Advances in Computational Design

  • 제5권1호
  • /
  • Pages.55-89
  • /
  • 2020
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  • 2383-8477(pISSN)
  • /
  • 2466-0523(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Analytical free vibration solution for angle-ply piezolaminated plate under cylindrical bending: A piezo-elasticity approach

  • Singh, Agyapal (Department of Mechanical Engineering, Indian Institute of Technology Guwahati) ;
  • Kumari, Poonam (Department of Mechanical Engineering, Indian Institute of Technology Guwahati)
  • 투고 : 2019.05.21
  • 심사 : 2019.08.28
  • 발행 : 2020.01.25
⟨ 이전 논문 다음 논문 ⟩

초록

For the first time, an accurate analytical solution, based on coupled three-dimensional (3D) piezoelasticity equations, is presented for free vibration analysis of the angle-ply elastic and piezoelectric flat laminated panels under arbitrary boundary conditions. The present analytical solution is applicable to composite, sandwich and hybrid panels having arbitrary angle-ply lay-up, material properties, and boundary conditions. The modified Hamiltons principle approach has been applied to derive the weak form of governing equations where stresses, displacements, electric potential, and electric displacement field variables are considered as primary variables. Thereafter, multi-term multi-field extended Kantorovich approach (MMEKM) is employed to transform the governing equation into two sets of algebraic-ordinary differential equations (ODEs), one along in-plane (x) and other along the thickness (z) direction, respectively. These ODEs are solved in closed-form manner, which ensures the same order of accuracy for all the variables (stresses, displacements, and electric variables) by satisfying the boundary and continuity equations in exact manners. A robust algorithm is developed for extracting the natural frequencies and mode shapes. The numerical results are reported for various configurations such as elastic panels, sandwich panels and piezoelectric panels under different sets of boundary conditions. The effect of ply-angle and thickness to span ratio (s) on the dynamic behavior of the panels are also investigated. The presented 3D analytical solution will be helpful in the assessment of various 1D theories and numerical methods.

