Advances in nano research

  • 제16권1호
  • /
  • Pages.27-40
  • /
  • 2024
  • /
  • 2287-237X(pISSN)
  • /
  • 2287-2388(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Nonlocal Mindlin plate theory with the application for vibration and bending analysis of nanoplates with the flexoelectricity effect

  • Pham Ba Khien (HUTECH institute of Engineering, HUTECH University) ;
  • Du Dinh Nguyen (Faculty of Civil Engineering, Lac Hong University) ;
  • Abdelouahed Tounsi (Department of Civil and Environmental Engineering, King Fahd University of Petroleum & Minerals) ;
  • Bui Van Tuyen (Faculty of Mechanical Engineering, Thuyloi University)
  • 투고 : 2022.12.22
  • 심사 : 2023.08.28
  • 발행 : 2024.01.25
⟨ 이전 논문 다음 논문 ⟩

초록

This work is the first of its kind to integrate Mindlin's theory with analytical methods in order to produce an exact solution to a specific vibration issue as well as a bending problem involving a nanoplate that is supported by a viscoelastic foundation. The plate is exposed to the simultaneous effects of a compressive load in the plate plane and a force operating perpendicular to the plane of the nanoplate. In addition, the flexoelecity effect is included into the plate. The strain gradient component is taken into consideration while calculating the plate equilibrium equation using the nonlocal theory and Hamilton's principle. The free vibration and static responses of the nanoplate seem to be both real and imaginary components because of the appearance of the viscoelastic drag coefficient of the viscoelastic foundation. This study also shows that when analyzing the mechanical response for nanostructure, taking into account the flexoelectricity effect and the influence of the nonlocal parameter, the results will be completely different from the case in which this parameter is ignored. This indicates that it is vital to take into consideration the effects of nonlocal parameters on the nanosheet structure while also taking into consideration the effect of flexoelectricity.

키워드

참고문헌

  1. Abdelaziz, T. (2021), "A cylindrical shell model for nonlocal buckling behavior of CNTs embedded in an elastic foundation under the simultaneous effects of magnetic field, temperature change, and number of walls", Adv. Nano. Res., 11(6), 581-593. https://doi.org/10.12989/anr.2021.11.6.581
  2. Aissa, K., Rabbab, B.B., Attia, B., Mohamed, S., Samir, B., Mahmoud, M.S.S., Tounsi, A. and Muzamal, H. (2022), "Study on the Mechanical Instability of Bidirectional Imperfect FG Sandwich Plates Subjected to In-Plane Loading", Arab. J. Sci. Eng., 47, 13655-13672. https://doi.org/10.1007/s13369-022-07203-8
  3. 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.
  4. Ahmed-Amine, D., Mohamed-Ouejdi, B., Drai, A., Mohamed, S. A.H., Mehmet, A., Tounsi, A., Mohamed, A.E. (2022), "Static analysis of functionally graded plate structures resting on variable elastic foundation under various boundary conditions", Acta Mech., 2022. https://doi.org/10.1007/s00707-022-03405-1
  5. Akbas, S.D. (2016), "Forced vibration analysis of viscoelastic nanobeams embedded in an elastic medium", Smart Struct. Syst., 18(6), 1125-1143. https://doi.org/10.12989/sss.2016.18.6.1125.
  6. Akbas, S.D. (2017), "Forced vibration analysis of functionally graded nanobeams", Int. J. Appl. Mech., 9(7), 1750100. https://doi.org/10.1142/S1758825117501009
  7. Akbas, S.D. (2018), "Forced vibration analysis of cracked nanobeams", J. Brazilian Soc. Mech. Sci. Eng., 40, 392. https://doi.org/10.1007/s40430-018-1315-1.
