Advances in nano research

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Nonlinear bending analysis of porous sigmoid FGM nanoplate via IGA and nonlocal strain gradient theory

  • Cuong-Le, Thanh (Faculty of Civil Engineering, Ho Chi Minh City Open University) ;
  • Nguyen, Khuong D. (Department of Engineering Mechanics, Faculty of Applied Science, Ho Chi Minh City University of Technology (HCMUT)) ;
  • Le-Minh, Hoang (Faculty of Civil Engineering, Ho Chi Minh City Open University) ;
  • Phan-Vu, Phuong (Faculty of Civil Engineering, Ho Chi Minh City Open University) ;
  • Nguyen-Trong, Phuoc (Faculty of Civil Engineering, Ho Chi Minh City Open University) ;
  • Tounsi, Abdelouahed (YFL (Yonsei Frontier Lab), Yonsei University)
  • Received : 2021.09.26
  • Accepted : 2022.02.04
  • Published : 2022.05.25
⟨ Previous Next ⟩

Abstract

This study explores the linear and nonlinear solutions of sigmoid functionally graded material (S-FGM) nanoplate with porous effects. A size-dependent numerical solution is established using the strain gradient theory and isogeometric finite element formulation. The nonlinear nonlocal strain gradient is developed based on the Reissner-Mindlin plate theory and the Von-Karman strain assumption. The sigmoid function is utilized to modify the classical functionally graded material to ensure the constituent volume distribution. Two different patterns of porosity distribution are investigated, viz. pattern A and pattern B, in which the porosities are symmetric and asymmetric varied across the plate's thickness, respectively. The nonlinear finite element governing equations are established for bending analysis of S-FGM nanoplates, and the Newton-Raphson iteration technique is derived from the nonlinear responses. The isogeometric finite element method is the most suitable numerical method because it can satisfy a higher-order derivative requirement of the nonlocal strain gradient theory. Several numerical results are presented to investigate the influences of porosity distributions, power indexes, aspect ratios, nonlocal and strain gradient parameters on the porous S-FGM nanoplate's linear and nonlinear bending responses.

Keywords

Acknowledgement

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02-2020.27.

References

  1. Ahmed, R.A., Al-Maliki, A.F. and Faleh, N.M. (2020), "Dynamic characteristics of multi-phase crystalline porous shells with using strain gradient elasticity", Adv. Nano Res., 8(2), 157-167. https://doi.org/10.12989/anr.2020.8.2.157.
  2. Ansari, R. and Norouzzadeh, A. (2016), "Nonlocal and surface effects on the buckling behavior of functionally graded nanoplates: An isogeometric analysis", Physica E, 84, 84-97. https://doi.org/10.1016/j.physe.2016.05.036.
  3. Ansari, R., Torabi, J. and Norouzzadeh, A. (2018), "Bending analysis of embedded nanoplates based on the integral formulation of Eringen's nonlocal theory using the finite element method", Physica B, 534, 90-97. https://doi.org/10.1016/j.physb.2018.01.025.
  4. Arefi, M., Kiani, M. and Zamani, M.H. (2018), "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.
  5. Abouelregal, A.E., Mohammad-Sedighi, H., Faghidian, S.A. and Shirazi, A.H. (2021), "Temperature-dependent physical characteristics of the rotating nonlocal nanobeams subject to a varying heat source and a dynamic load", Facta Universitatis Mech. Eng., 19(4), 633-656. https://doi.org/10.22190/FUME201222024A.
  6. Abdulrazzaq, M.A., Fenjan, R.M., Ahmed, R.A. and Faleh, N.M. (2020), "Thermal buckling of nonlocal clamped exponentially graded plate according to a secant function based refined theory", Steel Compos. Struct., 35(1), 147-57. http://doi.org/10.12989/scs.2020.35.1.147.
  7. Babu, B. and Patel, B.P. (2019), "A new computationally efficient finite element formulation for nanoplates using second-order strain gradient Kirchhoff's plate theory", Compos. Part B Eng., 168, 302-311. https://doi.org/10.1016/j.compositesb.2018.12.066.
