Advances in nano research

  • 제12권5호
  • /
  • Pages.467-482
  • /
  • 2022
  • /
  • 2287-237X(pISSN)
  • /
  • 2287-2388(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

A Fourier sine series solution of static and dynamic response of nano/micro-scaled FG rod under torsional effect

  • Civalek, Omer (Akdeniz University, Faculty of Engineering, Department of Civil Engineering) ;
  • Uzun, Busra (Bursa Uludag University, Faculty of Engineering, Department of Civil Engineering, Gorukle Campus) ;
  • Yayli, M. Ozgur (Bursa Uludag University, Faculty of Engineering, Department of Civil Engineering, Gorukle Campus)
  • 투고 : 2021.11.04
  • 심사 : 2022.02.21
  • 발행 : 2022.05.25
⟨ 이전 논문 다음 논문 ⟩

초록

In the current work, static and free torsional vibration of functionally graded (FG) nanorods are investigated using Fourier sine series. The boundary conditions are described by the two elastic torsional springs at the ends. The distribution of functionally graded material is considered using a power-law rule. The systems of equations of the mechanical response of nanorods subjected to deformable boundary conditions are achieved by using the modified couple stress theory (MCST) and taking the effects of torsional springs into account. The idea of the study is to construct an eigen value problem involving the torsional spring parameters with small scale parameter and functionally graded index. This article investigates the size dependent free torsional vibration based on the MCST of functionally graded nano/micro rods with deformable boundary conditions using a Fourier sine series solution for the first time. The eigen value problem is constructed using the Stokes' transform to deformable boundary conditions and also the convergence and accuracy of the present methodology are discussed in various numerical examples. The small size coefficient influence on the free torsional vibration characteristics is studied from the point of different parameters for both deformable and rigid boundary conditions. It shows that the torsional vibrational response of functionally graded nanorods are effected by geometry, small size effects, boundary conditions and material composition. Furthermore, for all deformable boundary conditions in the event of nano-sized FG nanorods, the incrementing of the small size parameters leads to increas the torsional frequencies.

키워드

참고문헌

  1. Abouelregal, A.E. (2020), "A novel model of nonlocal thermoelasticity with time derivativesof higher order", Math. Method Appl. Sci., 43(11), 6746-6760. https://doi.org/10.1002/mma.6416.
  2. Abouelregal, A.E. and Mohammed, W.W. (2020), "Effects of nonlocal thermoelasticity on nanoscale beams based on couple stress theory", Math. Method Appl. Sci., Special Issue Paper. https://doi.org/10.1002/mma.6764.
  3. Akbas, S.D., Ersoy, H., Akgoz, B. and Civalek, O. (2021), "Dynamic analysis of a fiber-reinforced composite beam under a moving load by the ritz method", Mathematics, 9, 1048. https://doi.org/10.3390/math9091048.
  4. Akgoz, B., amd Civalek, O . (2013), "A size-dependent shear deformation beam model based on the strain gradient elasticity theory", Int. J. Eng. Sci., 70, 1-14. https://doi.org/10.1016/j.ijengsci.2013.04.004
  5. Akgoz, B. and Civalek, O . (2014), "Longitudinal vibration analysis for microbars based on strain gradient elasticity theory", J. Vib. Control, 20, 606-616. https://doi.org/10.1177%2F1077546312463752. https://doi.org/10.1177%2F1077546312463752
  6. Ali, Z., Khadimallah, M. A., Hussain, M., Asghar, S., Al-Thobiani, F., Elbahar, M., Elimame, E. and Tounsi, A. (2021), "Propagation of waves with nonlocal effects for vibration response of armchair double-walled CNTs", Adv. Nano Res., 11(2), 183-192. https://doi.org/10.12989/anr.2021.11.2.183.
  7. Ansari, R., Oskouie, M.F., Roghani, M. and Rouhi, H. (2021), "Nonlinear analysis of laminated FG-GPLRC beams resting on an elastic foundation based on the two-phase stress-driven nonlocal model", Acta Mechanica, 232, 2183-2199. https://doi.org/10.1007/s00707-021-02935-4.
  8. Aydogdu, M. and Elishakoff, I. (2014), "On the vibration of nanorods restrained by a linear springin-span", Mech. Res. Commun., 57, 90-96. https://doi.org/10.1016/j.mechrescom.2014.03.003.
