Requirements on Periodic Micromorphic Media

  • Ralf Jänicke
  • Stefan Diebels
Part of the Advances in Mechanics and Mathematics book series (AMMA, volume 21)


In order to investigate the properties of microstructured materials, the underlying heterogeneous material is commonly replaced by a homogeneous material involving additional degrees of freedom. Making use of an appropriate homogenization methodology, the present contribution compares deformation states predicted by the homogenization technique to the deformation state within a reference solution. The results indicate on what terms the predicted deformation modes can be clearly interpreted from the physical point of view.


Deformation Mode Kinematic Quantity Computational Homogenization Micromorphic Continuum Projection Rule 
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  1. 1.
    Cosserat, E., Cosserat, F.: Théorie des corps déformables. Hermann et Fils, Paris (1909) Google Scholar
  2. 2.
    Diebels, S., Steeb, H.: The size effect in foams and its theoretical and numerical investigation. Proc. R. Soc. Lond. A 458, 2869–2883 (2002) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Eringen, A.C.: Microcontinuum Field Theories, vol. I: Foundations and Solids. Springer, New York (1999) Google Scholar
  4. 4.
    Feyel, F., Chaboche, J.L.: FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fiber SiC/Ti composite materials. Comput. Methods Appl. Mech. Eng. 183, 309–330 (2000) MATHCrossRefGoogle Scholar
  5. 5.
    Forest, S.: Homogenization methods and the mechanics of generalized continua—Part 2. Theor. Appl. Mech. 28, 113–143 (2002) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Forest, S.: Nonlinear microstrain theories. Int. J. Solids Struct. 43, 7224–7245 (2006) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Forest, S., Sab, K.: Cosserat overall modeling of heterogeneous materials. Mech. Res. Commun. 25, 449–454 (1998) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Germain, P.: La méthode des puissances virtuelles en mécanique des milieux continus. Première partie: Théorie du second gradient. J. Mec. 12, 235–274 (1973) MATHMathSciNetGoogle Scholar
  9. 9.
    Jänicke, R., Diebels, S.: A numerical homogenisation strategy for micromorphic continua. Nuovo Cim. Soc. Ital. Fis. C 31(1), 121–132 (2009) Google Scholar
  10. 10.
    Jänicke, R., Diebels, S., et al.: Two-scale modelling of micromorphic continua. Contin. Mech. Therm. 21, 297–315 (2009) CrossRefGoogle Scholar
  11. 11.
    Kouznetsova, V.G.: Computational homogenization for the multi-scale analysis of multi-phase material. PhD thesis, Technische Universiteit Eindhoven, The Netherlands (2002) Google Scholar
  12. 12.
    Kouznetsova, V.G., Geers, M.G.D., Brekelmans, W.A.M.: Size of a representative volume element in a second-order computational homogenization framework. Int. J. Multiscale Comput. Eng. 2(4), 575–598 (2004) CrossRefGoogle Scholar
  13. 13.
    Larsson, R., Diebels, S.: A second order homogenization procedure for multi-scale analysis based on micropolar kinematics. Int. J. Numer. Meth. Eng. 69, 2485–2512 (2006) CrossRefMathSciNetGoogle Scholar
  14. 14.
    Maugin, G.A.: Nonlocal theories or gradient-type theories: a matter of convenience? Acta Mater. 31, 15–26 (1979) MATHMathSciNetGoogle Scholar
  15. 15.
    Miehe, C., Koch, A.: Computational micro-to-macro transitions of discretized microstructures. Arch. Appl. Mech. 72, 300–317 (2002) MATHCrossRefGoogle Scholar
  16. 16.
    Sab, K., Pradel, F.: Homogenisation of periodic Cosserat media. Int. J. Comput. Appl. Technol. 34(1), 60–71 (2009) CrossRefGoogle Scholar
  17. 17.
    Tekoglu, C., Onck, P.R.: Size effects in the mechanical behaviour of cellular materials. J. Math. Sci. 40, 5911–5917 (2005) CrossRefGoogle Scholar
  18. 18.
    Tekoglu, C., Onck, P.R.: Size effects in two-dimensional Voronoi foams: a comparison between generalized continua and discrete models. J. Mech. Phys. Solids 56, 3541–3564 (2008) MATHCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Chair of Applied MechanicsSaarland UniversitySaarbrückenGermany

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