arash, B. (2010), "Nonlocal plate model for free vibrations of single-layered graphene sheets", Phys. Lett. A., 375(1), 53-62. https://doi.org/10.1016/j.physleta.2010.10.028.
  9. Aristodemo, M. (1985), "A high-continuity finite element model for two dimensional elastic problems", Comput. Struct., 21(5), 987-993. https://doi.org/10.1016/0045-7949(85)90211-1.
  10. Bilotta, A., Formica, G. and Turco, E. (2010), "Performance of a high-continuity finite element in three-dimensional elasticity", Int. J. Numer. Method. Biomed. Eng., 26(9), 1155-1175. https://doi.org/10.1002/cnm.1201.
  11. Bu, H., Chen, Y., Zou, M., Yi, H., Bi, K. and Ni, Z. (2009), "Atomistic simulations of mechanical properties of graphene nanoribbons", Phys. Lett. A, 373(37), 3359-3362. https://doi.org/10.1016/j.physleta.2009.07.048.
  12. Chen, C.Q., Shi, Y., Zhang, Y.S., Zhu, J. and Yan, Y.J. (2006), "Size Dependence of Young's Modulus in ZnO Nanowires", Phys. Rev. Lett., 96(7).
  13. Chu, L., Shi, J. and Souza de Cursi, E. (2018), "Vibration analysis of vacancy defected graphene sheets by monte carlo based finite element method", Nanomaterials, 8(7), 489. https://doi.org/10.3390/nano8070489.
  14. Compagnini, G., Giannazzo, F., Sonde, S., Raineri, V. and Rimini, E. (2009), "Ion irradiation and defect formation in single layer graphene", Carbon, 47(14), 3201-3207. https://doi.org/10.1016/j.carbon.2009.07.033.
  15. Dastjerdi, S., Lot, M. and Jabbarzadeh, M. (2016), "The effect of vacant defect on bending analysis of graphene sheets based on the Mindlin nonlocal elasticity theory", Compos. Part B Eng., 98(1), 78-87. https://doi.org/10.1016/j.compositesb.2016.05.009.
  16. Eringen, A.C. (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54, 4703-4710. https://doi.org/10.1063/1.332803.
  17. Fadaee, M. (2016), "Buckling analysis of a defective annular graphene sheet in elastic medium", Appl. Math. Model, 40(3), 1863-1872. https://doi.org/10.1016/j.apm.2015.09.029.
  18. Georgantzinos, S.K., Giannopoulos, G.I. and Anifantis, N.K. (2010), "Numerical investigation of elastic mechanical properties of graphene structures", Mater. Des., 31(10), 4646-4654. https://doi.org/10.1016/j.matdes.2010.05.036.
  19. Guo, Y., Ruess, M. and Gurdal, Z. (2014), "A contact extended isogeometric layerwise approach for the buckling analysis of delaminated composites", Compos. Struct., 116(1), 55-66. https://doi.org/10.1016/j.compstruct.2014.05.006.
  20. Hosseini Hashemi, S., Mehrabani, H. and Ahmadi-Savadkoohi, A. (2015), "Exact solution for free vibration of coupled double viscoelastic graphene sheets by visco pasternak medium", Compos. Part B-Eng., 78, 377-383. https://doi.org/10.1016/j.compositesb.2015.04.008.
  21. 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., 27(3), 255-271. https://doi.org/10.12989/scs.2018.27.3.255.
  22. Hughes, T.J.R., Cottrell, J.A. and Bazilevs, Y. (2005), "Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement", Comput. Method. Appl. M., 194(39-41), 4135-4195. https://doi.org/10.1016/j.cma.2004.10.008.
  23. Jalali, S.K., Jomehzadeh, E. and Pugno, N.M. (2016), "Influence of out-of-plane defects on vibration analysis of graphene: Molecular Dynamics and Non-local Elasticity approaches", Superlattices Microstruct., 91, 331-344. https://doi.org/10.1016/j.spmi.2016.01.023.
  24. Javani, R., Bidgoli, M.R. and Kolahchi, R. (2019), "Buckling analysis of plates reinforced by Graphene platelet based on Halpin-Tsai and Reddy theories", Steel Compos. Struct., 31(4), 419-427. https://doi.org/10.12989/scs.2019.31.4.419.
