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Micromechanical Model of a Polycrystalline Ferroelectrelastic Material with Consideration of Defects

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Journal of Applied Mechanics and Technical Physics Aims and scope

Abstract

Constitutive equations for describing the nonlinear behavior of a polycrystalline ferroelectroelastic material are proposed which take into account the dissipative movement of domain walls, the presence of point defects, and their effect on switching processes in the temperature range not accompanied by phase transitions. The method of two-level homogenization is used to describe the behavior of a polycrystalline ferroelectroelastic material at the macro-level. Accounting for defects in the micromechanical model of ferroelectroelastic materials has significantly improved the predictive ability of the model under multiaxial loading. Comparison of the results of computations with experimental data on dielectric hysteresis curves and switching surfaces under nonproportional loading of PZT-4D, PZT-5H, and BaTiO3 polycrystalline piezoelectric ceramics shows that the proposed model has good prediction accuracy.

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References

  1. G. A. Smolenskii and N. N. Krainik, Ferroelectrics and Antiferroelectrics (Nauka, Moscow, 1968) [in Russian].

    Google Scholar 

  2. M. E. Lines and A.M. Glass, Principles and Applications of Ferroelectrics and Related Materials (Oxford, New York, 1977).

    Google Scholar 

  3. S. A. Gridnev, “Ferroelastic Crystals: Basic Properties, Influence of Defects,” Priroda, No. 6, 22–29 (2002).

    Google Scholar 

  4. B. Jaffe, W. R. Cook, and H. Jaffe, Piezoelectric Ceramics (Academic Press, New York, 1971).

    Google Scholar 

  5. A. V. Belokon’ and A. S. Skaliukh, Mathematical Modeling of Irreversible Polarization Processes (Fizmatlit, Moscow, 2010) [in Russian].

    Google Scholar 

  6. L. Pardo and J. Ricote, Multifunctional Polycrystalline Ferroelectric Materials (Springer, Dordrecht, 2011).

    Book  Google Scholar 

  7. K. Worden, New Smart Materials and Structures (Tekhnosfera, Moscow, 2003) [Russian translation].

    Google Scholar 

  8. R. C. Smith, Smart Material Systems (SIAM, Philadelphia, 2005).

    Book  MATH  Google Scholar 

  9. K. Uchino, Ferroelectric Devices (CRC Press, New York, 2009).

    Google Scholar 

  10. C. M. Landis, “Fully Coupled, Multi-Axial, Symmetric Constitutive Laws for Polycrystalline Ferroelectric Ceramics,” J. Mech. Phys. Solids 50, 127–152 (2002).

    Article  ADS  MATH  Google Scholar 

  11. R. M. McMeeking and C. M. Landis, “A Phenomenological Multi-Axial Constitutive Law for Switching in Polycrystalline Ferroelectric Ceramics,” Int. J. Eng. Sci. 40, 1553–1577 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  12. M. Kamlah and C. Tsakmakis, “Phenomenological Modeling of the Non-Linear Electromechanical Coupling in Ferroelectrics,” Int. J. Solids Struct. 36, 669–695 (1999).

    Article  MATH  Google Scholar 

  13. M. Elhadrouz, T. B. Zineb, and E. Patoor, “Constitutive Law for Ferroelastic and Ferroelectric Piezoceramics,” J. Intell. Mat. Syst. Struct. 16, 221–236 (2005).

    Article  Google Scholar 

  14. S. C. Hwang, C. S. Lynch, and R. M. McMeeking, “Ferroelectric/Ferroelastic Interactions and a Polarization Switching Model,” Acta Metallurg. 43, 2073–2084 (1995).

    Article  Google Scholar 

  15. J. E. Huber and N. A. Fleck, “Multi-Axial Electrical Switching of a Ferroelectric: Theory Versus Experiment,” J. Mech. Phys. Solids 49, 785–811 (2001).

    Article  ADS  MATH  Google Scholar 

  16. A. Y. Belov and W. S. Kreher, “Viscoplastic Models for Ferroelectric Ceramics,” J. Europ. Ceram. Soc. 25, 2567–2571 (2005).

