Journal of Elasticity

, 82:31 | Cite as

On Transverse and Rotational Symmetries in Elastic Rods

Article

Abstract

The influences of transverse and rotational symmetries on the strain-energy functions of elastic rods are discussed. Complete function bases are presented and, for some constrained theories, these bases are also proven to be irreducible. The treatment of symmetry is based on a reformulation of a recent work by Luo and O’Reilly. It is also shown how this work relates to existing treatments by Antman and Healey for a particular constrained rod theory.

Mathematics Subject Classifications (2000)

74K10 74A35 74E10 74L15 

Key words

material symmetry elastic rods Cosserat theory of rods 

References

  1. 1.
    S.S. Antman, Nonlinear Problems of Elasticity. Springer, Berlin Heidelberg New York (1995).MATHGoogle Scholar
  2. 2.
    J.-P. Boehler, A simple derivation of representations for non-polynomial constitutive equations in some cases of anisotropy. Z. Angew. Math. Mech. 59(4) (1979) 157–167.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    H. Cohen, Homogeneous monotropic elastic rods: Normal uniform configurations and universal solutions. Meccanica 31 (1996) 527–546.CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    A.E. Green and N. Laws, A general theory of rods. Proc. R. Soc. Lond. A293 (1966) 145–155.ADSGoogle Scholar
  5. 5.
    A.E. Green and N. Laws, Remarks on the theory of rods. J. Elast. 3 (1973) 179–184.CrossRefGoogle Scholar
  6. 6.
    A.E. Green and P.M. Naghdi, On thermal effects in the theory of rods. Int. J. Solids Struct. 15(11) (1979) 829–853.CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    T.J. Healey, Material symmetry and chirality in nonlinearly elastic rods. Math. Mech. Solids 7(4) (2002) 405–420.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    I.-S. Liu, On representations of anisotropic invariants. Int. J. Eng. Sci. 20(10) (1982) 1099–1109.CrossRefMATHGoogle Scholar
  9. 9.
    T.A. Lauderdale and O.M. O’Reilly, On the restrictions imposed by non-affine material symmetry groups for elastic rods: Application to helical substructures, Preprint, Department of Mechanical Engineering, University of California at Berkeley, June 2005.Google Scholar
  10. 10.
    C. Luo and O.M. O’Reilly, On the material symmetry of elastic rods. J. Elast. 60(1) (2000) 35–56.CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    P.M. Naghdi, Finite deformation of elastic rods and shells. In: D.E. Carlson and R.T. Shield (eds.), Proceedings of the IUTAM Symposium on Finite Elasticity, Bethlehem PA 1980, Martinus Nijhoff, The Hague (1982) pp. 47–104.Google Scholar
  12. 12.
    O.M. O’Reilly, On constitutive relations for elastic rods. Int. J. Solids Struct. 35(11) (1998) 1009–1024.CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    O.M. O’Reilly and J.S. Turcotte, Some remarks on invariance requirements for constrained rods. Math. Mech. Solids 1(3) (1996) 343–348.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    S. Pennisi and M. Trovato, On the irreducibility of Professor G. F. Smith’s representations for isotropic functions. Int. J. Eng. Sci. 25(8) (1987) 1059–1065.CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    M.B. Rubin, Restrictions on nonlinear constitutive relations for elastic rods. J. Elast. 44 (1996) 9–36.CrossRefMATHGoogle Scholar
  16. 16.
    G.F. Smith, On tranversely isotropic functions of vectors, symmetric second-order tensors and skew-symmetric second-order tensors. Q. Appl. Math. 39(1) (1982) 509–516.MATHADSGoogle Scholar
  17. 17.
    Q.-S. Zheng, On transversely isotropic, orthotropic and relative isotropic functions of symmetric tensors, skew-symmetric tensors and vectors. Part I: Two dimensional othotropic and relative isotropic functions and three dimensional relative isotropic functions. Int. J. Eng. Sci. 31(10) (1993) 1399–1409.CrossRefMATHGoogle Scholar
  18. 18.
    Q.-S. Zheng, On transversely isotropic, orthotropic and relative isotropic functions of symmetric tensors, skew-symmetric tensors and vectors. Part II: The representations for three dimensional transversely isotropic functions. Int. J. Eng. Sci. 31(10) (1993) 1411–1423.CrossRefGoogle Scholar
  19. 19.
    Q.-S. Zheng, On transversely isotropic, orthotropic and relative isotropic functions of symmetric tensors, skew-symmetric tensors and vectors. Part III: The irreducibility of the representations for three dimensional transversely isotropic functions. Int. J. Eng. Sci. 31(10) (1993) 1425–1433.CrossRefGoogle Scholar
  20. 20.
    Q.-S. Zheng, Theory of representations for tensor functions – A unified invariant approach to constitutive equations. ASME Appl. Mech. Rev. 47(11) (1994) 545–587.Google Scholar
  21. 21.
    Q.-S. Zheng and A.J.M. Spencer, Tensors which characterize anisotropies. Int. J. Eng. Sci. 31(5) (1993) 679–693.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of California at BerkeleyBerkeleyUSA

Personalised recommendations