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Nonlocal strain gradient model for thermal stability of FG nanoplates integrated with piezoelectric layers

  • Karami, Behrouz (Department of Mechanical Engineering, Marvdasht Branch, Islamic Azad University) ;
  • Shahsavari, Davood (Department of Mechanical Engineering, Marvdasht Branch, Islamic Azad University)
  • Received : 2018.05.13
  • Accepted : 2019.02.23
  • Published : 2019.03.25
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Abstract

In the present paper, the nonlocal strain gradient refined model is used to study the thermal stability of sandwich nanoplates integrated with piezoelectric layers for the first time. The influence of Kerr elastic foundation is also studied. The present model incorporates two small-scale coefficients to examine the size-dependent thermal stability response. Elastic properties of nanoplate made of functionally graded materials (FGMs) are supposed to vary through the thickness direction and are estimated employing a modified power-law rule in which the porosity with even type of distribution is approximated. The governing differential equations of embedded sandwich piezoelectric porous nanoplates under hygrothermal loading are derived through Hamilton's principle where the Galerkin method is applied to solve the stability problem of the nanoplates with simply-supported edges. It is indicated that the thermal stability characteristics of the porous nanoplates are obviously influenced by the porosity volume fraction and material variation, nonlocal parameter, strain gradient parameter, geometry of the nanoplate, external voltage, temperature and humidity variations, and elastic foundation parameters.

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References

  1. Adpakpang, K., Patil, S.B., Oh, S.M., Kang, J.H., Lacroix, M. and Hwang, S.J. (2016), "Effective chemical route to 2D nanostructured silicon electrode material: phase transition from exfoliated clay nanosheet to porous Si nanoplate", Electrochimica Acta, 204, 60-68. https://doi.org/10.1016/j.electacta.2016.04.043
  2. Atmane, H.A., Tounsi, A., Bernard, F. and Mahmoud, S. (2015), "A computational shear displacement model for vibrational analysis of functionally graded beams with porosities", Steel Compos. Struct., 19(2), 369-384. https://doi.org/10.12989/scs.2015.19.2.369
  3. Barati, M., Sadr, M. and Zenkour, A. (2016), "Buckling analysis of higher order graded smart piezoelectric plates with porosities resting on elastic foundation", Int. J. Mech. Sci., 117, 309-320. https://doi.org/10.1016/j.ijmecsci.2016.09.012
  4. Barati, M.R. (2017a), "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
  5. Barati, M.R. (2017b), "On wave propagation in nanoporous materials", Int. J. Eng. Sci., 116, 1-11. https://doi.org/10.1016/j.ijengsci.2017.03.007
  6. Barati, M.R. (2017c), "Vibration analysis of FG nanoplates with nanovoids on viscoelastic substrate under hygro-thermomechanical loading using nonlocal strain gradient theory", Struct. Eng, Mech., 64(6), 683-693. https://doi.org/10.12989/SEM.2017.64.6.683
  7. Barati, M.R. (2018), "A general nonlocal stress-strain gradient theory for forced vibration analysis of heterogeneous porous nanoplates", Eur. J. Mech. -A/Solids, 67, 215-230. https://doi.org/10.1016/j.euromechsol.2017.09.001
  8. Belkorissat, I., Houari, M.S.A., Tounsi, A., Bedia, E. and Mahmoud, S. (2015), "On vibration properties of functionally graded nano-plate using a new nonlocal refined four variable model", Steel Compos. Struct., 18(4), 1063-1081. https://doi.org/10.12989/scs.2015.18.4.1063
