Structural Engineering and Mechanics

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Bending and free vibration analysis of laminated piezoelectric composite plates

  • Zhang, Pengchong (School of Civil and Transportation Engineering, Beijing University of Civil Engineering and Architecture) ;
  • Qi, Chengzhi (School of Civil and Transportation Engineering, Beijing University of Civil Engineering and Architecture) ;
  • Fang, Hongyuan (College of Water Conservancy & Environmental Engineering, Zhengzhou University) ;
  • Sun, Xu (School of Civil and Transportation Engineering, Beijing University of Civil Engineering and Architecture)
  • Received : 2019.07.02
  • Accepted : 2020.03.19
  • Published : 2020.09.25
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Abstract

This paper provides a semi-analytical approach to investigate the variations of 3D displacement components, electric potential, stresses, electric displacements and transverse vibration frequencies in laminated piezoelectric composite plates based on the scaled boundary finite element method (SBFEM) and the precise integration algorithm (PIA). The proposed approach can analyze the static and dynamic responses of multilayered piezoelectric plates with any number of laminae, various geometrical shapes, boundary conditions, thickness-to-length ratios and stacking sequences. Only a longitudinal surface of the plate is discretized into 2D elements, which helps to improve the computational efficiency. Comparing with plate theories and other numerical methods, only three displacement components and the electric potential are set as the basic unknown variables and can be represented analytically through the transverse direction. The whole derivation is built upon the three dimensional key equations of elasticity for the piezoelectric materials and no assumptions on the plate kinematics have been taken. By virtue of the equilibrium equations, the constitutive relations and the introduced set of scaled boundary coordinates, three-dimensional governing partial differential equations are converted into the second order ordinary differential matrix equation. Furthermore, aided by the introduced internal nodal force, a first order ordinary differential equation is obtained with its general solution in the form of a matrix exponent. To further improve the accuracy of the matrix exponent in the SBFEM, the PIA is employed to make sure any desired accuracy of the mechanical and electric variables. By virtue of the kinetic energy technique, the global mass matrix of the composite plates constituted by piezoelectric laminae is constructed for the first time based on the SBFEM. Finally, comparisons with the exact solutions and available results are made to confirm the accuracy and effectiveness of the developed methodology. What's more, the effect of boundary conditions, thickness-to-length ratios and stacking sequences of laminae on the distributions of natural frequencies, mechanical and electric fields in laminated piezoelectric composite plates is evaluated.

Keywords

Acknowledgement

This research is supported by Grants 2018M641168 from China Postdoctoral Science Foundation, Grants 51908022, 2015CB57800 and 51774018 from the National Natural Science Foundation of China, Grant IRT_17R06 from program for Changjiang Scholars and Innovative Research Team, Grants 19YJC630148 from the Humanity and Social Science Youth foundation of Ministry of Education of China, for which the authors are gratefully acknowledged.

References

  1. Akhras, G., and Li, W.C. (2007), "Three-dimensional static, vibration and stability analysis of piezoelectric composite plates using a finite layer method", Smart Mater. Struct., 16(3), 561-569. http://dx.doi.org/10.1088/0964-1726/16/3/002.
  2. Akhras, G., and Li, W.C. (2007), "Three-dimensional static, vibration and stability analysis of piezoelectric composite plates using a finite layer method", Smart Mater. Struct., 16(3), 561-569. http://dx.doi.org/10.1088/0964-1726/16/3/002.
  3. Akhras, G., and Li, W.C. (2011), "Stability and free vibration analysis of thick piezoelectric composite plates using spline finite strip method", Int. J. Mech. Sci., 53(8), 575-584. http://dx.doi.org/10.1016/j.ijmecsci.2011.05.004.
  4. Askari, M., Saidi, A.R., and Rezaei, A.S. (2018), "An investigation over the effect of piezoelectricity and porosity distribution on natural frequencies of porous smart plates", J. Sandwich Struct. Mater.. http://dx.doi.org/10.1177/1099636218791092.
  5. Baghaee, M., Farrokhabadi, A., and Jafari-Talookolaei, R.A. (2019), "A solution method based on Lagrange multipliers and Legendre polynomial series for free vibration analysis of laminated plates sandwiched by two MFC layers", J. Sound Vibr., 447, 42-60. https://doi.org/10.1016/j.jsv.2019.01.037.
  6. Balamurugan, V., and Narayanan, S. (2007), "A piezoelectric higher-order plate element for the analysis of multi-layer smart composite laminates", Smart Mater. Struct., 16(6), 2026-2039. http://dx.doi.org/10.1088/0964-1726/16/6/005.
