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Effects of hygro-thermal environment on dynamic responses of variable thickness functionally graded porous microplates

  • Quoc-Hoa Pham (Advanced Structural Engineering Laboratory, Department of Structural Engineering, Faculty of Civil Engineering, Ho Chi Minh City Open University) ;
  • Phu-Cuong Nguyen (Advanced Structural Engineering Laboratory, Department of Structural Engineering, Faculty of Civil Engineering, Ho Chi Minh City Open University) ;
  • Van-Ke Tran (Department of Mechanics, Le Quy Don Technical University)
  • 투고 : 2022.03.13
  • 심사 : 2024.02.20
  • 발행 : 2024.03.10

초록

This paper presents a novel finite element model for the free vibration analysis of variable-thickness functionally graded porous (FGP) microplates resting on Pasternak's medium in the hygro-thermal environment. The governing equations are established according to refined higher-order shear deformation plate theory (RPT) in construction with the modified couple stress theory. For the first time, three-node triangular elements with twelve degrees of freedom for each node are developed based on Hermitian interpolation functions to describe the in-plane displacements and transverse displacements of microplates. Two laws of variable thickness of FGP microplates, including the linear law and the nonlinear law in the x-direction are investigated. Effects of thermal and moisture changes on microplates are assumed to vary continuously from the bottom surface to the top surface and only cause tension loads in the plane, which does not change the material's mechanical properties. The numerical results of this work are compared with those of published data to verify the accuracy and reliability of the proposed method. In addition, the parameter study is conducted to explore the effects of geometrical and material properties such as the changing law of the thickness, length-scale parameter, and the parameters of the porosity, temperature, and humidity on the free vibration response of variable thickness FGP microplates. These results can be applied to design of microelectromechanical structures in practice.

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참고문헌

  1. Abdelkader, B., Hassaine, D.T., Hassen, A.A., Abdelouahed, T. and Meftah, S.A. (2011), "A four variable refined plate theory for free vibrations of functionally graded plates with arbitrary gradient", Compos Part: B Eng., 42, 1386-1394. https://doi.org/10.1016/j.compositesb.2011.05.032.
  2. Ansari, R., Gholami, R., Shojaei, M.F., Mohammadi, V. and Sahmani, S. (2013), "Size-dependent bending, buckling and free vibration of functionally graded Timoshenko microbeams based on the most general strain gradient theory", Compos Struct., 100, 385-397. https://doi.org/10.1016/j.compstruct.2012.12.048.
  3. Appl, F.C. and Byers, N.R. (1965), "Fundamental frequency of simply supported rectangular plates with linearly varying thickness", J. Appl. Mech., 32, 163-168. https://doi.org/10.1115/1.3625713.
  4. Azhari, M., Shahidi, A.R. and Saadatpour, M.M. (2005), "Local and post-local buckling of stepped and perforated thin plates", Appl. Math. Model., 29, 633-652. https://doi.org/10.1016/j.apm.2004.10.004.
  5. Bounouara, F., Benrahou, K.H., Belkorissat, I. and Touns, A. (2016), "A nonlocal zeroth-order shear deformation theory for free vibration of functionally graded nanoscale plates resting on elastic foundation", Steel Comp. Struct., 20(2), 227-249. https://doi.org/10.12989/scs.2016.20.2.227.
  6. Chong, A.C.M., Yang, F., Lam, D.C.C. and Tong, P. (2001), "Torsion and bending of micron-scaled structures", J Mater Res., 16, 1052-1058. https://doi.org/10.1557/JMR.2001.0146.
  7. Chong, A.C.M., Yang, F., Lam, D.C.C. and Tong, P. (2003), "Experiments and theory in strain gradient elasticity", J. Mech. Phys. Solids., 51, 1477-1508. https://doi.org/10.1016/S0022-5096(03)00053-X.
  8. Chopra, I. and Durvasula, S. (1971), "Natural frequencies and modes of tapered skew plates", Int J Mech Sci., 13, 935-944. https://doi.org/10.2514/3.5296.
