Effective Electromechanical Properties of Heterogeneous Piezoelectrics

  • Marc-André Keip
  • Jörg Schröder
Conference paper
Part of the Lecture Notes in Applied and Computational Mechanics book series (LNACM, volume 59)


The present contribution discusses a two-scale homogenization procedure for the continuum mechanical modeling of heterogeneous electro-mechanically coupled materials. The direct meso-macro formulation is implemented into an FE2-homogenization environment, which allows for the computation of a macroscopic boundary value problem in consideration of attached heterogeneous representative volume elements at each macroscopic point. The resulting homogenization approach is capable of computing the effective elastic, piezoelectric, and dielectric properties of electro-mechanically coupled materials in consideration of arbitrary mesostructures.


Representative Volume Element Piezoelectric Material Electric Displacement Macroscopic Strain Piezoelectric Composite 
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  1. 1.
    Benveniste, Y.: Exact results in the micromechanics of fibrous piezoelectric composites exhibiting pyroelectricity. Proceedings of the Royal Society London A 441(1911), 59–81 (1993)CrossRefGoogle Scholar
  2. 2.
    Benveniste, Y.: Universal relations in piezoelectric composites with eigenstress and polarization fields, Part I: Binary media: Local fields and effective behavior. Journal of Applied Mechanics 60, 265–269 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Benveniste, Y.: Universal relations in piezoelectric composites with eigenstress and polarization fields, Part II: Multiphase mediaeffective behavior. Journal of Applied Mechanics 60, 270–275 (1993)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Benveniste, Y.: Piezoelectric inhomogeneity problems in anti-plane shear and in-plane electric fields – how to obtain the coupled fields from the uncoupled dielectric solution. Mechanics of Materials 25(1), 59–65 (1997)CrossRefGoogle Scholar
  5. 5.
    Budiansky, B.: On the elastic moduli of some heterogeneous materials. Journal of the Mechanics and Physics of Solids 13, 223–227 (1965)CrossRefGoogle Scholar
  6. 6.
    Chen, T.: Piezoelectric properties of multiphase fibrous composites: Some theoretical results. Journal of the Mechanics and Physics of Solids 41(11), 1781–1794 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Chen, T.: Micromechanical estimates of the overall thermoelectroelastic moduli of multiphase fibrous composites. International Journal of Solids and Structures 31(22), 3099–3111 (1994)CrossRefzbMATHGoogle Scholar
  8. 8.
    Chung, D.H.: Elastic moduli of single crystal and polycrystalline MgO. Philosophical Magazine 8(89), 833–841 (1963)CrossRefGoogle Scholar
  9. 9.
    Dunn, M.L., Taya, M.: An analysis of piezoelectric composite materials containing ellipsoidal inhomogeneities. Proceedings of the Royal Society London A 443(1918), 265–287 (1993)CrossRefzbMATHGoogle Scholar
  10. 10.
    Dunn, M.L., Taya, M.: Micromechanics predictions of the effective electroelastic moduli of piezoelectric composites. International Journal of Solids and Structures 30, 161–175 (1993)CrossRefzbMATHGoogle Scholar
  11. 11.
    Eshelby, J.: The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proceedings of the Royal Society London A 241, 376–396 (1957)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Fang, D.N., Jiang, B., Hwang, K.C.: A model for predicting effective properties of piezocomposites with non-piezoelectric inclusions. Journal of Elasticity 62(2), 95–118 (2001)CrossRefzbMATHGoogle Scholar
  13. 13.
    Francfort, G.A., Murat, F.: Homogenization and optimal bounds in linear elasticity. Archive for Rational Mechanics and Analysis 94, 307–334 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Hashin, Z.: The differential scheme and its application to cracked materials. Journal of the Mechanics and Physics of Solids 36(6), 719–734 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Hashin, Z., Shtrikman, S.: On some variational principles in anisotropic and nonhomogeneous elasticity. Journal of the Mechanics and Physics of Solids 10, 335–342 (1962)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Hashin, Z., Shtrikman, S.: A variational approach to the theory of the elastic behaviour of polycrystals. Journal of the Mechanics and Physics of Solids 10(4), 343–352 (1962)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Hashin, Z., Shtrikman, S.: A variational approach to the theory of the elastic behaviour of multiphase materials. Journal of the Mechanics and Physics of Solids 11(2), 127–140 (1963)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Hill, R.: The elastic behaviour of a crystalline aggregate. Proceedings of the Royal Society London A 65(5), 349–354 (1952)CrossRefGoogle Scholar
  19. 19.
    Hill, R.: Elastic properties of reinforced solids: Some theoretical principles. Journal of the Mechanics and Physics of Solids 11, 357–372 (1963)CrossRefzbMATHGoogle Scholar
  20. 20.
    Hill, R.: A self-consistent mechanics of composite materials. Journal of the Mechanics and Physics of Solids 13, 213–222 (1965)CrossRefGoogle Scholar
  21. 21.
    Hill, R.: On constitutive macro-variables for heterogeneous solids at finite strain. Proceedings of the Royal Society London A 326(1565), 131–147 (1972)CrossRefzbMATHGoogle Scholar
  22. 22.