  7. Benjeddou, A. (2000) "Advances in piezoelectric finite element modeling of adaptive structural elements: a survey", Comput. Struct., 76(1-3), 347-363. https://doi.org/10.1016/S0045-7949(99)00151-0.
  8. Benjeddou, A., and Deu, J.F. (2002a), "A two-dimensional closed-form solution for the free-vibrations analysis of piezoelectric sandwich plates", Int. J. Solids Struct., 39(6), 1463-1486. https://doi.org/10.1016/S0020-7683(01)00287-6.
  9. Benjeddou, A., Deu, J.F., and Letombe, S. (2002b), "Free vibrations of simply-supported piezoelectric adaptive plates: an exact sandwich formulation", Thin-Walled Struct., 40(7-8), 573-593. https://doi.org/10.1016/S0263-8231(02)00013-7.
  10. Bian, Z.G., Ying, J., Chen, W.Q., and Ding, H.J. (2006), "Bending and free vibration analysis of a smart functionally graded plate", Struct. Eng. Mech., 23(1), 97-113. https://doi.org/10.12989/sem.2006.23.1.097.
  11. Birk, C., and Song, C. (2009), "A continued-fraction approach for transient diffusion in unbounded medium", Comput. Meth. Appl. Mech. Eng., 198(33-36), 2576-2590. https://doi.org/10.1016/j.cma.2009.03.002.
  12. Carrera, E., and Nali, P. (2009), "Mixed piezoelectric plate elements with direct evaluation of transverse electric displacement", Int. J. Numer. Methods Eng., 80(4), 403-424. https://doi.org/10.1002/nme.2641.
  13. Carrera, E., Buttner, A., and Nali, P. (2010a), "Mixed elements for the analysis of anisotropic multilayered piezoelectric plates", J. Intell. Mater. Syst. Struct., 21(7), 701-717. https://doi.org/10.1177/1045389X10364864.
  14. Carrera, E., and Robaldo, A. (2010b), "Hierarchic finite elements based on a unified formulation for the static analysis of shear actuated multilayered piezoelectric plates", Multidiscipline Modeling in Mater. Struct., 6(1), 45-77. https://doi.org/10.1108/15736101011055266.
  15. Cen, S., Soh, A.K., Long, Y.Q., and Yao, Z. H. (2002), "A new 4-node quadrilateral FE model with variable electrical degrees of freedom for the analysis of piezoelectric laminated composite plates", Compos. Struct., 58(4), 583-599. https://doi.org/10.1016/S0263-8223(02)00167-8.
  16. Chen, D., Birk, C., Song, C., and Du, C. (2014), "A high-order approach for modelling transient wave propagation problems using the scaled boundary finite element method", Int. J. Numer. Methods Eng., 97(13), 937-959. https://doi.org/10.1002/nme.4613.
  17. Cheung, Y.K., and Jiang, C.P. (2001), "Finite layer method in analyses of piezoelectric composite laminates", Comput. Meth. Appl. Mech. Eng., 191(8-10), 879-901. https://doi.org/10.1016/S0045-7825(01)00285-7.
  18. Ding H.J., Xu R.Q., Chi Y.W., and Chen W.Q. (1999), "Free axisymmetric vibration of transversely isotropic piezoelectric circular plates", Int. J. Solids Struct., 36(30), 4629-4652. https://doi.org/10.1016/S0020-7683(98)00206-6.
  19. Duan, W.H., Quek, S.T., and Wang, Q. (2005), "Free vibration analysis of piezoelectric coupled thin and thick annular plate", J. Sound Vibr., 281(1-2), 119-139. https://doi.org/10.1016/j.jsv.2004.01.009.
  20. Dube, G.P., Upadhyay, M.M., Dumir, P.C., and Kumar, S. (1998), "Piezothermoelastic solution for angle-ply laminated plate in cylindrical bending", Struct. Eng. Mech., 6(5), 529-542. https://doi.org/10.12989/SEM.1998.6.5.529.
  21. Garcao, J.S., Soares, C.M., Soares, C.M., and Reddy, J.N. (2004), "Analysis of laminated adaptive plate structures using layerwise finite element models", Comput. Struct., 82(23-26), 1939-1959. https://doi.org/10.1016/j.compstruc.2003.10.024.
