Coupling electro-mechanical behaviors in the interdigital electrode device of ferroelectrics


The electro-mechanical coupling behaviors of ferroelectric devices with interdigital electrodes may become complicated due to the material inhomogeneity and local field concentration under the complex working conditions. In this paper, a ferroelectric model, drawn from the typical interdigital electrode structure of a ferroelectric sensor, is established based on phase field theory, to study the polarization evolution and explore the evolution laws in ferroelectrics. Numerical results show that there appears ferroelectric creep even under an applied electric field below the coercive field value. Also, the configurational force theory is introduced to investigate the mechanical behaviors related to polarization switching in the ferroelectric samples with interdigital electrodes. It is found that configurational force and polarization have similar evolution laws in both time evolving and space distribution. And considering the configurational force as the driving force, it is possible to predict the potential direction of polarization evolution and explore its evolution mechanism in ferroelectrics, demonstrating the configurational force as a useful parameter for describing mechanical behavior during the polarization evolution and a powerful tool for investigating the evolution mechanism of microstructure with coupling effects in ferroelectric materials.

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  1. 1.

    Zhang, T.Y., Zhao, M.H., Tong, P.: Fracture of piezoelectric ceramics. Adv. Appl. Mech. 38, 147–289 (2002)

    Article  Google Scholar 

  2. 2.

    Chen, Y.H., Hasebe, N.: Current understanding on fracture behaviors of ferroelectric/piezoelectric materials. J. Intell. Mat. Syst. Str. 3, 673–687 (2006)

    Google Scholar 

  3. 3.

    Jiang, Y.J., Fang, D.N., Li, F.X.: In situ observation of electric-field-induced domain switching near a crack tip in poled PMNT62/38 single crystal. Appl. Phys. Lett. 90, 222907 (2007)

    Article  Google Scholar 

  4. 4.

    Scott, J.F.: Applications of modern ferroelectrics. Science 315, 954–959 (2007)

    Article  Google Scholar 

  5. 5.

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

    Article  Google Scholar 

  6. 6.

    Shu, Y.C., Bhattacharya, K.: Domain patterns and macroscopic behavior of ferroelectric materials. Philos. Mag. B 8, 2021–2054 (2001)

    Article  Google Scholar 

  7. 7.

    Zeng, X., Rajapakse, R.K.N.D.: Domain switching induced fracture toughness variation in ferroelectrics. Smart. Mater. Struct. 10, 203–211 (2001)

    Article  Google Scholar 

  8. 8.

    Sheng, J.S., Landis, C.M.: Toughening due to domain switching in single crystal ferroelectric materials. Int. J. Fract. 143, 161–175 (2007)

    Article  Google Scholar 

  9. 9.

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

    MathSciNet  Article  Google Scholar 

  10. 10.

    Mao, G.Z., Fang, D.N.: Fatigue crack growth induced by domain switching under electromechanical load in ferroelectrics. Theor. Appl. Fract. Mech. 41, 115–123 (2004)

    Article  Google Scholar 

  11. 11.

    Kamlah, M., Liskowsky, A., McMeeking, R., et al.: Finite element simulation of a polycrystalline ferroelectric based on a multidomain single crystal switching model. Int. J. Solids. Struct. 42, 2949–2964 (2005)

    Article  Google Scholar 

  12. 12.

    Li, F.X., Rajapakse, R.K.N.D.: A constrained domain-switching model for polycrystalline ferroelectric ceramics. Part I: Model formulation and application to tetragonal materials. Acta. Mater. 55, 6472–6480 (2007)

    Article  Google Scholar 

  13. 13.

    Xue, F., Wang, J.J., Sheng, G., et al.: Phase field simulations of ferroelectrics domain structures in PbZrxTi1xO3 bilayers. Acta. Mater. 61, 2909–2918 (2013)

    Article  Google Scholar 

  14. 14.

    Fedeli, P., Frangi, A., Auricchio, F., et al.: Phase-field modeling for polarization evolution in ferroelectric materials via an isogeometric collocation method. Comput. Methods Appl. Mech. Eng. 351, 789–807 (2019)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Ji, Y., Chen, W.J., Zheng, Y.: Crossover of polar and toroidal orders in ferroelectric nanodots with a morphotropic phase boundary and nonvolatile polar-vortex transformations. Phys. Rev. B 100, 014101.1-014101.13 (2019)

    MathSciNet  Google Scholar 

  16. 16.

    Muench, I., Renuka, B.A., Huber, J.E.: Periodic boundary conditions for the simulation of 3D domain patterns in tetragonal ferroelectric material. Arch. Appl. Mech. 89, 955–972 (2019)

    Article  Google Scholar 

  17. 17.

    Chen, L.Q.: Phase-field models for microstructure evolution. Annu. Rev. Mater. Res. 32, 113–140 (2002)

    Article  Google Scholar 

  18. 18.

    Zhang, W., Bhattacharya, K.: A computational model of ferroelectric domains. Part II: grain boundaries and defect pinning. Acta. Mater. 53, 199–209 (2005)

    Article  Google Scholar 

  19. 19.