  9. Bower, C., Rosen, R., Jin, L., Han, J. and Zhou, O. (1999), "Deformation of carbon nanotubes in nanotube polymer composites", Appl. Phys. Lett., 74, 3317-3319. https://doi.org/10.1063/1.123330.
  10. Chang, T.P. (2012), "Small scale effect on axial vibration of nonuniform and nonhomogeneous nanorods", Comput. Mater. Sci., 54, 23-27. https://doi.org/10.1016/j.commatsci.2011.10.033.
  11. Chen, Y., Lee, J.D. and Eskandarian, A. (2004), "Atomistic viewpoint of the applicability of microcontinuum theories", Int. J. Solids Struct., 41, 2085-2097. https://doi.org/10.1016/j.ijsolstr.2003.11.030.
  12. Civalek, O., Uzun, B., Yayli, M.O. and Akgoz, B. (2020a), "Size-dependent transverse and longitudinal vibrations of embedded carbon and silica carbide nanotubes by nonlocal finite element method", Eur. Phys. J. Plus, 135, 381. https://doi.org/10.1140/epjp/s13360-020-00385-w.
  13. Civalek, O., Uzun, B. and Yayli, M.O. (2020b), "Stability analysis of nanobeams placed in electromagnetic field using a finite element method", Arab. J. Geosci., 13, 1-9. https://doi.org/10.1007/s12517-020-06188-8.
  14. Civalek, O., Uzun, B. and Yayli, M.O. (2020c), "Frequency, bending and buckling loads ofnanobeams with different cross sections", Adv. Nano Res., 9(2), 91-104. https://doi.org/10.12989/anr.2020.9.2.091.
  15. Civalek, O., Dastjerdi, S., Akbas, S.D. and Akgoz, B. (2021a), "Vibration analysis of carbonnanotube-reinforced composite microbeams", Math. Method Appl. Sci., Special Issue Paper. https://doi.org/10.1002/mma.7069.
  16. Civalek, O., Uzun, B. and Yayli, M.O. (2021b), "Buckling analysis of nanobeams with deformable boundaries via doublet mechanics", Arch. Appl. Mech., 14-32. https://doi.org/10.1007/s00419-021-02032-x.
  17. Civalek, O ., Uzun, B. and Yayli, M.O . (2022), "An effective analytical method for buckling solutions of a restrained FGM nonlocal beam", Comput. Appl. Math., 41(2), 1-20. https://doi.org/10.1007/s40314-022-01761-1.
  18. Danesh, M., Farajpour, A. and Mohammadi, M. (2012), "Axial vibration analysis of a tapered nanorod based on nonlocal elasticity theory and differential quadrature method", Mech. Res. Commun., 39, 23-27. https://doi.org/10.1016/j.mechrescom.2011.09.004.
  19. Dastjerdi, S. and Beni, Y.T. (2019), "A novel approach for nonlinear bending response of macro-and nanoplates with irregular variable thickness under nonuniform loading in thermal environment", Mech. Based Des. Struct., 453-478. https://doi.org/10.1080/15397734.2018.1557529.
  20. Delfani, M.R. (2017), "Extended theory of elastica for free torsional, longitudinal and radialbreathing vibrations of single-walled carbon nanotubes", J. Sound Vib., 403, 104-128. https://doi.org/10.1016/j.jsv.2017.05.020.
  21. Demir, C. and Civalek, O. (2017), "On the analysis of microbeams", Int. J. Eng. Sci., 121, 14-33. https://doi.org/10.1016/j.ijengsci.2017.08.016.
  22. Ebrahimi, F., Barati, M.R. and Civalek, O. (2020), "Application of ChebyshevRitz method forstatic stability and vibration analysis of nonlocal microstructure-dependent nanostructures", Eng. Comput., 36, 953-964. https://doi.org/10.1007/s00366-019-00742-z.
  23. Ebrahimi, N. and Beni, Y.T. (2016), "Electro-mechanical vibration of nanoshells using consistent size-dependent piezoelectric theory", Steel Compos. Struct., 22(6), 1301-1336. http://doi.org/10.12989/scs.2016.22.6.1301.