  8. Baferani, A.H., Saidi, A.R. and Ehteshami, H. (2011) "Accurate solution for free vibration analysis of functionally graded thick rectangular plates resting on elastic foundation", Compos. Struct., 93(7), 1842-1853. https://doi.org/10.1016/j.compstruct.2011.01.020
  9. Bui, V.T. (2022), "Free vibration behaviors of nanoplates resting on viscoelastic medium", Ara. J. Scie. Eng., 2022. https://doi.org/10.1007/s13369-022-07500-2
  10. Bui, V.T. and Gia, T.L. (2023), "Static Buckling Analysis of FG Sandwich Nanobeams", J. Vib. Eng. Technol., 2023. https://doi.org/10.1007/s42417-023-01081-6
  11. Cuong-Le, T., Khuong, D.N., Jaehong, L., Timon, R. and Hung, N.X. (2021), "A 3D nano scale IGA for free vibration and buckling analyses of multi-directional FGM nanoshells", Nanot., 33, 065703. https://doi.org/10.1088/1361-6528/ac32f9
  12. Demir, C. and Oz, F.E. (2014), "Free vibration analysis of a functionally graded viscoelastic supported beam", J. Vib. Control, 20(16), 2464-2486. https://doi.org/10.1177/1077546313479634.
  13. Deng, Q., Kammoun, M., Erturk, A. and Sharma, P. (2014), "Nanoscale flexoelectric energy harvesting", Int. J. Solids Struct., 51(18), 3218-3225. https://doi.org/10.1016/j.ijsolstr.2014.05.018.
  14. Doan, H.D, Do, V.T., Pham, H.C., Phung, V.M., Nguyen, X.N. (2022), "Vibration and static buckling behavior of variable thickness flexoelectric nanoplates", Mech. Bas. Des. Struct. Mach., 2022, 1-19. https://doi.org/10.1080/15397734.2022.2088558.
  15. Duc, H.D., Ashraf, M.Z. and Do, V.T. (2022), "Finite element modeling of free vibration of cracked nanoplates with flexoelectric effects", The Euro. Phys. J. Plus., 137, 447. https://doi.org/10.1140/epjp/s13360-022-02631-9
  16. Duc, H. D., Thom, V. D., Phuc, M. P. and Nguyen, D. D. (2018a), "Validation simulation for free vibration and buckling of cracked Mindlin plates using phase-field method", Mech. Adv. Mat. Struct., 26(12), 1018-1027. https://doi.org/10.1080/15376494.2018.1430262
  17. Duc, N. D., Trinh, T. D., Do, T. V. and Doan, D. H. (2018b), "On the buckling behavior of multi-cracked FGM plates", Proc. Int. Conf. Adv. Comp. Mech., 29-45. https://doi.org/10.1007/978-981-10-7149-2_3
  18. Ebrahimi, F., Karimiasl, M., Civalek, O., Vinyas, M. (2019a), "Surface effects on scale-dependent vibration behavior of flexoelectric sandwich nanobeams", Adv. Nano. Res., 7(2), 77-88. https://doi.org/10.12989/anr.2019.7.2.077
  19. Ebrahimi, F., Barati, M.R. (2019b), "On static stability of electro-magnetically affected smart magneto-electro-elastic nanoplates", Adv. Nano. Res., 63-75. https://doi.org/10.12989/anr.2019.7.1.063
  20. Ebrahimi, F., Barati, M.R., Mahesh, V. (2019c), "Dynamic modeling of smart magneto-electro-elastic curved nanobeams", Adv. Nano. Res., 7(3), 145-155. https://doi.org/10.12989/anr.2019.7.3.145
  21. Ebrahimi, F., Fardshad, R.E. and Mahesh, V. (2019d), "Frequency response analysis of curved embedded magneto-electro-viscoelastic functionally graded nanobeams", Adv. Nano Res., 7(6), 391-403. https://doi.org/10.12989/anr.2019.7.6.391.
  22. Eichler, A., Moser, J., Chaste, J., Zdrojek, M., Wilson-Rae, I. and Bachtold, A. (2011), "Nonlinear damping in mechanical resonators made from carbon nanotubes and graphene", Nat. Nanotechnol., 6(6), 339-342. https://doi.org/10.1038/nnano.2011.71.
  23. Foroutan, K. and Dai, L. (2022), "Post-buckling analysis of sandwich FG porous cylindrical shells with a viscoelastic core", Steel Compose. Struct., 43(3), 349-367. https://doi.org/10.12989/scs.2022.45.3.349.