  9. Chunwei, Z., Qiao, J., Yansheng, S., Jingli, W., Li, S., Haicheng, L., Limin, D., He, T., Xiaodong, Y., Hongmei, X., Limeng, Z. and Songlin, G. (2021), "Vibration analysis of a sandwich cylindrical shell in hygrothermal environment", Nanotechnol. Rev., 10(1), 414-430. https://doi.org/10.1515/ntrev-2021-0026
  10. Civalek, O. (2009), "Fundamental frequency of isotropic and orthotropic rectangular plates with linearly varying thickness by discrete singular convolution method", Appl. Math. Model., 33, 3825-3835. https://doi.org/10.1016/j.apm.2008.12.019.
  11. El-Zafrany, A. and Cookson, R.A. (1986), "Derivation of Lagrangian and Hermitian shape functions for triangular elements", Int. J. Numer. Meth. Eng., 23, 275-285. https://doi.org/10.1002/nme.162023021.
  12. Eringen, A.C. (1972), "Nonlocal polar elastic continua", Int. J. Eng. Sci., 10, 1-16. https://doi.org/10.1016/0020-7225(72)90070-5.
  13. Fleck, N.A., Muller, G.M., Ashby, M.F. and Hutchinson, J.W. (1994), "Strain gradient plasticity: theory and experiment", Acta Metall. Mater., 42, 475-487. https://doi.org/10.1016/0956-7151(94)90502-9.
  14. Ghandourah, E.E and Abdraboh, A.M. (2020), "Dynamic analysis of functionally graded nonlocal nanobeam with different porosity models", Steel Comp. Struct., 36(3), 293-305. https://doi.org/10.12989/scs.2020.36.3.293.
  15. Haiping, Z., Yang, L. and Yang, D. (2021), "Temperature gradient modeling of a steel box-girder suspension bridge using Copulas probabilistic method and field monitoring", Adv. Struct. Eng., 24(5), 947-961. https://doi.org/10.1177/1369433220971779
  16. Hans-Jurgen, B., Brunero, C. and Michael, K. (2005), "Force measurements with the atomic force microscope: Technique, interpretation and applications", Surf Sci Rep., 59, 1-152. https://doi.org/10.1016/j.surfrep.2005.08.003.
  17. He, L., Lou, J., Zhang, E., Wang, Y. and Bai, Y. (2015), "A sizedependent four variable refined plate model for functionally graded microplates based on modified couple stress theory", Compos. Struct., 130, 107-115. https://doi.org/10.1016/j.compstruct.2015.04.033.
  18. Hosseini, M., Shishesaz, M. and Hadi, A. (2019), "Thermoelastic analysis of rotating functionally graded micro/nanodisks of variable thickness", Thin-Wall. Struct., 134, 508-523. https://doi.org/10.1016/j.tws.2018.10.030.
  19. Huang, X.L. and Shen, H.S. (2014), "Nonlinear vibration and dynamic response of functionally graded plates in thermal environments", Int J Solids Struct., 41, 2403-2427. https://doi.org/10.1016/j.ijsolstr.2003.11.012.
  20. Jouneghani, F.Z., Dimitri, R. and Tornabene, F. (2018), "Structural response of porous FG nanobeams under hygro-thermomechanical loadings", Compos Part: B Eng., 152, 71-78. https://doi.org/10.1016/j.compositesb.2018.06.023.
  21. Karamanli, A. and Vo, T.P. (2018), "Size dependent bending analysis of two directional functionally graded microbeams via a quasi-3D theory and finite element method", Compos Part: B Eng., 144, 171-183. https://doi.org/10.1016/j.compositesb.2018.02.030.
  22. Karami, B., Janghorban, M. and Tounsi, A. (2018a), "Nonlocal strain gradient 3D elasticity theory for anisotropic spherical nanoparticles", Steel Comp. Struct., 27(2), 201-216. https://doi.org/10.12989/scs.2018.27.2.201.
  23. Karami, B., Janghorban, M., Shahsavari, D. and Tounsi, A. (2018b), "A size-dependent quasi-3D model for wave dispersion analysis of FG nanoplates", Steel Comp. Struct., 28(1), 99-110. https://doi.org/10.12989/scs.2018.28.1.099.
  24. Ke, L.L. and Wang, Y.S. (2011), "Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory", Compos Struct., 93, 342-350. https://doi.org/10.1016/j.compstruct.2010.09.008.
  25. 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 Mech., 226, 2973-2998. https://doi.org/10.1007/s00707-015-1370-y.