  24. Esmaeili, M. and Tadi Beni, Y. (2019), "Vibration and buckling analysis of functionally graded flexoelectric smart beam", J. Appl. Comput. Mech., 5(5), 900-917. http://doi.org/10.22055/JACM.2019.27857.1439.
  25. 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.
  26. Eringen A.C. and Edelen, D.G.B. (1972), "On nonlocal elasticity". Int. J. Eng. Sci., 10, 233-248. https://doi.org/10.1016/0020-7225(72)90039-0.
  27. Eringen, A.C. and Suhubi, E.S. (1964), "Nonlinear theory of simple micro-elastic solids-I", Int. J. Eng. Sci., 2, 189-203. https://doi.org/10.1016/0020-7225(64)90004-7.
  28. Faghidian, S.A. (2020), "Two phase local/nonlocal gradient mechanics of elastic torsion", Math. Method Appl. Sci., Special Issue Paper. https://doi.org/10.1002/mma.6877.
  29. Feng, T., Liu, N., Wang, S., Qin, C., Shi, S., Zeng, X. and Liu, G. (2021), "Research on the dispersion of carbon nanotubes and their application in solution-processed polymeric matrix composites: A review", Adv. Nano Res., 10(6), 559-576. https://doi.org/10.12989/anr.2021.10.6.559.
  30. Forsat, M., Musharavati, F., Eltai, E., Zain, A.M., Mobayen, S. and Mohamed, A.M. (2021), "Vibration characteristics of microplates with GNPs-reinforced epoxy core bonded to piezoelectric-reinforced CNTs patches", Adv. Nano Res., 11(2), 115-140. https://doi.org/10.12989/anr.2021.11.2.115.
  31. Gheshlaghi, B., Hasheminejad, S.M and Abbasion, S. (2010), "Size dependent torsional vibration of nanotubes", Physica E, 43, 45-48. https://doi.org/10.1016/j.physe.2010.06.015.
  32. Gorman, D.J. (1975), Free Vibration Analysis of Beams and Shafts, Wiley, New York, U.S.A.
  33. Guo, S., He, Y., Liu, D., Lei, J., Shen, L. and Li, Z. (2016), "Torsional vibration of carbon nanotube with axial velocity and velocity gradient effect", Int. J. Mech. Sci., 119, 88-96. https://doi.org/10.1016/j.ijmecsci.2016.09.036.
  34. Hadji, L. and Avcar, M. (2021), "Nonlocal free vibration analysis of porous FG nanobeams using hyperbolic shear deformation beam theory", Adv. Nano Res., 10(3), 281-293. https://doi.org/10.12989/anr.2021.10.3.281.
  35. Huang, Z. (2012), "Nonlocal effects of longitudinal vibration in nanorod with internal longrange interactions", Int. J. Solid. Struct., 49, 2150-2154. https://doi.org/10.1016/j.ijsolstr.2012.04.020.
  36. Jalaei, M. and Civalek, O. (2019), "On dynamic instability of magnetically embedded viscoelastic porous FG nanobeam", Int. J. Eng. Sci., 143, 14-32. https://doi.org/10.1016/j.ijengsci.2019.06.013.
  37. Kiani, K. (2013), "Longitudinal, transverse and torsional vibrations and stabilities of axially moving single-walled carbon nanotubes". Curr. Appl. Phys., 13(8), 1651-1660. https://doi.org/10.1016/j.cap.2013.05.008.
  38. Khadimallah, M.A., Hussain, M., Elbahar, M., Ghandourah, E., Elimame, E. and Tounsi, A. (2021), "The effects of ring and fraction laws: Vibration of rotating isotropic cylindrical shell", Adv. Nano Res., 11(1), 19-26. https://doi.org/10.12989/anr.2021.11.1.019.
  39. Khosravi, F., Simyari, M., Hosseini, S.A. and Tounsi, A. (2020), "Size dependent axial free and forced vibration of carbon nanotube via different rod models", Adv. Nano Res., 9(3), 157-172. https://doi.org/10.12989/anr.2020.9.3.157.
  40. Kumar, Y., Gupta, A. and Tounsi, A. (2021), "Size-dependent vibration response of porous graded nanostructure with FEM and nonlocal continuum model", Adv. Nano Res., 11(1), 1-17. https://doi.org/10.12989/anr.2021.11.1.001.