  24. Ghobadi, A., Golestanian, H., Beni, Y.T. and Zur, K.K. (2021), "On the size-dependent nonlinear thermo-electro-mechanical free vibration analysis of functionally graded flexoelectric nano-plate", Commun. Nonlinear Sci. Numer. Simul., 95, 105585. https://doi.org/10.1016/j.cnsns.2020.105585.
  25. Hashemi, S.H., Mehrabani, H. and Ahmadi-Savadkoohi, A. (2015), "Exact solution for free vibration of coupled double viscoelastic graphene sheets by viscoPasternak medium", Compos. Part B Eng., 78, 377-383. https://doi.org/10.1016/j.compositesb.2015.04.008.
  26. Hoang-Nam, N., Tran, C.T., Doan, T.L., Van-Duc, P., Do, V.T. and Phung, V.M. (2019a), "Research on the buckling behavior of functionally graded plates with stiffeners based on the third-order shear deformation theory", Materials, 12(8), 1262. https://doi.org/10.3390/ma12081262.
  27. Hoang-Nam, N., Tan-Y, N., Ke, V.T., Thanh, T.T., Truong-Thinh, N., Van-Duc, P. and Thom, V.D. (2019b), "A finite element model for dynamic analysis of triple-layer composite plates with layers connected by shear connectors subjected to moving load", Materials, 12(4), 598. https://doi.org/10.3390/ma12040598
  28. Hieu, N.T., Do, V.T., Thai, N.D., Long, T.D. and Minh, P.V. (2020), "Enhancing the quality of the characteristic transmittance curve in the infrared region of range 2.5-7 ㎛ of the optical magnesium fluoride (MgF2) ceramic using the hot-pressing technique in a vacuum environment", Adv. Mater. Sci. Eng., 2020. https://doi.org/10.1155/2020/7258431.
  29. Jinxuan, Z., Zohre, M., Maryam, S. and Mohamed, A. K. (2022), "Intelligent modeling to investigate the stability of a two-dimensional functionally graded porosity-dependent nanobeam", Comput. Concr., 30(2), 85-97. https://doi.org/10.12989/cac.2022.30.2.085
  30. Jing-Lei, Z., Wu, F., Ru-Qing, B. and Wang, S. (2022), "Guided waves of porous FG nanoplates with four edges clamped", Adv. Nano. Res., 13(5), 465-474. https://doi.org/10.12989/anr.2022.13.5.465
  31. Karlicic, D., Kozic, P. and Pavlovic, R. (2014), "Free transverse vibration of nonlocal viscoelastic orthotropic multi-nanoplate system (MNPS) embedded in a viscoelastic medium", Compos. Struct., 115 (1), 89-99. https://doi.org/10.1016/j.compstruct.2014.04.002.
  32. Karlicic, D., Cajic, M., Murmu, T. and Adhikari, S. (2015), "Nonlocal longitudinal vibration of viscoelastic coupled double-nanorod systems", Eur. J. Mech. A Solids, 49, 183-196. https://doi.org/10.1016/j.euromechsol.2014.07.005.
  33. Liang, X., Hu, S. and Shen, S. (2014), "Effects of surface and flexoelectricity on a piezoelectric nanobeam," Smart Mater. Struct., 23 (3), 035020. https://doi.org/10.1088/0964-1726/23/3/035020.
  34. Liang, X., Hu, S. and Shen, S. (2015), "Size-dependent buckling and vibration behaviors of piezoelectric nanostructures due to flexoelectricity", Smart Mater. Struct., 24(10), 105012. https://doi.org/10.1088/0964-1726/24/10/105012.
  35. Liang, X., Yang, W., Hu, S. and Shen, S. (2016), "Buckling and vibration of flexoelectric nanofilms subjected to mechanical loads", J. Phys. D. Appl. Phys., 49(11), 115307. https://doi.org/10.1088/0022-3727/49/11/115307.
  36. Li, A., Zhou, S. and Qi, L. (2016), "Size-dependent electro-mechanical coupling behaviors of circular micro-plate due to flexoelectricity", Appl. Phys. A Mater. Sci. Proc., 122, 918. https://doi.org/10.1007/s00339-016-0455-3.
