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Constitutive modeling of polycrystalline multiconstituent and multiphase ferroic materials based on a condensed approach

  • A. Ricoeur
  • S. Lange
Original
  • 25 Downloads

Abstract

The constitutive behavior of polycrystalline ferroelectric or ferromagnetic materials is essentially determined by ferroelectric domain and ferromagnetic Bloch or Neel wall motions, respectively, on the one hand and grain interactions on the other. In physically motivated models, domain switching and the rotation of elementary magnets, respectively, are directly considered coping with the first issue. In phenomenological macroscale approaches, both domain processes and grain interactions are merged in one constitutive model. The implementation of constitutive equations into a finite element code provides intrinsic interactions due to the coupling of physical quantities on the node- and element-level. The condensed approach is based on microphysical considerations of domains and accounts for grain interaction on the level of generalized residual stresses due to mismatches of individual grains and the associated effective medium. The approach is first applied to the prediction of magnetoelectric coupling in a multiferroic ferroelectric–ferromagnetic compound, where its basic idea is translated to model interactions of the constituents without going back to any discretization scheme. Second, a tetragonal–rhombohedral ferroelectric composition is considered with associated interactions in multiphase grains.

Keywords

Multiferroics Ferroelectrics Morphotropic phase boundary Rhombohedral unit cell Magnetoelectric coupling Mechanical stress 