  26. Kobayashi, H. and Sonoda, K. (1991), "Vibration and buckling of tapered rectangular plates with two opposite edges simply supported and the other two edges elastically restrained against rotation", J. Sound Vib., 146, 323-337. https://doi.org/10.1016/0022-460X(91)90766-D.
  27. Koiter, W. (1964), Couple Stresses in the Theory of Easticity, I and II. Philos Trans R. Soc London B. 67, 17-4.
  28. Kukreti, A.R. and Bert, C.W. (1996), "Differential quadrature and Rayleigh-Ritz method to determine the fundamental frequencies of simply supported rectangular plates with linearly varying thickness", J. Sound Vib., 189, 103-122. https://doi.org/10.1006/jsvi.1996.0008.
  29. Lazopoulos, K.A. (2009), "On bending of strain gradient elastic micro-plates", Mech. Res. Commun., 36, 777-783. https://doi.org/10.1016/j.mechrescom.2009.05.005.
  30. Lee, Z., Ophus, C., Fischer, L., Nelson-Fitzpatrick, N., Westra, K.L., Evoy, S., Radmilovic, V., Dahmen, U. and Mitlin, M. (2006), "Metallic NEMS components fabricated from nanocomposite Al-Mo films", Nanotechnology., 17, 3063-3070. https://doi.org/10.1088/0957-4484/17/12/042.
  31. Li, S. (2000), "The micromechanics theory of classical plates: a congruous estimate of overall elastic stiffness", Int. J. Solids Struct., 37, 5599-5628. https://doi.org/10.1016/S0020-7683(99)00239-5.
  32. Lu, C., Wu, D. and Chen, W. (2011), "Non-linear responses of nano-scale FGM films including the effects of surface energies, IEEE", Trans Nanotechnol., 10, 1321-1327. https://doi.org/10.1109/TNANO.2011.2139223.
  33. Ma, H.M., Gao, X.L. and Reddy, J.N. (2011), "A non-classical Mindlin plate model based on a modified couple stress theory", Acta Mech., 220, 217-235. https://doi.org/10.1007/s00707-011-0480-4.
  34. Marie, K.T., Christoph, S., David, C.M., Thomas, H., Cari, F.H., Christofe, H., Ken, G., Steven, M.G. and Victor, M.B. (2006), "The mechanical properties of atomic layer deposited alumina for use in micro- and nano-electromechanical systems", Sens. Actuat. A-Phys., 130-131, 419-429. https://doi.org/10.1016/j.sna.2006.01.029.
  35. Mechab, B., Mechab, I., Benaissa, S., Ameri, M. and Serier, B. (2005), "Probabilistic analysis of effect of the porosities in functionally graded material nanoplate resting on Winkler-Pasternak elastic foundations", Appl Math Model., 0, 0-12. https://doi.org/10.1016/j.apm.2015.09.093.
  36. Mindlin, R.D. and Tiersten, H.F. (1962), "Effects of couplestresses in linear elasticity", Arch Ration Mech Anal., 11, 415-448. https://doi.org/10.1007/BF00253946.
  37. Najafi, M. and Ahmadi. I. (2021), "A nonlocal Layerwise theory for free vibration analysis of nanobeams with various boundary conditions on Winkler-Pasternak foundation", Steel Comp. Struct., 40(1), 101-119. https://doi.org/10.12989/scs.2021.40.1.101.
  38. Van Minh, P. and Van Ke, T. (2023), "A comprehensive study on mechanical responses of non-uniform thickness piezoelectric nanoplates taking into account the flexoelectric effect", Arab. J. Sci. Eng., 48(9), 11457-11482. https://doi.org/10.1007/s13369-022-07362-8
  39. Pham, Q.H., Nguyen, P.C., Tran, V.K. and Nguyen-Thoi, T (2021), "Finite element analysis for functionally graded porous nanoplates resting on elastic foundation", Steel Comp. Struct., 41(2),149-166. https://doi.org/10.12989/scs2021.41.2.149.
  40. Pham, Q.H., Tran, V.K., Tran, T.T., Nguyen-Thoi, T., Nguyen, P.C. and Pham, V.D. (2021), "A nonlocal quasi-3D theory for thermal free vibration analysis of functionally graded material nanoplates resting on elastic foundation", Case Stud Therm Eng., 26,101170. https://doi.org/10.1016/j.csite.2021.101170.
