Abstract
The coupling of magnetic and electric fields due to the constitutive behavior of a material is commonly denoted as magnetoelectric (ME) effect. The latter is only observed in a few crystal classes exhibiting a very weak coupling which can hardly be exploited for technical applications. Much larger coupling coefficients are obtained in composite materials, where ferroelectric and ferromagnetic constituents are embedded in a matrix. The ME effect is then induced by the strain of the matrix converting electrical and magnetic energies based on the ferroelectric and magnetostrictive effects. In this paper, the theoretical background of linear and nonlinear constitutive multifield behavior as well as the finite element implementation is presented. A nonlinear material model describing the magnetoferroelectric behavior is presented. On this basis, polarization switching in the ferroelectric phase is simulated and the influence on stress distribution and ME coupling is analyzed. Numerical homogenization is performed, in order to supply ME-coupling constants, which are compared to experimental results for both the perfect poling state of a linear calculation and the more realistic case of nonlinear magnetoferroelectricity. The numerical tools supply useful means for the optimization of multiferroic composites with respect to strength and functionality.
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Lu X.Y., Li H., Wang B.: Theoretical analysis of electric, magnetic and magnetoelectric properties of nano-structured multiferroic composites. J. Mech. Phys. Solids 59(10), 1966–1977 (2011)
Lu X.Y., Wang B., Zheng Y., Ryba E.: Phenomenological theory of 1–3 type multiferroic composite thin film: thickness effect. J. Phys. D: Appl. Phys. 42(1), 15309 (2009)
Ma F.D., Jin Y.M., Wang Y.U., Kampe S.L., Dong S.: Effect of magnetic domain structure on longitudinal and transverse magnetoelectric response of particulate magnetostrictive–piezoelectric composites. Appl. Phys. Lett. 104(11), 112903 (2014)
Fina I., Dix N., Rebled J. M., Gemeiner P., Martí X., Peiró F., Dkhil B., Sánchez F., Fà àbrega L., Fontcuberta J.: The direct magnetoelectric effect in ferroelectric–ferromagnetic epitaxial heterostructures. Nanoscale 5(17), 8037 (2013)
Auslender M., Liverts E., Zadov B., Elmalem A., Zhdanov A., Grosz A., Paperno E.: Inverse effect of magnetostriction in magnetoelectric laminates. Appl. Phys. Lett. 103(2), 22907 (2013)
Scott J.F.: Data storage: multiferroic memories. Nat. Mater. 6(4), 256–257 (2007)
Fiebig M.: Revival of the magnetoelectric effect. J. Phys. D: Appl. Phys. 38(8), R123–R152 (2005)
Aboudi J.: Micromechanical analysis of fully coupled electro-magneto-thermo-elastic multiphase composites. Smart Mater. Struct. 10(5), 867–877 (2001)
Buchanan G.R.: Layered versus multiphase magneto-electro-elastic composites. Compos. Part B: Eng. 35(5), 413–420 (2004)
Huang J.H., Kuo W.-S.: The analysis of piezoelectric/piezomagnetic composite materials containing ellipsoidal inclusions. J. Appl. Phys. 81(3), 1378–1386 (1997)
Kuo H.-Y., Slinger A., Bhattacharya K.: Optimization of magnetoelectricity in piezoelectric–magnetostrictive bilayers. Smart Mater. Struct. 19(12), 125010–125022 (2010)
Li J.Y., Dunn M.L.: Micromechanics of magnetoelectroelastic composite materials: average fields and effective behavior. J. Int. Mat. Syst. Struct. 9(6), 404–416 (1998)
Nan C.-W.: Magnetoelectric effect in composites of piezoelectric and piezomagnetic phases. Phys. Rev. B 50(9), 6082–6088 (1994)
