DOI QR코드

DOI QR Code

Nonlinear modelling and analysis of thin piezoelectric plates: Buckling and post-buckling behaviour

  • Krommer, Michael (Institute of Mechanics and Mechatronics, Vienna University of Technology) ;
  • Vetyukova, Yury (Institute of Mechanics and Mechatronics, Vienna University of Technology) ;
  • Staudigl, Elisabeth (Institute of Mechanics and Mechatronics, Vienna University of Technology)
  • Received : 2015.09.26
  • Accepted : 2016.05.06
  • Published : 2016.07.25

Abstract

In the present paper we discuss the stability and the post-buckling behaviour of thin piezoelastic plates. The first part of the paper is concerned with the modelling of such plates. We discuss the constitutive modelling, starting with the three-dimensional constitutive relations within Voigt's linearized theory of piezoelasticity. Assuming a plane state of stress and a linear distribution of the strains with respect to the thickness of the thin plate, two-dimensional constitutive relations are obtained. The specific form of the linear thickness distribution of the strain is first derived within a fully geometrically nonlinear formulation, for which a Finite Element implementation is introduced. Then, a simplified theory based on the von Karman and Tsien kinematic assumption and the Berger approximation is introduced for simply supported plates with polygonal planform. The governing equations of this theory are solved using a Galerkin procedure and cast into a non-dimensional formulation. In the second part of the paper we discuss the stability and the post-buckling behaviour for single term and multi term solutions of the non-dimensional equations. Finally, numerical results are presented using the Finite Element implementation for the fully geometrically nonlinear theory. The results from the simplified von Karman and Tsien theory are then verified by a comparison with the numerical solutions.

Acknowledgement

Supported by : Linz Center of Mechatronics (LCM)

