Acta Mechanica Sinica

, Volume 25, Issue 4, pp 499–506 | Cite as

A version of Hill’s lemma for Cosserat continuum

  • Xikui LiEmail author
  • Qipeng Liu
Research Paper


On the basis of Hill’s lemma for classical Cauchy continuum, a version of Hill’s lemma for micro–macro homogenization modeling of heterogeneous Cosserat continuum is presented in the frame of average-field theory. The admissible boundary conditions required to prescribe on the representative volume element for the modeling are extracted and discussed to ensure the satisfaction of Hill–Mandel energy condition and the first-order average field theory.


Hill’s lemma Hill–Mandel condition Cosserat continuum Average-field theory RVE boundary conditions 


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  1. 1.
    Hill R.: Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids 11, 357–372 (1963)zbMATHCrossRefGoogle Scholar
  2. 2.
    Qu J., Cherkaoui M.: Fundamentals of Micromechanics of Solids. Wiley, Hoboken (2006)CrossRefGoogle Scholar
  3. 3.
    Chen S.H., Wang T.: Advances in strain gradient theory. Adv. Mech. 33(2), 207–216 (2003) (in Chinese)Google Scholar
  4. 4.
    Zhang H.W., Wang H., Liu Z.G.: Quadrilateral isoparametric finite elements for plane elastic Cosserat bodies. Acta Mech. Sin. 21(4), 388–394 (2005)CrossRefGoogle Scholar
  5. 5.
    Dai T.M.: Renewal of basic laws and principles for polar continuum theories (XI)—consistency problems. Appl. Math. Mech. 28(2), 147–155 (2007)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Li X.K., Tang H.X.: A consistent return mapping algorithm for pressure-dependent elastoplastic Cosserat continua and modeling of strain localization. Comp. Struct. 83, 1–10 (2005)CrossRefGoogle Scholar
  7. 7.
    Suquet P.M.: Local and global aspects in the mathematical theory of plasticity. In: Sawczuk, A., Bianchi, G. (eds) Plasticity Today: Modelling, Methods and Applications, pp. 279–310. Elsevier, London (1985)Google Scholar
  8. 8.
    Michel J.C., Moulinec C., Suquet P.M.: Effective properties of composite materials with periodic macrostructure: a computational approach. Comput. Methods Appl. Mech. Eng. 172, 109–143 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Hu G.K., Zheng Q.S., Huang Z.P.: Micromechanics methods for effective elastic properties of composite materials. Adv. Mech. 31(3), 361–393 (2001) (in Chinese)Google Scholar
  10. 10.
    Forest S., Sab K.: Cosserat overall modeling of heterogeneous materials. Mech. Res. Comm. 25(4), 449–454 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Forest S., Dendievel R., Canova G.R.: Estimating the overall properties of heterogeneous Cosserat materials. Model. Simul. Mater. Sci. Eng. 7, 829–840 (1999)CrossRefGoogle Scholar
  12. 12.
    Yuan X., Tomita Y.: Effective properties of Cosserat composites with periodic microstructure. Mech. Res. Comm. 28(3), 265–270 (2001)zbMATHCrossRefGoogle Scholar
  13. 13.
    Hu G.K., Liu X.N., Lu T.J.: A variational method for non-linear micopolar composites. Mech. Mater. 37, 407–425 (2005)CrossRefGoogle Scholar
  14. 14.
    Chang C.S., Kuhn M.R.: On virtual work and stress in granular media. Int. J. Solids Struct. 42, 3773–3793 (2005)zbMATHCrossRefGoogle Scholar
  15. 15.
    Onck P.R.: Cosserat modeling of cellular solids. C. R. Mecanique 330, 717–722 (2002)zbMATHCrossRefGoogle Scholar
  16. 16.
    Van der Sluis O., Schreurs P.J.G., Brekelmans W.A.M., Meijer H.E.H.: Overall behaviour of heterogeneous elastoviscoplastic materials: effect of microstructural modeling. Mech. Mater. 32, 449–462 (2000)CrossRefGoogle Scholar
  17. 17.
    Terada K., Hori M., Kyoya T., Kikuchi N.: Simulation of the multi-scale convergence in computational homogenization approach. Int. J. Solids Struct. 37, 2285–2311 (2000)zbMATHCrossRefGoogle Scholar
  18. 18.
    Kouznetsova V., Brekelmans W.A.M., Baaijens F.P.T.: An approach to micro–macro modeling of heterogeneous materials. Comput. Mech. 27, 37–48 (2001)zbMATHCrossRefGoogle Scholar
  19. 19.
    Kouznetsova V., Geers M.G.D., Brekelmans W.A.M.: Multi-scale constitutive modeling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. Int. J. Numer. Meth. Eng. 54, 1235–1260 (2002)zbMATHCrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH 2009

Authors and Affiliations

  1. 1.The State Key Laboratory for Structural Analysis of Industrial EquipmentDalian University of TechnologyDalianChina

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