  8. Chen, D., Feng, K. and Zheng, S. (2019), "Flapwise vibration analysis of rotating composite laminated Timoshenko microbeams with geometric imperfection based on a re-modified couple stress theory and isogeometric analysis", Eur. J. Mech. A Solids, 76, 25-35. https://doi.org/10.1016/j.euromechsol.2019.03.002.
  9. Chung, Y.L. and Chi, S. (2001), "The residual stress of functionally graded materials", J. Chin. Inst. Civil Hydraulic Eng., 13, 1-9.
  10. Chai, Q. and Wang, Y.Q. (2022), "Traveling wave vibration of graphene platelet reinforced porous joined conical-cylindrical shells in a spinning motion", Eng. Struct., 252, 113718. https://doi.org/10.1016/j.engstruct.2021.113718.
  11. Daikh, A.A., Houari, M.S.A. and Eltaher, M.A. (2021), "A novel nonlocal strain gradient Quasi-3D bending analysis of sigmoid functionally graded sandwich nanoplates", Compos. Struct., 262, 113347. https://doi.org/10.1016/j.compstruct.2020.113347.
  12. Ebrahimi, F. and Barati, M.R. (2019), "Dynamic modeling of embedded nanoplate systems incorporating flexoelectricity and surface effects", Microsyst. Technol., 25(1), 175-187. https://doi.org/10.1007/s00542-018-3946-7.
  13. Ebrahimi, F. and Seyfi, A. (2020). "Studying propagation of wave in metal foam cylindrical shells with graded porosities resting on variable elastic substrate", Eng. Comput., 1-17. https://doi.org/10.1007/s00366-020-01069-w.
  14. Ebrahimi, F., Jafari, A. and Selvamani, R. (2020), "Thermal buckling analysis of magneto-electro-elastic porous FG beam in thermal environment", Adv. Nano Res., 8(1), 83-94. https://doi.org/10.12989/anr.2020.8.1.083.
  15. Ebrahimi, F., Dabbagh, A. and Taheri, M. (2021), "Vibration analysis of porous metal foam plates rested on viscoelastic substrate", Eng. Comput., 37(4), 3727-3739. https://doi.org/10.1007/s00366-020-01031-w.
  16. Emadi, M., Nejad, M.Z., Ziaee, S. and Hadi, A. (2021b), "Buckling analysis of arbitrary two-directional functionally graded nano-plate based on nonlocal elasticity theory using generalized differential quadrature method", Steel Compos. Struct., 39(5), 565-581. https://doi.org/10.12989/scs.2021.39.4.419.
  17. Eringen, A.C. and Edelen, D.G.B. (1972), "On nonlocal elasticity", Int. J. Eng. Sci., 10(3), 233-248. https://doi.org/10.1016/0020-7225(72)90039-0.
  18. Eringen, A.C. (1972), "Nonlocal polar elastic continua", Int. J. Eng. Sci., 10(1), 1-16. https://doi.org/10.1016/0020-7225(72)90070-5.
  19. Fan, F., Safaei, B. and Sahmani, S. (2021), "Buckling and postbuckling response of nonlocal strain gradient porous functionally graded micro/nano-plates via NURBS-based isogeometric analysis", Thin Wall. Struct., 159, 107231. https://doi.org/10.1016/j.tws.2020.107231.
  20. Fattahi, A., 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.
  21. Faleh, N.M., Fenjan, R.M. and Ahmed, R.A. (2018), "Dynamic analysis of graded small-scale shells with porosity distributions under transverse dynamic loads", Eur. Phys. J. Plus, 133(9), 1-11. https://doi.org/10.1140/epjp/i2018-12152-5.
  22. Fenjan, R.M., Moustafa, N.M. and Faleh, N.M. (2020a), "Scale-dependent thermal vibration analysis of FG beams having porosities based on DQM", Adv. Nano Res, 8(4), 283-292. https://doi.org/10.12989/anr.2020.8.4.283.