  41. Lam, D.C., Yang, F., Chong, A.C.M., Wang, J. and Tong, P. (2003), "Experiments and theory in strain gradient elasticity", J. Mech. Phys. Solid, 51(8), 1477-1508. https://doi.org/10.1016/S0022-5096(03)00053-X.
  42. Lazar, M. (2021), "Incompatible strain gradient elasticity of Mindlin type: screw and edge dislocations", Acta Mechanica, 232(9), 3471-3494. https://doi.org/10.1007/s00707-021-02999-2
  43. Lei, J., He, Y., Guo, S., Li, Z., Liu, D. (2016), "Size-dependent vibration of nickel cantilever microbeams: experiment and gradient elasticity", AIP Adv., 6(10), 105202. https://doi.org/10.1063/1.4964660.
  44. Li, C. (2014), "Torsional vibration of carbon nanotubes: comparison of two nonlocal models and a semi-continuum model", Int. J. Mech. Sci., 82, 25-31. https://doi.org/10.1016/j.ijmecsci.2014.02.023.
  45. Liebold, C. and Muller, W.H. (2016), "Comparison of gradient elasticity models for the bending of micromaterials", Comput. Mater. Sci., 116, 52-61. https://doi.org/10.1016/j.commatsci.2015.10.031.
  46. Lim, C.W., Li, C. and Yu, J.L. (2012), "Free torsional vibration of nanotubes based on nonlocal stress theory", J. Sound Vib., 331(12), 2798-2808. https://doi.org/10.1016/j.jsv.2012.01.016.
  47. Lim, C.W., Zhang, G. and Reddy, J.N. (2015), "A higher-order nonlocal elasticity and straingradient theory and its applications in wave propagation", J. Mech. Phys. Solid, 78, 298-313. https://doi.org/10.1016/j.jmps.2015.02.001.
  48. Loya, J.A., Aranda-Ruiz, J. and Fernandez-Saez, J. (2014), "Torsion of cracked nanorods using anonlocal elasticity model", J. Phys. D, 47 (3), 115304. https://doi.org/10.1088/0022-3727/47/11/115304.
  49. Luat, D.T., Van Thom, D., Thanh, T.T., Van Minh, P., Van Ke, T. and Van Vinh, P. (2021), "Mechanical analysis of bi-functionally graded sandwich nanobeams", Adv. Nano Res., 11(1), 55-71. https://doi.org/10.12989/anr.2021.11.1.055.
  50. 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.
  51. Madenci, E. (2021), "Free vibration analysis of carbon nanotube RC nanobeams with variational approaches", Adv. Nano Res., 11(2), 157-171. https://doi.org/10.12989/anr.2021.11.2.157.
  52. Mehralian, F. and Beni, Y.T. (2016), "Size-dependent torsional buckling analysis of functionally graded cylindrical shell", Compos. Part B Eng., 94, 11-25. https://doi.org/10.1016/j.compositesb.2016.03.048.
  53. Mehar, K., Panda, S.K. and Patle, B.K. (2017), "Thermoelastic vibration and flexural behavior of FG-CNT reinforced composite curved panel", Int. J. Appl. Mech., 9(4), 1750046. https://doi.org/10.1142/S1758825117500466.
  54. Murmu, T., Adhikari S. and Wang, C. (2011), "Torsional vibration of carbon nanotube buckyball systems based on nonlocal elasticity theory", Physica E, 43, 127680. https://doi.org/10.1016/j.physe.2011.02.017.
  55. Murmu, T., Adhikari, S. and McCarthy, M.A. (2014), "Axial vibration of embedded nanorods undertransverse magnetic field effects via nonlocal elastic continuum theory", J. Comput. Theor. Nanosci., 11, 1230-1236. https://doi.org/10.1166/jctn.2014.3487.
  56. Nejadi, M.M., Mohammadimehr, M. and Mehrabi, M. (2021), "Free vibration and bucklingof functionally graded carbon nanotubes/graphene platelets Timoshenko sandwich beam resting on variable elastic foundation", Adv. Nano Res., 10(6), 539-548. https://doi.org/10.12989/anr.2021.10.6.539.
