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.
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References
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
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
Chen X., Fang D.N. and Hwang K.C. (1997). Micromechanics simulation of ferroelectric polarization switching. Acta Mater. 45: 3181–3189
Huo Y. and Jiang Q. (1998). Modeling of domain switching in ferroelektric ceramics: an example. Int. J. Solids Struct. 35: 1339–1353
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
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
Huber J.E. and Fleck N.A. (2004). Ferroelectric switching: a micromechanics model versus measured behaviour. Eur. J. Mech. A/Solids 23: 203–217
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
Chen W. and Lynch C.S. (1998). A micro-electro-mechanical model for polarization switching of ferroelectric materials. Acta Mater. 46: 5303–5311
Chen W. and Lynch C.S. (1999). Finite element analysis of cracks in ferroelectric ceramic materials. Eng. Fract. Mech. 64: 539–562
Steinkopff T. (1999). Finite-element modeling of ferroelectric domain switching in piezoelectric ceramics. J. Eur. Ceramic Soc. 19: 1247–1249
Chen P.J. and Peercy P.S. (1979). One dimensional dynamic electromechanical constitutive relations of ferroelectric materials. Acta Mech. 31: 231–241
Chen P.J. (1980). Three dimensional dynamic electromechanical constitutive relations for ferroelectric materials. Int. J. Solids Struct. 16: 1059–1067
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
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
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
Kamlah M. and Tsakmakis C. (1999). Phenomenological modeling of the non-linear electromechanical coupling in ferroelectrics. Int. J. Solids Struct. 36: 669–695
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
Kamlah M. and Wang Z. (2003). A thermodynamically and microscopically motivated constitutive model for piezo-ceramics. Comput. Mater. Sci. 28: 409–418
Cocks A.C.F. and McMeeking R.M. (1999). A phenomenological constitutive law for the behavior of ferroelectric ceramics. Ferroelectrics 228: 219–228
Landis C.M. (2002). Fully coupled, multi-axial, symmetric constitutive laws for polycrystalline ferroelectric ceramics. J. Mech. Phys. Solids 50: 127–152
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
Landis C.M. (2004). Non-linear constitutive modeling of ferroelectrics. Curr. Opin. Solid State Mater. Sci. 8: 59–69
Miller R.C. and Weinreich G. (1960). Mechanism for the sidewise motion of 180° domain walls in barium titanate. Phys. Rev. 117: 1460–1466
Hayashi M. (1972). Kinetics of domain wall motion in ferroelectric switching. I. General Formulation. J. Phys. Soc. Jpn. 33: 616–628
Zhong W.L. (1996). Physics of Ferroelectrics (in Chinese). Science Press, Beijing
Zhang Y. (2000). On the spontaneous configuration of ferroelectric-ferroelastic materials (in Chinese). Chin. J. Theor. Appl. Mech. 32(2): 213–222
Hannes K. and Herbert B. (2001). On the local and average energy release in polarization switching phenomena. J. Mech. Phys. Solids 49: 953–978
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
Lynch C.S. (1996). The effect of uniaxial stress on the electro-mechanical response of 8/65/35 PLZT. Acta Mater. 44: 4137–4148
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
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
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The project supported by the National Natural Science Foundation of China (10572138).
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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
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DOI: https://doi.org/10.1007/s10409-007-0095-0