Abbreviations

CM

Condensed method

FE

Finite element/Ferroelectric

FM

Ferromagnetic

FEL

Ferroelectric material

FMA

Ferromagnetic material

MaMP

Macroscopic material point

MCC

Magnetoelectric coupling coefficient

ME

Magnetoelectric

MiMP

Microscopic material point

MPB

Morphotropic phase boundary

PZT

Lead zirconate titanate

RVE

Representative volume element

Notes

References

  1. 1.
    Abed-Meraim, F., Nguyen, Q.S.: A quasi-static stability analysis for Biot’s equation and standard dissipative systems. Eur. J. Mech. A/Solids 26, 383–393 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Avakian, A., Ricoeur, A.: Constitutive modeling of nonlinear reversible and irreversible ferromagnetic behaviors and application to multiferroic composites. J. Intell. Mater. Syst. Struct. 27(18), 2536–2554 (2016)CrossRefGoogle Scholar
  3. 3.
    Avakian, A., Gellmann, R., Ricoeur, A.: Nonlinear modeling and finite element simulation of magnetoelectric coupling and residual stress in multiferroic composites. Acta Mech. 226, 2789–2806 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Beom, H.G., Atluri, S.N.: Effect of electric fields on fracture behavior of ferroelectric ceramics. J. Mech. Phys. Solids 51, 1107–1125 (2003)CrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, W., Lynch, C.S.: A micro-electro-mechanical model for polarization switching of ferroelectric materials. Acta Mater. 46, 5303–5311 (1998)CrossRefGoogle Scholar
  6. 6.
    Franzbach, D.J., Seo, Y.H., Studer, A.J., Zhang, Y., Glaum, J., Daniels, J.E., Koruza, J., Benčan, A., Malič, B., Webber, K.G.: Electric-field-induced phase transitions in co-doped Pb(\(\text{ Zr }_{\text{1-x }}{\text{ T } \text{ i }}_{\text{ x }})\text{ O }_{3}\) at the morphotropic phase boundary. Sci. Technol. Adv. Mater. 15(1), 015,010 (2014)CrossRefGoogle Scholar
  7. 7.
    Gellmann, R., Ricoeur, A.: Continuum damage model for ferroelectric materials and its application to multilayer actuators. Smart Mater. Struct. 25, 055045 (2016)CrossRefGoogle Scholar
  8. 8.
    Hoffmann, M.J., Hammer, M., Endriss, A., Lupascu, D.C.: Correlation between microstructure, strain behavior and acoustic emission of soft PZT ceramics. Acta Mater. 49, 1301–1310 (2001)CrossRefGoogle Scholar
  9. 9.
    Huber, J.E., Fleck, N.A., Landis, C.M., McMeeking, R.M.: A constitutive model for ferroelectric polycrystals. J. Mech. Phys. Solids 47, 1663–1697 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hwang, S.C., Lynch, C.S., McMeeking, R.M.: Ferroelectric/ferroelastic interactions and a polarization switching model. Acta Metall. Mater. 43, 2073–2084 (1995)CrossRefGoogle Scholar
  11. 11.
    Kamlah, M., Liskowsky, A.C., McMeeking, R.M., Balke, H.: Finite element simulation of a polycrystalline ferroelectric based on a multidomain single crystal switching model. Int. J. Solids Struct. 42, 2949–2964 (2005)CrossRefzbMATHGoogle Scholar
  12. 12.
    Kellogg, R.A., Flatau, A., Clark, A.E., Wun-Fogle, M., Lograsso, T.: Quasi-static transduction characterization of galfenol. J. Intell. Mater. Syst. Struct. 16, 471–479 (2005)CrossRefGoogle Scholar
  13. 13.
    Kessler, H., Balke, H.: On the local and average energy release in polarization switching phenomena. J. Mech. Phys. Solids 49, 953–978 (2001)CrossRefzbMATHGoogle Scholar
  14. 14.
    Kungl, H., Hoffmann, M.J.: Temperature dependence of poling strain and strain under high electric fields in LaSr-doped morphotropic PZT and its relation to changes in structural characteristics. Acta Mater. 55(17), 5780–5791 (2007)CrossRefGoogle Scholar
  15. 15.
    Kungl, H., Theissmann, R., Knapp, M., Baehtz, C., Fuess, H., Wagner, S., Fett, T., Hoffmann, M.J.: Estimation of strain from piezoelectric effect and domain switching in morphotropic PZT by combined analysis of macroscopic strain measurements and synchrotron X-ray data. Acta Mater. 55(6), 1849–1861 (2007)CrossRefGoogle Scholar
  16. 16.
    Labusch, M., Etier, M., Lupascu, D.C., Schröder, J., Keip, M.A.: Product properties of a two-phase magneto-electric composite: synthesis and numerical modeling. Comput. Mech. 54, 71–83 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lange, S.: Untersuchung des konstitutiven Verhaltens ferroelektrischer, ferromagnetischer und multiferroischer Festkörper bei mehrachsiger multiphysikalischer Beanspruchung auf Basis kondensierter Modelle. PhD thesis, Universität Kassel, Kassel (2017)Google Scholar
  18. 18.
    Lange, S., Ricoeur, A.: A condensed microelectromechanical approach for modeling tetragonal ferroelectrics. Int. J. Solids Struct. 54, 100–110 (2015)CrossRefGoogle Scholar
  19. 19.
    Lange, S., Ricoeur, A.: High cycle fatigue damage and life time prediction for tetragonal ferroelectrics under electromechanical loading. Int. J. Solids Struct. 80, 181–192 (2016)CrossRefGoogle Scholar
  20. 20.
    Lee, J., Boyd, J.G., Lagoudas, D.C.: Effective properties of three-phase electro-magneto-elastic composites. Int J Eng Sci 43(10), 790–825 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    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 (2007a)CrossRefGoogle Scholar
  22. 22.
    Li, F.X., Rajapakse, R.K.N.D.: A constrained domain-switching model for polycrystalline ferroelectric ceramics. Part II: Combined switching and application to rhombohedral materials. Acta Mater. 55, 6481–6488 (2007b)CrossRefGoogle Scholar
  23. 23.
    Li, Q., Ricoeur, A., Enderlein, M., Kuna, M.: Evaluation of electromechanical coupling effect by microstructural modeling of domain switching in ferroelectrics. Mech. Res. Commun. 37, 332–336 (2010)CrossRefzbMATHGoogle Scholar
  24. 24.
    Magnetfabrik Bonn GmbH (2009) Datenblatt für AlNiCo 35/5Google Scholar
  25. 25.
    Michelitsch, T., Kreher, W.S.: A simple model for the nonlinear material behavior of ferroelectrics. Acta Mater. 46, 5085–5094 (1998)CrossRefGoogle Scholar
  26. 26.
    Ricoeur, A., Lange, S., Gellmann, R.: Modeling approaches to predict damage evolution and life time of brittle ferroelectrics. In: Hütter, G., Zybell, L. (eds.) Recent Trends in Fracture and Damage Mechanics, pp. 257–282. Springer, Berlin (2015)Google Scholar
  27. 27.
    Ricoeur, A., Avakian, A., Lange, S.: Microstructured multiferroic materials: modeling approaches towards efficiency and durability. In: Altenbach, H., Jablonski, F., Müller, W.H., Naumenko, K., Schneider, P. (eds.) Advances in Mechanics of Materials and Structural Analysis, pp. 297–328. Springer, Berlin (2018)CrossRefGoogle Scholar
  28. 28.
    Schneider, G.A.: Influence of electric field and mechanical stress on the fracture of ferroelectrics. Annu. Rev. Mater. Res. 37, 491–538 (2007)CrossRefGoogle Scholar
  29. 29.
    Scholehwar, T.: Charakterisierung der Struktur–Gefüge–Eigenschaftsbeziehung von piezokeramischen Werkstoffen des Systems PZT/SKN. PhD thesis, Technische Universität Dresden, Dresden (2010)Google Scholar
  30. 30.
    Schröder, J., Labusch, M., Keip, M.A., Kiefer, B., Brands, D., Lupascu, D.C.: Computation of non-linear magneto-electric product properties of 0–3 composites. GAMM-Mitteilung 38, 8–24 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of MechanicsUniversity of KasselKasselGermany

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