  22. Ghasemabadian, M.A., and Saidi, A.R. (2017), "Stability analysis of transversely isotropic laminated Mindlin plates with piezoelectric layers using a Levy-type solution", Struct. Eng. Mech., 62(6), 675-693. https://doi.org/10.12989/SEM.2017.62.6.675.
  23. Hashemi, S.H., Es'haghi, M., and Karimi, M. (2010), "Closed-form solution for free vibration of piezoelectric coupled annular plates using Levinson plate theory", J. Sound Vibr., 329(9), 1390-1408. https://doi.org/10.1016/j.jsv.2009.10.043.
  24. Heyliger, P. (1994), "Static behavior of laminated elastic/piezoelectric plates", AIAA J., 32(12), 2481-2484. https://doi.org/10.2514/3.12321.
  25. Heyliger, P., and Brooks, S. (1995a), "Free vibration of piezoelectric laminates in cylindrical bending", Int. J. Solids Struct., 32(20), 2945-2960. https://doi.org/10.1016/0020-7683(94)00270-7.
  26. Heyliger, P., and Saravanos, D.A. (1995b), "Exact free-vibration analysis of laminated plates with embedded piezoelectric layers", J. Acoust. Soc. Am., 98(3), 1547-1557. https://doi.org/10.1121/1.413420.
  27. Heyliger, P. (1997), "Exact solutions for simply supported laminated piezoelectric plates", J. Appl. Mech., 64(2), 299-306. https://doi.org/10.1115/1.2787307.
  28. Heyliger, P.R., and Ramirez, G. (2000), "Free vibration of laminated circular piezoelectric plates and discs", J. Sound Vibr., 229(4), 935-956. https://doi.org/10.1006/jsvi.1999.2520.
  29. Hosseini-Hashemi, S., Es'haghi, M., and Taher, H. R. D. (2010), "An exact analytical solution for freely vibrating piezoelectric coupled circular/annular thick plates using Reddy plate theory", Compos. Struct., 92(6), 1333-1351. https://doi.org/10.1016/j.compstruct.2009.11.006.
  30. Kapuria, S. (2004), "A coupled zig-zag third-order theory for piezoelectric hybrid cross-ply plates", J. Appl. Mech., 71(5), 604-614. https://doi.org/10.1115/1.1767170.
  31. Kapuria, S., and Kulkarni, S.D. (2008), "An efficient quadrilateral element based on improved zigzag theory for dynamic analysis of hybrid plates with electroded piezoelectric actuators and sensors", J. Sound Vibr., 315(1-2), 118-145. https://doi.org/10.1016/j.jsv.2008.01.053.
  32. Kapuria, S., and Kulkarni, S.D. (2009), "Static electromechanical response of smart composite/sandwich plates using an efficient finite element with physical and electric nodes", Int. J. Mech. Sci., 51(1), 1-20. https://doi.org/10.1016/j.ijmecsci.2008.11.005.
  33. Kapuria, S., Kumari, P., and Nath, J.K. (2010), "Efficient modeling of smart piezoelectric composite laminates: a review", Acta Mech., 214(1-2), 31-48. https://doi.org/10.1007/s00707-010-0310-0.
  34. Khandelwal, R.P., Chakrabarti, A., and Bhargava, P. (2013), "An efficient hybrid plate model for accurate analysis of smart composite laminates", J. Intell. Mater. Syst. Struct., 24(16), 1927-1950. https://doi.org/10.1177/1045389X13486713.
  35. Khandelwal, R.P., Chakrabarti, A., and Bhargava, P. (2014), "Static and dynamic control of smart composite laminates", AIAA J., 52(9), 1896-1914. https://doi.org/10.2514/1.J052666.
  36. Kulikov, G.M., and Plotnikova, S.V. (2013), "Three-dimensional exact analysis of piezoelectric laminated plates via a sampling surfaces method", Int. J. Solids Struct., 50(11-12), 1916-1929. https://doi.org/10.1016/j.ijsolstr.2013.02.015.
  37. Kulikov, G.M., and Plotnikova, S.V. (2017), "Benchmark solutions for the free vibration of layered piezoelectric plates based on a variational formulation", J. Intell. Mater. Syst. Struct., 28(19), 2688-2704. https://doi.org/10.1177/1045389X17698241.
  38. Lage, R.G., Soares, C.M., Soares, C.M., and Reddy, J. N. (2004), "Modelling of piezolaminated plates using layerwise mixed finite elements", Comput. Struct., 82(23-26), 1849-1863. https://doi.org/10.1016/j.compstruc.2004.03.068.
