Skip to main content
Log in

Energy principle of ferroelectric ceramics and single domain mechanical model

  • Research Paper
  • Published:
Acta Mechanica Sinica Aims and scope Submit manuscript

Abstract

Many physical experiments have shown that the domain switching in a ferroelectric material is a complicated evolution process of the domain wall with the variation of stress and electric field. According to this mechanism, the volume fraction of the domain switching is introduced in the constitutive law of ferroelectric ceramic and used to study the nonlinear constitutive behavior of ferroelectric body in this paper. The principle of stationary total energy is put forward in which the basic unknown quantities are the displacement u i , electric displacement D i and volume fraction ρ I of the domain switching for the variant I. Mechanical field equation and a new domain switching criterion are obtained from the principle of stationary total energy. The domain switching criterion proposed in this paper is an expansion and development of the energy criterion. On the basis of the domain switching criterion, a set of linear algebraic equations for the volume fraction ρ I of domain switching is obtained, in which the coefficients of the linear algebraic equations only contain the unknown strain and electric fields. Then a single domain mechanical model is proposed in this paper. The poled ferroelectric specimen is considered as a transversely isotropic single domain. By using the partial experimental results, the hardening relation between the driving force of domain switching and the volume fraction of domain switching can be calibrated. Then the electromechanical response can be calculated on the basis of the calibrated hardening relation. The results involve the electric butterfly shaped curves of axial strain versus axial electric field, the hysteresis loops of electric displacement versus electric filed and the evolution process of the domain switching in the ferroelectric specimens under uniaxial coupled stress and electric field loading. The present theoretic prediction agrees reasonably with the experimental results given by Lynch.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Hwang S.C., Lynch C.S. and McMeeking R.M. (1995). Ferroelectric/ferroelastic interactions and a polarization switching model. Acta Metall. Mater. 43: 2073–2084

    Article  Google Scholar 

  2. Lu W., Fang D.N. and Hwang K.C. (1999). Nonlinear electric-mechanical behavior and micormechanics modeling of ferroelectric domain evolution. Acta Mater. 47: 2913–2926

    Article  Google Scholar 

  3. Chen X., Fang D.N. and Hwang K.C. (1997). Micromechanics simulation of ferroelectric polarization switching. Acta Mater. 45: 3181–3189

    Article  Google Scholar 

  4. Huo Y. and Jiang Q. (1998). Modeling of domain switching in ferroelektric ceramics: an example. Int. J. Solids Struct. 35: 1339–1353

    Article  MATH  Google Scholar 

  5. Huber J.E., Fleck N.A., Landis C.M. and McMeeking R.M. (1999). A constitutive model for ferroelectric polycrystals. J. Mech. Phys. Solids 47: 1663–1697

    Article  MATH  MathSciNet  Google Scholar 

  6. Huber J.E. and Fleck N.A. (2001). Multi-axial electrical switching of a ferroelectric: theory versus experiment. J. Mech. Phys. Solids 49: 785–811

    Article  MATH  Google Scholar 

  7. Huber J.E. and Fleck N.A. (2004). Ferroelectric switching: a micromechanics model versus measured behaviour. Eur. J. Mech. A/Solids 23: 203–217

    Article  MATH  Google Scholar 

  8. Hwang S.C., Huber J.E., McMeeking R.M. and Fleck N.A. (1998). The simulation of switching in polycrystalline ferroelectric ceramics. J. Appl. Phys. 84: 1530–1540

    Article  Google Scholar 

  9. Chen W. and Lynch C.S. (1998). A micro-electro-mechanical model for polarization switching of ferroelectric materials. Acta Mater. 46: 5303–5311

    Article  Google Scholar 

  10. Chen W. and Lynch C.S. (1999). Finite element analysis of cracks in ferroelectric ceramic materials. Eng. Fract. Mech. 64: 539–562

    Article  Google Scholar 

  11. Steinkopff T. (1999). Finite-element modeling of ferroelectric domain switching in piezoelectric ceramics. J. Eur. Ceramic Soc. 19: 1247–1249

    Article  Google Scholar 

  12. Chen P.J. and Peercy P.S. (1979). One dimensional dynamic electromechanical constitutive relations of ferroelectric materials. Acta Mech. 31: 231–241

    Article  MATH  MathSciNet  Google Scholar 

  13. Chen P.J. (1980). Three dimensional dynamic electromechanical constitutive relations for ferroelectric materials. Int. J. Solids Struct. 16: 1059–1067