  9. Bellifa, H., Benrahou, K.H., Bousahla, A.A., Tounsi, A. and Mahmoud, S. (2017), "A nonlocal zeroth-order shear deformation theory for nonlinear postbuckling of nanobeams", Struct. Eng. Mech., 62(6), 695-702. https://doi.org/10.12989/SEM.2017.62.6.695
  10. Bouafia, K., Kaci, A., Houari, M.S.A., Benzair, A. and Tounsi, A. (2017), "A nonlocal quasi-3D theory for bending and free flexural vibration behaviors of functionally graded nanobeams", Smart Struct. Syst., 19(2), 115-126. https://doi.org/10.12989/sss.2017.19.2.115
  11. Bounouara, F., Benrahou, K. H., Belkorissat, I. and Tounsi, A. (2016), "A nonlocal zeroth-order shear deformation theory for free vibration of functionally graded nanoscale plates resting on elastic foundation", Steel Compos. Struct., 20(2), 227-249. https://doi.org/10.12989/scs.2016.20.2.227
  12. Chaht, F.L., Kaci, A., Houari, M.S.A., Tounsi, A., Beg, O.A. and Mahmoud, S. (2015), "Bending and buckling analyses of functionally graded material (FGM) size-dependent nanoscale beams including the thickness stretching effect", Steel Compos. Struct., 18(2), 425-442. https://doi.org/10.12989/scs.2015.18.2.425
  13. Ebrahimi, F. and Daman, M. (2017), "Dynamic characteristics of curved inhomogeneous nonlocal porous beams in thermal environment", Struct. Eng. Mech., 64(1), 121-133. https://doi.org/10.12989/sem.2017.64.1.121
  14. Ebrahimi, F. and Salari, E. (2017), "Semi-analytical vibration analysis of functionally graded size-dependent nanobeams with various boundary conditions", Smart Struct. Syst., 19(3), 243-257. https://doi.org/10.12989/sss.2017.19.3.243
  15. Ebrahimi, F., Daman, M. and Jafari, A. (2017), "Nonlocal strain gradient-based vibration analysis of embedded curved porous piezoelectric nano-beams in thermal environment", Smart Struct. Syst., 20(6), 709-728. https://doi.org/10.12989/SSS.2017.20.6.709
  16. Eringen, A.C. and Edelen, D. (1972), "On nonlocal elasticity", Int. J. Eng. Sci., 10(3), 233-248. https://doi.org/10.1016/0020-7225(72)90039-0
  17. Farajpour, A., Ghayesh, M.H. and Farokhi, H. (2018), "Largeamplitude coupled scale-dependent behaviour of geometrically imperfect NSGT nanotubes", Int. J. Mech. Sci., 150, 510-525. https://doi.org/10.1016/j.ijmecsci.2018.09.043
  18. Ghadiri, M., Shafiei, N. and Babaei, R. (2017), "Vibration of a rotary FG plate with consideration of thermal and Coriolis effects", Steel Compos. Struct., 25(2), 197-207. https://doi.org/10.12989/SCS.2017.25.2.197
  19. Ghayesh, M.H. and Farokhi, H. (2017), "Global dynamics of imperfect axially forced microbeams", Int. J. Eng. Sci., 115, 102-116. https://doi.org/10.1016/j.ijengsci.2017.01.005
  20. Ghayesh, M.H., Farokhi, H. and Gholipour, A. (2017a), "Oscillations of functionally graded microbeams", Int. J. Eng. Sci., 110, 35-53. https://doi.org/10.1016/j.ijengsci.2016.09.011
  21. Ghayesh, M.H., Farokhi, H., Gholipour, A. and Tavallaeinejad, M. (2017c), "Nonlinear bending and forced vibrations of axially functionally graded tapered microbeams", Int. J. Eng. Sci., 120, 51-62. https://doi.org/10.1016/j.ijengsci.2017.03.010
  22. Ghayesh, M.H., Farokhi, H., Gholipour, A., Hussain, S. and Arjomandi, M. (2017b), "Resonance responses of geometrically imperfect functionally graded extensible microbeams", J. Comput. Nonlinear Dynam., 12(5), 051002. https://doi.org/10.1115/1.4035214
