Steel and Composite Structures

  • 제47권6호
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  • Pages.795-811
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  • 2023
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  • 1229-9367(pISSN)
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  • 1598-6233(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Nonlinear resonances of nonlocal strain gradient nanoplates made of functionally graded materials considering geometric imperfection

  • Jia-Qin Xu (College of Mechanical and Vehicle Engineering, Chongqing University) ;
  • Gui-Lin She (College of Mechanical and Vehicle Engineering, Chongqing University) ;
  • Yin-Ping Li (College of Mechanical and Vehicle Engineering, Chongqing University) ;
  • Lei-Lei Gan (College of Mechanical and Vehicle Engineering, Chongqing University)
  • 투고 : 2023.01.09
  • 심사 : 2023.03.20
  • 발행 : 2023.06.25
⟨ 이전 논문 다음 논문 ⟩

초록

When studying the resonance problem of nanoplates, the existing papers do not consider the influences of geometric nonlinearity and initial geometric imperfection, so this paper is to fill this gap. In this paper, based on the nonlocal strain gradient theory (NSGT), the nonlinear resonances of functionally graded (FG) nanoplates with initial geometric imperfection under different boundary conditions are established. In order to consider the small size effect of plates, nonlocal parameters and strain gradient parameters are introduced to expand the assumptions of the first-order shear deformation theory. Subsequently, the equations of motion are derived using the Euler-Lagrange principle and solved with the help of perturbation method. In addition, the effects of initial geometrical imperfection, functionally graded index, strain gradient parameter, nonlocal parameter and porosity on the nonlinear forced vibration behavior of nanoplates under different boundary conditions are discussed.

키워드

과제정보

The authors acknowledge this work is supported by the talent introduction project of Chongqing University (02090011044159), and Fundamental Research Funds for the Central Universities (2022CDJXY-005).

참고문헌

  1. Ahmed, R.A., Khalaf, B.S., Raheef, K.M., Fenjan, R.M., Faleh, N.M. (2021), "Investigating dynamic response of nonlocal functionally graded porous piezoelectric plates in thermal environment", Steel Compos. Struct., 40(2), 243-254. https://doi.org/10.12989/scs.2021.40.2.243.
  2. Alazwari, M.A., Daikh, A.A., Houari, M.S., Tounsi, A. and Eltaher, M.A. (2021), "On static buckling of multilayered carbon nanotubes reinforced composite nanobeams supported on non-linear elastic foundations", Steel Compos. Struct., 40(3), 389-404. https://doi.org/10.12989/scs.2021.40.3.389.
  3. Allahyari, E., Asgari, M. and Jafari, A.A. (2020), "Nonlinear size-dependent vibration behavior of graphene nanoplate considering surfaces effects using a multiple-scale technique", Mech. Adv. Mater. Struct., 27(9), 697-706. https://doi.org/10.1080/15376494.2018.1494870.
  4. Arani, A.G. and Zamani, M.H. (2018), "Nonlocal free vibration analysis of FG-porous shear and normal deformable sandwich nanoplate with piezoelectric face sheets resting on silica aerogel foundation", Arab. J. Sci. Eng., 43(9), 4675-4688. https://doi.org/10.1007/s13369-017-3035-8
  5. Arefi, M., Kiani, M. and Zamani, M.H. (2020), "Nonlocal strain gradient theory for the magneto-electro-elastic vibration response of a porous FG-core sandwich nanoplate with piezomagnetic face sheets resting on an elastic foundation", J. Sandw. Struct. Mater., 22(7), 2157-2185. https://doi.org/10.1177/1099636218795378.
  6. Arefi, M. and Zenkour, A.M. (2017), "Size-dependent free vibration and dynamic analyses of piezo-electro-magnetic sandwich nanoplates resting on viscoelastic foundation", Physica B-Condensed Matter., 521, 188-197. https://doi.org/10.1016/j.physb.2017.06.066.