  23. Fenjan, R.M., Ahmed, R.A., Faleh, N.M. and Hani, F.M. (2020b), "Static stability analysis of smart nonlocal thermo-piezo-magnetic plates via a quasi-3D formulation", Smart Struct. Syst., 26(1), 77-87. http://doi.org/10.12989/sss.2020.26.1.077.
  24. Gain, A.K., Zhang, L. and Liu, W. (2015), "Microstructure and material properties of porous hydroxyapatite-zirconia nanocomposites using polymethyl methacrylate powders", Mater. Des., 67, 136-144. https://doi.org/10.1016/j.matdes.2014.11.028.
  25. He, L., Xu, J., Dekai, Z.H.O.U., Qinghai, Y.A.N.G. and Longqiu, L. (2018), "Potential application of functional micro-nano structures in petroleum", Petrol. Explo. Develop., 45(4), 745-753. https://doi.org/10.1016/S1876-3804(18)30077-6.
  26. Hughes, T.J.R., Cottrell, J.A. and Bazilevs, Y. (2005), "Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement", Comput. Meth. Appl. Mech. Eng., 194(39), 4135-4195. https://doi.org/10.1016/j.cma.2004.10.008.
  27. Ishihara, M., Yoshida, T., Ootao, Y. and Kameo, Y. (2020), "Hygrothermoelasticity in a porous cylinder under nonlinear coupling between heat and moisture", Struct. Eng. Mech., 75(1), 59-69. https://doi.org/10.12989/sem.2020.75.1.059.
  28. Jung, W.Y. and Han, S.C. (2013), "Analysis of sigmoid functionally graded material (S-FGM) nanoscale plates using the nonlocal elasticity theory", Math. Probl. Eng., 2013, 476131. https://doi.org/10.1155/2013/476131.
  29. Jung, W.Y. and Han, S.C. (2015), "Static and eigenvalue problems of sigmoid functionally graded materials (S-FGM) micro-scale plates using the modified couple stress theory", Appl. Math. Model., 39(12), 3506-3524. https://doi.org/10.1016/j.apm.2014.11.056.
  30. Kaci, A., Bakhti, K., Hebali, H. and Tounsi, A. (2013), "Mathematical solution for nonlinear cylindrical bending of sigmoid functionally graded plates", J. Appl. Mech. Tech. Phys., 54(1), 124-131. https://doi.org/10.1134/S002189441301015X.
  31. Karami, B. and Karami, S. (2019), "Buckling analysis of nanoplate-type temperature-dependent heterogeneous materials", Adv. Nano Res., 7(1), 51-61. https://doi.org/10.12989/anr.2019.7.1.051.
  32. Kim, W.J., Lee, W.H., Park, W.T. and Han, S.C. (2014), "Nonlocal elasticity effects on free vibration properties of sigmoid functionally graded material nano-scale plates", J. Korea Acad. Ind. Coop. Soc., 15(2), 1109-1117. https://doi.org/10.5762/KAIS.2014.15.2.1109.
  33. Kim, J. and Reddy, J.N. (2015), "A general third-order theory of functionally graded plates with modified couple stress effect and the von Karman nonlinearity: theory and finite element analysis", Acta Mechanica, 226(9), 2973-2998. https://doi.org/10.1007/s00707-015-1370-y.
  34. Koiter, W. (1964), "Couple-stresses in the theory of elasticity, I and II, Prec", Roy. Netherlands Acad. Sci. B, 67, 0964.
  35. Koizumi, M. (1993), "The concept of FGM", Ceram. Transact., 34, 3-10.
  36. Koochi, A. and Goharimanesh, M. (2021), "Nonlinear oscillations of cnt nano-resonator based on nonlocal elasticity: The energy balance method", Rep. Mech. Eng., 2(1), 41-50. https://doi.org/10.31181/rme200102041g.
  37. Khalaf, B.S., Fenjan, R.M. and Faleh, N.M. (2019), "Analyzing nonlinear mechanical-thermal buckling of imperfect micro-scale beam made of graded graphene reinforced composites", Adv. Mater. Res., 8(3), 219-235. https://doi.org/10.12989/amr.2019.8.3.219.
  38. 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. Solids, 51(8), 1477-1508. https://doi.org/10.1016/S0022-5096(03)00053-X.
