Skip to main content
SpringerLink
Log in

On the impossibility of linear Cauchy and Piola-Kirchhoff constitutive theories for stress in solids

  • Published:
Journal of Elasticity Aims and scope Submit manuscript

Abstract

We investigate the possibility of linear elasticity as an infinitesimal theory based on a genuinely linear response function which retains its validity even for finite deformations. Careful consideration of the domain of definition of the stress response function, the definition of linearity and the notion of material frame-indifference leads to our main result that an exact linear constitutive theory for elastic solids is impossible. We then generalize our result to viscoelasticity theory where the stress response is dependent on deformation gradient histories.

Zusammenfassung

Verf. betrachten die Möglichkeit linearer Elastizität als infinitesimale Theorie begründet auf einer echt linearen Reaktionsfunktion die ihre Gültigkeit sogar für endliche Deformationen behält. Genaue Betrachtung des Definitionsbereiches der Spannungsreaktionsfunktion, der Definition von Linearität, und des Objektivitätsbegriffes führen zum Hauptresultat dass eine echt lineare Theorie für elastische Körper unmöglich ist. Das Resultat wird dann auf viskoelastische Theorie verallgemeinert, wobei die Spannungsreaktion von der Vorgeschichte des Deformationsgradienten abhängt.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Love, A. E. H.,A Treatise on the mathematical Theory of Elasticity. 4th ed. Cambridge. Reprinted by Dover Publications, New York 1944.

  2. Sokolnikoff, I. S.,Mathematical Theory of Elasticity. 2nd ed. McGraw-Hill, New York-Toronto-London, 1956.

    Google Scholar 

  3. Gurtin, M. E., The Linear Theory of Elasticity.Flügge's Handbuch der Physik VI/a/2, Berlin-Heidelberg-New York, Springer, 1972.

    Google Scholar 

  4. Truesdell, C. and Toupin, R. A., The Classical Field Theories.Flügge's Handbuch der Physik, III/1, Berlin-Heidelberg-New York: Springer 1960.

    Google Scholar 

  5. Truesdell, C. and Noll, W. The Non-Linear Field Theories of mechanics.Flügge's Handbuch der Physik, III/3, Berlin-Heidelberg-New York: Springer 1965.

    Google Scholar 

  6. Truesdell, C.,A First course in Rational Continuum Mechanics, Vol. 1. Academic Press, New York-San Francisco-London, 1977.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Minnesota, 55455, Minneapolis, Minnesota, U.S.A.

    R. L. Fosdick & J. Serrin

Authors
  1. R. L. Fosdick
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Serrin
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Department of Aerospace Engineering and Mechanics

Department of Mathematics

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fosdick, R.L., Serrin, J. On the impossibility of linear Cauchy and Piola-Kirchhoff constitutive theories for stress in solids. J Elasticity 9, 83–89 (1979). https://doi.org/10.1007/BF00040982

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00040982

Keywords

Search

Navigation