  37. Lieu, P.V. and Gia, T.L. (2023), "Static bending, free and forced vibration responses of organic nanobeams in a temperature environment", Arch. Appl. Mech., 2023. https://doi.org/10.1007/s00419-023-02469-2
  38. Luat, D.T, Do, V.T., Thanh, T.T., Minh, P.V, Ke, T.V, and Vinh, P.V. (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
  39. Mahmoud, S.R., Ghandourah, E., Ali, A., Mohammed, B., Tounsi, A. and Fouad, B. (2022), "On thermo-mechanical bending response of porous functionally graded sandwich plates via a simple integral plate model", Arch. Civil Mech. Eng., 22, 186. https://doi.org/10.1007/s43452-022-00506-5
  40. Majdoub, M.S., Sharma, P. and Cagin, T. (2008), "Enhanced size-dependent piezoelectricity and elasticity in nanostructures due to the flexoelectric effect", Phys. Rev. B, 77, 119904. https://doi.org/10.1103/PhysRevB.77.125424.
  41. Minh, T.T. and Thanh, C.L. (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), 2250193. https://doi.org/10.1142/S0219455422501930
  42. Mohammadi, P., Liu, L.P. and Sharma, P. (2014), "A theory of flexoelectric membranes and effective properties of heterogeneous membranes", J. Appl. Mech. Trans. ASME, 81(1), 011007. https://doi.org/10.1115/1.4023978.
  43. Nam, V.H, Pham, V.V., Nguyen, V.C., Do, V.T. and Tran, T.H. (2019), "A new beam model for simulation of the mechanical behaviour of variable thickness functionally graded material beams based on modified first order shear deformation theory", Materials, 12(3), 404. https://doi.org/10.3390/ma12030404
  44. Nguyen, C.T., Nguyen, T.T. and Do, V.T. (2019), "New numerical results from simulations of beams and space frame systems with a tuned mass damper", Materials, 12(8), 1329. https://doi.org/10.3390/ma12081329
  45. Pham, T.D., Doan, T.L. and Do, V.T. (2016), "Free vibration of functionally graded sandwich plates with stiffeners based on the third-order shear deformation theory", Viet. J. Mech., 38(2), 103-122. https://doi.org/10.15625/0866-7136/38/2/6730
  46. Pham, H.C., Doan, H.D. and Do, V.T. (2022), "Phase field model for fracture based on modified couple stress", Eng. Fract. Mech., 269, 108534. https://doi.org/10.1016/j.engfracmech.2022.108534
  47. Phung, M.V, Nguyen, D.T., Doan, L.T., Nguyen, D.V. and Duong, T.V. (2022), "Numerical investigation on static bending and free vibration responses of two-layer variable thickness plates with shear connectors", Iran. J. Sci. Technol. Trans. Mech. Eng., 46, 1047-1065. https://doi.org/10.1007/s40997-021-00459-9.
  48. Phung, V.M., Le, M.T., Doan, T.L. and Nguyen, D.A.V. (2022), "Static bending analysis of nanoplates on discontinuous elastic foundation with flexoelectric effect". J. Scie. Technol., 17(5). https://doi.org/10.56651/lqdtu.jst.v17.n05.529
  49. Pouresmaeeli, S., Ghavanloo, E. and Fazelzadeh, S.A. (2013), "Vibration analysis of viscoelastic orthotropic nanoplates resting on viscoelastic medium", Compos. Struct., 96, 405-410. https://doi.org/10.1016/j.compstruct.2012.08.051.
  50. Qi, L., Zhou, S. and Li, A. (2016), "Size-dependent bending of an electro-elastic bilayer nanobeam due to flexoelectricity and strain gradient elastic effect", Compos. Struct., 135, 167-175. https://doi.org/10.1016/j.compstruct.2015.09.020.
  51. Qi, L., Huang, S., Fu, G., Zhou, S. and Jiang, X. (2018), "On the mechanics of curved flexoelectric microbeams", Int. J. Eng. Sci., 124, 1-15. https://doi.org/10.1016/j.ijengsci.2017.11.022.