  39. Lee, W.H., Han, S.C. and Park, W.T. (2012), "Nonlocal elasticity theory for bending and free vibration analysis of nano plates", J. Korea Acad. Ind. Coop. Soc., 13(7), 3207-3215. https://doi.org/10.5762/KAIS.2012.13.7.3207.
  40. Lee, W.H., Han, S.C. and Park, W.T. (2015), "A refined higher order shear and normal deformation theory for E-, P-, and S-FGM plates on Pasternak elastic foundation", Compos. Struct., 122, 330-342. https://doi.org/10.1016/j.compstruct.2014.11.047.
  41. Levy, S. and Chiarito, P.T. (1942), Square Plate with Clamped Edges Under Normal Pressure Producing Large Deflections, National Advisory Committee for Aeronautics, U.S. Government Printing Office, U.S.A.
  42. Lim, C.W., Zhang, G. and Reddy, J.N. (2015), "A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation", J. Mech. Phys. Solids, 78, 298-313. https://doi.org/10.1016/j.jmps.2015.02.001.
  43. Liu, S., Yu, T., Bui, T.Q. and Xia, S. (2020), "Size-dependent analysis of homogeneous and functionally graded microplates using IGA and a non-classical Kirchhoff plate theory", Compos. Struct., 172, 34-44. https://doi.org/10.1016/j.compstruct.2017.03.067.
  44. Liu, H., Zhang, Q., Yang, X. and Ma, J. (2021), "Size-dependent vibration of laminated composite nanoplate with piezo-magnetic face sheets", Eng. Comput., 1-17. https://doi.org/10.1007/s00366-021-01285-y.
  45. Ma, L.H., Ke, L.L., Wang, Y.Z. and Wang, Y.S. (2018), "Wave propagation analysis of piezoelectric nanoplates based on the nonlocal theory", Int. J. Struct. Stabil. Dynam., 18(4), 1850060. https://doi.org/10.1142/S0219455418500608.
  46. Mechab, I., Mechab, B., Benaissa, S., Serier, B. and Bouiadjra, B.B. (2016), "Free vibration analysis of FGM nanoplate with porosities resting on Winkler Pasternak elastic foundations based on two-variable refined plate theories", J. Brazil. Soc. Mech. Sci. Eng., 38(8), 2193-2211. https://doi.org/10.1007/s40430-015-0482-6.
  47. Mindlin, R.D. (1963), "Microstructure in linear elasticity", Research Report No. AD0424156; Department of Civil Engineering and Engineering Mechanics, Columbia University of New York, New York, U.S.A.
  48. Mindlin, R.D. and N. (1968), "Eshel, On first strain-gradient theories in linear elasticity", Int. J. Solid. Struct., 4(1), 109-124. https://doi.org/10.1016/0020-7683(68)90036-X.
  49. Miyamoto, Y., Kaysser, W.A., Rabin, B.H., Kawasaki, A. and Ford, R.G. (2013), "Functionally graded materials: design, processing and applications", Springer Science & Business Media, 5.
  50. Mueller, E., Drasar, C., Schilz, J. and Kaysser, W.A, (2003), "Functionally graded materials for sensor and energy applications", Mater. Sci. Eng. A, 362(1-2), 17-39. https://doi.org/10.1016/S0921-5093(03)00581-1.
  51. Moory-Shirbani, M., Sedighi, H.M., Ouakad, H.M. and Najar, F, (2018), "Experimental and mathematical analysis of a piezoelectrically actuated multilayered imperfect microbeam subjected to applied electric potential", Compos. Struct., 184, 950-960. https://doi.org/10.1016/j.compstruct.2017.10.062.
  52. Natarajan, S., Chakraborty, S., Thangavel, M., Bordas, S. and Rabczuk, T. (2012), "Size-dependent free flexural vibration behavior of functionally graded nanoplates", Comput. Mater. Sci., 65, 74-80. https://doi.org/10.1016/j.commatsci.2012.06.031.
  53. 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. Meth. Appl. Mech. Eng., 297, 191-218. https://doi.org/10.1016/j.cma.2015.07.021.