  7. Assie, A.E., Mohamed, S.A., Shanab, R.A., Abo-bakr, R.M. and Eltaher, M.A. (2023), "Static buckling of 2D FG porous plates resting on elastic foundation based on unified shear theories", J. Appl. Comput. Mech., 9(1), 239-258. https://doi.org/10.22055/jacm.2022.41265.3723.
  8. Bai, Y.H., Liu, R.M. and Wang, L.F. (2021), "Nonlinear thermal vibration of a nanoplate attached to a cavity", Mater. Res. Express, 8(11), 115009. https://doi.org/10.1088/2053-1591/ac36fc.
  9. Babaei, H. (2022a), "Nonlinear analysis of size‑dependent frequencies in porous FG curved nanotubes based on nonlocal strain gradient theory", Eng. Struct., 38(Suppl3), S1717-S1734. https://doi.org/10.1007/s00366-021-01317-7.
  10. Babaei, H. (2022b), "Free vibration and snap-through instability of FG-CNTRC shallow arches supported on nonlinear elastic foundation", Appl. Math. Comput., 413, 126606. https://doi.org/10.1016/j.amc.2021.126606.
  11. Barati, M.R. (2017), "Nonlocal-strain gradient forced vibration analysis of metal foam nanoplates with uniform and graded porosities", Adv. Nano Res., 5(4), 393-414 https://doi.org/10.12989/anr.2017.5.4.393.
  12. Barati, M.R. (2018), "A general nonlocal stress-strain gradient theory for forced vibration analysis of heterogeneous porous nanoplates", Europ. J. Mech. A-Solids, 67, 215-230. https://doi.org/10.1016/j.euromechsol.2017.09.001.
  13. Basha, M., Daikh, A.A., Melaibari, A., Wagih, A., Othman, R., Almitani, K.H., Hamed, M.A., Abdelrahman, A. and Eltaher, M.A. (2022), "Nonlocal strain gradient theory for buckling and bending of FG-GRNC laminated sandwich plates", Steel Compos. Struct., 43(5), 639-660. https://doi.org/10.12989/scs.2022.43.5.639.
  14. Chen, T., Chen, H.X. and Liu, L.M. (2020), "Vibration energy flow analysis of periodic nanoplate structures under thermal load using fourth-order strain gradient theory", Acta Mechanica, 231(10), 4365-4379. https://doi.org/10.1007/s00707-020-02765-w.
  15. Chen, X., Zhao, J.L., She, G.L., Jing, Y., Luo, J. and Pu, H.Y. (2022a), "On wave propagation of functionally graded CNT strengthened fluid-conveying pipe in thermal environment", Eur. Phys. J. Plus., 137(10), 1158. https://doi.org/10.1140/epjp/s13360-022-03234-0.
  16. Chen, X., Zhao, J.L., She, G.L., Jing, Y., Pu, H.Y. and Luo, J. (2022b), "Nonlinear free vibration analysis of functionally graded carbon nanotube reinforced fluid-conveying pipe in thermal environment", Steel Compos. Struct., 45(5), 641-652. https://doi.org/10.12989/scs.2022.45.5.641.
  17. Cuong-Le, T., Nguyen, K.D.D., Hoang-Le, M., Sang-To, T., PhanVu, P. and Magd, A.W. (2022), "Nonlocal strain gradient IGA numerical solution for static bending, free vibration and buckling of sigmoid FG sandwich nanoplate", Physica B-Condensed Matter., 631, 413726. ttps://doi.org/10.1016/j.physb.2022.41372.
  18. Dangi, C. and Lal, R. (2022), "Nonlinear thermal effect on free vibration of FG rectangular mindlin nanoplate of bilinearly varying thickness via Eringen's nonlocal theory", J. Vib. Eng. Technol., https://doi.org/10.1007/s42417-022-00531-x.
  19. Ding, H.X. and She, G.L. (2021), "A higher-order beam model for the snap-buckling analysis of FG pipes conveying fluid", Struct. Eng. Mech., 80(1), 63-72. http://dx.doi.org/10.12989/sem.2021.80.1.063.