  52. Quang, D.V., Doan T.N., Luat, D.T. and Thom, D.V. (2022), "Static analysis and boundary effect of FG-CNTRC cylindrical shells with various boundary conditions using quasi-3D shear and normal deformations theory", Struct., 44, 828-850. https://doi.org/10.1016/j.istruc.2022.08.039
  53. Reissner, E. (1985), "Reflections on the theory of elastic plates", Appl. Mech. Rev., 38(11) 1453-1464. https://doi.org/10.1115/1.3143699
  54. Rupa N.S. and Ray, M.C. (2017), "Analysis of flexoelectric response in nanobeams using nonlocal theory of elasticity", Int. J. Mech. Mater. Des., 13(3), 453-467. https://doi.org/10.1007/s10999-016-9347-0.
  55. Shen, S. and Hu, S. (2010) , "A theory of flexoelectricity with surface effect for elastic dielectrics", J. Mech. Phys. Solids, 58(5), 665-677. https://doi.org/10.1016/j.jmps.2010.03.001.
  56. Singh, P.P., 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
  57. Shu, L., Liang, R., Rao, Z., Fei, L., Ke, S. and Wang, Y. (2019), "Flexoelectric materials and their related applications: A focused review", J. Adv. Ceram., 8(2), 153-173. https://doi.org/10.1007/s40145-018-0311-3.
  58. Shu, L., Wei, X., Pang, T., Yao, X. and Wang, C. (2011) , "Symmetry of flexoelectric coefficients in crystalline medium", J. Appl. Phys., 110, 104106. https://doi.org/10.1063/1.3662196.
  59. Su, Y., Hao,W., Rungang, G., Zhi, Y., Jing, Z., Zhaohui, Z. and Yafei, Z. (2012), "Exceptional negative thermal expansion and viscoelastic properties of graphene oxide paper", Carbon, 50(8), 2804-2809. https://doi.org/10.1016/j.carbon.2012.02.045.
  60. Srivastava, I., Z. Yu, Z. and Koratkar, N. (2012), "Viscoelastic Properties of Graphene-Polymer Composites," Adv. Sci. Eng. Med., 4 (1), 10-14. https://doi.org/10.1166/asem.2012.1127.
  61. Tinh, Q.B., Duc, H.D., Thom, V.D., Sohichi, H. and Nguyen, D.D. (2016), "High frequency modes meshfree analysis of Reissner-Mindlin plates", J. Sci. Adv. Mat. Develop., 1(3), 400-412. https://doi.org/10.1016/j.jsamd.2016.08.005
  62. Tien, D.M, Thom, D.V, Minh, P.V., Tho, N.C., Doan, T.N. and Mai, D.N. (2023), "The application of the nonlocal theory and various shear strain theories for bending and free vibration analysis of organic nanoplates", Mech. Based Des. Struct., 2023. https://doi.org/10.1080/15397734.2023.2186893
  63. Thai, L.M., Luat, D.T., Phung, V.B., Minh, P.V. and Thom, D.V. (2022), "Finite element modeling of mechanical behaviors of piezoelectric nanoplates with flexoelectric effects", Arch. Appl. Mech., 92(1), 163-182. https://doi.org/10.1007/s00419-021-02048-3.
  64. Thanh, C. L., Khuong, D. N., Minh, H. L., Thanh, S. T., Phan-Vu, P. and Magd, A. W. (2022), "Nonlocal strain gradient IGA numerical solution for static bending, free vibration and buckling of sigmoid FG sandwich nanoplate", Phys. B Cond. Mat., 631, 413726. https://doi.org/10.1016/j.physb.2022.413726
  65. Tho, N.C., Thom, D. Van, Cong, P.H., Zenkour, M.A., Doan, D.H. and Minh, P.V. (2023), "Finite element modeling of the bending and vibration behavior of three-layer composite plates with a crack in the core layer", Compos. Struct., 305, 116529. https://doi.org/10.1016/j.compstruct.2022.116529
  66. Thom, V.D., Duc, H.D., Nguyen, D.D. and Tinh, Q.B. (2017), "Phase-field thermal buckling analysis for cracked functionally graded composite plates considering neutral surface", Compos. Struct., 182, 542-548. https://doi.org/10.1016/j.compstruct.2017.09.059
  67. Thom, V.D., Duc, H.D., Nguyen, C.T. and Nguyen, D.D. (2022), "Thermal buckling analysis of cracked functionally graded plates", Int. J. Struct. Stab. Dyn., 22(8), 2250089. https://doi.org/10.1142/S0219455422500894.