    Article  MATH  Google Scholar 

  14. Chen P.J. and Madsen M.M. (1981). One dimensional polar response of the electrooptic PLZT 7/65/35 due to domain switching. Acta Mech. 41: 255–264

    Article  Google Scholar 

  15. Bassiouny A.F., Ghaleb G. and Maugin G. (1988). Thermodynamical formulation for coupled electromechanical hysteresis effects—I basic equations. Int. J. Eng. Sci. 26: 1279–1295

    Article  MATH  MathSciNet  Google Scholar 

  16. Bassiouny A.F., Ghaleb G. and Maugin G. (1988). Thermodynamical formulation for coupled electromechanical hysteresis effects—II poling of ceramics. Int. J. Eng. Sci. 26: 1297–1306

    Article  MATH  MathSciNet  Google Scholar 

  17. Kamlah M. and Tsakmakis C. (1999). Phenomenological modeling of the non-linear electromechanical coupling in ferroelectrics. Int. J. Solids Struct. 36: 669–695

    Article  MATH  Google Scholar 

  18. Kamlah M. and Böhle U. (2001). Finite element analysis of piezoceramics components taking into account ferroelectric hysteresis behavior. Int. J. Solids Struct. 38: 605–633

    Article  MATH  Google Scholar 

  19. Kamlah M. and Wang Z. (2003). A thermodynamically and microscopically motivated constitutive model for piezo-ceramics. Comput. Mater. Sci. 28: 409–418

    Article  Google Scholar 

  20. Cocks A.C.F. and McMeeking R.M. (1999). A phenomenological constitutive law for the behavior of ferroelectric ceramics. Ferroelectrics 228: 219–228

    Article  Google Scholar 

  21. Landis C.M. (2002). Fully coupled, multi-axial, symmetric constitutive laws for polycrystalline ferroelectric ceramics. J. Mech. Phys. Solids 50: 127–152

    Article  MATH  Google Scholar 

  22. McMeeking R.M. and Landis C.M. (2002). A phenomenological multi-axial constitutive law for switching in polycrystalline ferroelectric ceramics. Int. J. Eng. Sci. 40: 1553–1577

    Article  MathSciNet  Google Scholar 

  23. Landis C.M. (2004). Non-linear constitutive modeling of ferroelectrics. Curr. Opin. Solid State Mater. Sci. 8: 59–69

    Article  Google Scholar 

  24. Miller R.C. and Weinreich G. (1960). Mechanism for the sidewise motion of 180° domain walls in barium titanate. Phys. Rev. 117: 1460–1466

    Article  Google Scholar 

  25. Hayashi M. (1972). Kinetics of domain wall motion in ferroelectric switching. I. General Formulation. J. Phys. Soc. Jpn. 33: 616–628

    Article  Google Scholar 

  26. Zhong W.L. (1996). Physics of Ferroelectrics (in Chinese). Science Press, Beijing

    Google Scholar 

  27. Zhang Y. (2000). On the spontaneous configuration of ferroelectric-ferroelastic materials (in Chinese). Chin. J. Theor. Appl. Mech. 32(2): 213–222

    Google Scholar 

  28. Hannes K. and Herbert B. (2001). On the local and average energy release in polarization switching phenomena. J. Mech. Phys. Solids 49: 953–978

    Article  MATH  Google Scholar 

  29. Li F.X. and Fang D.N. (2005). Effect of lateral pressure on the non-linear behaviour of PZT ceramics. Sci. China Ser. E 35(11): 1193–1201

    Google Scholar 

  30. Lynch C.S. (1996). The effect of uniaxial stress on the electro-mechanical response of 8/65/35 PLZT. Acta Mater. 44: 4137–4148

    Article  Google Scholar 

  31. Fang D.N., Mao G.Z. and Li F.X. (2005). Experimental study on electro-magneto-mechanical coupling behavior of smart materials. J. Mech. Strength 27(2): 217–226

    Google Scholar 

  32. Wang Q., Chen C.Q. and Shen Y.P. (2005). An experimental investigation into the complex electromechanical behavior of PZT53. Chin J. Theor. Appl. Mech. 37(4): 413–420

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to T. C. Wang.

Additional information

The project supported by the National Natural Science Foundation of China (10572138).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, F., Li, H. & Wang, T.C. Energy principle of ferroelectric ceramics and single domain mechanical model. Acta Mech Sin 23, 531–543 (2007). https://doi.org/10.1007/s10409-007-0095-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10409-007-0095-0

Keywords

Navigation