Material Symmetry Group and Consistently Reduced Constitutive Equations of the Elastic Cosserat Continuum

  • Victor A. Eremeyev
  • Wojciech Pietraszkiewicz
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 22)


We discuss the material symmetry group of the polar-elastic continuum and related consistently simplified constitutive equations. Following [1] we extent the definition of the group proposed by Eringen and Kafadar [2] by taking into account the microstructure curvature tensor as well as different transformation properties of polar and axial tensors. Our material symmetry group consists of ordered triples of tensors which make the strain energy density of the polar-elastic continuum invariant under change of the reference placement. Within the polar-elastic solids we discuss the isotropic, hemitropic, orthotropic, transversely isotropic and cubic-symmetric materials and give explicitly the consistently reduced representations of the strain energy density.


Strain Energy Density Deformation Gradient Tensor Cosserat Continuum Invariant Transformation Micropolar Continuum 
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The first author was supported by the DFG grant No. AL 341/33-1 and by the RFBR with the grant No. 12-01-00038.


  1. 1.
    Eremeyev, V.A., Pietraszkiewicz, W.: Material symmetry group of the non-linear polar-elastic continuum. Int. J. Solids Struct. 49(14), 1993–2005 (2012)CrossRefGoogle Scholar
  2. 2.
    Eringen, A.C., Kafadar, C.B.: Polar field theories. In: Eringen, A.C. (ed.) Continuum Physics, vol. IV, pp. 1–75. Academic Press, New York (1976)Google Scholar
  3. 3.
    Cosserat, E., Cosserat, F.: Théorie des corps déformables. Herman et Fils, Paris (1909)Google Scholar
  4. 4.
    Eringen, A.C.: Microcontinuum Field Theory I. Foundations and Solids. Springer, New York (1999)CrossRefGoogle Scholar
  5. 5.
    Eringen, A.C.: Microcontinuum Field Theory II. Fluent Media. Springer, New York (2001)Google Scholar
  6. 6.
    Nowacki, W.: Theory of Asymmetric Elasticity. Pergamon-Press, Oxford (1986)zbMATHGoogle Scholar
  7. 7.
    Eremeyev, V.A., Lebedev, L.P., Altenbach, H.: Foundations of Micropolar Mechanics. Springer, Heidelberg (2012)Google Scholar
  8. 8.
    Pietraszkiewicz, W., Eremeyev, V.A.: On natural strain measures of the non-linear micropolar continuum. Int. J. Solids Struct. 46(3–4), 774–787 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Truesdell, C., Noll, W.: The nonlinear field theories of mechanics. In: Flügge, S. (ed.) Handbuch der Physik, Vol III/3, pp 1–602. Springer, Berlin (1965)Google Scholar
  10. 10.
    Wang, C.C., Truesdell, C.: Introduction to Rational Elasticity. Noordhoof Int. Publishing, Leyden (1973)zbMATHGoogle Scholar
  11. 11.
    Spencer, A.J.M.: Isotropic integrity bases for vectors and second-order tensors. Part II. Arch. Ration. Mech. Anal. 18(1), 51–82 (1965)zbMATHGoogle Scholar
  12. 12.
    Spencer, A.J.M.: Theory of invariants. In: Eringen, A.C. (ed.) Continuum Physics, vol. 1, pp. 239–353. Academic Press, New York (1971)Google Scholar
  13. 13.
    Zheng, Q.S.: Theory of representations for tensor functions–a unified invariant approach to constitutive equations. Appl. Mech. Rev. 47(11), 545–587 (1994)CrossRefGoogle Scholar
  14. 14.
    Ramezani, S., Naghdabadi, R., Sohrabpour, S.: Constitutive equations for micropolar hyper-elastic materials. Int. J. Solids Struct. 46(14–15), 2765–2773 (2009)zbMATHCrossRefGoogle Scholar
  15. 15.
    Kafadar, C.B., Eringen, A.C.: Micropolar media–I. The classical theory. Int. J. Eng. Sci. 9, 271–305 (1971)zbMATHCrossRefGoogle Scholar
  16. 16.
    Smith, M.M., Smith, R.F.: Irreducable expressions for isotropic functions of two tensors. Int. J. Eng. Sci. 19(6), 811–817 (1971)CrossRefGoogle Scholar
  17. 17.
    Zheng, Q.S., Spencer, A.J.M.: On the canonical representations for Kronecker powers of orthogonal tensors with application to material symmetry problems. Int. J. Eng. Sci. 31(4), 617–635 (1993)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Victor A. Eremeyev
    • 2
    • 1
  • Wojciech Pietraszkiewicz
    • 3
  1. 1.Faculty of Mechanical EngineeringOtto-von-Guericke-UniversityMagdeburgGermany
  2. 2.South Scientific Center of RASci and South Federal UniversityRostov on DonRussia
  3. 3.Institute of Fluid-Flow Machinery, PASciGdańskPoland

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