JOURNAL BROWSE
Search
Advanced SearchSearch Tips
Nonlinear modelling and analysis of thin piezoelectric plates: Buckling and post-buckling behaviour
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
  • Journal title : Smart Structures and Systems
  • Volume 18, Issue 1,  2016, pp.155-181
  • Publisher : Techno-Press
  • DOI : 10.12989/sss.2016.18.1.155
 Title & Authors
Nonlinear modelling and analysis of thin piezoelectric plates: Buckling and post-buckling behaviour
Krommer, Michael; Vetyukova, Yury; Staudigl, Elisabeth;
 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.
 Keywords
piezoelastic plates;geometrical nonlinearity;buckling and post-buckling behaviour;nonlinear Finite Elements;
 Language
English
 Cited by
1.
Hybrid asymptotic–direct approach to finite deformations of electromechanically coupled piezoelectric shells, Acta Mechanica, 2017  crossref(new windwow)
2.
Finite deformations of thin plates made of dielectric elastomers: Modeling, numerics, and stability, Journal of Intelligent Material Systems and Structures, 2017, 1045389X1773305  crossref(new windwow)
 References
1.
Alkhatib, R. and Golnaraghi, M.F. (2003), "Active Structural Vibration Control: A Review", Shock Vib. Dig., 35(5), 367-383. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

8.
Crawley, E.F. (1994), "Intelligent structures for aerospace: A Technology Overview and Assessment", AIAA J., 32(8), 1689-1699. crossref(new window)

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. crossref(new window)

12.
Heuer, H., and Ziegler, F. (2004), "Thermoelastic stability of layered shallow shells", Int. J. Solids Struct., 41, 2111-2120. crossref(new window)

13.
Heuer, R. (1994), "Large flexural vibrations of thermally stressed layered shallow shells", Nonlinear Dynamics, 5(1), 25-38.

14.
Heuer, R., Irschik, H. and Ziegler, F. (1993), "Nonlinear random vibrations of thermally buckled skew plates", Probabilist. Eng. Mech., 8, 265-271. crossref(new window)

15.
Irschik, H. (1986), "Large thermoelastic deflections and stability of simply supported polygonal panels", Acta Mech., 59, 31-46. crossref(new window)

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. crossref(new window)

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. crossref(new window)

18.
Kamlah, M. (2001), "Ferroelectric and ferroelastic piezoceramics - modeling of electromechanical hysteresis phenomena", Continuum Mech. Therm., 13, 219-268. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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.

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. crossref(new window)

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. crossref(new window)

33.
Tani, J., Takagi, T., and Qiu, J. (1998), "Intelligent material systems: application of functional materials", Appl. Mech. Rev., 51, 505-521. crossref(new window)

34.
Tauchert, T.R. (1991), "Thermally induced flexure, buckling, and vibration", Appl. Mech. Rev., 44, 347-360. crossref(new window)

35.
Tauchert, T.R. (1992), "Piezothermoelastic Behavior of a Laminated Plate", J. Therm. Stresses, 15, 25-37. crossref(new window)

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. crossref(new window)

38.
Vetyukov, Y. (2014a), "Finite element modeling of Kirchhoff-Love shells as smooth material surfaces", ZAMM, 94, 150-163. crossref(new window)

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. crossref(new window)

41.
von Karman, T. and Tsien, H.S. (1941), "The buckling of thin cylindrical shells under axial compression", J. Aeronaut. Sci., 8, 303-312. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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.