  41. Pham, Q.H., Nguyen, P.C., Tran, V.K., Lieu, Q.X. and Tran, T.T. (2023), "Modified nonlocal couple stress isogeometric approach for bending and free vibration analysis of functionally graded nanoplates", Eng. Comput., 39(1), 993-1018. https://doi.org/10.1007/s00366-022-01726-2
  42. Pham, Q.H., Tran, V.K. and Nguyen, P.C. (2023), "Nonlocal strain gradient finite element procedure for hygro-thermal vibration analysis of bidirectional functionally graded porous nanobeams", Waves Random Complex Media. 1-32.
  43. QH Pham, VK Tran, TT Tran, PC Nguyen, P Malekzadeh (2022a). Dynamic instability of magnetically embedded functionally graded porous nanobeams using the strain gradient theory. Alexandria Engineering Journal. 61 (12), 10025-10044 https://doi.org/10.1016/j.aej.2022.03.007
  44. Reddy, J.N. and Kim, J. (2012), "A nonlinear modified couple stress-based third-order theory of functionally graded plates", Compos Struct., 94, 1128-1143. https://doi.org/10.1016/j.compstruct.2011.10.006.
  45. Reddy, J.N., Romanoff, J. and Loya, J.A. (2016), "Nonlinear Finite Element Analysis of Functionally Graded Circular Plates with Modified Couple Stress Theory", Eur J Mech- A/Solids., 56, 92-104. https://doi.org/10.1016/j.euromechsol.2015.11.001.
  46. Roque, C.M.C., Fidalgo, D.S., Ferreira, A.J.M. and Reddy, J.N. (2013), "A study of a microstructure-dependent composite laminated Timoshenko beam using a modified couple stress theory and a meshless method", Compos Struct., 96, 532-537. https://doi.org/10.1016/j.compstruct.2012.09.011.
  47. Sahmani, S. and Safaei, B. (2020), "Influence of homogenization models on size-dependent nonlinear bending and postbuckling of bi-directional functionally graded micro/nano-beams", Appl Math Model., 82, 336 -358. https://doi.org/10.1016/j.apm.2020.01.051.
  48. Shahidi, A.R., Anjomshoa, A., Shahidi, S.H. and Kamrani, M. (2013), "Fundamental size dependent natural frequencies of non-uniform orthotropic nano scaled plates using nonlocal variational principle and finite element method", Appl. Math. Model., 37, 7047-7061. https://doi.org/10.1016/j.apm.2013.02.015.
  49. Shahsavari, D., Karami, D., Fahham, H.R. and Li, L. (2018), "On the shear buckling of porous nanoplates using a new sizedependent quasi-3D shear deformation theory", Acta Mech., 229, 4549-4573. https://doi.org/10.1007/s00707-018-2247-7.
  50. Shen, H.S., Xiang, Y. and Lin, F. (2017), "Thermal buckling and postbuckling of functionally graded graphene-reinforced composite laminated plates resting on elastic foundations", Thin-Wall. Struct., 118, 229-237. https://doi.org/10.1016/j.tws.2017.05.006.
  51. Shimpi, S.P. (2002), "Refined plate theory and its variants", AIAA. J. 40,137-146. https://doi.org/abs/10.2514/2.1622.
  52. Shimpi, S.P. and Patel, H.G. (2006a), "A two variable refined plate theory for orthotropic plate analysis", Int. J. Solids Struct., 43, 6783-6799. https://doi.org/10.1016/j.ijsolstr.2006.02.007.
  53. Shimpi, S.P. and Patel, H.G. (2006b), "Free vibrations of plate using two variable refined plate theory", J. Sound Vib., 296, 979-999. https://doi.org/10.1016/j.jsv.2006.03.030.
  54. Sobhy, M. (2017), "A new Quasi 3D nonlocal plate theory for vibration and buckling of FGM nanoplates", Int. J. Appl. Mech., 9(1), 1750008.
  55. Stolken, J.S. and Evans, A.G. (1998), "A microbend test method for measuring the plasticity length scale", Acta Mater., 46, 5109-5115. https://doi.org/10.1016/S1359-6454(98)00153-0.
  56. Thai, H.T. and Choi, D.H. (2013), "Finite element formulation of various four unknown shear deformation theories for functionally graded plates", Finite Elem Anal Des., 75, 50-61. https://doi.org/10.1016/j.finel.2013.07.003.
