Unified solutions for piezoelectric bilayer cantilevers and solution modifications



Wang, Xianfeng;Shi, Zhifei

  • 투고 : 2014.10.31
  • 심사 : 2015.02.17
  • 발행 : 2015.11.25


Based on the theory of piezoelasticity, the static performance of a piezoelectric bilayer cantilever fully covered with electrodes on the upper and lower surfaces is studied. Three models are considered, i.e., the sensor model, the driving displacement model and the blocking force model. By establishing suitable boundary conditions and proposing an appropriate Airy stress function, the exact solutions for piezoelectric bilayer cantilevers are obtained, and the effect of ambient thermal excitation is taken into account. Since the layer thicknesses and material parameters are distinguished in different layers, this paper gives unified solutions for composite piezoelectric bilayer cantilevers including piezoelectric bimorph and piezoelectric heterogeneous bimorph, etc. For some special cases, the simplifications of the present results are compared with other solutions given by other researches based on one-dimensional constitutive equations, and some amendments have been found. The present investigation shows: (1) for a PZT-4 piezoelectric bimorph, the amendments of tip deflections induced by an end shear force, an end moment or an external voltage are about 19.59%, 23.72% and 7.21%, respectively; (2) for a PZT-4-Al piezoelectric heterogeneous bimorph with constant layer thicknesses, the amendments of tip deflections induced by an end shear force, an end moment or an external voltage are 9.85%, 11.78% and 4.07%, respectively, and the amendments of the electrode charges induced by an end shear force or an end moment are both 1.04%; (3) for a PZT-4-Al piezoelectric heterogeneous bimorph with different layer thicknesses, the maximum amendment of tip deflection approaches 23.72%, and the maximum amendment of electrode charge approaches 31.09%. The present solutions can be used to optimize bilayer devices, and the Airy stress function can be used to study other piezoelectric cantilevers including multi-layered piezoelectric cantilevers under corresponding loads.