  23. Hadji, L. (2017), "Analysis of functionally graded plates using a sinusoidal shear deformation theory", Smart Struct. Syst., 19(4), 441-448. https://doi.org/10.12989/sss.2017.19.4.441
  24. Jandaghian, A.A. and Rahmani, O. (2017), "Vibration analysis of FG nanobeams based on third-order shear deformation theory under various boundary conditions", Steel Compos. Struct., 25(1), 67-78. https://doi.org/10.12989/SCS.2017.25.1.067
  25. Kaghazian, A., Hajnayeb, A. and Foruzande, H. (2017), "Free vibration analysis of a piezoelectric nanobeam using nonlocal elasticity theory", Struct. Eng. Mech., 61(5), 617-624. https://doi.org/10.12989/sem.2017.61.5.617
  26. Kar, V.R. and Panda, S.K. (2015), "Nonlinear flexural vibration of shear deformable functionally graded spherical shell panel", Steel Compos. Struct., 18(3), 693-709. https://doi.org/10.12989/scs.2015.18.3.693
  27. Karami, B. and Janghorban, M. (2016), "Effect of magnetic field on the wave propagation in nanoplates based on strain gradient theory with one parameter and two-variable refined plate theory", Modern Phys. Lett. B, 30(36), 1650421. https://doi.org/10.1142/S0217984916504212
  28. Karami, B. and Janghorban, M. (2019), "On the dynamics of porous nanotubes with variable material properties and variable thickness", Int. J. Eng. Sci., 136, 53-66. https://doi.org/10.1016/j.ijengsci.2019.01.002
  29. 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
  30. Karami, B., Janghorban, M. and Li, L. (2018a), "On guided wave propagation in fully clamped porous functionally graded nanoplates", Acta Astronautica, 143, 380-390. https://doi.org/10.1016/j.actaastro.2017.12.011
  31. Karami, B., Janghorban, M. and Tounsi, A. (2017), "Effects of triaxial magnetic field on the anisotropic nanoplates", Steel Compos. Struct., 25(3), 361-374. https://doi.org/10.12989/SCS.2017.25.3.361
  32. Karami, B., Janghorban, M. and Tounsi, A. (2018c), "Galerkin's approach for buckling analysis of functionally graded anisotropic nanoplates/different boundary conditions", Engineering with Computers.
  33. Karami, B., Janghorban, M. and Tounsi, A. (2018d), "Nonlocal strain gradient 3D elasticity theory for anisotropic spherical nanoparticles", Steel Compos. Struct., 27(2), 201-216. https://doi.org/10.12989/SCS.2018.27.2.201
  34. Karami, B., Janghorban, M. and Tounsi, A. (2018e), "Variational approach for wave dispersion in anisotropic doubly-curved nanoshells based on a new nonlocal strain gradient higher order shell theory", Thin-Wall. Struct., 129, 251-264. https://doi.org/10.1016/j.tws.2018.02.025
  35. Karami, B., janghorban, M. and Tounsi, A. (2019a), "On exact wave propagation analysis of triclinic material using threedimensional bi-Helmholtz gradient plate model", Struct. Eng. Mech., 69(5), 487-497. https://doi.org/10.12989/sem.2019.69.5.487
  36. Karami, B., Janghorban, M., Shahsavari, D. and Tounsi, A. (2018b), "A size-dependent quasi-3D model for wave dispersion analysis of FG nanoplates", Steel Compos. Struct., 28(1), 99-110. https://doi.org/10.12989/SCS.2018.28.1.099
  37. Karami, B., Shahsavari, D. and Janghorban, M. (2018f), "A comprehensive analytical study on functionally graded carbon nanotube-reinforced composite plates", Aerosp. Sci. Technol., 82, 499-512. https://doi.org/10.1016/j.ast.2018.10.001
  38. Karami, B., Shahsavari, D. and Janghorban, M. (2018g), "Wave propagation analysis in functionally graded (FG) nanoplates under in-plane magnetic field based on nonlocal strain gradient theory and four variable refined plate theory", Mech. Adv. Mater. Struct., 25(12), 1047-1057. https://doi.org/10.1080/15376494.2017.1323143