References

  1. Alkhatib, R. and Golnaraghi, M.F. (2003), "Active Structural Vibration Control: A Review", Shock Vib. Dig., 35(5), 367-383. https://doi.org/10.1177/05831024030355002
  2. Arefi, M. and Rahimi, G.H. (2012), "Studying the nonlinear behavior of the functionally graded annular plates with piezoelectric layers as a sensor and actuator under normal pressure", Smart Struct. Syst., 9(2), 127-143. https://doi.org/10.12989/sss.2012.9.2.127
  3. Ashwell, D.G. (1962), "Nonlinear problems", in Handbook of Engineering Mechanics, (Ed., W. Flugge), McGraw Hill, New York, NY, USA.
  4. Batra, R.C. and Vidoli, S. (2002), "Higher order piezoelectric plate theory derived from a three dimensional variational principle", AIAA J, 40, 91-104. https://doi.org/10.2514/2.1618
  5. Berger, H.M. (1955), "A new approach to the analysis of large deflections of plates", J. Appl. Mech. - ASCE, 77, 465-472.
  6. Bonet, J. and Wood, R.D. (2008), Nonlinear Continuum Mechanics for Finite Element Analysis, 2nd Ed., Cambridge University Press, Cambridge, England.
  7. Carrera, E. and Boscolo, M. (2007), "Classical and mixed finite elements for static and dynamic analysis of piezoelectric plates", Int. J. Numer Meth. Eng., 70(10), 1135-1181. https://doi.org/10.1002/nme.1901
  8. Crawley, E.F. (1994), "Intelligent structures for aerospace: A Technology Overview and Assessment", AIAA J., 32(8), 1689-1699. https://doi.org/10.2514/3.12161
  9. Dorfmann, A, and Ogden, R.W. (2005), "Nonlinear Electroelasticity", Acta Mech., 17, 167-183.
  10. Eringen, A.C. and Maugin, G.A. (1990), Electrodynamics of Continua I: Foundations and Solid Media, Springer, New York, NY, USA.
  11. Hause, T., Librescu, L. and Johnson, T.F. (1998), "Thermomechanical load-carrying capacity of sandwich flat panels", J. Therm. Stresses, 21(6), 627-653. https://doi.org/10.1080/01495739808956166
  12. Heuer, H., and Ziegler, F. (2004), "Thermoelastic stability of layered shallow shells", Int. J. Solids Struct., 41, 2111-2120. https://doi.org/10.1016/j.ijsolstr.2003.11.032
  13. Heuer, R. (1994), "Large flexural vibrations of thermally stressed layered shallow shells", Nonlinear Dynamics, 5(1), 25-38. https://doi.org/10.1007/BF00045078
  14. Heuer, R., Irschik, H. and Ziegler, F. (1993), "Nonlinear random vibrations of thermally buckled skew plates", Probabilist. Eng. Mech., 8, 265-271. https://doi.org/10.1016/0266-8920(93)90020-V
  15. Irschik, H. (1986), "Large thermoelastic deflections and stability of simply supported polygonal panels", Acta Mech., 59, 31-46. https://doi.org/10.1007/BF01177058
  16. Jabbaria, M., Farzaneh Joubaneha, E., Khorshidvanda A.R. and Eslamib, M.R. (2013), "Buckling analysis of porous circular plate with piezoelectric actuator layers under uniform radial compression", Int. J. Mech. Sci., 70, 50-56. https://doi.org/10.1016/j.ijmecsci.2013.01.031
  17. Jadhav, P.A. and Bajoria, K.M. (2012), "Buckling of piezoelectric functionally graded plate subjected to electro-mechanical loading", Smart Mat. Struct., 21(10), 105005. https://doi.org/10.1088/0964-1726/21/10/105005
  18. Kamlah, M. (2001), "Ferroelectric and ferroelastic piezoceramics - modeling of electromechanical hysteresis phenomena", Continuum Mech. Therm., 13, 219-268. https://doi.org/10.1007/s001610100052
  19. Klinkel, S. and Wagner, W. (2006), "A geometrically non-linear piezoelectric solid shell element based on a mixed multi-field variational formulation", Int. J. Numer Meth. Eng., 65, 349-382. https://doi.org/10.1002/nme.1447
  20. Klinkel, S. and Wagner, W. (2008), "A piezoelectric solid shell element based on a mixed variational formulation for geometrically linear and nonlinear applications", Comput. Struct., 86, 38-46. https://doi.org/10.1016/j.compstruc.2007.05.032
  21. Krommer, M. (2003), "The significance of non-local constitutive relations for composite thin plates including piezoelastic layers with prescribed electric charge", Smart Mater. Struct., 12(3), 318-330. https://doi.org/10.1088/0964-1726/12/3/302
  22. Krommer, M. and Irschik, H. (2015), "Post-buckling of piezoelectric thin plates", Int. J. Str. Stab. Dyn., 15(7), 1540020, 21pp.
  23. Lentzen, S., Klosowski, P. and Schmidt, R. (2007), "Geometrically nonlinear finite element simulation of smart piezolaminated plates and shells", Smart Mater. Struct., 16, 2265-2274. https://doi.org/10.1088/0964-1726/16/6/029
  24. Liu, S.C., Tomizuka, M. and Ulsoy, G. (2005), "Challenges and opportunities in the engineering of intelligent structures", Smart Struct. Syst., 1(1), 1-12. https://doi.org/10.12989/sss.2005.1.1.001
  25. Marcus, H. (1932), Die Theorie elastischer Gewebe, 2nd edn., Springer, Berlin, Germany.