  25. Jiang, J.W., Wang, J.S. and Li, B. (2009), "Young's modulus of graphene: A molecular dynamics study", Phys. Rev. B, 80(11), 113405-113408. https://doi.org/10.1103/physrevb.80.113405
  26. Kapoor, H. and Kapania, R. (2012), "Geometrically nonlinear NURBS isogeometric finite element analysis of laminated composite plates", Compos. Struct., 94(12), 3434-3447. https://doi.org/10.1016/j.compstruct.2012.04.028.
  27. Karami, B., Janghorban, M., Shahsavari, D. and Touns, A. (2018), "A size-dependent quasi-3D model for wave dispersion analysis of FG nanoplates", Steel Compos. Struct., 28(1), 99-110. https://doi.org/10.12989/scs.2018.28.1.099.
  28. Karami, B., Janghorban, M. and Tounsi, A. (2017), "Effects of triaxial magnetic field on the anisotropic nanoplates", Steel Compos. Struct., 25(3), 361-374. https://doi.org/10.12989/scs.2017.25.3.361.
  29. Karami, B., Janghorban, M. and Tounsi, A. (2018), "Nonlocal strain gradient 3D elasticity theory for anisotropic spherical nanoparticles", Steel Compos. Struct., 27(2), 201-216 https://doi.org/10.12989/scs.2018.27.2.201.
  30. Kiendl, J., Bletzinger, K. U., Linhard, J. and Wuchner, R. (2009), "Isogeometric shell analysis with Kirchhoff-Love elements", Comput. Method Appl. M., 198(49-52), 3902-3914. https://doi.org/10.1016/j.cma.2009.08.013.
  31. Kitipornchai, S., He, X. Q. and Liew, K. M. (2005), "Continuum model for the vibration of multilayered graphene sheets", Phys. Rev. B., 72(7), 075443-Published 29 August 2005. https://doi.org/10.1103/physrevb.72.075443
  32. Kumar, D. and Srivastava, A. (2016), "Elastic properties of CNT- and Graphene-reinforced nanocomposites using RVE", Steel Compos. Struct., 21(5), 1085-1103. http://dx.doi.org/10.12989/scs.2016.21.5.1085.
  33. Kuzhir, P., Volynets, N., Maksimenko, S., Kaplas, T. and Svirko, Y. (2013), "Multilayered graphene in Ka-band: Nanoscale coating for aerospace applications", J. Nanosci. Nanotechnol., 13(8), 5864-5867. https://doi.org/10.1166/jnn.2013.7551.
  34. Kvashnin, A.G., Sorokin, P.B., Kvashnin, D.G. (2010), "The Theoretical Study of Mechanical Properties of Graphene Membranes", Fullerenes, Nanotubes, and Carbon Nanostructures, 18(4-6), 497-500. https://doi.org/10.1080/1536383X.2010.488160
  35. Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J. and Tong, P. (2003), "Experiments and theory in strain gradient elasticity", J. Mech. Phys. Solids, 51(8), 1477-1508. https://doi.org/10.1016/S0022-5096(03)00053-X.
  36. Lee, C., Wei, X.D., Kysar, J.W. and Hone, J. (2008), "Measurement of the elastic properties and intrinsic strength of monolayer graphene", Science, 321(5887), 385-388. DOI: 10.1126/science.1157996.
  37. Le-Manh, T. and Lee, J. (2014), "Postbuckling of laminated composite plates using NURBS-based isogeometric analysis", Compos. Struct., 109, 286-293. https://doi.org/10.1016/j.compstruct.2013.11.011.
  38. Le-Manh, T., Luu-Anh, T. and Lee, J. (2016), "Isogeometric analysis for flexural behavior of composite plates considering large deformation with small rotations", Mech. Adv. Mater. Struct., 23(3), 328-336. https://doi.org/10.1080/15376494.2014.981616.
  39. Lim, C.W.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.
  40. Liu, G.R. and Chen, X.L. (2001), "A mesh-free method for static and free vibration analyses of thin plates of complicated shape", J. Sound Vib., 241(5), 839-855. https://doi.org/10.1006/jsvi.2000.3330.