    Schrade, D., Mueller, R., Xu, B.X., et al.: Domain evolution in ferroelectric materials: a continuum phase field model and finite element implementation. Comput. Methods. Appl. Mech. Eng. 196, 4365–4374 (2007)

    Article  Google Scholar 

  20. 20.

    Shu, W.L., Wang, J., Zhang, T.Y.: Effect of grain boundary on the electromechanical response of ferroelectric polycrystals. J. Appl. Phys. 6, 064108.1-064108.16 (2012)

    Google Scholar 

  21. 21.

    Yu, H.J., Wang, J., Kozinov, S., et al.: Phase field analysis of crack tip parameters in ferroelectric polycrystals under large-scale switching. Acta. Mater. 154, 334–342 (2018)

    Article  Google Scholar 

  22. 22.

    Nadgir, O., Dornisch, W., Mueller, R., et al.: A phase-field model for transversely isotropic ferroelectrics. Arch. Appl. Mech. 89, 1057–1068 (2019)

    Article  Google Scholar 

  23. 23.

    Song, Y.C., Soh, A.K., Ni, Y.: Phase field simulation of crack tip domain switching in ferroelectrics. J. Phys. D 40, 1175–1182 (2007)

    Article  Google Scholar 

  24. 24.

    Li, W.Y., Landis, C.M.: Phase-field modeling of domain switching near crack tips in single crystal ferroelectrics. Proc. SPIE. 6929, 69290J (2008)

    Article  Google Scholar 

  25. 25.

    Muller, R., Gross, D., Schrade, D., et al.: Phase field simulation of domain structures in ferroelectric materials within the context of inhomogeneity evolution. Int. J. Fract. 147, 173–180 (2007)

    Article  Google Scholar 

  26. 26.

    Wang, J., Kamlah, M.: Three-dimensional finite element modeling of polarization switching in a ferroelectric single domain with an impermeable notch. Smart. Mater. Struct. 18, 104008 (2009)

    Article  Google Scholar 

  27. 27.

    Li, Q., Pan, S.X., Liu, Q.D., et al.: Domain switching emission from the mixed mode crack in ferroelectrics by birefringence measurement and phase field modeling. Smart. Mater. Struct. 25, 07LT01 (2016)

    Article  Google Scholar 

  28. 28.

    Pan, S.X., Li, Q., Liu, Q.D.: Ferroelectric creep associated with domain switching emission in the cracked ferroelectrics. Comput. Mater. Sci. 140, 244–252 (2017)

    Article  Google Scholar 

  29. 29.

    Su, Y., Weng, G.J.: The frequency dependence of microstructure evolution in a ferroelectric nano-film during AC dynamic polarization switching. Acta. Mech. 229, 795–805 (2018)

    Article  Google Scholar 

  30. 30.

    Fan, Z.M., Xue, F., Tutuncu, G., et al.: Interaction dynamics between ferroelectric and antiferroelectric domains in a PbZrO3-based ceramic. Phys. Rev. Appl. 11, 064005.1-064005.7 (2019)

    Google Scholar 

  31. 31.

    Saha, A.K., Ni, K., Dutta, S., et al.: Phase field modeling of domain dynamics and polarization accumulation in ferroelectric HZO. Appl. Phys. Lett. 114, 202903.1-202903.6 (2019)

    Article  Google Scholar 

  32. 32.

    Van Lich, L., Bui, T.Q., Shimada, T., et al.: Deterministic switching of polarization vortices in compositionally graded ferroelectrics using a mechanical field. Phys. Rev. Appl. 11, 054001.1-054001.13 (2019)

    Google Scholar 

  33. 33.

    Eshelby, J.D.: The force on an elastic singularity. Philos. Trans. R. Soc. A 244, 87–112 (1951)

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Pak, Y.E.: Crack extension force in a piezoelectric material. Appl. Mech. 57, 647–653 (1990)

    Article  Google Scholar 

  35. 35.

    Mueller, R., Kolling, S., Gross, D.: On configurational forces in the context of the finite element method. Int. J. Numer. Meth. Eng. 53, 1557–1574 (2002)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Li, Q., Kuna, M.: Inhomogeneity and material configurational forces in three dimensional ferroelectric polycrystals. Eur. J. Mech. A 31, 77–89 (2012)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Xu, B.X., Schrade, D., Gross, D., et al.: Phase field simulation of domain structures in cracked ferroelectrics. Int. J. Fract. 165, 163–173 (2010)

    Article  Google Scholar 

  38. 38.

    Guo, Y.L., Li, Q.: Material configurational forces applied to mixed mode crack propagation. Theor. Appl. Fract. Mech. 89, 147–157 (2017)

    Article  Google Scholar 

  39. 39.

    Li, Q., Lv, J.N., Guo, Y.L., et al.: A consistent framework of material configurational mechanics in piezoelectric materials. Acta Mech. 229, 299–322 (2018)

    MathSciNet  Article  Google Scholar 

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The authors are grateful for the support provided by the National Natural Science Foundation of China (Grant No. 11772245).

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Correspondence to Qun Li.

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Pan, S., Xie, S. & Li, Q. Coupling electro-mechanical behaviors in the interdigital electrode device of ferroelectrics. Acta Mech. Sin. (2021).

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  • Phase field modeling
  • Configurational force
  • Polarization evolution
  • Domain switching
  • Interdigital electrode