  26. Marinkovic, D., Koppe, H. and Gabbert, U. (2007), "Accurate modeling of the electric field within piezoelectric layers for active composite structures", J. Intel. Mat. Syst. Str., 18, 503-513. https://doi.org/10.1177/1045389X06067139
  27. Marinkovic, D., Koppe, H. and Gabbert, U. (2008), "Degenerated shell element for geometrically nonlinear analysis of thin-walled piezoelectric active structures", Smart Mat. Struct., 17(1), 10pp.
  28. Nader, M. (2008), Compensation of Vibrations in Smart Structures: Shape Control, Experimental Realization and Feedback Control, Trauner, Linz, Austria.
  29. Nestorovic, T., Trajkov, M. and Garmabi, S. (2015), "Optimal placement of piezoelectric actuators and sensors on a smart beam and a smart plate using multi-objective genetic algorithm", Smart Struct. Syst., 14(5), 1041-1062.
  30. Panahandeh-Shahraki, D., Mirdamadi H.R. and Vaseghi, O. (2014), "Thermoelastic buckling analysis of laminated piezoelectric composite plates", Int. J. Mech. Mater. Des., 11(4), 371-385. https://doi.org/10.1007/s10999-014-9284-8
  31. Stanciulescu, I., Mitchell, T., Chandra, Y., Eason T. and Spottswood, M. (2012), "A lower bound on snap-through instability of curved beams under thermomechanical loads", Int. J. Nonlinear Mech., 47(5), 561-575. https://doi.org/10.1016/j.ijnonlinmec.2011.10.004
  32. Tan, X. and Vu-Quoc, L. (2005), "Optimal solid shell element for large deformable composite structures with piezoelectric layers and active vibration control", Int. J. Numer Meth. Eng., 64, 1981-2013. https://doi.org/10.1002/nme.1433
  33. Tani, J., Takagi, T., and Qiu, J. (1998), "Intelligent material systems: application of functional materials", Appl. Mech. Rev., 51, 505-521. https://doi.org/10.1115/1.3099019
  34. Tauchert, T.R. (1991), "Thermally induced flexure, buckling, and vibration", Appl. Mech. Rev., 44, 347-360. https://doi.org/10.1115/1.3119508
  35. Tauchert, T.R. (1992), "Piezothermoelastic Behavior of a Laminated Plate", J. Therm. Stresses, 15, 25-37. https://doi.org/10.1080/01495739208946118
  36. Troger, H. and Steindl, A. (1991), Nonlinear Stability and Bifurcation Theory, An Introduction for Engineers and Applied Scientists, Springer, Vienna, Austria.
  37. Varelis, D. and Saravanos, D.A. (2002), "Nonlinear coupled mechanics and initial buckling of composite plates with piezoelectric actuators and sensors", Smart Mat. Struct., 11, 330-336. https://doi.org/10.1088/0964-1726/11/3/302
  38. Vetyukov, Y. (2014a), "Finite element modeling of Kirchhoff-Love shells as smooth material surfaces", ZAMM, 94, 150-163. https://doi.org/10.1002/zamm.201200179
  39. Vetyukov, Y. (2014b), Nonlinear Mechanics of Thin-Walled Structures: Asymptotics, Direct Approach and Numerical Analysis, Springer, Vienna, Austria.
  40. Vetyukov, Y., Kuzin, A. and Krommer, M. (2011), "Asymptotic splitting in the three-dimensional problem of elasticity for non-homogeneous piezoelectric plates", Int. J. Solids Struct., 48, 12-23. https://doi.org/10.1016/j.ijsolstr.2010.09.001
  41. von Karman, T. and Tsien, H.S. (1941), "The buckling of thin cylindrical shells under axial compression", J. Aeronaut. Sci., 8, 303-312. https://doi.org/10.2514/8.10722
  42. Wu, C.P. and Ding, S. (2015), "Coupled electro-elastic analysis of functionally graded piezoelectric material plates", Smart Struct. Syst., 16(5), 781-806. https://doi.org/10.12989/sss.2015.16.5.781
  43. Yaghoobi, H. and Rajabi, I. (2013), "Buckling analysis of three-layered rectangular plate with piezoelectric layers", J. Theor. Appl. Mech., 51(4), 813-826.
  44. Zenz, G., Berger, W., Gerstmayr, J., Nader, M. and Krommer, M. (2013), "Design of piezoelectric transducer arrays for passive and active modal control of thin plates", Smart Struct. Syst., 12(5), 547-577. https://doi.org/10.12989/sss.2013.12.5.547
  45. Zheng, S., Wang, X. and Chen, W. (2004), "The formulation of a refined hybrid enhanced assumed strain solid shell element and its application to model smart structures containing distributed piezoelectric sensors/ actuators", Smart Mater. Struct., 13, 43-50. https://doi.org/10.1088/0964-1726/13/4/N02
  46. Ziegler, F. (1998), Mechanics of Solids and Fluids, 2nd edn., Springer, New York, NY, USA.
  47. Ziegler, F. and Rammerstorfer, F.G. (1989), "Thermoelastic stability", in Thermal Stresses III, (Es., R.B. Hetnarski), Elsevier, Amsterdam, The Netherlands.

Cited by

  1. Hybrid asymptotic–direct approach to finite deformations of electromechanically coupled piezoelectric shells 2017, https://doi.org/10.1007/s00707-017-2046-6
  2. Finite deformations of thin plates made of dielectric elastomers: Modeling, numerics, and stability 2017, https://doi.org/10.1177/1045389X17733052
  3. Large deformation mixed finite elements for smart structures pp.1537-6532, 2019, https://doi.org/10.1080/15376494.2018.1536932