  39. Karami, B., Shahsavari, D. and Li, L. (2018j), "Hygrothermal wave propagation in viscoelastic graphene under in-plane magnetic field based on nonlocal strain gradient theory", Physica E: Low-dimensional Systems and Nanostructures, 97, 317-327. https://doi.org/10.1016/j.physe.2017.11.020
  40. Karami, B., Shahsavari, D. and Li, L. (2018k), "Temperaturedependent flexural wave propagation in nanoplate-type porous heterogenous material subjected to in-plane magnetic field", J. Therm. Stresses, 41(4), 483-499. https://doi.org/10.1080/01495739.2017.1393781
  41. Karami, B., Shahsavari, D., Janghorban, M. and Li, L. (2018h), "Wave dispersion of mounted graphene with initial stress", Thin-Wall. Struct., 122, 102-111. https://doi.org/10.1016/j.tws.2017.10.004
  42. Karami, B., shahsavari, D., janghorban, M. and Li, L. (2019b), "Influence of homogenization schemes on vibration of functionally graded curved microbeams", Compos. Struct., 216, 67-79. https://doi.org/10.1016/j.compstruct.2019.02.089
  43. Karami, B., Shahsavari, D., Janghorban, M., Dimitri, R. and Tornabene, F. (2019b), "Wave Propagation of Porous Nanoshells", Nanomaterials, 9(1), 22. https://doi.org/10.3390/nano9010022
  44. Karami, B., Shahsavari, D., Karami, M. and Li, L. (2018i), "Hygrothermal wave characteristic of nanobeam-type inhomogeneous materials with porosity under magnetic field", Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science.
  45. Karami, B., Shahsavari, D., Li, L., Karami, M. and Janghorban, M. (2019d), "Thermal buckling of embedded sandwich piezoelectric nanoplates with functionally graded core by a nonlocal second-order shear deformation theory", Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 233(1), 287-301. https://doi.org/10.1177/0954406218756451
  46. Karami, B., Shahsavari, D., Nazemosadat, S.M.R., Li, L. and Ebrahimi, A. (2018l), "Thermal buckling of smart porous functionally graded nanobeam rested on Kerr foundation", Steel Compos. Struct., 29(3), 349-362. https://doi.org/10.12989/SCS.2018.29.3.349
  47. Khetir, H., Bouiadjra, M.B., Houari, M.S.A., Tounsi, A. and Mahmoud, S. (2017), "A new nonlocal trigonometric shear deformation theory for thermal buckling analysis of embedded nanosize FG plates", Struct. Eng. Mech., 64(4), 391-402. https://doi.org/10.12989/SEM.2017.64.4.391
  48. Kneifati, M. C. (1985), "Analysis of plates on a Kerr foundation model", J. Eng. Mech., 111(11), 1325-1342. https://doi.org/10.1061/(ASCE)0733-9399(1985)111:11(1325)
  49. Lam, D.C., Yang, F., Chong, A., 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
  50. Lee, C.Y. and Kim, J.H. (2013), "Hygrothermal postbuckling behavior of functionally graded plates", Compos. Struct., 95, 278-282. https://doi.org/10.1016/j.compstruct.2012.07.010
  51. Li, L. and Hu, Y. (2015), "Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory", Int. J. Eng. Sci., 97, 84-94. https://doi.org/10.1016/j.ijengsci.2015.08.013
  52. Li, Q., Iu, V. and Kou, K. (2008), "Three-dimensional vibration analysis of functionally graded material sandwich plates", J. Sound Vib., 311(1-2), 498-515. https://doi.org/10.1016/j.jsv.2007.09.018
  53. Lim, C., Zhang, G. and Reddy, J. (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
  54. Lu, C., Chen, W. and Lim, C.W. (2009), "Elastic mechanical behavior of nano-scaled FGM films incorporating surface energies", Compos. Sci. Technol., 69(7-8), 1124-1130. https://doi.org/10.1016/j.compscitech.2009.02.005