  68. Tho, N.C., Thanh, N.T., Tho, T.D., Minh, P.V. and Hoa, L.K. (2021), "Modelling of the flexoelectric effect on rotating nanobeams with geometrical imperfection", J. Brazil. Soc. Mech. Sci. Eng., 43(11), 510. https://doi.org/10.1007/s40430-021-03189-w.
  69. Thom, D.V., Duc, D.H., Minh, P.V. and Tung, N.S. (2020), "Finite Element Modelling for Free Vibration Response of Cracked Stiffened Fgm Plates", Vietnam J. Sci. Technol., 58(1), 119. https://doi.org/10.15625/2525-2518/58/1/14278.
  70. Tuan, L.T., Dung, N. T., Thom, D. V., Minh, P. V. and Zenkour, A. (2021), "Propagation of non-stationary kinematic disturbances from a spherical cavity in the pseudo-elastic cosserat medium", Eur. Phys. J. Plus, 136, 1199. https://doi.org/10.1140/epjp/s13360-021-02191-4.
  71. Tuyen, B.V. (2023), "Vibration response of bamboo-reinforced composite beams", J. Vib. Eng. Technol., 2023. https://doi.org/10.1007/s42417-023-00998-2
  72. Tran, N.D., Thom, D.V., Thanh, N.T., Chuong, P.V., Tho, N.C., Ta, N.T. and Nam, N.H. (2020), "Analysis of stress concentration phenomenon of cylinder laminated shells using higher-order shear deformation Quasi-3D theory", Compos. Struct., 232, 111526. https://doi.org/10.1016/j.compstruct.2019.111526.
  73. Xiao, Z., Pinyi, W., Al-Dhaifallah, M., Muhyaddin, R. and Mohamed, A.K. (2022), "A machine learning-based model for the estimation of the critical thermo-electrical responses of the sandwich structure with magneto-electro-elastic face sheet", Adv. Nano. Res., 12(1), 81-99. https://doi.org/10.12989/anr.2022.12.1.081
  74. Xiang, S. and Li, X.F. (2018), "Elasticity solution of the bending of beams with the flexoelectric and piezoelectric effects", Smart Mater. Struct., 27(10), 105023. https://doi.org/10.1088/1361-665X/aadd5b.
  75. Xiang, S., Lee, K.Y. and Li, X.F. (2020), "Elasticity solution of functionally graded beams with consideration of the flexoelectric effect", J. Phys. D. Appl. Phys., 53(10), 105301. https://doi.org/10.1088/1361-6463/ab5cc1.
  76. Yan, Z. and Jiang, L.Y. (2013a), "Flexoelectric effect on the electroelastic responses of bending piezoelectric nanobeams", J. Appl. Phys., 113(19), 194102. https://doi.org/10.1063/1.4804949.
  77. Yan, Z. and Jiang, L. (2013b), "Size-dependent bending and vibration behaviour of piezoelectric nanobeams due to flexoelectricity", J. Phys. D. Appl. Phys., 46(35), 355502. https://doi.org/10.1088/0022-3727/46/35/355502.
  78. Yang, W., Liang, X. and Shen, S. (2015), "Electromechanical responses of piezoelectric nanoplates with flexoelectricity", Acta Mech., 226(9), 3097-3110. https://doi.org/10.1007/s00707-015-1373-8.
  79. Yue, Y.M., Xu, K.Y. and Chen, T. (2016), "A micro scale Timoshenko beam model for piezoelectricity with flexoelectricity and surface effects", Compos. Struct., 136, 278-286. https://doi.org/10.1016/j.compstruct.2015.09.046.
  80. Zenkour, A.M. (2016), "Nonlocal transient thermal analysis of a single-layered graphene sheet embedded in viscoelastic medium", Physica. E, 79, 87-97. https://doi.org/10.1016/j.physe.2015.12.003.
  81. Zhang, Z., Yan, Z. and Jiang, L. (2014), "Flexoelectric effect on the electroelastic responses and vibrational behaviors of a piezoelectric nanoplate", J. Appl. Phys., 116, 014307. https://doi.org/10.1063/1.4886315.