    Hill, R.: On the micro-to-macro transition in constitutive analyses of elastoplastic response at finite strain. Mathematical Proceedings of the Cambridge Philosophical Society 98, 579–590 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Hori, M., Nemat-Nasser, S.: Universal bounds for effective piezoelectric moduli. Mechanics of Materials 30(1), 1–19 (1998)CrossRefGoogle Scholar
  24. 24.
    Krawietz, A.: Materialtheorie: Mathematische Beschreibung des phänomenologischen thermomechanischen Verhaltens. Springer, Berlin (1986)CrossRefzbMATHGoogle Scholar
  25. 25.
    Kröner, E.: Bounds for effective elastic moduli of disordered materials. Journal of the Mechanics and Physics of Solids 25, 137–155 (1977)CrossRefzbMATHGoogle Scholar
  26. 26.
    Li, Z., Wang, C., Chen, C.: Effective electromechanical properties of transversely isotropic piezoelectric ceramics with microvoids. Computational Materials Science 27(3), 381–392 (2003)CrossRefGoogle Scholar
  27. 27.
    Lupascu, D.C., Schröder, J., Lynch, C.S., Kreher, W., Westram, I.: Mechanical properties of ferro-piezoceramics. In: Pardo, L., Ricote, J. (eds.) Multifunctional polycrystalline ferroelectric materials. Springer Series in Materials Science, vol. 140, pp. 485–559. Springer, Heidelberg (2011) ISBN 978-90-481-2874-7CrossRefGoogle Scholar
  28. 28.
    Markovic, D., Niekamp, R., Ibrahimbegovic, A., Matthies, H., Taylor, R.: Multi-scale modeling of heterogeneous structures with inelastic constitutive behavior. International Journal for Computer-Aided Engineering and Software 22(5/6), 664–683 (2005)CrossRefzbMATHGoogle Scholar
  29. 29.
    Michel, J., Moulinec, H., Suquet, P.: Effective properties of composite materials with periodic microstructure: a computational approach. Computer Methods in Applied Mechanics and Engineering 172, 109–143 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Miehe, C., Koch, A.: Computational micro-to-macro transitions of discretized microstructures undergoing small strains. Archive of Applied Mechanics 72(4), 300–317 (2002)CrossRefzbMATHGoogle Scholar
  31. 31.
    Miehe, C., Schotte, J., Schröder, J.: Computational micro-macro transitions and overall moduli in the analysis of polycrystals at large strains. Computational Materials Science 16(1-4), 372–382 (1999)CrossRefGoogle Scholar
  32. 32.
    Miehe, C., Schröder, J., Schotte, J.: Computational homogenization analysis in finite plasticity. Simulation of texture development in polycrystalline materials. Computer Methods in Applied Mechanics and Engineering 171, 387–418 (1999)CrossRefzbMATHGoogle Scholar
  33. 33.
    Mori, T., Tanaka, K.: Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Mechanica 21, 571–574 (1973)Google Scholar
  34. 34.
    Nemat-Nasser, S., Hori, M.: Micromechanics: Overall Properties of Heterogeneous Materials. North-Holland, London (1993)zbMATHGoogle Scholar
  35. 35.
    Norris, A.N.: A differential scheme for the effective moduli of composites. Mechanics of Materials 4(1), 1–16 (1985)CrossRefGoogle Scholar
  36. 36.
    Qin, Q.H., Yang, Q.S.: Macro-Micro Theory on Multifield Coupling Behavior of Heterogeneous Materials. Higher Education Press, Springer, Bejing, Berlin (2008)Google Scholar
  37. 37.
    Reuss, A.: Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle. Zeitschrift für angewandte Mathematik und Mechanik 9(1), 49–58 (1929)CrossRefzbMATHGoogle Scholar
  38. 38.
    Sanchez-Palencia, E.: Non-homogeneous media and vibration theory. Lecture Notes in Physics, vol. 127, pp. 46–293. Springer, Heidelberg (1980)zbMATHGoogle Scholar
  39. 39.
    Schröder, J.: Homogenisierungsmethoden der nichtlinearen Kontinuumsmechanik unter Beachtung von Instabilitäten. Bericht aus der Forschungsreihe des Instituts für Mechanik (Bauwesen), Lehrstuhl I, Universität Stuttgart (2000)Google Scholar
  40. 40.
    Schröder, J.: Derivation of the localization and homogenization conditions for electro-mechanically coupled problems. Computational Materials Science 46(3), 595–599 (2009)CrossRefGoogle Scholar
  41. 41.
    Schröder, J., Gross, D.: Invariant formulation of the electromechanical enthalpy function of transversely isotropic piezoelectric materials. Archive of Applied Mechanics 73(8), 533–552 (2004)CrossRefzbMATHGoogle Scholar
  42. 42.
    Schröder, J., Keip, M.A.: A framework for the two-scale homogenization of electro-mechanically coupled boundary value problems. In: Kuczma, M., Wilmanski, K. (eds.) Computer Methods in Mechanics, Advanced Structured Materials, vol. 1, pp. 311–329. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  43. 43.