  39. Li, B., Fang, H., He, H., Yang, K., Chen, C., and Wang, F. (2019), "Numerical simulation and full-scale test on dynamic response of corroded concrete pipelines under Multi-field coupling", Constr. Build. Mater., 200, 368-386. https://doi.org/10.1016/j.conbuildmat.2018.12.111.
  40. Li, C., Song, C., Man, H., Ooi, E.T., and Gao, W. (2014), "2D dynamic analysis of cracks and interface cracks in piezoelectric composites using the SBFEM", Int. J. Solids Struct., 51(11-12), 2096-2108. https://doi.org/10.1016/j.ijsolstr.2014.02.014.
  41. Li, G., Dong, Z.Q., and Li, H.N. (2018), "Simplified Collapse-Prevention Evaluation for the Reserve System of Low-Ductility Steel Concentrically Braced Frames", J. Struct. Eng., 144(7), 04018071. https://doi.org/10.1061/(ASCE)ST.1943-541X.0002062.
  42. Li, J., Shi, Z., and Ning, S. (2017), "A two-dimensional consistent approach for static and dynamic analyses of uniform beams", Eng. Anal. Bound. Elem., 82, 1-16. https://doi.org/10.1016/j.enganabound.2017.05.009.
  43. Li, S., Huang, L., Jiang, L., and Qin, R. (2014), "A bidirectional B-spline finite point method for the analysis of piezoelectric laminated composite plates and its application in material parameter identification", Compos. Struct., 107, 346-362. https://doi.org/10.1016/j.compstruct.2013.08.007.
  44. Lin, G., Zhang, P., Liu, J., and Li, J. (2018), "Analysis of laminated composite and sandwich plates based on the scaled boundary finite element method", Compos. Struct., 187, 579-592. https://doi.org/10.1016/j.compstruct.2017.11.001.
  45. Liu, C.F., Chen, T.J., and Chen, Y.J. (2008), "A modified axisymmetric finite element for the 3-D vibration analysis of piezoelectric laminated circular and annular plates" J. Sound Vibr., 309(3-5), 794-804. https://doi.org/10.1016/j.jsv.2007.07.048.
  46. Mackerle, J. (2003), "Smart materials and structures-a finite element approach-an addendum: a bibliography (1997-2002)", Model. Simul. Mater. Sci. Eng., 11(5), 707-744. https://doi.org/10.1088/0965-0393/11/5/302.
  47. Man, H., Song, C., Gao, W., and Tin-Loi, F. (2012), "A unified 3D-based technique for plate bending analysis using scaled boundary finite element method", Int. J. Numer. Methods Eng., 91(5), 491-515. https://doi.org/10.1002/nme.4280.
  48. Man, H., Song, C., Xiang, T., Gao, W., and Tin-Loi, F. (2013), "High-order plate bending analysis based on the scaled boundary finite element method", Int. J. Numer. Methods Eng., 95(4), 331-360. https://doi.org/10.1002/nme.4519.
  49. Man, H., Song, C., Gao, W., and Tin-Loi, F. (2014), "Semi-analytical analysis for piezoelectric plate using the scaled boundary finite-element method", Comput. Struct., 137, 47-62. https://doi.org/10.1016/j.compstruc.2013.10.005.
  50. Mauritsson, K., and Folkow, P.D. (2015), "Dynamic equations for a fully anisotropic piezoelectric rectangular plate", Comput. Struct., 153, 112-125. https://doi.org/10.1016/j.compstruc.2015.02.023.
  51. Messina, A., and Carrera, E. (2015), "Three-dimensional free vibration of multi-layered piezoelectric plates through approximate and exact analyses", J. Intell. Mater. Syst. Struct., 26(5), 489-504. https://doi.org/10.1177/1045389X14529611.
  52. Messina, A., and Carrera, E. (2016), "Three-dimensional analysis of freely vibrating multilayered piezoelectric plates through adaptive global piecewise-smooth functions", J. Intell. Mater. Syst. Struct., 27(20), 2862-2876. https://doi.org/10.1177/1045389X16642303.
  53. Moleiro, F., Soares, C.M., Soares, C.M., and Reddy, J.N. (2012), "Assessment of a layerwise mixed least-squares model for analysis of multilayered piezoelectric composite plates", Comput. Struct., 108, 14-30. https://doi.org/10.1016/j.compstruc.2012.04.002.