  41. Liu, F., Ming, P. and Li, J. (2007), "Ab initio calculation of ideal strength and phonon instability of graphene under tension", Phys. Rev. B, 76(6), 064120- 064127. https://doi.org/10.1103/physrevb.76.064120
  42. Liu, G.R. (2003), "Meshfree Methods: Moving Beyond the Finite Element Method", (CRC Press, USA).
  43. Liu, X., Metcalf, T.H., Robinson, J.T., Houston, B.H. and Scarpa, F. (2012), "Shear modulus of monolayer graphene prepared by chemical vapor deposition", Nano. Lett., 12(2), 1013-1017. https://doi.org/10.1021/nl204196v.
  44. Ma, J., Alfe, D., Michaelides, A. and Wang, E. (2009), "Stone-Wales defects in graphene and other planar sp 2 -bonded materials", Phys. Rev. B, 80, 1-4.
  45. Malagu, M., Benvenuti, E. and Simone, A. (2012), "A Finite element and b-spline methods for one-dimensional non-local elasticity", Proceedings of the ECCOMAS 2012: 6th European Congress on computational methods in applied sciences and engineering, September 2012, Vienna University of Technology, Vienna, Austria.
  46. Marina, P.E., Ali, G.A.M., See, L.M., Teo, E.Y.L., Ng, E.P. and Chong, K.F. (2016), "In situ growth of redox-active iron-centered nanoparticles on graphene sheets for specific capacitance enhancement", Arab. J. Chem., https://doi.org/10.1016/j.arabjc.2016.02.006.
  47. Marin, M. (2008), "Weak Solutions in Elasticity of Dipolar Porous Materials", Math. Probl. Eng., 2008, http://dx.doi.org/10.1155/2008/158908, 1-8.
  48. Marin, M. (2016), "An approach of a heat-flux dependent theory for micropolar porous media", Meccanica, 51(5), 1127-1133. https://doi.org/10.1007/s11012-015-0265-2.
  49. Marin, M. and Baleanu, D. (2016), "On vibrations in thermoelasticity without energy dissipation for micropolar bodies", Bound. Value Probl., 2016(1), 1-19. https://doi.org/10.1186/s13661-016-0620-9.
  50. Martinez-Asencio, J. and Caturla, M.J. (2015), "Molecular dynamics simulations of defect production in graphene by carbon irradiation", Nucl. Instruments Methods Phys. Res. Sect. B Beam Interact with Mater. Atoms, 352, 225-228. https://doi.org/10.1016/j.nimb.2014.12.010.
  51. Min, K. and Aluru, N.R. (2011), "Mechanical properties of graphene under shear deformation", Appl. Phys. Lett., 98(1), 013113. https://doi.org/10.1063/1.3534787
  52. Mirakhory, M., Khatibi, M.M. and Sadeghzadeh, S. (2018), "Vibration analysis of defected and pristine triangular single-layer graphene nanosheets", Curr. Appl. Phys., 18(11), 1327-1337. https://doi.org/10.1016/j.cap.2018.07.014.
  53. Mirzaei, M. and Kiani, Y. (2017), "Isogeometric thermal buckling analysis of temperature dependent FG graphene reinforced laminated plates using NURBS formulation", Compos. Struct.., 180, 606-616. https://doi.org/10.1016/j.compstruct.2017.08.057.
  54. Moradi-Dastjerdi, R. and Behdinan, K. (2019), "Thermoelastic static and vibrational behaviors of nanocomposite thick cylinders reinforced with Graphene", Steel Compos. Struct., 31(5), 529-539. https://doi.org/10.12989/scs.2019.31.5.529.
  55. Nguyen, N.T., Hui, D., Lee, J. and Nguyen-Xuan, H. (2015), "An efficient computational approach for size-dependent analysis of functionally graded nanoplates", Comput. Method. Appl. M., 297, 191-218. https://doi.org/10.1016/j.cma.2015.07.021.
