Computational Mechanics

, Volume 50, Issue 2, pp 229–244 | Cite as

Two-scale homogenization of electromechanically coupled boundary value problems

Consistent linearization and applications
  • Jörg SchröderEmail author
  • Marc-André Keip
Original Paper


The contribution addresses a direct micro-macro transition procedure for electromechanically coupled boundary value problems. The two-scale homogenization approach is implemented into a so-called FE2-method which allows for the computation of macroscopic boundary value problems in consideration of microscopic representative volume elements. The resulting formulation is applicable to the computation of linear as well as nonlinear problems. In the present paper, linear piezoelectric as well as nonlinear electrostrictive material behavior are investigated, where the constitutive equations on the microscale are derived from suitable thermodynamic potentials. The proposed direct homogenization procedure can also be applied for the computation of effective elastic, piezoelectric, dielectric, and electrostrictive material properties.


Homogenization FE2-method Effective properties Electromechanical coupling Piezoelectricity Electrostriction 


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  1. 1.
    Berger H, Kari S, Gabbert U, Rodríguez-Ramos R, Guinovart-Díaz R, Otero JA, Bravo-Castillero J (2005) An analytical and numerical approach for calculating effective material coefficients of piezoelectric fiber composites. Int J Solids Struct 42(21–22): 5692–5714zbMATHCrossRefGoogle Scholar
  2. 2.
    Debus J-C, Dubus B, Coutte J (1998) Finite element modeling of lead magnesium niobate electrostrictive materials: static analysis. J Acoust Soc Am 103(6): 3336–3343CrossRefGoogle Scholar
  3. 3.
    Geers MGD, Kouznetsova VG, Brekelmans WAM (2010) Multi-scale computational homogenization: trends and challenges. Fourth Int Conf Adv Comput Methods Eng 234(7): 2175–2182zbMATHGoogle Scholar
  4. 4.
    Haun MJ, Furman E, Jang SJ, Cross LE (1989) Modeling of the electrostrictive, dielectric, and piezoelectric properties of ceramic PbTiO3. IEEE Trans Ultrason Ferroelectr Freq Control 36: 393–401CrossRefGoogle Scholar
  5. 5.
    Hill R (1963) Elastic properties of reinforced solids—some theoretical principles. J Mech Phys Solids 11: 357–372zbMATHCrossRefGoogle Scholar
  6. 6.
    Hom CL, Shankar N (1994) A fully coupled constitutive model for electrostrictive ceramic materials. J Intell Mater Syst Struct 5: 795–801CrossRefGoogle Scholar
  7. 7.
    Jang SJ, Uchino K, Nomura S, Cross LE (1980) Electrostrictive behavior of lead magnesium niobate based ceramic dielectrics. Ferroelectrics 27(1): 31–34CrossRefGoogle Scholar
  8. 8.
    Kay HF (1955) Electrostriction. Rep Prog Phys 18(1): 230–250MathSciNetCrossRefGoogle Scholar
  9. 9.
    Keip M-A (2012) Modeling of electro-mechanically coupled materials on multiple scales. PhD Thesis. Faculty of Engineering, Department Civil Engineering, Institute of Mechanics, University of Duisburg-EssenGoogle Scholar
  10. 10.
    Keip M-A, Schröder J (2011) Effective electromechanical properties of heterogeneous piezoelectrics. In: Markert B (ed) Advances in extended and multifield theories for continua of lecture notes in applied and computational mechanics, vol 59. Springer, Berlin, Heidelberg, pp 109–128CrossRefGoogle Scholar
  11. 11.
    Knops RJ (1963) Two-dimensional electrostriction. The Quarterly Journal of Mechanics and Applied Mathematics 16(3): 377–388MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kouznetsova V, Geers MGD, Brekelmans WAM (2002) Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. Int J Numer Methods Eng 54(8): 1235–1260zbMATHCrossRefGoogle Scholar
  13. 13.
    Li Z, Wang C, Chen C (2003) Effective electromechanical properties of transversely isotropic piezoelectric ceramics with microvoids. Comput Mater Sci 27(3): 381–392CrossRefGoogle Scholar
  14. 14.
    Markovic D, Niekamp R, Ibrahimbegovic A, Matthies HG, Taylor RL (2005) Multi-scale modeling of heterogeneous structures with inelastic constitutive behavior. Int J Comput Aid Eng Softw 22(5/6): 664–683zbMATHCrossRefGoogle Scholar
  15. 15.
    Maugin GA, Pouget J, Drouot R, Collet B (1992) Nonlinear electromechanical couplings. Wiley, New YorkGoogle Scholar
  16. 16.