  55. 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. Braz. Soc. Mech. Sci. Eng., 38(8), 2193-2211. https://doi.org/10.1007/s40430-015-0482-6
  56. Nami, M.R. and Janghorban, M. (2014), "Resonance behavior of FG rectangular micro/nano plate based on nonlocal elasticity theory and strain gradient theory with one gradient constant", Compos. Struct., 111, 349-353. https://doi.org/10.1016/j.compstruct.2014.01.012
  57. Nejad, M.Z. and Hadi, A. (2016), "Non-local analysis of free vibration of bi-directional functionally graded Euler-Bernoulli nano-beams", Int. J. Eng. Sci., 105, 1-11. https://doi.org/10.1016/j.ijengsci.2016.04.011
  58. Nejad, M.Z., Hadi, A. and Rastgoo, A. (2016), "Buckling analysis of arbitrary two-directional functionally graded Euler-Bernoulli nano-beams based on nonlocal elasticity theory", Int. J. Eng. Sci., 103, 1-10. https://doi.org/10.1016/j.ijengsci.2016.03.001
  59. Rahmani, O., Refaeinejad, V. and Hosseini, S. (2017), "Assessment of various nonlocal higher order theories for the bending and buckling behavior of functionally graded nanobeams", Steel Compos. Struct., 23(3), 339-350. https://doi.org/10.12989/scs.2017.23.3.339
  60. Saadatfar, M. and Aghaie-Khafri, M. (2015), "Electromagnetothermoelastic behavior of a rotating imperfect hybrid functionally graded hollow cylinder", Smart Struct. Syst., 15(6), 1411-1437. https://doi.org/10.12989/sss.2015.15.6.1411
  61. Sedighi, H.M., Daneshmand, F. and Abadyan, M. (2015a), "Modified model for instability analysis of symmetric FGM double-sided nano-bridge: corrections due to surface layer, finite conductivity and size effect", Compos. Struct., 132, 545-557. https://doi.org/10.1016/j.compstruct.2015.05.076
  62. Sedighi, H.M., Keivani, M. and Abadyan, M. (2015b), "Modified continuum model for stability analysis of asymmetric FGM double-sided NEMS: corrections due to finite conductivity, surface energy and nonlocal effect", Compos. Part B: Eng., 83, 117-133. https://doi.org/10.1016/j.compositesb.2015.08.029
  63. Shafiei, N. and She, G.L. (2018), "On vibration of functionally graded nano-tubes in the thermal environment", Int. J. Eng. Sci., 133, 84-98. https://doi.org/10.1016/j.ijengsci.2018.08.004
  64. Shahsavari, D. and Janghorban, M. (2017), "Bending and shearing responses for dynamic analysis of single-layer graphene sheets under moving load", J. Braz. Soc. Mech. Sci. Eng., 39(10), 3849-3861. https://doi.org/10.1007/s40430-017-0863-0
  65. Shahsavari, D., Karami, B. and Li, L. (2018b), "Damped vibration of a graphene sheet using a higher-order nonlocal straingradient Kirchhoff plate model", Comptes Rendus Mecanique, 346(12), 1216-1232. https://doi.org/10.1016/j.crme.2018.08.011
  66. Shahsavari, D., Karami, B. and Li, L. (2018c), "A high-order gradient model for wave propagation analysis of porous FG nanoplates", Steel Compos. Struct., 29(1), 53-66. https://doi.org/10.12989/scs.2018.29.1.053
  67. Shahsavari, D., Karami, B. and Mansouri, S. (2018d), "Shear buckling of single layer graphene sheets in hygrothermal environment resting on elastic foundation based on different nonlocal strain gradient theories", Eur. J. Mech.-A/Solids, 67, 200-214. https://doi.org/10.1016/j.euromechsol.2017.09.004
  68. Shahsavari, D., Karami, B., Fahham, H.R. and Li, L. (2018a), "On the shear buckling of porous nanoplates using a new sizedependent quasi-3D shear deformation theory", Acta Mechanica, 229(11), 4549-4573. https://doi.org/10.1007/s00707-018-2247-7