  56. Neek-Amal, M., Peeters, F.M. (2010), "Linear reduction of stiffness and vibration frequencies in defected circular monolayer graphene", Phys. Rev. B, 81(23), 235437. https://doi.org/10.1103/physrevb.81.235437
  57. 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, 405(5), 1301-1306. https://doi.org/10.1016/j.physb.2009.11.071.
  58. Norouzzadeh, A. and Ansari, R. (2018), "Isogeometric vibration analysis of functionally graded nanoplates with the consideration of nonlocal and surface effects", Thin-Wall. Struct., 127, 354-372. https://doi.org/10.1016/j.tws.2017.11.040.
  59. Piegl, L. and Tiller, W. (1997), "The NURBS Book", (Monographs in Visual Communication) (Springer-Verlag, New York).
  60. Plimpton, S. (1995), "Fast parallel algorithms for short-range molecular dynamics", J. Comput. Phys., 117(1), 1-19. https://doi.org/10.1006/jcph.1995.1039.
  61. Pradhan, S.C. and Kumar, A. (2010), "Vibration analysis of orthotropic graphene sheets using nonlocal elasticity theory and differential quadrature method", Compos. Struct., 93(2), 774-779. https://doi.org/10.1016/j.compstruct.2010.08.004.
  62. Rajasekaran, G., Narayanan, P. and Parashar, A. (2016), "Effect of point and line defects on mechanical and thermal properties of graphene", Crit. Rev. Solid State Mater. Sci., 41(1), 47-71. https://doi.org/10.1080/10408436.2015.1068160
  63. Reddy, C.D., Rajendran, S. and Liew, K.M. (2006), "Equilibrium configuration and continuum elastic properties of finite sized graphene", Nanotechnology, 17(3), 864-870. https://doi.org/10.1088/0957-4484/17/3/042
  64. Ribeiro, P. and Chuaqui, T.R.C. (2019), "Non-linear modes of vibration of single-layer non-local graphene sheets", Int. J. Mech. Sci., 150, 727-743. https://doi.org/10.1016/j.ijmecsci.2018.10.068.
  65. Roh, H.Y. and Cho, M. (2004), "The application of geometrically exact shell elements to B-spline surfaces", Comput. Method Appl. M., 193(23-26), 2261-2299. https://doi.org/10.1016/j.cma.2004.01.019.
  66. Rouhi, S., Ansari, R. (2012), "Atomistic finite element model for axial buckling and vibration analysis of single-layered graphene sheets", Phys. E. Low-Dimensional Syst. Nanostructures, 44(4), 764-772. https://doi.org/10.1016/j.physe.2011.11.020.
  67. Sadeghi, M. and Naghdabadi, R. (2010), "Nonlinear vibrational analysis of single-layer graphene sheets", Nanotechnology, 21(10).
  68. Sakhaee-Pour, A. (2009), "Elastic buckling of single-layered graphene sheet", Comput. Mater. Sci., 45(2), 266-270. https://doi.org/10.1016/j.commatsci.2008.09.024.
  69. Sakhaee-Pour A. (2009), "Elastic properties of single-layered graphene sheet", Solid State Commun., 149(1-2), 91-95. https://doi.org/10.1016/j.ssc.2008.09.050.
  70. Sakhaee-Pour, A., Ahmadian, M.T. and Naghdabadi, R. (2008), "Vibrational analysis of single-layered graphene sheets", Nanotechnology, 19(8), 85702. doi:10.1088/0957-4484/19/8/085702
  71. Shahsavari, D., Karami, B. and Li, L. (2018), "A high-order gradient model for wave propagation analysis of porous FG nanoplates", Steel Compos. Struct., 29(1), 53-66. https://doi.org/10.12989/scs.2018.29.1.053.
  72. Shen, L., Shen, H.S. and Zhang, C.L. (2010), "Temperature-dependent elastic properties of single layer graphene sheets", Mater. Des., 31(9), 4445-4449. https://doi.org/10.1016/j.matdes.2010.04.016.