키워드

과제정보

연구 과제 주관 기관 : Science and Engineering Research Board

참고문헌

  1. Abad, F. and Rouzegar, J. (2019), "Exact wave propagation analysis of moderately thick Levytype plate with piezoelectric layers using spectral element method", Thin-Walled Struct., 141, 319-331. https://doi.org/10.1016/j.tws.2019.04.007.
  2. Abaqus, A. (2013), "6.13 Analysis Users Manual", SIMULIA, Providence, IR, USA.
  3. Balabaev, S. and Ivina, N. (2014), "A three-dimensional analysis of natural vibrations of rectangular piezoelectric transducers", Russian J. Nondestructive Testing, 50(10), 602-606. https://doi.org/10.1134/S1061830914100027.
  4. Barati, M.R. and Zenkour, A.M. (2018), "Electro-thermoelastic vibration of plates made of porous functionally graded piezoelectric materials under various boundary conditions", J. Vib. Control, 24(10), 1910-1926. https://doi.org/10.1177/1077546316672788.
  5. Batra, R. and Liang, X. (1997), "The vibration of a rectangular laminated elastic plate with embedded piezoelectric sensors and actuators", Comput. Struct., 63(2), 203-216. https://doi.org/10.1016/S0045-7949(96)00349-5.
  6. Behera, S. and Kumari, P. (2018), "Free vibration of Levy-type rectangular laminated plates using efficient zig-zag theory", Adv. Comput. Design, 3(3), 213-232. https://doi.org/10.12989/acd.2018.3.3.213.
  7. Behera, S. and Kumari, P. (2019), "Analytical Piezoelasticity Solution for Natural Frequencies of Levy-Type Piezolaminated Plates", J. Appl. Mech., 11(03), 1950023. https://doi.org/10.1142/S1758825119500236.
  8. Chen, W. and Ding, H. (2002), "On free vibration of a functionally graded piezoelectric rectangular plate", Acta Mechanica, 153(3-4), 207-216. https://doi.org/10.1007/BF01177452.
  9. Chen, W. and Lee, K.Y. (2004), "On free vibration of cross-ply laminates in cylindrical bending", J. Sound Vib., 3(273), 667-676. https://doi.org/10.1016/j.jsv.2003.08.003.
  10. Chen, W. and Lu, C. (2005), "3D free vibration analysis of cross-ply laminated plates with one pair of opposite edges simply supported", Compos. Struct., 69(1), 77-87. https://doi.org/10.1016/j.compstruct.2004.05.015.
  11. Cheng, Z.Q. and Batra, R. (2000a), "Three-dimensional asymptotic analysis of multiple-electroded piezoelectric laminates", AIAA J., 38(2), 317-324. https://doi.org/10.2514/2.959.
  12. Cheng, Z.Q. and Batra, R. (2000b), "Three-dimensional asymptotic scheme for piezothermoelastic laminates", J. Thermal Stresses, 23(2), 95-110. https://doi.org/10.1080/014957300280470.
  13. Cheng, Z.Q., Lim, C. and Kitipornchai, S. (1999), "Three-dimensional exact solution for inhomogeneous and laminated piezoelectric plates", J. Eng. Sci., 37(11), 1425-1439. https://doi.org/10.1016/S0020-7225(98)00125-6.
  14. Cheng, Z.Q., Lim, C. and Kitipornchai, S. (2000), "Three-dimensional asymptotic approach to inhomogeneous and laminated piezoelectric plates", J. Solids Struct., 37(23), 3153-3175. https://doi.org/10.1016/S0020-7683(99)00036-0.
  15. Dube, G., Kapuria, S. and Dumir, P. (1996a), "Exact piezothermoelastic solution of simplysupported orthotropic circular cylindrical panel in cylindrical bending", Archive Appl. Mech., 66(8), 537-554. https://doi.org/10.1007/BF00808143.
  16. Dube, G., Kapuria, S. and Dumir, P. (1996b), "Exact piezothermoelastic solution of simply supported orthotropic flat panel in cylindrical bending", J. Mech. Sci., 38(11), 1161-1177. https://doi.org/10.1016/0020-7403(96)00020-3.
  17. Ebrahimi, F. and Barati, M.R. (2016), "Size-dependent thermal stability analysis of graded piezomagnetic nanoplates on elastic medium subjected to various thermal environments", Appl. Phys., 122(10), 910. https://doi.org/10.1007/s00339-016-0441-9.
  18. Ebrahimi, F. and Barati, M.R. (2017a), "Damping vibration analysis of smart piezoelectric polymeric nanoplates on viscoelastic substrate based on nonlocal strain gradient theory", Smart Mater. Struct., 26(6), 065018. https://doi.org/10.1088/1361-665X/aa6eec.
  19. Ebrahimi, F. and Barati, M.R. (2017b), "Magnetic field effects on dynamic behavior of inhomogeneous thermo-piezo-electrically actuated nanoplates", J. Brazilian Soc. Mech. Sci. Eng., 39(6), 2203-2223. https://doi.org/10.1007/s40430-016-0646-z.
  20. Heyliger, P. and Brooks, S. (1995), "Free vibration of piezoelectric laminates in cylindrical bending", J. Solids Struct., 32(20), 2945-2960. https://doi.org/10.1016/0020-7683(94)00270-7.
  21. Heyliger, P. and Brooks, S. (1996), "Exact solutions for laminated piezoelectric plates in cylindrical bending", J. Appl. Mech., 63(4), 903-910. https://doi.org/10.1115/1.2787245