  20. Ding, H.X., She, G.L. and Zhang, Y.W. (2022a), "Nonlinear buckling and resonances of functionally graded fluid-conveying pipes with initial geometric imperfection", Eur. Phys. J. Plus, 137, 1329. https://doi.org/10.1140/epjp/s13360-022-03570-1.
  21. Ding, H.X., Zhang, Y.W. and She, G.L. (2022b), "On the resonance problems in FG-GPLRC beams with different boundary conditions resting on elastic foundations", Comput. Concrete, 30(6),433-443. https://doi.org/10.12989/cac.2022.30.6.433.
  22. Ding, H.X. and She, G.L. (2023), "Nonlinear resonance of axially moving graphene platelet reinforced metal foam cylindrical shells with geometric imperfection", Archiv. Civil Mech. Eng., http://dx.doi.org/10.1007/s43452-023-00634-6
  23. Ebrahimi, F. and Barati, M.R. (2017), "Damping vibration analysis of smart piezoelectric polymeric nanoplates on viscoelastic substrate based on nonlocal strain gradient theory", Smart Mater. Struct., 26(6), 065018. https://doi.org/10.1088/1361-665X/aa6eec.
  24. Ebrahimi, F. and Hosseini, S.H. (2020), "Double harmonically excited nonlinear vibration of viscoelastic piezoelectric nanoplates subjected to thermo-electro-mechanical forces", J. Vib. Control, 26, 430-446. https://doi.org/10.1177/1077546319889785.
  25. Emam, S.A., Eltaher, M.A., Khater, M.E. and Abdalla, W.S. (2018), "Postbuckling and free vibration of multilayer imperfect nanobeams under a pre-stress load", Appl. Sci., 8(11), 2238. https://doi.org/10.3390/app8112238.
  26. Emadi, M., Nejad, M.Z., Ziaee, S. and Hadi, A. (2021), "Buckling analysis of arbitrary two-directional functionally graded nanoplate based on nonlocal elasticity theory using generalized differential quadrature method", Steel Compos. Struct., 39(5) 565-581. https://doi.org/10.12989/scs.2021.39.5.565.
  27. Eskandari, S.M., Shariati, M., Asiaban, N. and Eskandari, J.J. (2021), "Bending, buckling and vibrations analysis of the graphene nanoplate using the modified couple stress theory", Mechanika, 27(5), 376-384. https://doi.org/10.5755/j02.mech.25299.
  28. Gan, L.L. and She, G.L. (2023), "Nonlinear snap-buckling and resonance of FG-GPLRC curved beams with different boundary conditions", Geomech. Eng., 32(5), 541-551. https://doi.org/10.12989/gae.2023.32.5.541.
  29. Gan, L.L., Xu, J.Q. and She, G.L. (2023), "Wave propagation of graphene platelets reinforced metal foams circular plates", Struct. Eng. Mech., 85(5), 645-654. https://doi.org/10.12989/sem.2023.85.5.645.
  30. Hendi, A., Eltaher, M.A, Mohamed, S.A. and Attia, M. (2022), "Nonlinear thermal vibration of pre/post-buckled two-dimensional FGM tapered microbeams based on a higher order shear deformation theory", Steel Compos. Struct., 41(6), 787-802. http://doi.org/DOI10.12989/scs.2021.41.6.787.
  31. Ke, L.L., Liu, C. and Wang, Y.S. (2015), "Free vibration of nonlocal piezoelectric nanoplates under various boundary conditions", Physica E-Low-Dimensional Syst. Nanostruct., 66, 93-106.https://doi.org/10.1016/j.physe.2014.10.002.
  32. Li, C., Liu, J.J., Cheng, M. and Fan, X.L. (2017), "Nonlocal vibrations and stabilities in parametric resonance of axially moving viscoelastic piezoelectric nanoplate subjected to thermo-electro-mechanical forces", Compos. Part B-Eng., 116, 153-169. https://doi.org/10.1016/j.compositesb.2017.01.071.