piezoelectric;bimorph;unimorph;sensor;actuator;thermal effect


  1. Ashida, F. and Tauchert, T.R. (1997), "Temperature determination for a contacting body based on an inverse piezothermoelastic problem", Int. J. Solids Struct., 34(20), 2549-2561. https://doi.org/10.1016/S0020-7683(96)00135-7
  2. Ashida, F. and Tauchert, T.R. (1998), "Transient response of a piezothermoelastic circular disk under axisymmetric heating", Acta Mech., 128(1-2), 1-14. https://doi.org/10.1007/BF01463155
  3. Chen, Y. and Shi, Z.F. (2005a), "Double-layered piezothermoelastic hollow cylinder under thermal loading", Key Eng Mater., 302, 684-692.
  4. Chen, Y. and Shi, Z.F. (2005b), "Exact solutions of functionally gradient piezothermoelastic cantilevers and parameter identification", J. Intel. Mat. Syst. Str., 16(6), 531-539. https://doi.org/10.1177/1045389X05053208
  5. Erturk, A. (2011), "Piezoelectric energy harvesting for civil infrastructure system applications: Moving loads and surface strain fluctuations", J. Intel. Mat. Syst. Str., 22(17), 1959-1973. https://doi.org/10.1177/1045389X11420593
  6. Erturk, A. and Inman, D.J. (2009), "An experimentally validated bimorph cantilever model for piezoelectric energy harvesting from base excitations", Smart Mater Struct., 18(2), 25009. https://doi.org/10.1088/0964-1726/18/2/025009
  7. Gehring, G.A., Cooke, M.D., Gregory, I.S., Karl, W.J. and Watts, R. (2000), "Cantilever unified theory and optimization for sensors and actuators", Smart Mater Struct., 9(6), 918-931. https://doi.org/10.1088/0964-1726/9/6/324
  8. Hauke, T., Kouvatov, A., Steinhausen, R., Seifert, W., Beige, H. and Theo, H. et al. (2000), "Bending behavior of functionally gradient materials", Ferroelectrics, 238(1), 195-202. https://doi.org/10.1080/00150190008008784
  9. Kapuria, S. and Achary, G. (2005), "A coupled consistent third-order theory for hybrid piezoelectric plates", Compos Struct., 70(1), 120-133. https://doi.org/10.1016/j.compstruct.2004.08.018
  10. Kapuria, S., Bhattacharyya, M. and Kumar, A.N. (2006), "Assessment of coupled 1D models for hybrid piezoelectric layered functionally graded beams", Compos Struct., 72(4), 455-468. https://doi.org/10.1016/j.compstruct.2005.01.015
  11. Malgaca, L. and Karaguelle, H. (2009), "Simulation and experimental analysis of active vibration control of smart beams under harmonic excitation", Smart Struct. Syst., 5(1), 55-68. https://doi.org/10.12989/sss.2009.5.1.055
  12. Peng, W.Y., Xiao, Z.X. and Farmer, K.R. (2003), "Optimization of thermally actuated bimorph cantilevers for maximum deflection", Nanotechnology Conference and Trade Show (Nanotech 2003), San Francisco, USA, February.
  13. Ray, M.C. and Reddy, J.N. (2005), "Active control of laminated cylindrical shells using piezoelectric fiber reinforced composites", Compos Sci. Technol., 65(7-8), 1226-1236. https://doi.org/10.1016/j.compscitech.2004.12.027
  14. Schoeftner, J. and Irschik, H. (2011), "Passive shape control of force-induced harmonic lateral vibrations for laminated piezoelastic Bernoulli-Euler beams-theory and practical relevance", Smart Struct. Syst., 7(5), 417-432. https://doi.org/10.12989/sss.2011.7.5.417
  15. Shi, Z.F. (2002), "General solution of a density functionally gradient piezoelectric cantilever and its applications", Smart Mater Struct., 11(1), 122-129. https://doi.org/10.1088/0964-1726/11/1/314
  16. Shi, Z.F. (2005), "Bending behavior of piezoelectric curved actuator", Smart Mater Struct., 14(4), 835-842. https://doi.org/10.1088/0964-1726/14/4/043
  17. Smits, J.G. and Choi, W. (1991), "The constituent equations of piezoelectric heterogeneous bimorphs", IEEE T. Ultrason Ferr., 38(3), 256-270. https://doi.org/10.1109/58.79611
  18. Smits, J.G. and Choi, W. (1993), "Equations of state including the thermal domain of piezoelectric and pyroelectric heterogeneous bimorphs", Ferroelectrics, 141(1), 271-276. https://doi.org/10.1080/00150199308223454
  19. Smits, J.G., Dalke, S.I. and Cooney, T.K. (1991), "The constituent equations of piezoelectric bimorphs", Sensor Actuat. A-Phys., 28(1), 41-61. https://doi.org/10.1016/0924-4247(91)80007-C
  20. Tzou, H.S. and Bao, Y. (1995), "A theory on anisotropic piezothermoelastic shell laminates with sensor/actuator applications", J. Sound Vib., 184(3), 453-473. https://doi.org/10.1006/jsvi.1995.0328
  21. Tzou, H.S. and Howard, R.V. (1994), "A piezothermoelastic thin shell theory applied to active structures", J. Vib. Acoust., 116(3), 295-302. https://doi.org/10.1115/1.2930428
  22. Xiang, H.J. and Shi, Z.F. (2008), "Static analysis for multi-layered piezoelectric cantilevers", Int. J. Solids Struct., 45(1), 113-128. https://doi.org/10.1016/j.ijsolstr.2007.07.022
  23. Xiang, H.J. and Shi, Z.F. (2009), "Static analysis for functionally graded piezoelectric actuators or sensors under a combined electro-thermal load", Eur. J.Mech. A-Solid., 28(2), 338-346. https://doi.org/10.1016/j.euromechsol.2008.06.007
  24. Zhang, T.T. and Shi, Z.F. (2006), "Two-dimensional exact analysis for piezoelectric curved actuators", J Micromech. Microeng., 16(3), 640-647. https://doi.org/10.1088/0960-1317/16/3/020


연구 과제 주관 기관 : National Natural Science Foundation of China