  22. Heyliger, P. and Saravanos, D. (1995), "Exact free-vibration analysis of laminated plates with embedded piezoelectric layers", J. Acoustical Soc. America, 98(3), 1547-1557. https://doi.org/10.1121/1.413420.
  23. Hussein, M. and Heyliger, P. (1998), "Three-dimensional vibrations of layered piezoelectric cylinders", J. Eng. Mech., 124(11), 1294-1298. https://doi.org/10.1061/(ASCE)0733-9399(1998)124:11(1294).
  24. Jones, A.T. (1970), "Exact natural frequencies for cross-ply laminates", J. Compos. Mater., 4(4), 476-491. https://doi.org/10.1177/002199837000400404.
  25. Jones, A.T. (1971), "Exact natural frequencies and modal functions for a thick off-axis lamina", J. Compos. Mater., 5(4), 504-520. https://doi.org/10.1177/002199837100500409.
  26. Kapuria, S. and Dhanesh, N. (2017), "Free edge stress field in smart piezoelectric composite structures and its control: An accurate multiphysics solution", J. Solids Struct., 126, 196-207. https://doi.org/10.1016/j.ijsolstr.2017.08.007.
  27. Kapuria, S. and Kumari, P. (2011), "Extended Kantorovich method for three-dimensional elasticity solution of laminated composite structures in cylindrical bending", J. Appl. Mech., 78(6), 061004. https://doi.org/10.1115/1.4003779.
  28. Kapuria, S. and Kumari, P. (2012), "Multiterm extended Kantorovich method for three-dimensional elasticity solution of laminated plates", J. Appl. Mech., 79(6), 061018. https://doi.org/10.1115/1.4006495.
  29. Kapuria, S. and Kumari, P. (2013), "Extended Kantorovich method for coupled piezoelasticity solution of piezolaminated plates showing edge effects", Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 469(2151), 20120565. https://doi.org/10.1098/rspa.2012.0565.
  30. Kapuria, S., Kumari, P. and Nath, J. (2010), "Efficient modeling of smart piezoelectric composite laminates: a review", Acta Mechanica, 214(1-2), 31-48. https://doi.org/10.1007/s00707-010-0310-0.
  31. Kerr, A.D. (1969), "An extended Kantorovich method for the solution of eigenvalue problems", J. Solids Struct., 5(6), 559-572. https://doi.org/10.1016/0020-7683(69)90028-6.
  32. Kerr, A.D. and Alexander, H. (1968), "An application of the extended Kantorovich method to the stress analysis of a clamped rectangular plate", Acta Mechanica, 6(2-3), 180-196. https://doi.org/10.1007/BF01170382.
  33. Khdeir, A. (2001), "Free and forced vibration of antisymmetric angle-ply laminated plate strips in cylindrical bending", J. Vib. Control, 7(6), 781-801. https://doi.org/10.1177/107754630100700602.
  34. Kim, J.S. (2007), "Free vibration of laminated and sandwich plates using enhanced plate theories", J. Sound Vib., 308(1-2), 268-286. https://doi.org/10.1016/j.jsv.2007.07.040.
  35. Kumari, P. and Behera, S. (2017a), "Three-dimensional free vibration analysis of levy-type laminated plates using multi-term extended Kantorovich method", Compos. Part B Eng., 116, 224-238. https://doi.org/10.1016/j.compositesb.2017.01.057.
  36. Kumari, P. and Behera, S. (2017b), "Three-dimensional free vibration analysis of levy-type laminated plates using multi-term extended Kantorovich method", Compos. Part B Eng., 116, 224-238. https://doi.org/10.1016/j.compositesb.2017.01.057.
  37. Kumari, P., Kapuria, S. and Rajapakse, R. (2014), "Three-dimensional extended Kantorovich solution for Levy-type rectangular laminated plates with edge effects", Compos. Struct., 107, 167-176. https://doi.org/10.1016/j.compstruct.2013.07.053.
  38. Kumari, P., Nath, J., Dumir, P. and Kapuria, S. (2007), "2D exact solutions for flat hybrid piezoelectric and magnetoelastic angle-ply panels under harmonic load", Smart Mater. Struct., 16(5), 1651. https://doi.org/10.1088/0964-1726/16/5/018.
  39. Kumari, P., Singh, A., Rajapakse, R. and Kapuria, S. (2017), "Three-dimensional static analysis of Levy-type functionally graded plate with in-plane stiffness variation", Compos. Struct., 168, 780-791. https://doi.org/10.1016/j.compstruct.2017.02.078.
  40. Messina, A. (2001), "Two generalized higher order theories in free vibration studies of multilayered plates", J. Sound Vib., 242(1), 125-150. https://doi.org/10.1006/jsvi.2000.3364.
  41. Messina, A. and Soldatos, K.P. (2002), "A general vibration model of angle-ply laminated plates that accounts for the continuity of interlaminar stresses", J. Solids Struct., 39(3), 617-635. https://doi.org/10.1016/S0020-7683(01)00169-X.
  42. Pan, E. and Heyliger, P.R. (2003), "Exact solutions for magneto-electro-elastic laminates in cylindrical bending", J. Solids Struct., 40(24), 6859-6876. https://doi.org/10.1016/j.ijsolstr.2003.08.003.
