Abstract
A computational homogenization analysis for the simulation of porous magneto-electric composite materials is presented. These materials combine two or more ferroic states with each other enabling a coupling between magnetization and electric polarization. This magneto-electric coupling finds application in sensor technology or data storage devices. Since most single-phase multiferroics show coupling at very low temperatures beyond technically relevant applications, two-phase composites, consisting of a ferroelectric and a ferromagnetic phases, are manufactured. They generate a strain-induced magneto-electric coupling at room temperature. The performance and reliability of these materials is influenced by defects or pores, which can arise during the manufacturing process. We analyze the impact of pores on the magnitude of the magneto-electric coupling coefficient. In order to determine the effective properties of the composite, a two-scale finite element (\(\hbox {FE}^2\)) homogenization approach is performed. It combines the macroscopic and microscopic scale by direct incorporation of the microscopic morphology. We derive the basic equations for the localization and the homogenization of the individual field variables and give an algorithmic expression for the effective tangent moduli. We discuss the influence of pores on the magneto-electric coupling in two-phase composites by analyzing numerical examples.
Similar content being viewed by others
References
Astrov, D.N.: The magnetoelectric effect in antiferromagnetics. Soviet Phys. JETP 38, 984–985 (1960)
Astrov, D.N.: Magnetoelectric effect in chromium oxide. J. Exp. Theor. Phys. 40, 1035–1041 (1961)
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)
Bibes, M., Barthélémy, A.: Multiferroics: towards a magnetoelectric memory. Nat. Mater. 7(6), 425–426 (2008). ISSN 1476-1122
Brown, W.F., Hornreich, R.M., Shtrikman, S.: Upper bound on the magnetoelectric susceptibility. Phys. Rev. 168(2), 574–577 (1968)
Cheong, S.-W., Mostovoy, M.: Multiferroics: a magnetic twist for ferroelectricity. Nat. Mater. 6(1), 13–20 (2007). ISSN 1476-1122
Crottaz, O., Rivera, J.-P., Revaz, B., Schmid, H.: Magnetoelectric effect of \(\text{ Mn }_3\text{ B }_7\text{ O }_{13}{\text{ I }}\) boracite. Ferroelectrics 204, 125–133 (1997)
Eerenstein, W., Mathur, N.D., Scott, J.F.: Multiferroic and magnetoelectric materials. Nature 442(7104), 759–765 (2006)
Eerenstein, W., Wiora, M., Prieto, J.L., Scott, J.F., Mathur, N.D.: Giant sharp and persistent converse magnetoelectric effects in multiferroic epitaxial heterostructures. Nat. Mater. 6(5), 348–351 (2007)
Etier, M., Shvartsman, V.V., Gao, Y., Landers, J., Wende, H., Lupascu, D.C.: Magnetoelectric effect in (0–3) \(\text{ CoFe }_2{\text{ O }}_4\)–\(\text{ BaTiO }_3\) (20/80) composite ceramics prepared by the organosol route. Ferroelectrics 448, 77–85 (2013)
Fiebig, M.: Revival of the magnetoelectric effect. J. Phys. D: Appl. Phys. 38, R123–R152 (2005)
Hill, R.: Elastic properties of reinforced solids—some theoretical principles. J. Mech. Phys. Solids 11, 357–372 (1963)
Hill, N.A.: Why are there so few magnetic ferroelectrics? J. Phys. Chem. B 104, 6694–6709 (2000)
Hwang, S.C., Lynch, C.S., McMeeking, R.M.: Ferroelectric/ferroelastic interaction and a polarization switching model. Acta Metall. Mater. 43, 2073–2084 (1995)
Keip, M.-A.: Modeling of electro-mechanically coupled materials on multiple scales. PhD thesis, University of Duisburg-Essen (2012)
Khomskii, D.: Classifying multiferroics: mechanisms and effects. Physics 2, 20 (2009)
Kouznetsova, V., Geers, M.G.D., Brekelmans, W.A.M.: Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. Int. J. Numer. Methods Eng. 54(8), 1235–1260 (2002)
Kumar, M.M., Srinivas, A., Kumar, G.S., Suryanarayana, S.V.: Investigation of the magnetoelectric effect in \(\text{ BiFeO }_3\)–\(\text{ BaTiO }_3\) solid solutions. J. Phys. Condens. Matter 11, 8131–8139 (1999)