    Article  Google Scholar 

  17. A. V. Belokon’ and A. S. Skaliukh, “On the Constitutive Relations in Three-Dimensional Polarization Models,” Vestn. Perm. Gos. Tekh. Univ., Mat. Model. Sist. Prots., No. 16, 10–16 (2008).

    Google Scholar 

  18. J. E. Huber, N. A. Fleck, C. M. Landis, and R. M. McMeeking, “A Constitutive Model for Ferroelectric Polycrystals,” J. Mech. Phys. Solids 47, 1663–1697 (1999).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. C. M. Landis and R. M. McMeeking, “A Self-Consistent Constitutive Model for Switching in Polycrystalline Barium Titanate,” Ferroelectrics 255, 13–34 (2001).

    Article  Google Scholar 

  20. A. C. Liskowsky, A. S. Semenov, H. Balke, and R. M. McMeeking “Finite Element Modeling of the Ferroelectroelastic Material Behavior in Consideration of Domain Wall Motions,”in Coupled Nonlinear Phenomena-Modeling and Simulation for Smart, Ferroic and Multiferroic Materials: Proc. of the 2005 MRS Spring Meeting, San Francisco (USA), Mar. 29–31, 2005, Vol. 881E, CC4.2.

    Google Scholar 

  21. A. Pathak and R. M. McMeeking, “Three-Dimensional Finite Element Simulations of Ferroelectric Polycrystals under Electrical and Mechanical Loading,” J. Mech. Phys. Solids 56, 663–683 (2008).

    Article  ADS  MATH  Google Scholar 

  22. A. S. Semenov, H. Balke, and B. E. Melnikov, “Modeling of Polycrystalline Piezoelectric Ceramics by Finite Element Homogenization,” Mor. Intell. Tekhnol., No. S3, 106–112 (2011).

    Google Scholar 

  23. P. Neumeister and H. Balke, “Micromechanical Modelling of the Remanent Properties of Morphotropic PZT,” J. Mech. Phys. Solids 59, 1794–1807 (2011).

    Article  ADS  MATH  Google Scholar 

  24. A. S. Semenov, A. C. Liskowsky, and H. Balke, “Return Mapping Algorithms and Consistent Tangent Operators in Ferroelectroelasticity,” Int. J. Num. Meth. Eng. 81, 1298–1340 (2010).

    MathSciNet  MATH  Google Scholar 

  25. A. S. Semenov, A. C. Liskowsky, P. Neumeister, et al., “Effective Methods for Solving Nonlinear Boundary-Value Problems of Ferroelectroelasticity,” Mor. Intell. Tekhnol., No. 1, 55–61 (2010).

    Google Scholar 

  26. G. A. Smolenskii, V. A. Bokov, V. A. Isupov, et al., Physics of Ferroelectric Phenomena (Nauka, Leningrad, 1985) [in Russian].

    Google Scholar 

  27. V. A. Golovnin, A. A. Movchikova, B. B. Ped’ko, I. A. Kaplunov, and O. V. Malyshkina, Physical Foundations, Research Methods and Practical Application of Piezomaterials (Tekhnosphera, Moscow, 2013) [in Russian].

    Google Scholar 

  28. B. A. Strukov, “Phase Transitions in Ferroelectric Crystals with Defects,” Soros. Obraz. Zh., No. 12, 95–101 (1996).

    Google Scholar 

  29. L. He and D. Vanderbilt, “First-Principles Study of Oxygen-Vacancy Pinning of Domain Walls in PbTiO3,” Phys. Rev. B 68, 134103 (2003).

    Article  ADS  Google Scholar 

  30. E. T. Keve, K. L. Bye, P. W. Whipps, and A. D. Annis, “Structural Inhibition of Ferroelectric Switching in Triglycine Sulphate. 1. Additives,” Ferroelectrics 3 (1), 39–48 (1972).

    Article  Google Scholar 

  31. B. D. Laikhtman, “Bending Vibrations of Domain Walls and Dielectric Dispersion in Ferroelectrics,” Fiz. Tv. Tela 15 (1), 93–102 (1973).

    Google Scholar 

  32. G. E. Pike, W. L. Warren, D. Dimos, et al., “Voltage Offsets in (Pb,La)(Zr,Ti)O3 Thin Films,” Appl. Phys. Lett. 66 (4), 484–486 (1995).