  54. Norouzzadeh, A., Ansari, R. and Rouhi, H. (2019), "Nonlinear bending analysis of nanobeams based on the nonlocal strain gradient model using an isogeometric finite element approach", Iran. J. Sci. Technol. Transact. Civil Eng., 43(1), 533-547. https://doi.org/10.1007/s40996-018-0184-2.
  55. Ouakad, H.M., Valipour, A., Zur, K.K., Sedighi, H.M. and Reddy, J.N. (2020), "On the nonlinear vibration and static deflection problems of actuated hybrid nanotubes based on the stress-driven nonlocal integral elasticity", Mech. Mater., 148, 103532. https://doi.org/10.1016/j.mechmat.2020.103532.
  56. Pompe, W., Worch, H., Epple, M., Friess, W., Gelinsky, M., Greil, P., Hempel, U., Scharnweber, D. and Schulte, K.J.M.S. (2003), "Functionally graded materials for biomedical applications", Mater. Sci. Eng. A, 362(1-2), 40-60. https://doi.org/10.1016/S0921-5093(03)00580-X.
  57. Reddy, J.N. (2004), "Mechanics of laminated composite plates and shells theory and analysis", CRC Press, New York, U.S.A.
  58. Samaniego, E., Anitescu, C., Goswami, S., Nguyen-Thanh, V.M., Guo, H., Hamdia, K., Zhuang, X. and Rabczuk, T. (2020), "An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications", Comput. Method. Appl. Mech. Eng., 362, 112790. https://doi.org/10.1016/j.cma.2019.112790.
  59. 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.
  60. Shariati, A., Mohammad-Sedighi, H., Zur, K.K., Habibi, M. and Safa, M. (2020a), "Stability and dynamics of viscoelastic moving rayleigh beams with an asymmetrical distribution of material parameters", Symmetry, 12(4), 586. https://doi.org/10.3390/sym12040586.
  61. Shariati, A., Ebrahimi, F., Karimiasl, M., Vinyas, M. and Toghroli, A. (2020b), "On transient hygrothermal vibration of embedded viscoelastic flexoelectric/piezoelectric nanobeams under magnetic loading", Adv. Nano Res., 8(1), 49-58. https://doi.org/10.12989/anr.2020.8.1.049.
  62. Sedighi, H.M., Ouakad, H.M., Dimitri, R. and Tornabene, F. (2020), "Stress-driven nonlocal elasticity for the instability analysis of fluid-conveying C-BN hybrid-nanotube in a magneto-thermal environment", Phys. Scripta, 95(6), 065204. https://doi.org/10.1088/1402-4896/ab793f.
  63. Sedighi, H.M. and Malikan, M. (2020), "Stress-driven nonlocal elasticity for nonlinear vibration characteristics of carbon/boron-nitride hetero-nanotube subject to magneto-thermal environment", Phys. Scripta, 95(5), 055218. https://doi.org/10.1088/1402-4896/ab7a38.
  64. Singh, P.P. and Azam, M.S. (2020), "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. https://doi.org/10.1177/1077546320966932.
  65. Singh, S.J. and Harsha, S.P. (2019), "Nonlinear dynamic analysis of sandwich S-FGM plate resting on pasternak foundation under thermal environment", Eur. J. Mech. A Solids, 76, 155-179. https://doi.org/10.1016/j.euromechsol.2019.04.005.
  66. Singh, S.J. and Harsha, S.P. (2020), "Analysis of porosity effect on free vibration and buckling responses for sandwich sigmoid function based functionally graded material plate resting on Pasternak foundation using Galerkin Vlasov's method", J. Sandw. Struct. Mater., 23(5), 1717-1760. https://doi.org/10.1177/1099636220904340.
  67. Sobhy, M.A. (2015), "Comprehensive study on FGM nanoplates embedded in an elastic medium", Compos. Struct., 134, 966-980. https://doi.org/10.1016/j.compstruct.2015.08.102.
  68. Thai, C.H., Ferreira, A.J.M. and Phung-Van, P. (2020), "A nonlocal strain gradient isogeometric model for free vibration and bending analyses of functionally graded plates", Compos. Struct., 251, 112634. https://doi.org/10.1016/j.compstruct.2020.112634.
