Advertisement

General Constitutive Equations for Simple and Non-Simple Materials

  • P. R. Gummert
Conference paper
Part of the International Centre for Mechanical Sciences book series (CISM, volume 399)

Abstract

Stress and deformation are related by constitutive equations. Using the symmetric stress tensor field S(X, t) at the place of X and at the present time t and a (symmetric) deformation field, e.g. B(X, τ) at the same place of X and the past time τ, it is necessary and possible to postulate six equations S(B). Together with three equations of NEWTON’s law (resp. with three equilibrium conditions) and six equations of displacement-deformation conditions between B(X, τ) and the displacement field u(X, τ) an array of fifteen equations is generated to solve the fifteen unknowns as scalar components of the two tensor fields S(X, t), B(X, τ) and the vector field u(X, τ). Embedded in a consistent mathematical and physical frame theory, this paper is an attempt to derive constitutive laws in a general way and to classify the materials in a systematical way. The knowledge about a material is complete, if in addition to the constitutive equations a procedure of determination of the appropriate material functions and/or parameters is provided (material identification).

Keywords

Constitutive Equation Reference Configuration Deformation Tensor Incompressible Material Material Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Truesdell C. and Toupin R.: Classical Field Theories, in: Encyclopedia of Physics (Edited by W. Flügge), volume III/1, Springer Verlag, Berlin et al., 1960.Google Scholar
  2. 2.
    Truesdell C. and Noll W.: Non—Linear Field Theories, in: Encyclopedia of Physics (Edited by W. Flügge), volume III/3, Springer Verlag, Berlin et al., 1965.Google Scholar
  3. 3.
    Eringen C.A.: Mechanics of Continua, J. Wiley & Sons, Inc., New York et al., 1965.Google Scholar
  4. 4.
    Eringen C.A.: Nonlinear Theory of Continuous Media, McGraw-Hill Book Co., New York et al., 1962.Google Scholar
  5. 5.
    Leigh D.C.: Nonlinear Continuum Mechanics, McGraw-Hill Book Co., New York, 1968.Google Scholar
  6. 6.
    Truesdell C.: Rational Continuum Mechanics, volume 1, Academic Press, New York et al., 1977.MATHGoogle Scholar
  7. 7.
    Truesdell C. and Bell J.F.: Mechanics of Solids, in: Encyclopedia of Physics (Edited by W. Flügge), volume VIa/1, Springer Verlag, Berlin et al., 1973.Google Scholar
  8. 8.
    Trostel R.: Gedanken zur Konstruktion mechanischer Theorien I, II, Institut für Mechanik Technißche Universität Berlin, Forschungsberichte 1985/1988.Google Scholar
  9. 9.
    Trostel R.: Mathematische Grundlagen der Technischen Mechanik I, Vektor-und Tensor-Algebra, Vieweg Verlag, Braunschweig, 1993.CrossRefMATHGoogle Scholar
  10. 10.
    de Boer R.: Vektor- und Tensorrechnung für Ingenieure, Springer Verlag, Berlin et al., 1988.Google Scholar
  11. 11.
    Bertram A.: Axiomatische Einführung in die Kontinuumsmechanik, B.I. Wissenschaftsverlag, Mannheim et al., 1989.MATHGoogle Scholar
  12. 12.
    Haupt P.: Viskoelastizität und Plastizität, Springer Verlag, Berlin et al., 1977.CrossRefMATHGoogle Scholar
  13. 13.
    Haupt P.: Viskoelastizität inkompressibler isotroper Stoffe, Diss., TU Berlin D 83, 1972.Google Scholar
  14. 14.
    Silber G.: Eine Systematik nichtlokaler kelvinhafter Fluide vom Grade drei auf der Basis eines klassischen Kontinuummodelles, volume Fortschrittberichte VDI, Reihe 18/Nr. 26 VDI Verlag, Düsseldorf, TU Berlin, 1986.Google Scholar
  15. 15.
    Alexandru C.: Systematik nichtlokaler kelvinhafter Fluide vom Grade 2 auf der Basis eines COSSERAT Kontinuumsmodells, Fortschrittberichte VDI, Reihe 18/Nr. 61, VDI Verlag, Düsseldorf, 1989.Google Scholar
  16. 16.
    Imiela C. Ein Beitrag zur materialtheoretischen Interpretation der Schüdigungsme- chanik, Diss., Institut für Mechanik TU Berlin, 1998.Google Scholar
  17. 17.
    Gummert P.. Reckling, K.-A. Mechanik VIEWEG Verlag, Braunschweig 1986/1989/1993.Google Scholar
  18. 18.
    Gummert P.: Materialgesetze des Kriechens und der Relaxation Zur Problematik des zeitabhüngigen Stoffverhaltens, Fortschrittberichte VDI, Reihe 38/Nr. 7 VDI Verlag, Düsseldorf, 1987.Google Scholar
  19. 19.
    Gummert P.: A Constitutive Equation for Non-Linear Visco-Elastic Materials with Fading Memory, in: Proceedings of IUTAM-conference: ‘Creep in Structures’, Cracow 1990, Springer Verlag, New York et al., 1990.Google Scholar
  20. 20.
    Altenbach J. and Altenbach H.: Einführung in die Kontinuumsmechanik, TEUBNER Studienbücher B.G. TEUBNER, Stuttgart, 1994.Google Scholar

Copyright information

© Springer-Verlag Wien 1999

Authors and Affiliations

  • P. R. Gummert
    • 1
  1. 1.Technical University BerlinBerlinGermany

Personalised recommendations