  54. Moleiro, F., Soares, C.M., Soares, C.M., and Reddy, J.N. (2014), "Benchmark exact solutions for the static analysis of multilayered piezoelectric composite plates using PVDF", Compos. Struct., 107, 389-395. https://doi.org/10.1016/j.compstruct.2013.08.019.
  55. Moleiro, F., Soares, C.M., Soares, C.M., and Reddy, J. N. (2015), "Layerwise mixed models for analysis of multilayered piezoelectric composite plates using least-squares formulation", Compos. Struct., 119, 134-149. https://doi.org/10.1016/j.compstruct.2014.08.031.
  56. Moleiro, F., Araujo, A.L., and Reddy, J.N. (2017), "Benchmark exact free vibration solutions for multilayered piezoelectric composite plates", Compos. Struct., 182, 598-605. https://doi.org/10.1016/j.compstruct.2017.09.035.
  57. Pendhari, S.S., Sawarkar, S., and Desai, Y.M. (2015), "2D semi-analytical solutions for single layer piezoelectric laminate subjected to electro-mechanical loading", Compos. Struct., 120, 326-333. https://doi.org/10.1016/j.compstruct.2014.10.018.
  58. Plagianakos, T.S., and Papadopoulos, E.G. (2015), "Higher-order 2-D/3-D layerwise mechanics and finite elements for composite and sandwich composite plates with piezoelectric layers", Aerosp. Sci. Technol., 40, 150-163. https://doi.org/10.1016/j.ast.2014.10.015.
  59. Rezaiee-Pajand, M., and Sadeghi, Y. (2013), "A bending element for isotropic, multilayered and piezoelectric plates", Lat. Am. J. Solids Struct., 10(2), 323-348. http://dx.doi.org/10.1590/S1679-78252013000200006.
  60. Saravanos, D.A., Heyliger, P.R., and Hopkins, D.A. (1997), "Layerwise mechanics and finite element for the dynamic analysis of piezoelectric composite plates", Int. J. Solids Struct., 34(3), 359-378. https://doi.org/10.1016/S0020-7683(96)00012-1.
  61. Saravanos, D.A., and Heyliger, P.R. (1999), "Mechanics and computational models for laminated piezoelectric beams, plates, and shells", Appl. Mech. Rev., 52(10), 305-320. https://doi.org/10.1115/1.3098918.
  62. Sawarkar, S., Pendhari, S., and Desai, Y. (2016), "Semi-analytical solutions for static analysis of piezoelectric laminates", Compos. Struct., 153, 242-252. https://doi.org/10.1016/j.compstruct.2016.05.106.
  63. Shiyekar, S.M., and Kant, T. (2011), "Higher order shear deformation effects on analysis of laminates with piezoelectric fibre reinforced composite actuators", Compos. Struct., 93(12), 3252-3261. https://doi.org/10.1016/j.compstruct.2011.05.016.
  64. Singh, A.K., Chaki, M.S., Hazra, B., and Mahto, S. (2017), "Influence of imperfectly bonded piezoelectric layer with irregularity on propagation of Love-type wave in a reinforced composite structure", Struct. Eng. Mech., 62(3), 325-344. https://doi.org/10.12989/SEM.2017.62.3.325.
  65. Song, C., and Wolf, J.P. (1997), "The scaled boundary finite-element method-alias consistent infinitesimal finite-element cell method-for elastodynamics", Comput. Meth. Appl. Mech. Eng., 147(3-4), 329-355. https://doi.org/10.1016/S0045-7825(97)00021-2.
  66. Song, C., and Wolf, J.P. (1999), "The scaled boundary finite element method-alias consistent infinitesimal finite element cell method-for diffusion", Int. J. Numer. Methods Eng., 45(10), 1403-1431. https://doi.org/10.1002/(SICI)1097-0207(19990810)45:10<1403::AID-NME636>3.0.CO;2-E.
  67. Song, C., and Wolf, J.P. (2000), "The scaled boundary finite-element method-a primer: solution procedures", Comput. Struct., 78(1-3), 211-225. https://doi.org/10.1016/S0045-7949(00)00100-0.
  68. Song, C., and Vrcelj, Z. (2008), "Evaluation of dynamic stress intensity factors and T-stress using the scaled boundary finite-element method", Eng. Fract. Mech., 75(8), 1960-1980. https://doi.org/10.1016/j.engfracmech.2007.11.009.
  69. Song, C. (2009), "The scaled boundary finite element method in structural dynamics", Int. J. Numer. Methods Eng., 77(8), 1139-1171. https://doi.org/10.1002/nme.2454.