  33. Li, C.L., Tian, X.G. and He, T.H. (2021), "Nonlocal thermoviscoelasticity and its application in size-dependent responses of bi-layered composite viscoelastic nanoplate under nonuniform temperature for vibration control", Mech. Adv. Mater. Struct., 28 (17), 1797-1811. https://doi.org/10.1080/15376494.2019.1709674.
  34. Li, M.G., Zhang, Q.Y., Wang, B.B. and Zhao, M.H. (2021), "Analysis of flexural vibrations of a piezoelectric semiconductor nanoplate driven by a time-harmonic force", Materials, 14(14), 3926. https://doi.org/10.3390/ma14143926.
  35. Li, Y.P., She, G.L., Gan, L.L. and Liu, H.B. (2023), "Nonlinear thermal post-buckling analysis of graphene platelets reinforced metal foams plates with initial geometrical imperfection", Steel. Compos, Struct., 46(5) 649-658. https://doi.org/10.12989/scs.2023.46.5.649.
  36. Lu, L., She, G.L. and Guo, X. (2021), "Size-dependent postbuckling analysis of graphene reinforced composite microtubes with geometrical imperfection", Int. J. Mech. Sci., 199, 106428. https://doi.org/10.1016/j.ijmecsci.2021.
  37. Mahmoudpour, E., Hosseini-Hashemi, S.H. and Faghidian, S.A. (2019), "Nonlinear resonant behaviors of embedded thick FG double layered nanoplates via nonlocal strain gradient theory", Microsyst. Technologies-Micro Nanosystems-Inform. Storage Processing Syst., 25(3) 951-964. https://doi.org/10.1007/s00542-018-4198-2.
  38. Melaibari, A., Mohamed, S.A., Assie, A.E., Shanab, R.A. and Eltaher, M.A. (2023), "Static response of 2D FG porous plates resting on elastic foundation using midplane and neutral surfaces with movable constraints", Mathematics, 10(24), 4784. http://dx.doi.org/10.3390/math10244784.
  39. Mohamed, N., Mohamed, S.A. and Eltaher, M. A. (2021), "Buckling and post-buckling behaviors of higher order carbon nanotubes using energy-equivalent model", Eng. Comput., 37(4), 2823-2836. http://dx.doi.org/10.1007/s00366-020-00976-2.
  40. Mohamed, N., Eltaher, M.A., Mohamed, S.A. and Seddek, L.F. (2019), "Energy equivalent model in analysis of postbuckling of imperfect carbon nanotubes resting on nonlinear elastic foundation", Struct. Eng. Mech., 70(6), 737-750. http://dx.doi.org/10.12989/sem.2019.70.6.737.
  41. Selim, M.M. and El-Safty, S.A. (2021), "Vibration analysis of nanoplate with the effects of surface irregularity and initial stresses", J. Nanoelectron. Optoelectronic., 16(1),48-53. https://doi.org/10.1166/jno.2021.2901.
  42. She, G.L. (2020), "Wave propagation of FG polymer composite nanoplates reinforced with GNPs", Steel Compos. Struct., 37 (1), 27-35. https://doi.org/10.12989/scs.2020.37.1.027.
  43. She, G.L. (2021), "Guided wave propagation of porous functionally graded plates: The effect of thermal loadings", J. Thermal Stresses, 44(10), 1289-1305. https://doi.org/10.1080/01495739.2021.1974323.
  44. She, G.L. and Ding, H.X. (2023), "Nonlinear primary resonance analysis of initially stressed graphene platelet reinforced metal foams doubly curved shells with geometric imperfection", Acta Mech. Sin., 39, 522392. https://doi.org/10.1007/s10409-022-22392-x.
  45. She, G.L., Ding, H.X. and Zhang, Y.W. (2022), "Wave propagation in a FG circular plate via the physical neutral surface concept", Struct. Eng. Mech., 82(2), 225-232. https://doi.org/10.12989/sem.2022.82.2.225.