  73. Siddique, J.A., Attia, N.F. and Geckeler, K.E. (2015), "Polymer nanoparticles as a tool for the exfoliation of graphene sheets", Mater. Lett., 158(1), 186-189. https://doi.org/10.1016/j.matlet.2015.05.134
  74. Sobhy, M. (2014), "Thermomechanical bending and free vibration of single-layered graphene sheets embedded in an elastic medium", Phys. E. Low-Dimensional Syst. Nanostructures, 56, 400-409. https://doi.org/10.1016/j.physe.2013.10.017.
  75. 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., 30(6), 517-534. https://doi.org/10.12989/scs.2019.30.6.517.
  76. Soleimani, A., Naei, M.H. and Mashhadi, M.M. (2017), "Buckling analysis of graphene sheets using nonlocal isogeometric finite element method for NEMS applications", Microsyst. Technol., 23(7), 2859-2871. https://doi.org/10.1007/s00542-016-3098-6.
  77. Song M, Kitipornchai S. and Yang, J. (2017), "Free and forced vibrations of functionally graded polymer composite plates reinforced with graphene nanoplatelets", Compos. Struct., 159, 579-588. https://doi.org/10.1016/j.compstruct.2016.09.070.
  78. Stan, G., Ciobanu, C.V., Parthangal, P.M. and Cook, R.F. (2007), "Diameter-Dependent Radial and Tangential Elastic Moduli of ZnO Nanowires", Nano Lett., 7(12), 3691-3697. https://doi.org/10.1021/nl071986e.
  79. Stuart, S.J., Tutein, A.B. and Harrison, J.A. (2000), "A reactive potential for hydrocarbons with intermolecular interactions", J. Chem. Phys., 112, 6472-6487. https://doi.org/10.1063/1.481208.
  80. Subramaniyan, A.K. and Sun, C.T. (2008), "Continuum interpretation of virial stress in molecular simulations", J. Solids Struct., 45(14-15), 4340-4346. https://doi.org/10.1016/j.ijsolstr.2008.03.016.
  81. Sun, S., Wang, C., Chen, M. and Zheng, J. (2013), "A novel method to control atomic defects in graphene sheets by selective surface reactions", Appl. Surf. Sci., 283, 566-570. https://doi.org/10.1016/j.apsusc.2013.06.146.
  82. Sun, X.Y., Hu, H., Caob, C. and Xua, Y.J. (2015), "Anisotropic vacancy-defect-induced fracture strength loss of graphene", RSC Adv., 5(2), 13623-13627. https://doi.org/10.1039/C4RA14044C
  83. Sun, X., Fu, Z. and Xia, M. (2014), "Effects of vacancy defect on the tensile behavior of graphene", Theor. Appl. Mech. Lett., 4(5), 51002. https://doi.org/10.1063/2.1405102.
  84. Tasis, D., Papagelis, K., Spiliopoulos, P. and Galiotis, C. (2013), "Efficient exfoliation of graphene sheets in binary solvents", Mater. Lett., 94, 47-50. https://doi.org/10.1016/j.matlet.2012.12.027
  85. Tahouneh, V., Naei, M.H. and Mosavi Mashhadi, M. (2018), "The effects of temperature and vacancy defect on the severity of the SLGS becoming anisotropic", Steel Compos. Struct., 29(5), 647-657. https://doi.org/10.12989/scs.2018.29.5.647.
  86. Thai, C.H., Nguyen-Xuan, H., Nguyen-Thanh, N., Le, T.H., Nguyen-Thoi, T. and Rabczuk, T. (2012), "Static, free vibration, and buckling analysis of laminated composite Reissner-Mindlin plates using NURBS-based isogeometric approach", Int. J. Numer. Method. Eng., 91(6), 571-603. https://doi.org/10.1002/nme.4282.
  87. Tornabene, F., Bacciocchi, M., Fantuzzi, N. and Reddy, J.N. (2018), "Multiscale approach for three-phase CNT/Polymer/Fiber laminated nanocomposite structures", Polymer Composites, In Press, DOI: 10.1002/pc.24520.
  88. Tornabene, F., Fantuzzi, N., Ubertini, F. and Viola, E. (2015), "Strong formulation finite element method based on differential quadrature: a survey", Appl. Mech. Rev., 67(2), 1-55.