  57. Thai, H.T. and Kim, S.E. (2013), "A size-dependent functionally graded Reddy plate model based on a modified couple stress theory", Compos. Part: B Eng., 45, 1636-1645. https://doi.org/10.1016/j.compstruct.2012.09.025.
  58. Thai, H.T., Vo, T.P., Nguyen, T.K. and Lee, J. (2015), "Sizedependent behavior of functionally graded sandwich microbeams based on the modified couple stress theory", Compos Struct., 123, 337-349. https://doi.org/10.1016/j.compstruct.2014.11.065.
  59. Toupin, R.A. (1962), "Elastic materials with couple-stresses", Arch Ration Mech Anal., 11, 385-414. https://doi.org/10.1007/BF00253945.
  60. Tran, T.T., Tran, V.K., Pham, Q.H. and Zenkour, A.M. (2021), "Extended four-unknown higher-order shear deformation nonlocal theory for bending, buckling and free vibration of functionally graded porous nanoshell resting on elastic foundation", Compos Struct., 113737. https://doi.org/10.1016/j.compstruct.2021.113737.
  61. Tran, V.K., Pham, Q.H. and Nguyen-Thoi, T. (2020), "A finite element formulation using four-unknown incorporating nonlocal theory for bending and free vibration analysis of functionally graded nanoplates resting on elastic medium foundations", Eng Comput-Germany., 1-26. https://doi.org/10.1007/s00366-020-01107-7.
  62. Tsiatas, G.C. (2009), "A new Kirchhoff plate model based on a modified couple stress theory", Int. J. Solids Struct., 46, 2757-2764. https://doi.org/10.1016/j.ijsolstr.2009.03.004.
  63. Wu, H., Yang, J. and Kitipornchai, S. (2018), "Parametric instability of thermo-mechanically loaded functionally graded graphene reinforced nanocomposite plates", Int J Mech Sci., 135, 431-40. https://doi.org/10.1016/j.ijmecsci.2017.11.039.
  64. Xia, W., Wang, L. and Yin, L. (2010), "Nonlinear non-classical microscale beams: Static bending, postbuckling and free vibration", Int J Eng Sci., 48, 2044-2053. https://doi.org/10.1016/j.ijengsci.2010.04.010.
  65. Xiao, X., Bu, G. Ou, Z. and Li, Z. (2022), "Nonlinear in-plane instability of the confined FGP arches with nanocomposites reinforcement under radially-directed uniform pressure", Eng. Struct., 252, 113670. https://doi.org/10.1016/j.engstruct.2021.113670.
  66. Xue, Y., Jin, G., Ma, X., Chen, H., Ye, T., Chen, M. and Zhan, Y. (2019), "Free vibration analysis of porous plates with porosity distributions in the thickness and in-plane directions using isogeometric approach", Int J Mech Sci., 152, 346-362. https://doi.org/10.1016/j.ijmecsci.2019.01.004.
  67. Yang, F., Chong, A.C.M., Lam, D.C.C. and Tong, P. (2002), "Couple stress based strain gradient theory for elasticity", Int. J. Solids Struct., 39, 2731-2743. https://doi.org/10.1016/S0020-7683(02)00152-X.
  68. Yang, J. and Shen, H.S. (2002), "Vibration characteristic and transient response of shear-deformable functionally graded plates in thermal environments", J. Sound Vib., 255, 579-602. https://doi.org/10.1006/jsvi.2001.4161.
  69. Yin, L., Qian, Q., Wang, L. and Xia, W. (2010), 'Vibration analysis of microscale plates based on modified couple stress theory", Acta Mech. Solida Sin., 23, 386-393. https://doi.org/10.1016/S0894-9166(10)60040-7.
  70. Chu, Y.M., Nazir, U., Sohail, M., Selim, M.M. and Lee, J.R. (2021), "Enhancement in thermal energy and solute particles using hybrid nanoparticles by engaging activation energy and chemical reaction over a parabolic surface via finite element approach", Fractal Fract., 5(3), 119.
  71. Zhang, B., He, Y., Liu, D., Gan, Z. and Shen, L. (2013), "A nonclassical Mindlin plate finite element based on a modified couple stress theory couple stress theory", Eur J Mech - A/Solids., 42, 63-80. https://doi.org/10.1016/j.euromechsol.2013.04.005.