  43. Qing, G., Qiu, J. and Liu, Y. (2006), "A semi-analytical solution for static and dynamic analysis of plates with piezoelectric patches", J. Solids Struct., 43(6), 1388-1403. https://doi.org/10.1016/j.ijsolstr.2005.03.048.
  44. 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.
  45. Sayyad, A.S. and Ghugal, Y.M. (2015), "On the free vibration analysis of laminated composite and sandwich plates: A review of recent literature with some numerical results", Compos. Struct., 129, 177-201. https://doi.org/10.1016/j.compstruct.2015.04.007.
  46. Sheng, H., Wang, H. and Ye, J. (2007), "State space solution for thick laminated piezoelectric plates with clamped and electric open-circuited boundary conditions", J. Mech. Sci., 49(7), 806-818. https://doi.org/10.1016/j.ijmecsci.2006.11.012.
  47. Shu, X. (2005a), "Free vibration of laminated piezoelectric composite plates based on an accurate theory", Compos. Struct., 67(4), 375-382. https://doi.org/10.1016/j.compstruct.2004.01.022.
  48. Shu, X. (2005b), "Modelling of cross-ply piezoelectric composite laminates in cylindrical bending with interfacial shear slip", J. Mech. Sci., 47(11), 1673-1692. https://doi.org/10.1016/j.ijmecsci.2005.07.003.
  49. Singh, A., Kumari, P. and Hazarika, R. (2018), "Analytical Solution for Bending Analysis of Axially Functionally Graded Angle-Ply Flat Panels", Math. Problems Eng., 2018. https://doi.org/10.1155/2018/2597484.
  50. Singhatanadgid, P. and Singhanart, T. (2017), "The Kantorovich method applied to bending, buckling, vibration, and 3D stress analyses of plates: A literature review", Mech. Adv. Mater. Struct., pages 1-19. https://doi.org/10.1080/15376494.2017.1365984.
  51. Singhatanadgid, P. and Singhanart, T. (2019), "The Kantorovich method applied to bending, buckling, vibration, and 3D stress analyses of plates: A literature review", Mech. Adv. Mater. Struct., 26(2), 170-188. https://doi.org/10.1080/15376494.2017.1365984.
  52. Udayakumar, B. and Gopal, K.N. (2017), "A modified state space differential quadrature method for free vibration analysis of soft-core sandwich panels", J. Sandwich Struct. Mater., https://doi.org/10.1177/1099636217727801.
  53. Vel, S.S., Mewer, R. and Batra, R. (2004), "Analytical solution for the cylindrical bending vibration of piezoelectric composite plates", J. Solids Struct., 41(5-6), 1625-1643. https://doi.org/10.1016/j.ijsolstr.2003.10.012.
  54. Wu, C.P., Chiu, K.H. and Wang, Y.M. (2008), "A review on the three-dimensional analytical approaches of multilayered and functionally graded piezoelectric plates and shells", Comput. Mater. Continua, 8(2), 93- 132.
  55. Wu, C.P. and Liu, Y.C. (2016), "A review of semi-analytical numerical methods for laminated composite and multilayered functionally graded elastic/piezoelectric plates and shells", Compos. Struct., 147, 1-15. https://doi.org/10.1016/j.compstruct.2016.03.031.
  56. Xu, K., Noor, A.K. and Tang, Y.Y. (1997), "Three-dimensional solutions for free vibrations of initially-stressed thermoelectroelastic multilayered plates", Comput. Methods Appl. Mech. Eng., 141(1-2), 125-139. https://doi.org/10.1016/S0045-7825(96)01065-1.
  57. Yan, W., Lv, T., Lei, T. and Zhi, J. (2019), "Exact analysis of imperfect angle-ply laminated panels with surface-bonded piezoelectric layers", Arch. Appl. Mech., 1-12. https://doi.org/10.1007/s00419-018-01502-z.
  58. Yang, J., Batra, R. and Liang, X. (1994), "The cylindrical bending vibration of a laminated elastic plate due to piezoelectric actuators", Smart Mater. Struct., 3(4), 485. https://doi.org/10.1088/0964-1726/3/4/011.
  59. Yang, J., Batra, R.C. and Liang, X. (1995), "Vibration of a simply supported rectangular elastic plate due to piezoelectric actuators", Smart Structures and Materials 1995: Mathematics and Control in Smart Structures, Volume 2442, International Society for Optics and Photonics, Bellingham, WA, USA. 168-181.
  60. Zhang, Z., Feng, C. and Liew, K. (2006), "Three-dimensional vibration analysis of multilayered piezoelectric composite plates", J. Eng. Sci., 44(7), 397-408. https://doi.org/10.1016/j.ijengsci.2006.02.002.
  61. Zhou, Y., Chen, W. and Lu, C. (2010), "Semi-analytical solution for orthotropic piezoelectric laminates in cylindrical bending with interfacial imperfections", Compos. Struct., 92(4), 1009-1018. https://doi.org/10.1016/j.compstruct.2009.09.048.
  62. Zhou, Y., Chen, W., Lu, C. and Wang, J. (2009), "Free vibration of cross-ply piezoelectric laminates in cylindrical bending with arbitrary edges", Compos. Struct., 87(1), 93-100. https://doi.org/10.1016/j.compstruct.2008.01.002.