    Schröder, J., Keip, M.A.: Multiscale modeling of electro-mechanically coupled materials: Homogenization procedure and computation of overall moduli. In: Kuna, M., Ricoeur, A. (eds.) IUTAM Symposium on Multiscale Modelling of Fatigue, Damage and Fracture in Smart Materials. IUTAM Bookseries, vol. 24, pp. 265–276. Springer, Netherlands (2011)CrossRefGoogle Scholar
  44. 44.
    Schröder, J., Romanowski, H., Kurzhöfer, I.: A computational meso-macro transition procedure for electro-mechanical coupled ceramics. In: Schröder, J., Lupascu, D., Balzani, D. (eds.) First Seminar on the Mechanics of Multifunctional Materials, Universität Duisburg-Essen, Bad Honnef, Germany (2007)Google Scholar
  45. 45.
    Silva, E., Fonseca, J., Kikuchi, N.: Optimal design of periodic piezocomposites. Computer Methods in Applied Mechanics and Engineering 159(1), 49–77 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  46. 46.
    Silva, E., Nishiwaki, S., Fonseca, J., Kikuchi, N.: Optimization methods applied to material and flextensional actuator design using the homogenization method. Computer Methods in Applied Mechanics and Engineering 172(1-4), 241–271 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  47. 47.
    Smit, R., Brekelmans, W., Meijer, H.: Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling. Computer Methods in Applied Mechanics and Engineering 155, 181–192 (1998)CrossRefzbMATHGoogle Scholar
  48. 48.
    Somer, D., de Souza Neto, E., Dettmer, W., Peric, D.: A sub-stepping scheme for multi-scale analysis of solids. Computer Methods in Applied Mechanics and Engineering 198(9-12), 1006–1016 (2009)CrossRefzbMATHGoogle Scholar
  49. 49.
    Suquet, P.M.: Elements of homogenization for inelastic solid mechanics. In: Suquet, P.M. (ed.) Homogenization Techniques for Composite Materials. Lecture Notes in Physics, vol. 272, pp. 193–278. Springer, Heidelberg (1986)CrossRefGoogle Scholar
  50. 50.
    Terada, K., Kikuchi, N.: A class of general algorithms for multi-scale analyses of heterogeneous media. Computer Methods in Applied Mechanics and Engineering 190(40-41), 5427–5464 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  51. 51.
    Terada, K., Saiki, I., Matsui, K., Yamakawa, Y.: Two-scale kinematics and linearization for simultaneous two-scale analysis of periodic heterogeneous solids at finite strain. Computer Methods in Applied Mechanics and Engineering 192(31-32), 3531–3563 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  52. 52.
    Topolov, V.Y., Bowen, C.R.: Electromechanical properties in composites based on ferroelectrics. Springer, Heidelberg (2009)Google Scholar
  53. 53.
    Uetsuji, Y., Nakamura, Y., Ueda, S., Nakamachi, E.: Numerical investigation on ferroelectric properties of piezoelectric materials using a crystallographic homogenization method. Modelling and Simulation in Material Science and Engineering 317, S303–S317 (2004)Google Scholar
  54. 54.
    Uetsuji, Y., Horio, M., Tsuchiya, K.: Optimization of crystal microstructure in piezoelectric ceramics by multiscale finite element analysis. Acta Materialia 56(9), 1991–2002 (2008)CrossRefGoogle Scholar
  55. 55.
    Voigt, W.: Lehrbuch der Kristallphysik. Teubner (1910)Google Scholar
  56. 56.
    Walpole, L.J.: On bounds for the overall elastic moduli of inhomogeneous system. Journal of the Mechanics and Physics of Solids 14, 151–162 (1966)CrossRefzbMATHGoogle Scholar
  57. 57.
    Willis, J.: Bounds and self-consistent estimates for the overall properties of anisotropic composites. Journal of the Mechanics and Physics of Solids 25, 185–202 (1977)CrossRefzbMATHGoogle Scholar
  58. 58.
    Xia, Z., Zhang, Y., Ellyin, F.: A unified periodical boundary conditions for representative volume elements of composites and applications. International Journal of Solids and Structures 40, 1907–1921 (2003)CrossRefzbMATHGoogle Scholar
  59. 59.
    Zgonik, M., Bernasconi, P., Duelli, M., Schlesser, R., Günter, P., Garrett, M.H., Rytz, D., Zhu, Y., Wu, X.: Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals. Physical Review B 50(9), 5941–5949 (1994)CrossRefGoogle Scholar
  60. 60.
    Zohdi, T.: On the computation of the coupled thermo-electromagnetic response of continua with particulate microstructure. International Journal for Numerical Methods in Engineering 76(8), 1250–1279 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  61. 61.
    Zohdi, T., Wriggers, P.: Introduction to computational micromechanics. In: Pfeiffer, F., Wriggers, P. (eds.) LNACM, vol. 20, Springer, Heidelberg (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Marc-André Keip
    • 1
  • Jörg Schröder
    • 1
  1. 1.Institute for Mechanics, Faculty of Engineering, Department of Civil EngineeringUniversity of Duisburg-EssenEssenGermany

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