Kurzhöfer, I.: Mehrskalen-Modellierung polykristalliner Ferroelektrika basierend auf diskreten Orientierungsverteilungsfunktionen. PhD thesis, University of Duisburg-Essen (2007)
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)
Labusch, M., Schröder, J., Lupascu, D.C.: Multiscale homogenization of magneto-electric porous two-phase composites. In: Zingoni, A. (ed.) Insights and Innovations in Structural Engineering, Mechanics and Computation: Proceedings of the Sixth International Conference on Structural Engineering, Mechanics and Computation (SEMC 2016). Taylor & Francis Group, London (2016)
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)
Mandel, J., Dantu, P.: Conribution à l’étude théorique et expérimentale du coefficient d’élasticité d’un milieu hétérogène mais statistiquement homogène. Annales des Ponts et Chaussées, Paris (1963)
Martin, L.W., Chu, Y.-H., Ramesh, R.: Advances in the growth and characterization of magnetic, ferroelectric, and multiferroic oxide thin films. Mater. Sci. Eng. R: Rep. 68(4–6), 89–133 (2010). ISSN 0927-796X
Michel, J.C., Moulinec, H., Suquet, P.: Effective properties of composite materials with periodic microstructure: a computational approach. Comput. Methods Appl. Mech. Eng. 172, 109–143 (1999)
Miehe, C., Bayreuther, C.G.: On mutiscale FE analyses of heterogeneous structures: from homogenization to multigrid solvers. Int. J. Numer. Methods Eng. 71, 1135–1180 (2007)
Miehe, C., Schotte, J., Schröder, J.: Computational micro–macro transitions and overall moduli in the analysis of polycrystals at large strains. Comput. Mater. Sci. 16(1–4), 372–382 (1999)
Nan, C.-W.: Magnetoelectric effect in composites of piezoelectric and piezomagnetic phases. Phys. Rev. B 50, 6082–6088 (1994)
Nan, C.-W., Liu, L., Cai, N., Zhai, J., Ye, Y., Lin, Y.H.: A three-phase magnetoelectric composite of piezoelectric ceramics, rare-earth iron alloys, and polymer. Appl. Phys. Lett. 81, 3831–3833 (2002)
Nan, C.-W., Bichurin, M.I., Dong, Shuxiang, Viehland, D., Srinivasan, G.: Multiferroic magnetoelectric composites: historical perspective, status, and future directions. J. Appl. Phys. 103(3), 031101 (2008)
Naveed-Ul-Haq, M., Shvartsman, V.V., Trivedi, H., Salamon, S., Webers, S., Wende, H., Hagemann, U., Schröder, J., Lupascu, D.C.: Strong converse magnetoelectric effect in \(\text{(Ba, } \text{ Ca)(Zr, } \text{ Ti)O }_3\)–\(\text{ NiFe }_2\text{ O }_4\) multiferroics: a relationship between phase-connectivity and interface coupling. Acta Mater. 144, 305–313 (2018)
Priya, S., Islam, R., Dong, S.X., Viehland, D.: Recent advancements in magnetoelectric particulate and laminate composites. J. Electroceram. 19(1), 147–164 (2007)
Rado, G.T., Ferrari, J.M., Maisch, W.G.: Magnetoelectric susceptibility and magnetic symmetry of magnetoelectrically annealed \(\text{ TbPO }_4\). Phys. Rev. B 29, 4041–4048 (1984)
Ramesh, R., Spaldin, N.A.: Multiferroics: progress and prospects in thin films. Nat. Mater. 6(1), 21–29 (2007). ISSN 1476-1122
Rivera, J.-P.: On definitions, units, measurements, tensor forms of the linear magnetoelectric effect and on a new dynamic method applied to Cr–Cl boracite. Ferroelectrics 161, 165–180 (1994a)
Rivera, J.-P.: The linear magnetoelectric effect in \(\text{ licopo }_4\) revisited. Ferroelectrics 161, 147–164 (1994b)
Rivera, J.-P., Schmid, H.: On the birefringence of magnetoelectric \(\text{ BiFeO }_3\). Ferroelectrics 204, 23–33 (1997)
Ryu, J., Carazo, A Vázquez, Uchino, K., Kim, H.-E.: Piezoelectric and magnetoelectric properties of lead zirconate titanate/ni-ferrite particulate composites. J. Electroceram. 7, 17–24 (2001)
Ryu, J., Priya, S., Uchino, K., Kim, H.E.: Magnetoelectric effect in composites of magnetostrictive and piezoelectric materials. J. Electroceram. 8, 107–119 (2002)
Schmid, H.: Multi-ferroic magnetoelectrics. Ferroelectrics 162, 317–338 (1994)