  82. Zhang, Z. and, Jin, C. (2022), "Axisymmetric vibration analysis of a sandwich porous plate in thermal environment rested on Kerr foundation", Steel Compos. Struct., 43(5), 581-601. https://doi.org/10.12989/scs.2022.43.5.581
  83. Zhou, L. (2022), "Computerized responses of spinning NEMS via numerical and mathematical modeling", Struct. Eng. Mech., 82(5), 629-641. https://doi.org/10.12989/sem.2022.82.5.629.
  84. Zhonghong, L. and Gongxing, Y. (2022), "Machine learning for structural stability: Predicting dynamics responses using physics-informed neural networks", Compit. Concr., 29(6), 419-432. https://doi.org/10.12989/cac.2022.29.6.419
  85. Zubko, P., Catalan, G. and Tagantsev, A.K. (2013) "Flexoelectric effect in solids", Annual Rev. Mater. Res., 43, 387-421. https://doi.org/10.1146/annurev-matsci-071312-121634.
  86. Wang, Y., Li, F.M. and Wang, Y.Z. (2015), "Nonlinear vibration of double layered viscoelastic nanoplates based on nonlocal theory", Physiaca E, 67, 65-76. https://doi.org/10.1016/j.physe.2014.11.007.
  87. Wang, X., Zhang, R. and Jiang, L. (2017), "A study of the flexoelectric effect on the electroelastic fields of a cantilevered piezoelectric nanoplate", Int. J. Appl. Mech., 9(4), 1750056. https://doi.org/10.1142/S1758825117500569.
  88. Wang, K.F. and Wang, B.L. (2018), "Energy gathering performance of micro/nanoscale circular energy harvesters based on flexoelectric effect", Energy, 149, 597-606. https://doi.org/10.1016/j.energy.2018.02.069.
  89. Wenjuan Y. (2022), "Intelligent computer modelling and simulation for the large amplitude of nano systems", Adv. Nano Res., 13(1), 63-75. https://doi.org/10.12989/anr.2022.13.1.063
  90. Zhe, Z., Qijian, Y. and Cong, J. (2022), "Axisymmetric vibration analysis of a sandwich porous plate in thermal environment rested on Kerr foundation", Steel Compos. Struct., 43(5), 581-601. https://doi.org/10.12989/scs.2022.43.5.581
  91. Smain, B., Aicha, B., Mohammed, S. A. H. and Marc, A. (2022), "A new quasi-3D plate theory for free vibration analysis of advanced composite nanoplates", Steel Compos. Struct., 45(6), 839-850. https://doi.org/10.12989/scs.2022.45.6.839
  92. Lingao, Z. (2022), "Computerized responses of spinning NEMS via numerical and mathematical modeling", Struct. Eng. Mech., 82(5), 629-641. https://doi.org/10.12989/sem.2022.82.5.629
  93. Zhonghong, L. and Gongxing, Y. (2022), "Machine learning for structural stability: Predicting dynamics responses using physics-informed neural networks", Comput. Concr., 29(6), 419-432. https://doi.org/10.12989/cac.2022.29.6.419
  94. Jinxuan, Z., Zohre, M., Maryam, S. and Mohamed, A.K. (2022), "Intelligent modeling to investigate the stability of a two-dimensional functionally graded porosity-dependent nanobeam", Comput. Concr., 30(2), 85-97. https://doi.org/10.12989/cac.2022.30.2.085
  95. Xiao, Z., Pinyi, W., Al-Dhaifallah, M., Muhyaddin, R. and Mohamed, A.K. (2022), "A machine learning-based model for the estimation of the critical thermo-electrical responses of the sandwich structure with magneto-electro-elastic face sheet", Adv. Nano. Res., 12(1), 81-99. https://doi.org/10.12989/anr.2022.12.1.081
  96. Jing-Lei, Z., Gui-Lin, S., Fei, W., Shu-Jin, Y., Ru-Qing, B., Hua-Yan, P., Shilong, W., Jun, L. (2022), "Guided waves of porous FG nanoplates with four edges clamped", Adv. Nano. Res., 13(5), 465-474. https://doi.org/10.12989/anr.2022.13.5.465