  57. Numanoglu, H.M, Akgoz, B. and Civalek, O. (2018), "On dynamic analysis of nanorods", Int. J. Eng. Sci., 130, 33-50. https://doi.org/10.1016/j.ijengsci.2018.05.001.
  58. Numanoglu, H. M., Ersoy, H., Akgoz, B. and Civalek, O . (2021), "A new eigenvalue problem solver for thermo-mechanical vibration of Timoshenko nanobeams by an innovative nonlocal finite element method", Math. Method Appl. Sci., 45(5), 2592-2614. https://doi.org/10.1002/mma.7942.
  59. Ozarpa, C., Esen, I. (2020), "Modelling the dynamics of a nanocapillary system with a moving mass using the nonlocal strain gradient theory", Math. Method Appl. Sci., Special Issue Paper. https://doi.org/10.1002/mma.6812.
  60. Park, S.K., Gao, X.L. (2006), "Bernoulli-Euler beam model based on a modified couple stress theory", J. Micromech. Microeng., 16(11), 2355-2359. http://doi.org/10.1088/0960-1317/16/11/015.
  61. Qian, D., Dickey, E.C., Andrews, R. and Rantell, T. (2000), "Load Transfer and deformation mechanisms in carbon nanotube-polystyrene composites", Appl. Phys. Lett., 76, 2868-2870. https://doi.org/10.1063/1.126500.
  62. Ramteke, P.M., Patel, B. and Panda, S.K. (2020), "Time-dependent deflection responses of porous FGM structure including pattern and porosity", Int. J. Appl. Mech., 12(9), 2050102. https://doi.org/10.1142/S1758825120501021.
  63. Ramteke, P.M., Patel, B. and Panda, S.K. (2021a), "Nonlinear eigenfrequency prediction of functionally graded porous structure with different grading patterns", Wave. Random Complex Med., 1-19. https://doi.org/10.1080/17455030.2021.2005850.
  64. Ramteke, P. M., Sharma, N., Choudhary, J., Hissaria, P. and Panda, S. K. (2021b), "Multidirectional grading influence on static/dynamic deflection and stress responses of porous FG panel structure: A micromechanical approach", Eng. Comput., 1-21. https://doi.org/10.1007/s00366-021-01449-w.
  65. Ramteke, P.M., Mehar, K., Sharma, N. and Panda, S.K. (2021c), "Numerical prediction of deflection and stress responses of functionally graded structure for grading patterns (power-law, sigmoid and exponential) and variable porosity (even/uneven)", Scientia Iranica, 28(2), 811-829. https://doi.org/10.24200/sci.2020.55581.4290.
  66. Ramteke, P.M. and Panda, S.K. (2021), "Free vibrational behaviour of multi-directional porous functionally graded structures", Arab. J. Sci. Eng., 46(8), 7741-7756. https://doi.org/10.1007/s13369-021-05461-6.
  67. Ramteke, P.M., Panda, S.K. and Patel, B. (2022), "Nonlinear eigenfrequency characteristics of multi-directional functionally graded porous panels", Compos. Struct., 279, 114707. https://doi.org/10.1016/j.compstruct.2021.114707.
  68. Ramezani, S., Naghdabadi, R. and Sohrabpour, S. (2009), "Analysis of micropolar elastic beams", Eur. J. Mech. A Solids, 28(2), 202-208. https://doi.org/10.1016/j.euromechsol.2008.06.006.
  69. Reddy, J.N. and Pang, S.D. (2008), "Nonlocal continuum theories of beams for the analysis of carbon nanotubes", J. Appl. Phys., 103, 023511-023526. https://doi.org/10.1063/1.2833431.
  70. Roostai, H. and Haghpanahi, M. (2014), "Vibration of nanobeams of different boundary conditions with multiple cracks based on nonlocal elasticity theory", Appl. Math. Modell., 38(3), 1159-1169. https://doi.org/10.1016/j.apm.2013.08.011.
  71. Ru, C.Q. (2001), "Axially compressed buckling of a double walled carbon nanotube embedded in an elastic medium", J. Mech. Phys. Solids, 49, 1265-1279. https://doi.org/10.1016/S0022-5096(00)00079-X.
