Formulation of Anisotropic Constitutive Equations

  • J. Betten
Part of the International Centre for Mechanical Sciences book series (CISM, volume 292)


This chapter is concerned with the formulation of constitutive expressions of the form
where dij = (vi,j + vj,i)/2 are the cartesian components of the so called “rate-of-deformation tensor” . Other common names are the “rate-of-strain” or “strain-rate-tensor”. Note that dij is linear in the velocity components vi, and that this linearity is exact and no approximation has been made in deriving it. Furthermore, the tensor is not to confuse with the “material time derivative” of the infinitesimal strain tensor εij = (ui,j + uj,i)/2, because we have [I]:


Constitutive Equation Tensor Function Integrity Basis Tensor Generator Undamaged State 
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  1. 1.
    BETTEN, J., Elasticitäts-und Plastizitätslehre, Vieweg-Verlag, Braun-schweig/Wiesbaden 1985.Google Scholar
  2. 2.
    BETTEN, J., Zur Aufstellung von Stoffgleichungen in der Kriechmechanik anisotroper Körper, Rheol. Acta 20 (1981): 527–535.MATHMathSciNetGoogle Scholar
  3. 3.
    BETTEN, J., Constitutive Equations of Isotropic and Anisotropic Materials in the Secondary and Tertiary Creep Stage, in: Creep and Fracture of Engineering Materials and Structures, Part II, (eds. B. Wilshire and D.R.J. Owen ), Pineridge Press, Swansea 1984, 1291–1305.Google Scholar
  4. 4.
    BETTEN, J., Materialgleichungen zur Beschreibung des sekundären und tertiären Kriechverhaltens anisotroper Stoffe, Z. Angew. Math. Mech. (ZAMM) 64 (1984): 211–220.CrossRefMATHGoogle Scholar
  5. 5.
    BETTEN, J., Applications of Tensor Functions to the Formulation of Constitutive Equations involving Damage and Initial Anisotropy, IUTAMSymposium on Mechanics of Damage and Fatigue, Haifa and Tel Aviv 1985, to be published in the proceedings (eds. S.R. Bodner and Z. Hashin).Google Scholar
  6. 6.
    KACHANOV, L.M., On the Time to Failure under Creep Conditions (in Russian), Izv. Ak. Nauk USSR Otdel. Tekh. Nauk, 8 (1958): 26–31.Google Scholar
  7. 7.
    RABOTNOV, Y.N., Creep Problems in Structural Members, North-Holland 1969.Google Scholar
  8. 8.
    BETTEN, J., Damage Tensors in Continuum Mechanics, Euromech Colloquium 147 on “Damage Mechanics”, Paris VI, Cachan 1981, published in Journal de Mécanique theórique et appliquée 2 (1983): 13–32.MATHGoogle Scholar
  9. 9.
    MURAKAMI, S. and OHNO, N., A Continuum Theory of Creep and Creep Damage, in: Creep in Structures (eds. A.R.S. Ponter and D.R. Hayhurst ), Springer-Verlag, Berlin/Heidelberg/New York 1981, 422–444.CrossRefGoogle Scholar
  10. 10.
    RABOTNOV, Yu. N., Creep Rupture, in: Applied Mechanics Conference (eds.M. Hetenyi and H. Vincenti ), Stanford University 1968, 342–349.Google Scholar
  11. 11.
    BETTEN, J., Net-Stress Analysis in Creep Mechanics, Ingenieur-Archiv 52 (1982): 405–419.CrossRefMATHGoogle Scholar
  12. 12.
    JOHNSON, A.E., Complex-Stress Creep of Metals, Metallurgical Reviews 5 (1960): 447–506.Google Scholar
  13. 13.
    SPENCER, A.J.M. and Rivlin, R.S., Finite Integrity Bases for Five or Fewer Symmetric 3 x 3 Matrices, Arch. Rational Mech. Anal. 2 (1958/59): 435–446.Google Scholar
  14. 14.
    SPENCER, A.J.M., Theory of Invariants, in: Continuum Physics. Vol. I, (ed. A.C. Eringen ), Academic Press, New York 1971, 239–353.Google Scholar
  15. 15.
    BETTEN, J., Integrity Basis for a Second-Order and a Fourth-Order Tensor, International J. Math. and Math. Sci. 5 (1982): 87–96.CrossRefGoogle Scholar
  16. 16.
    BETTEN, Berkeley 1985, to be published in the proceedings: Mathematical Modelling in Science and Technology (eds. X.J.R. Avula, G. Leitmann, C.D. Mote, Jr. and E.Y. Rodin), Pergamon Press, New York 1986.Google Scholar
  17. 17.
    RIVLIN, R.S. and SMITH, G.F., Orthogonal Integrity Basis for N Symmetric Matrices, in: Contributions to Mechanic (ed. D. Abir ), Pergamon Press, Oxford/…/Braunschweig 1969, 121–141.CrossRefGoogle Scholar
  18. 18.
    SPENCER, A.J.M. and RIVLIN, R.S., The Theory of Matrix Polynomials and its Application to the Mechanics of Isotropic Continua, Arch. Rational Mech. Anal. 2 (1958/59): 309–336.Google Scholar
  19. 19.
    SMITH, G.F., On Isotropic Functions of Symmetric Tensors, Skew-Symmetric Tensors, and Vectors, Int. J. Engng. Sc. 9 (1971): 899–916.CrossRefMATHGoogle Scholar
  20. 20.
    BETTEN, J., On the Representation of the Plastic Potential of Anisotropic Solids, Colloque International du CNRS no 319, Comportement plastique des solides anisotropes, Grenoble 1981, published in the the proceedings: Plastic Behavior of Anisotropie Solids (ed. J.P. Boehler ), CNRS, Paris 1985, 213–228.Google Scholar
  21. 21.
    WANG, C.C., On Representations for Isotropic Functions, Part I, Arch. Rational Mech. Anal. 33 (1969): 249–267.MATHGoogle Scholar
  22. 22.
    BOEHLER, J. P., On Irreducible Representations for Isotropic Scalar Functions, Z. Angew. Math. Mech. (ZAMM) 57 (1977): 323–327.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Wien 1987

Authors and Affiliations

  • J. Betten
    • 1
  1. 1.Technical University AachenF.R. Germany

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