Advertisement

A Framework for the Two-Scale Homogenization of Electro-Mechanically Coupled Boundary Value Problems

  • Jörg Schröder
  • Marc-André Keip
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 1)

Abstract

The contribution addresses the derivation of a meso-macro transition procedure for electro-mechanically coupled materials in two and three dimensions. In this two-scale homogenization approach piezoelectric material behavior will be analysed. In this context, a mesoscopic material model will be presented and implemented into an FE2-homogenization approach. The resulting model is able to capture macroscopic boundary value problems taking into account attached heterogeneous representative volume elements at each macroscopic point. The model is also applicable for the calculation of effective electro-mechanical material parameters, which are efficiently computed by means of the proposed direct homogenization procedure.

Keywords

Representative Volume Element Piezoelectric Material Macroscopic Strain Piezoelectric Composite Boundary Value Prob 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Benveniste, Y.: Exact results in the micromechanics of fibrous piezoelectric composites exhibiting pyroelectricity. Proceedings of the Royal Society London A 441, 59–81 (1911)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)CrossRefGoogle Scholar
  3. 3.
    Benveniste, Y.: Universal relations in piezoelectric composites with eigenstress and polarization fields, part ii: Multiphase media–effective behavior. Journal of Applied Mechanics 60, 270–275 (1993)CrossRefGoogle 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.
    Chen, T.: Piezoelectric properties of multiphase fibrous composites: Some theoretical results. Journal of the Mechanics and Physics of Solids 41(11), 1781–1794 (1993)CrossRefGoogle Scholar
  6. 6.
    Chen, T.: Micromechanical estimates of the overall thermoelectroelastic moduli of multiphase fibrous composites. International Journal of Solids and Structures 31(22), 3099–3111 (1994)CrossRefGoogle Scholar
  7. 7.
    Dunn, M.L., Taya, M.: An analysis of piezoelectric composite materials containing ellipsoidal inhomogeneities. Proceedings of the Royal Society London A 443, 265–287 (1918)CrossRefGoogle Scholar
  8. 8.
    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)CrossRefGoogle Scholar
  9. 9.
    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)CrossRefGoogle Scholar
  10. 10.
    Francfort, G.A., Murat, F.: Homogenization and optimal bounds in linear elasticity. Archive for Rational Mechanics and Analysis 94, 307–334 (1986)CrossRefGoogle Scholar
  11. 11.
    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)CrossRefGoogle Scholar
  12. 12.
    Hill, R.: Elastic properties of reinforced solids: Some theoretical principles. Journal of the Mechanics and Physics of Solids 11, 357–372 (1963)CrossRefGoogle Scholar
  13. 13.
    Hill, R.: On constitutive macro-variables for heterogeneous solids at finite strain. Proceedings of the Royal Society London A 326(1565), 131–147 (1972)CrossRefGoogle Scholar
  14. 14.
    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)CrossRefGoogle Scholar
  15. 15.
    Hori, M., Nemat-Nasser, S.: Universal bounds for effective piezoelectric moduli. Mechanics of Materials 30(1), 1–19 (1998)CrossRefGoogle Scholar
  16. 16.
    Krawietz, A.: Materialtheorie: Mathematische Beschreibung des phänomenologischen thermomechanischen Verhaltens. Springer, Berlin (1986)Google Scholar
  17. 17.
    Kröner, E.: Bounds for effective elastic moduli of disordered materials. Journal of the Mechanics and Physics of Solids 25, 137–155 (1977)CrossRefGoogle Scholar
  18. 18.
    Kurzhöfer, I.: Mehrskalen-Modellierung polykristalliner Ferroelektrika basierend auf diskreten Orientierungsverteilungsfunktionen. PhD thesis, Institut für Mechanik, Fakultät Ingenieurwissenschaften, Abteilung Bauwissenschaften, Universität Duisburg-Essen (2007)Google Scholar
  19. 19.
    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
  20. 20.
    Lupascu, D.C., Schröder, J., Lynch, C.S., Kreher, W., Westram, I.: Mechanical Properties of Ferro-Piezoceramics. In: Handbook of Multifunctional Polycrystalline Ferroelectric Materials. Springer, Heidelberg (2009) (in print)Google Scholar
  21. 21.
    Markovic, D., Niekamp, R., Ibrahimbegovic, A., Matthies, H.G., Taylor, R.L.: Multi-scale modeling of heterogeneous structures with inelastic constitutive behavior. International Journal for Computer-Aided Engineering and Software 22(5/6), 664–683 (2005)CrossRefGoogle Scholar
  22. 22.
    Michel, J.C., Moulinec, H., Suquet, P.: Effective properties of composite materials with periodic microstructure: a computational approach. Computer Methods in Applied Mechanics and Engineering 172(1-4), 109–143 (1999)CrossRefGoogle Scholar