    Michel JC, Moulinec H, Suquet P (1999) Effective properties of composite materials with periodic microstructure: a computational approach. Comput Methods Appl Mech Eng 172: 109–143MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Miehe C, Koch A (2002) Computational micro-to-macro transitions of discretized microstructures undergoing small strains. Arch Appl Mech 72(4): 300–317zbMATHCrossRefGoogle Scholar
  18. 18.
    Miehe C, Schotte J, Schröder J (1999) Computational micro-macro transitions and overall moduli in the analysis of polycrystals at large strains. Comput Mater Sci 16(1–4): 372–382CrossRefGoogle Scholar
  19. 19.
    Miehe C, Schröder J, Schotte J (1999) Computational homogenization analysis in finite plasticity. Simulation of texture development in polycrystalline materials. Comput Methods Appl Mech Eng 171: 387–418zbMATHCrossRefGoogle Scholar
  20. 20.
    Nan C-W, Weng GJ (2000) Theoretical approach to effective electrostriction in inhomogeneous materials. Phys Rev B 61(1): 258–265CrossRefGoogle Scholar
  21. 21.
    Poizat C, Sester M (1999) Effective properties of composites with embedded piezoelectric fibres. Comput Mater Sci 16(1-4): 89–97CrossRefGoogle Scholar
  22. 22.
    Schröder J (2000) Homogenisierungsmethoden der nichtlinearen Kontinuumsmechanik unter Beachtung von Instabilitäten. In: Habilitation, Bericht aus der Forschungsreihe des Instituts für Mechanik (Bauwesen), Lehrstuhl I. Universität Stuttgart, StuttgartGoogle Scholar
  23. 23.
    Schröder J (2009) Derivation of the localization and homogenization conditions for electro-mechanically coupled problems. Comput Mater Sci 46(3): 595–599CrossRefGoogle Scholar
  24. 24.
    Schröder J, Gross D (2004) Invariant formulation of the electromechanical enthalpy function of transversely isotropic piezoelectric materials. Arch Appl Mech 73(8): 533–552zbMATHCrossRefGoogle Scholar
  25. 25.
    Schröder J, Keip M-A (2010) A framework for the two-scale homogenization of electro-mechanically coupled boundary value problems. In: Kuczma M, Wilmanski K (eds) Computer methods in mechanics of advanced structured materials, vol 1. Springer, Berlin, Heidelberg, pp 311–329Google Scholar
  26. 26.
    Schröder J, Keip M-A (2011) 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 of IUTAM bookseries, vol 24. Springer, The Netherlands, pp 265–276CrossRefGoogle Scholar
  27. 27.
    Smit RJM, Brekelmans WAM, Meijer HEH (1998) Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling. Comput Methods Appl Mech Eng 155: 181–192zbMATHCrossRefGoogle Scholar
  28. 28.
    Somer DD, de Souza Neto EA, Dettmer WG, Peric D (2009) A sub-stepping scheme for multi-scale analysis of solids. Comput Methods Appl Mech Eng 198(9–12): 1006–1016zbMATHCrossRefGoogle Scholar
  29. 29.
    Sundar V, Newnham RE (1992) Electrostriction and polarization. Ferroelectrics 135: 431–446CrossRefGoogle Scholar
  30. 30.
    Suquet PM (1986) Elements of homogenization for inelastic solid mechanics. In: Sanchez-Palenzia E, Zaoui A (eds) Homogenization techniques for composite materials of lecture notes in Physics, vol 272. Springer, Berlin, pp 193–278Google Scholar
  31. 31.
    Terada K, Kikuchi N (2001) A class of general algorithms for multi-scale analyses of heterogeneous media. Comput Methods Appl Mech Eng 190(40–41): 5427–5464MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Terada K, Saiki I, Matsui K, Yamakawa Y (2003) Two-scale kinematics and linearization for simultaneous two-scale analysis of periodic heterogeneous solids at finite strain. Comput Methods Appl Mech Eng 192(31–32): 3531–3563MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Topolov VY, Bowen CR (2009) Electromechanical properties in composites based on ferroelectrics. Springer, BerlinGoogle Scholar
  34. 34.
    Yu N, Somphone T (2009) The inclusion and inhomogeneity problems of electrostrictive materials with microstructure. Mech Mater 41: 975–981CrossRefGoogle Scholar
  35. 35.
    Zgonik M, Bernasconi P, Duelli M, Schlesser R, Günter P, Garrett MH, Rytz D, Zhu Y, Wu X (1994) Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals. Phys Rev B 50(9): 5941–5949CrossRefGoogle Scholar
  36. 36.
    Zheludev IS (1971) Electrical properties of physics of polycrystalline dielectrics, vol 2. Plenum Press, New YorkGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of Civil Engineering, Faculty of EngineeringInstitute of MechanicsEssenGermany

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