  46. She, G.L. and Li, Y.P. (2022), "Wave propagation in an FG circular plate in thermal environment", Geomech. Eng., 31(6), 615-622. https://doi.org/10.12989/gae.2022.31.6.615.
  47. She, G.L., Liu, H.B. and Karami, B. (2021), "Resonance analysis of composite curved microbeams reinforced with graphene nanoplatelets", Thin Wall. Struct., 160, 107407. https://doi.org/10.1016/j.tws.2020.107407.
  48. Singh, P.P. and Azam, M.S. (2021), "Free vibration and buckling analysis of elastically supported transversely inhomogeneous functionally graded nanoplate in thermal environment using Rayleigh-Ritz method", J. Vib. Control, 27, 23-24, 2835-2847. 1077546320966932. https://doi.org/10.1177/1077546320966932.
  49. Singh, P.P. and Azam, M.S. (2021), "Size dependent vibration of embedded functionally graded nanoplate in hygrothermal environment by Rayleigh-Ritz method", Adv. Nano Res., 10 (1), 25-42. https://doi.org/10.12989/anr.2021.10.1.025.
  50. Tran, M.T. and Cuong-Le, T. (2022), "A nonlocal IGA numerical solution for free vibration and buckling analysis of Porous Sigmoid Functionally Graded (P-SFGM) nanoplate", Int. J. Struct. Stab. Dyn., 22(16). https://doi.org/10.1142/S0219455422501930.
  51. Wang, Y., Li, F.M., Wang, Y.Z. and Jing, X.J. (2017), "Nonlinear responses and stability analysis of viscoelastic nanoplate resting on elastic matrix under 3:1 internal resonances", Int. J. Mech. Sci., 128, 94-104. https://doi.org/10.1016/j.ijmecsci.2017.04.010.
  52. Xu, X.L., Zhang, C.W., Musharavati, F., Sebaey, T.A. and Khan, A. (2021), "Dispersion of waves characteristics of laminated composite nanoplate", Steel Compos. Struct., 40(3) 355-367. https://doi.org/10.12989/scs.2021.40.3.355.
  53. Arani, A.G., Arani, A.H. and Haghparast, E. (2021), "Flexoelectric and surface effects on vibration frequencies of annular nanoplate", Indian J. Phys., 95(10), 2063-2083. https://doi.org/10.1007/s12648-020-01854-9.
  54. Barati, M.R. (2018), "A general nonlocal stress-strain gradient theory for forced vibration analysis of heterogeneous porous nanoplates", Europ. J. Mech. A-Solids, 67, 215-230. https://doi.org/10.1016/j.euromechsol.2017.09.001.
  55. Barati, M.R. and Shahverdi, H. (2018), "Forced vibration of porous functionally graded nanoplates under uniform dynamic load using general nonlocal stress-strain gradient theory", J. Vib. Control, 24(20), 4700-4715. https://doi.org/10.1177/1077546317733832.
  56. Ebrahimi, F. and Hosseini, S.H. (2020), "Investigation of flexoelectric effect on nonlinear forced vibration of piezoelectric/functionally graded porous nanocomposite resting on viscoelastic foundation", J. Strain Anal. Eng. Des., 55(1-2), 53-68. https://doi.org/10.1177/0309324719890868.
  57. Eremeyev, V.A. (2022), "Advances in micro- and nanomechanics", Nanomaterials, 12(24). https://doi.org/10.3390/nano12244433.
  58. Fard, K.M., Kavanroodi, M.K., Malek-Mohammadi, H. and Pourmoayed, A.R. (2022), "Buckling and vibration analysis of a double-layer graphene sheet coupled with a piezoelectric nanoplate", J. Appl. Comput. Mech., 8(1), 129-143. https://doi.org/10.22055/JACM.2020.32145.1976.