  72. Sarparast, H., Ebrahimi Mamaghani, A., Safarpour, M., Ouakad, H.M., Dimitri, R. and Tornabene, F. (2020), "Nonlocal study of the vibration and stability response of smallscale axially moving supported beams on viscoelastic Pasternak foundation in a hygrothermal environment", Math. Method Appl. Sci., Special Issue Paper. https://doi.org/10.1002/mma.6859.
  73. Schadler, L.S., Giannaris S.C. and Ajayan P.M. (1998), "Load transfer in carbon nanotube epoxy composites", Appl. Phys. Lett., 73, 3842-3844. https://doi.org/10.1063/1.122911.
  74. Simsek, M. (2012), "Nonlocal effects in the free longitudinal vibration of axially functionally graded tapered nanorods", Comput. Mater. Sci., 61, 257-265. https://doi.org/10.1016/j.commatsci.2012.04.001.
  75. Swadener, J.G., George E.P. and Pharr G.M. (2002), "The correlation of the indentation size effect measured with indenters of various shapes", J. Mech. Phys. Solid, 50(4), 681-694. https://doi.org/10.1016/S0022-5096(01)00103-X.
  76. Tabassian, R. (2013), "Torsional vibration analysis of shafts based on Adomian decomposition method", Appl. Comput. Mech., 7(2), 205-222.
  77. Tadi Beni, Y. (2016), "Size-dependent electromechanical bending, buckling and free vibration analysis of functionally graded piezoelectric nanobeams", J. Intell. Mater. Syst. Struct., 27(16), 2199-2215. https://doi.org/10.1177/1045389X15624798.
  78. Toupin, R.A. (1962), "Elastic materials with couple-stresses", Arch. Ration. Mech. An., 11, 385-414. https://doi.org/10.1007/BF00253945.
  79. Uzun, B. and Yayli, M.O. (2020), "Nonlocal vibration analysis of Ti-6Al-4V/ZrO2 functionallygraded nanobeam on elastic matrix", Arab. J. Geosci., 13(4), 1-10. https://doi.org/10.1007/s12517-020-5168-4.
  80. Uzun, B., Kafkas, U. and Yayli, M.O. (2021), "Axial dynamic analysis of a Bishop nanorod with arbitrary boundary conditions", ZAMM J. Appl. Math. Mech., 100, 12. https://doi.org/10.21923/jesd.719059.
  81. Wagner, H.D., Lourie, O., Feldman, Y. and Tenne, R. (1998), "Stress-induced fragmentation of multiwall carbon nanotubes in a polymer matrix", Appl. Phys. Lett., 72, 188-190. https://doi.org/10.1063/1.120680.
  82. Wang, L., Ni, Q., Li, M. and Qian, Q. (2008), "The thermal effect on vibration and instability of carbon nanotubes conveying fluid", Physica E, 40, 3179-3182. https://doi.org/10.1016/j.physe.2008.05.009.
  83. Yang, J., Ke, L.L. and Kitipornchai, S. (2010), "Nonlinear free vibration of single-walled carbonnanotubes using nonlocal Timoshenko beam theory", Physica E, 42, 1727-1735. https://doi.org/10.1016/j.physe.2010.01.035.
  84. Yayli, M.O. (2011), "Stability analysis of a gradient elastic beam using finite element method", Int. J. Physical Science, 6(12), 2844-2851. https://doi.org/10.5897/IJPS11.361.
  85. Yayli, M.O. (2016), "A compact analytical method for vibration analysis of single-walledcarbon nanotubes with restrained boundary conditions", J. Vib. Control, 22(10), 2542-2555. https://doi.org/10.1177%2F1077546314549203. https://doi.org/10.1177%2F1077546314549203
  86. Yayli, M.O. (2017), "A compact analytical method for vibration of micro-sized beams with different boundary conditions", Mech. Adv. Mater. Struct., 24(6), 496508. https://doi.org/10.1080/15376494.2016.1143989.
  87. Yayli, M.O., Uzun, B., Deliktas, B. (2021), "Buckling analysis of restrained nanobeams using strain gradient elasticity", Wav. Random Complex Med., 1-20. https://doi.org/10.1080/17455030.2020.1871112.
  88. Zeverdejani, M.K. and Beni, Y.T. (2020), "Effect of laminate configuration on the free vibration/buckling of FG Graphene/PMMA composites", Adv. Nano Res., 8(2), 103-114. https://doi.org/10.12989/anr.2020.8.2.103.