  23. 23.
    Miehe, C., Koch, A.: Computational micro-to-macro transitions of discretized microstructures undergoing small strains. Archive of Applied Mechanics 72(4), 300–317 (2002)CrossRefGoogle Scholar
  24. 24.
    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
  25. 25.
    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(3-4), 387–418 (1999)CrossRefGoogle Scholar
  26. 26.
    Nemat-Nasser, S., Hori, M.: Micromechanics: Overall Properties of Heterogeneous Materials. North-Holland, London (1993)Google Scholar
  27. 27.
    Romanowski, H.: Kontinuumsmechanische Modellierung ferroelektrischer Materialien im Rahmen der Invariantentheorie. PhD thesis, Institut für Mechanik, Fakultät Ingenieurwissenschaften, Abteilung Bauwissenschaften, Universität Duisburg-Essen (2006)Google Scholar
  28. 28.
    Romanowski, H., Schröder, J.: Coordinate invariant modelling of the ferroelectric hysteresis within a thermodynamically consistent framework. A mesoscopic approach. In: Wang, Y., Hutter, K. (eds.) Trends in Applications of Mathematics and Mechanics, pp. 419–428. Shaker Verlag, Aachen (2005)Google Scholar
  29. 29.
    Sanchez-Palencia, E.: Non-Homogeneous Media and Vibration. Lecture Notes in Physics, vol. 172. Springer, Berlin (1980)Google Scholar
  30. 30.
    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
  31. 31.
    Schröder, J.: Derivation of the localization and homogenization conditions for electro-mechanically coupled problems. Computational Materials Science (Corrected Proof, 2009) (in press)Google Scholar
  32. 32.
    Schröder, J., Gross, D.: Invariant formulation of the electromechanical enthalpy function of transversely isotropic piezoelectric materials. Archive of Applied Mechanics 73, 533–552 (2004)CrossRefGoogle Scholar
  33. 33.
    Schröder, J., Keip, M.-A.: Computation of the overall properties of microheterogeneous piezoelectric materials (in preparation, 2009)Google Scholar
  34. 34.
    Schröder, J., Romanowski, H., Kurzhöfer, I.: Meso-macro-modeling of nonlinear ferroelectric ceramics. In: Ramm, E., Wall, W.A., Bletzinger, K., Bischoff, M. (eds.) Online Proceedings of the 5th International Conference on Computation of Shell and Spatial Structures, Salzburg, Austria, June 1-4 (2005)Google Scholar
  35. 35.
    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, Bad Honnef, Germany, May 7 - 10, Institut für Mechanik, Fakultät Ingenieurwissenschaften, Abteilung Bauwissenschaften, Universität Duisburg-Essen (2007)Google Scholar
  36. 36.
    Silva, E.C.N., Fonseca, J.S.O., Kikuchi, N.: Optimal design of periodic piezocomposites. Computer Methods in Applied Mechanics and Engineering 159(1-2), 49–77 (1998)CrossRefGoogle Scholar
  37. 37.
    Silva, E.C.N., Nishiwaki, S., Fonseca, J.S.O., 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)CrossRefGoogle Scholar
  38. 38.
    Smit, R.J.M., Brekelmans, W.A.M., Meijer, H.E.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)CrossRefGoogle Scholar
  39. 39.
    Somer, D.D., de Souza Neto, E.A., Dettmer, W.G., 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)CrossRefGoogle Scholar
  40. 40.
    Suquet, P.M.: Elements of homogenization for inelastic solid mechanics. In: Homogenization Techniques for Composite Materials. Lecture Notes in Physics, vol. 272, pp. 193–278. Springer, Heidelberg (1986)CrossRefGoogle Scholar
  41. 41.
    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)CrossRefGoogle Scholar
  42. 42.
    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)CrossRefGoogle Scholar
  43. 43.
    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
  44. 44.
    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 12, S303–S317 (2004)CrossRefGoogle Scholar
  45. 45.
    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)CrossRefGoogle Scholar
  46. 46.
    Willis, J.R.: Bounds and self-consistent estimates for the overall properties of anisotropic composites. Journal of the Mechanics and Physics of Solids 25, 185–202 (1966)CrossRefGoogle Scholar
  47. 47.
    Xia, Z., Zhang, Y., Ellyin, F.: A unified periodical boundary conditions for represantative colume elements of composites and applications. International Journal of Solids and Structures 40, 1907–1921 (2003)CrossRefGoogle Scholar
  48. 48.
    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
  49. 49.
    Zohdi, T., Wriggers, P.: Introduction to computational micromechanics. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  50. 50.
    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)CrossRefGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2010

Authors and Affiliations

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

Personalised recommendations