Lee J.S., Boyd J.G., Lagoudas D.C.: Effective properties of three-phase electro-magneto-elastic composites. Int. J. Eng. Sci. 43(10), 790–825 (2005)
Schröder J., Keip M.-A.: Two-scale homogenization of electromechanically coupled boundary value problems. Comput. Mech. 50(2), 229–244 (2012)
Miehe C., Rosato D., Kiefer B.: Variational principles in dissipative electro-magneto-mechanics: a framework for the macro-modeling of functional materials. Int. J. Numer. Methods Eng. 86(10), 1225–1276 (2011)
Moulson A.J., Herbert J.M.: Electroceramics. Materials, Properties, Applications. Wiley, Chichester, Hoboken (2003)
Hwang S.C., Lynch C.S., McMeeking R.M.: Ferroelectric/ferroelastic interactions and a polarization switching model. Acta Metall. Mater. 43(5), 2073–2084 (1995)
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(9–10), 2949–2964 (2005)
Arlt G., Sasko P.: Domain configuration and equilibrium size of domains in BaTiO3 ceramics. J. Appl. Phys. 51, 4956–4960 (1980)
Martin, H.-J.: Die Ferroelektrika. Akademische Verlagsgesellschaft Geest & Portig KG, Leipzig (1964)
Merz W.J.: The electrical and optical behavior of BaTiO3 single-domain crystals. Phys. Rev. 76(8), 1221–1225 (1949)
Merz W.J.: Domänenbildung und Domänenwandbewegung in ferroelektrischem BaTiO3. Phys. Rev. 95, 690–698 (1954)
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(3), 332–336 (2010)
Landau L.D., Lifschitz E.M.: Electrodynamics of Continua. Textbook of Theoretical Physics, vol. 8. Akademie, Berlin (1985)
Jackson J.D.: Classical Electrodynamics. Wiley, New York (1998)
Parton V.Z., Kudryavtsev B.A.: Electromagnetoelasticity, Piezoelectrics and Electrically Conductive Solids. Gordon and Breach Science Publishers, New York (1988)
Bathe K.-J.: Finite Elemente Methoden. Prentice Hall, Berlin (2006)
Katz V.J.: The history of Stokes’s theorem. Math. Mag. 52(3), 146–156 (1979)
Suquet P.: Elements of homogenization theory for inelastic solid mechanics. In: Sanchez-Palencia, E., Zaoui, A. (eds.) Homogenization Techniques for Composite Media, pp. 194–275. Springer, Berlin (1987)
Berger, H., Kari, S., Gabbert, U., Rodriguez-Ramos, R., Guinovart-Diaz, R., Otero, J.A., Bravo-Castillero, J.: An analytical and numerical approach for calculating effective material coefficients of piezoelectric fiber composites. Int. J. Solids Struct. 42, 5692–714 (2005)
Cocks A.C.F., McMeeking R.M.: A phenomenological constitutive law for the behaviour of ferroelectric ceramics. Ferroelectrics 228(1), 219–228 (1999)
Huber J.E., Fleck N.A., Landis C.M., McMeeking R.M.: A constitutive model for ferroelectric polycrystals. J. Mech. Phys. Solids 47(8), 1663–1697 (1999)
Kessler H., Balke H.: On the local and average energy release in polarization switching phenomena. J. Mech. Phys. Solids 49(5), 953–978 (2001)
Tang T., Yu W.: Micromechanical modeling of the multiphysical behavior of smart materials using the variational asymptotic method. Smart Mater. Struct. 18(12), 125026 (2009)
Etier, M., Gao, Y., Shvartsman, V.V., Lupascu, D.C., Landers, J., Wende, H.: Magnetoelectric properties of 0.2CoFe2O4–0.8BaTiO3 composite prepared by organic method. In: Joint 21st IEEE ISAF/11th IEEE ECAPD/IEEE PFM (ISAF/ ECAPD/PFM), Aveiro, Portugal, pp. 1–4 (2012)
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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). https://doi.org/10.1007/s00707-015-1336-0
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DOI: https://doi.org/10.1007/s00707-015-1336-0