Schmitz-Antoniak, C., Schmitz, D., Borisov, P., de Groot, F.M.F., Stienen, S., Warland, A., Krumme, B., Feyerherm, R., Dudzik, E., Kleemann, W., Wende, H.: Electric in-plane polarization in multiferroic \(\text{ CoFe }_2\text{ O }_4\)-\(\text{ BaTiO }_3\) nanocomposite tuned by magnetic fields. Nat. Commun. 4, 1–8 (2013)
Schröder, J.: Homogenisierungsmethoden der nichtlinearen Kontinuumsmechanik unter Beachtung von Instabilitäten. Habilitation, Bericht aus der Forschungsreihe des Instituts für Mechanik (Bauwesen), Lehrstuhl I, Universität Stuttgart (2000)
Schröder, J.: Derivation of the localization and homogenization conditions for electro-mechanically coupled problems. Comput. Mater. Sci. 46(3), 595–599 (2009)
Schröder, J., Gross, D.: Invariant formulation of the electromechanical enthalpy function of transversely isotropic piezoelectric materials. Arch. Appl. Mech. 73, 533–552 (2004)
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-Mitteilungen 38(1), 1–8 (2015)
Schröder, J., Labusch, M., Keip, M.-A.: Algorithmic two-scale transition for magneto–electro-mechanically coupled problems - \(\text{ FE }^2\)-scheme: localization and homogenization. Comput. Methods Appl. Mech. Eng. 302, 253–280 (2016)
Shvartsman, V.V., Alawneh, F., Borisov, P., Kozodaev, D., Lupascu, D.C.: Converse magnetoelectric effect in \(\text{ CoFe }_2\text{ O }_4\)-\(\text{ BaTiO }_3\) composites with a core-shell structure. Smart Mater. Struct. 20, 075006 (2011)
Spaldin, N.A., Fiebig, M.: The renaissance of magnetoelectric multiferroics. Science 309, 391–392 (2005)
Srinivasan, G.: Magnetoelectric composites. Ann. Rev. Mater. Res. 40, 1–26 (2010)
Van Den Boomgaard, J., Born, R.A.J.: A sintered magnetoelectric composite material \(\text{ BaTiO }_3\)–\(\text{ Ni(Co,Mn) }\) \(\text{ Fe }_2\text{ O }_4\). J. Mater. Sci. 13, 1538–1548 (1978)
Van den Boomgaard, J., Van Run, A.M.J.G., Van Suchtelen, J.: Magnetoelectricity in piezoelectric–magnetostrictive composites. Ferroelectrics 10, 295–298 (1976)
Van den Boomgard, J., Terrell, D.R., Born, R.A.J., Giller, H.F.J.I.: An in situ grown eutectic magnetoelectric composite material. J. Mater. Sci. 9, 1705–1709 (1974)
Van Run, A.M.J.G., Terrell, D.R., Scholing, J.H.: An in situ grown eutectic magnetoelectric composite material. J. Mater. Sci. 9, 1710–1714 (1974)
van Suchtelen, J.: Product properties: a new application of composite materials. Philips Res. Rep. 27, 28–37 (1972)
Vaz, C.A.F., Hoffman, J., Ahn, C.H., Ramesh, R.: Magnetoelectric coupling effects in multiferroic complex oxide composite structures. Adv. Mater. 22, 2900–2918 (2010)
Wang, Y., Hu, J., Lin, Y., Nan, C.-W.: Multiferroic magnetoelectric composite nanostructures. Nat. Asia-Pac. Asia Mater. 2, 61–68 (2010)
Ye, Z.-G., Rivera, J.-P., Schmid, H., Haida, M., Kohn, K.: Magnetoelectric effect and magnetic torque of chromium chlorine boracite \(\text{ Cr }_3{\text{ B }}_7{\text{ O }}_{13}\text{ Cl }\). Ferroelectrics 161, 99–110 (1994)
Zohdi, T.I.: Simulation of coupled microscale multiphysical-fields in particulate-doped dielectrics with staggered adaptive fdtd. Comput. Methods Appl. Mech. Eng. 199, 3250–3269 (2010)
Zohdi, T.I.: Electromagnetic Properties of Multiphase Dielectrics. Springer, Berlin (2012)
Acknowledgements
We gratefully acknowledge the financial support by the German Research Foundation (DFG) in the framework of the research unit 1509 “Ferroic Functional Materials—Multiscale Modeling and Experimental Characterization”, Projects SCHR 570/12-2 and LU 729/12-2. The authors sincerely acknowledge the image of a porous sample provided by Morad Etier.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Labusch, M., Schröder, J. & Lupascu, D.C. A two-scale homogenization analysis of porous magneto-electric two-phase composites. Arch Appl Mech 89, 1123–1140 (2019). https://doi.org/10.1007/s00419-018-01500-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-018-01500-1