  69. Shahsavari, D., Karami, B., Janghorban, M. and Li, L. (2017), "Dynamic characteristics of viscoelastic nanoplates under moving load embedded within visco-Pasternak substrate and hygrothermal environment", Mater. Res. Express, 4(8), 085013. https://doi.org/10.1088/2053-1591/aa7d89
  70. Shahsavari, D., Shahsavari, M., Li, L. and Karami, B. (2018e), "A novel quasi-3D hyperbolic theory for free vibration of FG plates with porosities resting on Winkler/Pasternak/Kerr foundation", Aerosp. Sci. Technol., 72, 134-149. https://doi.org/10.1016/j.ast.2017.11.004
  71. She, G.L., Ren, Y.R., Yuan, F.G. and Xiao, W.S. (2018a), "On vibrations of porous nanotubes", Int. J. Eng. Sci., 125, 23-35. https://doi.org/10.1016/j.ijengsci.2017.12.009
  72. She, G.L., Yan, K.M., Zhang, Y.L., Liu, H.B. and Ren, Y.R. (2018b), "Wave propagation of functionally graded porous nanobeams based on non-local strain gradient theory", Eur. Phys. J. Plus, 133(9), 368. https://doi.org/10.1140/epjp/i2018-12196-5
  73. She, G.L., Yuan, F.G. and Ren, Y.R. (2018c), "On wave propagation of porous nanotubes", Int. J. Eng. Sci., 130, 62-74. https://doi.org/10.1016/j.ijengsci.2018.05.002
  74. She, G.L., Yuan, F.G., Karami, B., Ren, Y.R. and Xiao, W.S. (2019), "On nonlinear bending behavior of FG porous curved nanotubes", Int. J. Eng. Sci., 135, 58-74. https://doi.org/10.1016/j.ijengsci.2018.11.005
  75. She, G.L., Yuan, F.G., Ren, Y.R. and Xiao, W.S. (2017), "On buckling and postbuckling behavior of nanotubes", Int. J. Eng. Sci., 121, 130-142. https://doi.org/10.1016/j.ijengsci.2017.09.005
  76. She, G.L., Yuan, F.G., Ren, Y.R., Liu, H.B. and Xiao, W.S. (2018d), "Nonlinear bending and vibration analysis of functionally graded porous tubes via a nonlocal strain gradient theory", Compos. Struct., 203, 614-623. https://doi.org/10.1016/j.compstruct.2018.07.063
  77. Shimpi, R.P. (2002), "Refined plate theory and its variants", AIAA J., 40(1), 137-146. https://doi.org/10.2514/2.1622
  78. Simsek, M. (2016), "Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach", Int. J. Eng. Sci., 105, 12-27. https://doi.org/10.1016/j.ijengsci.2016.04.013
  79. Sobhy, M. (2016), "An accurate shear deformation theory for vibration and buckling of FGM sandwich plates in hygrothermal environment", Int. J. Mech. Sci., 110, 62-77. https://doi.org/10.1016/j.ijmecsci.2016.03.003
  80. Sobhy, M. (2017), "Hygro-thermo-mechanical vibration and buckling of exponentially graded nanoplates resting on elastic foundations via nonlocal elasticity theory", Struct. Eng. Mech., 63(3), 401-415. https://doi.org/10.12989/SEM.2017.63.3.401
  81. Wattanasakulpong, N. and Ungbhakorn, V. (2014), "Linear and nonlinear vibration analysis of elastically restrained ends FGM beams with porosities", Aerosp. Sci. Technol., 32(1), 111-120. https://doi.org/10.1016/j.ast.2013.12.002
  82. Xiong, Q.L.. and Tian, X. (2017), "Transient thermo-piezo-elastic responses of a functionally graded piezoelectric plate under thermal shock", Steel Compos. Struct., 25(2), 187-196. https://doi.org/10.12989/SCS.2017.25.2.187
  83. Yu, J.C., Xu, A., Zhang, L., Song, R. and Wu, L. (2004), "Synthesis and characterization of porous magnesium hydroxide and oxide nanoplates", J. Phys. Chem. B, 108(1), 64-70. https://doi.org/10.1021/jp035340w
  84. Zhang, T. and Shi, Z. (2010), "Exact analyses for two kinds of piezoelectric hollow cylinders with graded properties", Smart Struct. Syst., 6(8), 975-989. https://doi.org/10.12989/sss.2010.6.8.975

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