  69. Thai, T.Q., Zhuang, X. and Rabczuk, T. (2021), "A nonlinear geometric couple stress based strain gradient Kirchhoff-Love shell formulation for microscale thin-wall structures", Int. J. Mech. Sci., 196, 106272. https://doi.org/10.1016/j.ijmecsci.2021.106272.
  70. Rabczuk, T., Ren, H., Zhuang, X. (2019), "A nonlocal operator method for partial differential equations with application to electromagnetic waveguide problem", Comput. Mater. Continua, 59(1), 31-55. https://doi.org/10.32604/cmc.2019.04567.
  71. Thang, P.T., Nguyen-Thoi, T. and Lee, J. (2016), "Closed-form expression for nonlinear analysis of imperfect sigmoid-FGM plates with variable thickness resting on elastic medium", Compos. Struct., 143, 143-150. https://doi.org/10.1016/j.compstruct.2016.02.002.
  72. Thanh, C.L., Tran, L.V., Vu-Huu, T., Nguyen-Xuan, H. and Abdel-Wahab, M. (2019a), "Size-dependent nonlinear analysis and damping responses of FG-CNTRC micro-plates", Comput. Methods Appl. Mech. Eng., 353, 253-276. https://doi.org/10.1016/j.cma.2019.05.002.
  73. Thanh, C.L., Tran, L.V., Bui, T.Q., Nguyen, H.X. and Abdel-Wahab, M. (2019b), "Isogeometric analysis for size-dependent nonlinear thermal stability of porous FG microplates", Compos. Struct., 221, 110838. https://doi.org/10.1016/j.compstruct.2019.04.010.
  74. Thanh, C.L., Ferreira, A. and Wahab, M.A. (2019c), "A refined size-dependent couple stress theory for laminated composite micro-plates using isogeometric analysis", Thin Wall. Struct., 145, 106427. https://doi.org/10.1016/j.cma.2019.05.002.
  75. Thanh, C. L., Nguyen, T.N., Vu, T.H., Khatir, S. and Wahab, M.A. (2020), "A geometrically nonlinear size-dependent hypothesis for porous functionally graded micro-plate", Eng. Comput., 1-12. https://doi.org/10.1007/s00366-020-01154-0.
  76. Thanh, C.L., Nguyen, K.D., Lee, J., Rabczuk, T. and Nguyen-Xuan, H. (2021), "A 3D nano scale IGA for free vibration and buckling analyses of multi-directional FGM nanoshells", Nanotechnology, 33(6), 065703. https://doi.org/10.1088/1361-6528/ac32f9.
  77. Toupin, R. (1962), "Elastic materials with couple-stresses", Arch. Ration. Mech. An., 11(1), 385-414. https://doi.org/10.1007/BF00253945
  78. Teng, M.W. and Wang, Y.Q. (2021), "Nonlinear forced vibration of simply supported functionally graded porous nanocomposite thin plates reinforced with graphene platelets", Thin Wall. Struct., 164, 107799. https://doi.org/10.1016/j.tws.2021.107799.
  79. Van Vinh, P. (2022), "Nonlocal free vibration characteristics of power-law and sigmoid functionally graded nanoplates considering variable nonlocal parameter", Physica E, 135, 114951. https://doi.org/10.1016/j.physe.2021.114951.
  80. Wang, Y.Q. and Zu, J.W. (2017a), "Vibration behaviors of functionally graded rectangular plates with porosities and moving in thermal environment", Aerosp. Sci. Technol., 69, 550-562. https://doi.org/10.1016/j.ast.2017.07.023.
  81. Wang, Y.Q. and Zu, J.W. (2017b), "Large-amplitude vibration of sigmoid functionally graded thin plates with porosities", Thin Wall. Struct., 119, 911-924. https://doi.org/10.1016/j.tws.2017.08.012.
  82. Wang, Y.Q. (2018), "Electro-mechanical vibration analysis of functionally graded piezoelectric porous plates in the translation state", Acta Astronautica, 143, 263-271. https://doi.org/10.1016/j.actaastro.2017.12.004.