  59. Fattahi, A.M., Safaei, B. and Moaddab, E. (2019), "The application of nonlocal elasticity to determine vibrational behavior of FG nanoplates", Steel Compos. Struct., 32(2), 281-292. https://doi.org/10.12989/scs.2019.32.2.281.
  60. Fazeli, H., Adamian, A. and Hosseini-Sianaki, A. (2021), "Influence of initial geometric imperfection on static and free vibration analyses of porous FG nanoplate using an isogeometric approach", J. Brazil. Soc. Mech. Sci. Eng., 43(4), 243-254. https://doi.org/10.12989/scs.2021.40.2.243.
  61. Ghadiri, M. and Hosseini, S.H. (2021), "Nonlinear forced vibration of graphene/piezoelectric sandwich nanoplates subjected to a mechanical shock", J. Sandw. Struct. Mater., 23(3), 956-987. https://doi.org/10.1177/1099636219849647.
  62. Jafari, E., Fakoor, M. and Karvand, E. (2019), "Hygrothermal free vibration of multiple magneto-electro-elastic nanoplate system via higher-order nonlocal strain gradient theory", Appl. Phys. A-Mater. Sci. Processing, 125(9), 607.https://doi.org/10.1007/s00339-019-2855-7,
  63. Karami, B., Shahsavari, D., Janghorban, M. and Tounsi, A. (2019), "Resonance behavior of functionally graded polymer composite nanoplates reinforced with graphene nanoplatelets", Int. J. Mech. Sci., 156, 94-105. https://doi.org/10.1016/j.ijmecsci.2019.03.036.
  64. Mahmoudpour, E. (2020), "Nonlinear resonant behavior of thick multilayered nanoplates via nonlocal strain gradient elasticity theory", Acta Mechanica, 231(6), 2651-2667. https://doi.org/10.1007/s00707-020-02636-4.
  65. Mazur, O. and Awrejcewicz, J. (2022), "The nonlocal elasticity theory for geometrically nonlinear vibrations of double-layer nanoplate systems in magnetic field", Meccanica, 57(11), 2835-2847. https://doi.org/10.1007/s11012-022-01602-9.
  66. Mohammadi, M. and Rastgoo, A. (2019), "Nonlinear vibration analysis of the viscoelastic composite nanoplate with three directionally imperfect porous FG core", Struct. Eng. Mech., 69(2), 131-143. https://doi.org/10.12989/sem.2019.69.2.131.
  67. Mohammadi, M. and Rastgoo, A. (2020), "Primary and secondary resonance analysis of FG/lipid nanoplate with considering porosity distribution based on a nonlinear elastic medium", Mech. Adv. Mater. Struct., 27(20), 1709-1730. https://doi.org/10.1080/15376494.2018.1525453.
  68. Sadoughifar, A., Farhatnia, F., Izadinia, M. and Talaeitaba, S.B. (2019), "Nonlinear bending analysis of porous FG thick annular/circular nanoplate based on modified couple stress and two-variable shear deformation theory using GDQM", Steel Compos. Struct., 33(2), 307-318. https://doi.org/10.12989/scs.2019.33.2.307.
  69. Sahmani, S., Fattahi, A.M. and Ahmed, N.A. (2020), "Analytical treatment on the nonlocal strain gradient vibrational response of postbuckled functionally graded porous micro-/nanoplates reinforced with GPL", Eng. Comput., 36(4), 1559-1578. https://doi.org/10.1007/s00366-019-00782-5.
  70. Selim, M.M., Gepreel, K.A., Mohammed, I.M.O. and Hussin, A.M. (2022), "Vibration of initially stressed nonlocal irregular nanoplate using wave propagation approach", Waves Random Complex Media. https://doi.org/10.1080/17455030.2022.2131936.
  71. Xu, J.Q. and She, G.L. (2022), "Thermal post-buckling analysis of porous functionally graded pipes with initial geometric imperfection", Geomech. Eng., 31(3), 329-337. https://doi.org/10.12989/gae.2022.31.3.329.