  89. Tran, L.V., Lee, J., Nguyen-Van, H., Nguyen-Xuan, H. and Wahab, M.A. (2015), "Geometrically nonlinear isogeometric analysis of laminated composite plates based on higher-order shear deformation theory", Int. J. Non-Linear Mech., 72, 42-52. https://doi.org/10.1016/j.ijnonlinmec.2015.02.007.
  90. Tran, L.V., Phung-Van, P., Lee, J., Wahab, M.A. and Nguyen-Xuan, H. (2016), "Isogeometric analysis for nonlinear thermomechanical stability of functionally graded plates", Compos. Struct., 140, 655-667. https://doi.org/10.1016/j.compstruct.2016.01.001.
  91. Tsai, D.H. (1979), "The virial theorem and stress calculation in molecular dynamics", J. Chem. Phys., 70, 1375-1382. https://doi.org/10.1063/1.437577.
  92. Tsai, J.L. and Tu, J.F. (2010), "Characterizing mechanical properties of graphite using molecular dynamics simulation", Mater. Des., 31(1), 194-199. https://doi.org/10.1016/j.matdes.2009.06.032.
  93. Udupa, A. and Martini, A. (2011), "Model predictions of shear strain-induced ridge defects in graphene", Carbon, 49(11), 3571-3578. https://doi.org/10.1016/j.carbon.2011.04.057.
  94. Wang, X., Zhu, X. and Hu, P. (2015), "Isogeometric finite element method for buckling analysis of generally laminated composite beams with different boundary conditions", Int. J. Mech. Sci., 104, 190-199. https://doi.org/10.1016/j.ijmecsci.2015.10.008.
  95. Wei, Y., Wang, B., Wu, J., Yang, R. and Dunn, M.L. (2012), "Bending rigidity and Gaussian bending stiffness of single-layered graphene", Nano Lett., 13(1), 26-30. https://doi.org/10.1021/nl303168w.
  96. Wenhu Wu, Y. D.,Yin, J., Xie, W., Zhang, W., Wu, B., Jiang, Y. and Zhang, P. (2015), "Effect of vacancy distribution on the relaxation properties of graphene : A molecular dynamics study", IET Micro. Nano Lett., 10(12).
  97. Xie, G., Shen, Y., Wei, X., Yang, L., Xiao, H., Zhong, J. and Zhang, G. (2014), "A bond-order theory on the phonon scattering by vacancies in two-dimensional materials", Electron. Spintron. devices, 1-23. https://doi.org/10.1038/srep05085.
  98. Yang, F., Chong, A.C.M., Lam, D.C.C. and Tong, P. (2002), "Couple stress based strain gradient theory for elasticity", Int. J. Solids Struct., 39(10), 2731-2743. https://doi.org/10.1016/S0020-7683(02)00152-X.
  99. Yanovsky, Y.G., Nikitina, E.A., Karnet, Y.N., Nikitin, S.M. (2009), "Quantum mechanics study of the mechanism of deformation and fracture of graphene", Phys. Mesomech., 12(5-6), 254-262. https://doi.org/10.1016/j.physme.2009.12.007.
  100. Yu, T.T., Yin, S., Bui, T.Q. and Hirose, S. (2015), "A simple FSDT-based isogeometric analysis for geometrically nonlinear analysis of functionally graded plates", Finite Elem. Anal. Des., 96, 1-10. https://doi.org/10.1016/j.finel.2014.11.003.
  101. Zhang, Y., Chen, Y., Zhou, K. and Liu, C. (2009), "Improving gas sensing properties of graphene by introducing dopants and defects : a first-principles study", Nanotechnology, 20(18).
  102. Zhang, Y., Lei, Z.X., Zhang, L.W., Liew, K.M. and Yu, J.L. (2015), "Nonlocal continuum model for vibration of single-layered graphene sheets based on the element-free kp-Ritz method", Eng. Anal. Bound. Elem., 56, 90-97. https://doi.org/10.1016/j.enganabound.2015.01.020.
  103. Zhou, M. (2003), "A new look at the atomic level virial stress: on continuum-molecular system equivalence", P. Roy. Soc. A-Math. Phy., 459(2), 2347-2392. https://doi.org/10.1098/rspa.2003.1127.