  83. Wang, Y.Q., Ye, C. and Zu, J.W. (2019), "Nonlinear vibration of metal foam cylindrical shells reinforced with graphene platelets", Aerosp. Sci. Technol., 85, 359-370. https://doi.org/10.1016/j.ast.2018.12.022.
  84. Yang, F.A.C.M., 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.
  85. Ye, C. and Wang, Y.Q. (2021), "Nonlinear forced vibration of functionally graded graphene platelet-reinforced metal foam cylindrical shells: Internal resonances", Nonlinear Dynam., 104(3), 2051-2069. https://doi.org/10.1007/s11071-021-06401-7.
  86. Xiang, T., Hou, J., Xie, H., Liu, X., Gong, T. and Zhou, S. (2020), "Biomimetic micro/nano structures for biomedical applications", Nano Today, 35, 100980. https://doi.org/10.1016/j.nantod.2020.100980.
  87. Xu, H., Wang, Y.Q. and Zhang, Y. (2021), "Free vibration of functionally graded graphene platelet-reinforced porous beams with spinning movement via differential transformation method", Arch. Appl. Mech., 91(12), 4817-4834. https://doi.org/10.1007/s00419-021-02036-7.
  88. Zhang, Z., Li, H.N. and Yao, L.Q. (2021), "Vibration analysis of flexoelectric nanoplates based on nonlocal theory", Proceedings of the 2020 15th Symposium on Piezoelectrcity, Acoustic Waves and Device Applications, Zhengzhou, Henan Province, China, April.
  89. Zhuang, X., Guo, H., Alajlan, N., Zhu, H. and Rabczuk, T. (2021), "Deep autoencoder based energy method for the bending, vibration, and buckling analysis of Kirchhoff plates with transfer learning", Eur. J. Mech. A Solids, 87, 104225. https://doi.org/10.1016/j.euromechsol.2021.104225.
  90. Zhou, C., Zhang, Z., Zhang, J., Fang, Y. and Tahouneh, V. (2020), "Vibration analysis of FG porous rectangular plates reinforced by graphene platelets", Steel Compos. Struct., 34(2), 215-226. https://doi.org/10.12989/scs.2020.34.2.215.
  91. Yan, K., Zhang, Y., Cai, H. and Tahouneh, V. (2020), "Vibrational characteristic of FG porous conical shells using Donnell's shell theory", Steel Compos. Struct., 35(2), 249-260. https://doi.org/10.12989/scs.2020.35.2.249.
  92. She, G.L., Liu, H.B. and Karami, B. (2020), "On resonance behavior of porous FG curved nanobeams", Steel Compos. Struct, 36(2), 179-186. https://doi.org/10.12989/scs.2020.36.2.179.
  93. Si, H., Shen, D., Xia, J. and Tahouneh, V. (2020), "Vibration behavior of functionally graded sandwich beam with porous core and nanocomposite layers", Steel Compos. Struct,, 36(1), 1-16. https://doi.org/10.12989/scs.2020.36.1.001.
  94. Khaniki, H.B., Ghayesh, M.H., Hussain, S. and Amabili, M. (2020), "Porosity, mass and geometric imperfection sensitivity in coupled vibration characteristics of CNT-strengthened beams with different boundary conditions", Eng. Comput., 1-27. https://doi.org/10.1007/s00366-020-01208-3.
  95. Khazaei, P. and Mohammadimehr, M. (2020), "Size dependent effect on deflection and buckling analyses of porous nanocomposite plate based on nonlocal strain gradient theory", Struct. Eng. Mech., 76(1), 27-56. https://doi.org/10.12989/sem.2020.76.1.027.
  96. Shan, W., Deng, Z., Zhong, H., Mo, H., Han, Z., Yang, Z., Xiang, C., Li, S. and Liu, P. (2020), "Propagation characteristics of longitudinal wave, shear wave and bending wave in porous circular nanoplates", Struct. Eng. Mech., 76(4), 551-559. https://doi.org/10.12989/sem.2020.76.4.551.