    Article  ADS  Google Scholar 

  33. D. Damjanovic, “Hysteresis in Piezoelectric and Ferroelectric Materials,” Sci. Hysteresis 3, 337–465 (2005).

    MATH  Google Scholar 

  34. T. J. Yang, V. Gopalan, P. J. Swart, and U. Mohadeen, “Direct Observation of Pinning and Bowing of a Single Ferroelectric Domain Wall,” Phys. Rev. Lett. 82, 4106–4109 (1999).

    Article  ADS  Google Scholar 

  35. U. Robels and G. Arlt, “Domain Wall Clamping in Ferroelectrics by Orientation of Defects,” J. Appl. Phys. 73, 3454–3460 (1993).

    Article  ADS  Google Scholar 

  36. A. S. Sidorkin, Domain Structure in Ferroelectrics and Related Materials (Fizmatlit, Moscow, 2000) [in Russian].

    Google Scholar 

  37. J. Shieh, J. E. Huber, and N. A. Fleck, “An Evaluation of Switching Criteria for Ferroelectrics under Stress and Electric Field,” Acta Mater. 51, 6123–6137 (2003).

    Article  Google Scholar 

  38. D. Zhou, M. Kamlah, and B. Laskewitz, “Multi-Axial Non-Proportional Polarization Rotation Tests of Soft PZT Piezoceramics under Electric Field Loading,” Proc. SPIE 6170, 617009 (2006).

    Article  Google Scholar 

  39. A. P. Levanyuk and A. S. Sigov, Defects and Structural Phase Transitions (Gordon and Bridge, New York, 1987).

    Google Scholar 

  40. B. A. Strukov and A. P. Levaniuk, Physical Foundations of Ferroelectric Phenomena in Crystals (Fizmatlit, Moscow, 1995) [in Russian].

    Google Scholar 

  41. R. Müller, J. Schröder, and D. C. Lupascu, “Thermodynamic Consistent Modelling of Defects and Microstructures in Ferroelectrics,” GAMM-Mitteilung 31 (2), 133–150 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  42. K. M. Rabe, C. H. Ahn, and J.-M. Triscone, Modern Physics of Ferroelectrics: Essential Background Physics of Ferroelectrics: A Modern View (Binom, Moscow, 2015) [Russian translation].

    Google Scholar 

  43. J. Eshelby, Continual Theory of Dislocations (Izd. Inostr. Lit., Moscow, 1963) [Russian translation].

    Google Scholar 

  44. C. C. Wang, “A New Representation Theorem for Isotropic Functions: An Answer to Prof. G. F. Smith’s Criticism of My Papers on Representations for Isotropic Functions,” Arch. Rational Mech. Anal. 36 (3), 166–197 (1970).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. A. S. Semenov, “PANTOCRATOR Finite Element Software Focused on the Solution of Nonlinear Problems in Mechanics,” in Scientific and Technical Problems of Forecasting the Reliability and Durability of Structures and Methods for Their Solution, Proc. 5th Int. Conf., St. Petersburg, October 14–17, 2003 (St. Petersburg State Polytech. Univ., St. Petersburg, 2003), pp. 466–480.

    Google Scholar 

  46. C. M. Landis, “A New Finite-Element Formulation for Electromechanical Boundary Value Problems,” Int. J. Numer. Methods Eng. 55, 613–628 (2002).

    Article  MATH  Google Scholar 

  47. A. S. Semenov, H. Kessler, A. Liskowsky, and H. Balke, “On a Vector Potential Formulation for 3D Electromechanical FE Analysis,” Comm. Numer. Meth. Eng. 22, 357–375 (2006).

    Article  MATH  Google Scholar 

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  1. Peter the Great Saint Petersburg Polytechnic University, Saint Petersburg, 195251, Russia

    A. S. Semenov

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Original Russian Text © A.S. Semenov.

__________

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 60, No. 6, pp. 173–191, November-December, 2019.

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Semenov, A.S. Micromechanical Model of a Polycrystalline Ferroelectrelastic Material with Consideration of Defects. J Appl Mech Tech Phy 60, 1125–1140 (2019). https://doi.org/10.1134/S002189441906018X

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  • DOI: https://doi.org/10.1134/S002189441906018X

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