  70. Song, C., Tin-Loi, F., and Gao, W. (2010), "A definition and evaluation procedure of generalized stress intensity factors at cracks and multi-material wedges", Eng. Fract. Mech., 77(12), 2316-2336. https://doi.org/10.1016/j.engfracmech.2010.04.032.
  71. Tanzadeh, H., and Amoushahi, H. (2019), "Buckling and free vibration analysis of piezoelectric laminated composite plates using various plate deformation theories", Eur. J. Mech. A-Solids, 74, 242-256. https://doi.org/10.1016/j.euromechsol.2018.11.013.
  72. Torres, D.A.F., and Mendonca, P.T.R. (2010a), "Analysis of piezoelectric laminates by generalized finite element method and mixed layerwise-HSDT models", Smart Mater. Struct., 19(3), 035004. http://iopscience.iop.org/0964-1726/19/3/035004. https://doi.org/10.1088/0964-1726/19/3/035004
  73. Torres, D.A.F., and Paulo de Tarso, R.M. (2010b), "HSDT-layerwise analytical solution for rectangular piezoelectric laminated plates", Compos. Struct., 92(8), 1763-1774. https://doi.org/10.1016/j.compstruct.2010.02.007.
  74. Torres, D.A.F., Paulo de Tarso, R.M., and De Barcellos, C.S. (2011), "Evaluation and verification of an HSDT-Layerwise generalized finite element formulation for adaptive piezoelectric laminated plates", Comput. Meth. Appl. Mech. Eng., 200(5-8), 675-691. https://doi.org/10.1016/j.cma.2010.09.014.
  75. Vel, S.S., Mewer, R.C., and Batra, R.C. (2004), "Analytical solution for the cylindrical bending vibration of piezoelectric composite plates", Int. J. Solids Struct., 41(5-6), 1625-1643. https://doi.org/10.1016/j.ijsolstr.2003.10.012.
  76. Vidal, P., Gallimard, L., and Polit, O. (2016), "Modeling of piezoelectric plates with variables separation for static analysis", Smart Mater. Struct., 25(5), 055043. https://doi.org/10.1088/0964-1726/25/5/055043.
  77. Wang, J., and Yang, J. (2000), "Higher-order theories of piezoelectric plates and applications", Appl. Mech. Rev., 53(4), 87-99. https://doi.org/10.1115/1.3097341.
  78. Wolf, J.P., and Song, C. (2000), "The scaled boundary finite-element method-a primer: derivations", Comput. Struct., 78(1-3), 191-210. https://doi.org/10.1016/S0045-7949(00)00099-7.
  79. Wu, L., Jiang, Z., and Feng, W. (2004), "An analytical solution for static analysis of a simply supported moderately thick sandwich piezoelectric plate", Struct. Eng. Mech., 17(5), 641-654. https://doi.org/10.12989/SEM.2004.17.5.641.
  80. Wu, N., Wang, Q., and Quek, S.T. (2010), "Free vibration analysis of piezoelectric coupled circular plate with open circuit", J. Sound Vibr., 329(8), 1126-1136. https://doi.org/10.1016/j.jsv.2009.10.040.
  81. Xiang, T., Natarajan, S., Man, H., Song, C., and Gao, W. (2014), "Free vibration and mechanical buckling of plates with in-plane material inhomogeneity-A three dimensional consistent approach", Compos. Struct., 118, 634-642. https://doi.org/10.1016/j.compstruct.2014.07.043.
  82. Zhang, P., Qi, C., Fang, H., Ma, C., and Huang, Y. (2019), "Semi-analytical analysis of static and dynamic responses for laminated magneto-electro-elastic plates", Compos. Struct., 222, 110933. https://doi.org/10.1016/j.compstruct.2019.110933.
  83. Zhang, P., Qi, C., Fang, H., and He, W. (2020), "Three dimensional mechanical behaviors of in-plane functionally graded plates", Compos. Struct., 112124. https://doi.org/10.1016/j.compstruct.2020.112124.
  84. Zhang, Z., Feng, C., and Liew, K.M. (2006), "Three-dimensional vibration analysis of multilayered piezoelectric composite plates", Int. J. Eng. Sci., 44(7), 397-408. https://doi.org/10.1016/j.ijengsci.2006.02.002.
  85. Zhong, W.X., Lin, J.H., and Gao, Q. (2004), "The precise computation for wave propagation in stratified materials", Int. J. Numer. Methods Eng., 60(1), 11-25. https://doi.org/10.1002/nme.952.