  72. Zhang, Y.W., Ding, H.X. and She, G.L. (2022), "Snap-buckling and resonance of functionally graded graphene reinforced composites curved beams resting on elastic foundations in thermal environment", J. Therm. Stresses, 45(12), 1029-1042. https://doi.org/10.1080/01495739.2022.2125137.
  73. Zhang, Y.W., Ding, H.X. and She, G.L. (2023a), "Wave propagation in spherical and cylindrical panels reinforced with carbon nanotubes", Steel Compos. Struct., 46(1), 133-141. https://doi.org/10.12989/scs.2023.46.1.133.
  74. Zhang, Y.W., She, G.L. and Ding, H.X. (2023b), "Nonlinear resonance of graphene platelets reinforced metal foams plates under axial motion with geometric imperfections", Eur. J. Mech. A-Solid., 98, 104887. https://doi.org/10.1016/j.euromechsol.2022.104887.
  75. Zhang, Y.W. and She, G.L. (2022), "Wave propagation and vibration of FG pipes conveying hot fluid", Steel. Compos, Struct., 42(3) 397-405. https://doi.org/10.12989/scs.2022.42.3.397.
  76. Zhang, Y.W. and She, G.L. (2023a), "Nonlinear low-velocity impact response of graphene platelet-reinforced metal foam cylindrical shells under axial motion with geometrical imperfection", Nonlinear Dyn., https://doi.org/10.1007/s11071-022-08186-9.
  77. Zhang, Y.W. and She, G.L. (2023b), "Nonlinear primary resonance of axially moving functionally graded cylindrical shells in thermal environment", Mech. Adv. Mater. Struct., https://doi.org/10.1080/15376494.2023.2180556.
  78. Zhang, Y.W., She, G.L., Gan, L.L. and Li, Y.P. (2023c), "Thermal post-buckling behavior of GPLRMF cylindrical shells with initial geometrical imperfection", Geomech. Eng., 32(6), 615-625. https://doi.org/10.12989/gae.2023.32.6.615.
  79. Zhang, Y.Y., Wang, X.Y., Zhang, X., Shen, H.M. and She, G.L. (2021), "On snap-buckling of FG-CNTRC curved nanobeams considering surface effects", Steel Compos. Struct., 38(3), 293-304. https://doi.org/10.12989/scs.2021.38.3.293.
  80. Zhao, J.L., Chen, X., She, G.L., Jing, Y., Bai, R.Q., Yi, J., Pu, H.Y. and Luo, J. (2022a), "Vibration characteristics of functionally graded carbon nanotube-reinforced composite double-beams in thermal environments", Steel. Compos. Struct., 43(6), 797-808. https://doi.org/10.12989/scs.2022.43.6.797.
  81. Zhao, J.L., She, G.L., Wu, F., Yuan, S.J., Bai, R.Q., Pu, H.Y., Wang, S.L. and Luo, J. (2022b), "Guided waves of porous FG nanoplates with four edges clamped", Adv. Nano. Res., 13(5), 465-474. https://10.12989/anr.2022.13.5.465
  82. Zhou, L., Wang, J., Li, X., Liu, C., Liu, P., Ren, S. and Li, M. (2021), "The magneto-electro-elastic multi-physics coupling element free Galerkin method for smart structures in statics and dynamics problems", Thin. Wall. Struct., 169, 108431. https://doi.org/10.1016/j.tws.2021.108431.
  83. Zhou, L., Wang, J., Liu, M., Li, M. and Chai, Y. (2022), "Evaluation of the transient performance of magneto-electro-elastic based structures with the enriched finite element method", Compos. Struct., 280, 114888. https://doi.org/10.1016/j.compstruct.2021.114888.
  84. Zhu, Y. and Chang, T.H. (2015), "A review of microelectromechanical systems for nanoscale mechanical characterization", J. Micromech. Microeng., 25(9), 093001. https://doi.org/10.1088/0960-1317/25/9/093001.