Advertisement

On the general solution of linear-elastic problems in isotropic and anisotropic Cosserat continua

  • H. Neuber

Abstract

The existence of couple stresses acting on the surfaces of elastic continua together with the usual force stresses had been object of numerous fundamental papers [1, 7–16]. For describing the complete deformation a geometrically independent rotation vector had been introduced besides the displacement vector. In this paper the basical equations of three-dimensional linear-anisotropic elasticity with all inertia effects are represented in general coordinates and their general solution is established by means of a six-functions-set. In this way all static and cinematic components can be derived directly. At isotropy the six-functions-set can be represented in analogy to the Papkovich-Netjbee set [2, 3, 4] of the classical theory of elasticity. The two-dimensional elastostatic case is equivalent to a set of H. Schäfer [12]. The theory also contains as limit case a set of Mindlin-Teuersten [14, 15], in which the curl of the displacement vector is used for the rotation vector. Some examples refer to the theory of stress concentration, where the existence of couple stresses leads to interesting aspects.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Cosserat, E., et F.: Théorie des corps déformables, Paris: Herman et fils 1909.Google Scholar
  2. [2]
    Papkovitoh, P. F.: Solution générale des équations différentielles fondamentales d’élasticité exprimée par trois fonctions harmoniques. C. R. Paris 195 (1932) 513–515.Google Scholar
  3. [3]
    Neubeb, H.: Ein neuer Ansatz zur Lösung räumlicher Probleme der Elastizitätstheorie. Der Hohlkegel unter Einzellast als Beispiel. Z. angew. Math. Mech. 14 (1934) 203–212.CrossRefGoogle Scholar
  4. [4]
    Neuber, H.: Kerbspannungslehre, Grundlagen für genaue Festigkeitsberechnung mit Berücksichtigung von Konstruktionsform und Werkstoff, Berlin/Göttingen/Heidelberg: Springer (1. Aufl.) 1937, (2. Aufl.) 1958.Google Scholar
  5. [5]
    Neuber, H.: Die Grundgleichungen der elastischen Stabilität in allgemeinen Koordinaten und ihre Integration. Z. angew. Math. Mech. 23 (1943) 321–330.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    Neuber, H.: Allgemeine Schalentheorie. Z. angew. Math. Mech. 29 (1949) 97–108, 142-146.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    Günther, W.: Zur Statik und Kinematik des Cosseratschen Kontinuums. Abh. Brschg. Wiss. Ges. X (1958) 195–213.Google Scholar
  8. [8]
    Kröner, E.: Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen. Arch. Rat. Mech. 4 (1959/60) 273–334.CrossRefGoogle Scholar
  9. [9]
    Grioli, G.: Elasticà asimmetrica. Ann. Mat. pura ed appl. Ser. IV, 50 (1960) 389–417.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    Truesdell, C., and R. A. Toupin: The classical field theories. Encyclopedia of Physics, Vol.III/1, sees. 200, 203, 205, Berlin/Göttingen/Heidelberg: Springer 1960.Google Scholar
  11. [11]
    Aero, E. L., and E. V. Kushinski: Fundamental equations of the theory of elastic media with rotationally interacting particles. Transi. Soviet Phys. Solid state 2 (1961) 1272–1281.Google Scholar
  12. [12]
    Schäfer, H.: Versuch einer Elastizitätstheorie des zweidimensionalen ebenen Cosserat-Kontinuums. Miszellaneen der angewandten Mechanik, Berlin: Akademie-Verlag 1962, S. 277–292.Google Scholar
  13. [13]
    Toupin, R. A.: Elastic materials with couple stresses. Arch. Rational Mech. Anal. 11 (1962) 385–414.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    Mindlin, R. D., and H. F. Thiersten: Effects of couple stresses in linear elasticity. Arch. Rational Mech. Anal. 11 (1962) 415–447.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    Mindlin, R. D.: Influence of couple-stresses stress concentrations. Exp. Mech. (1963) 1-7.Google Scholar
  16. [16]
    Koiter, W. T.: Couple stresses in the theory of elasticity. Proc. konigl. nederl. akad. wetensch. Amsterdam B 67 (1964) 17–44.MATHGoogle Scholar
  17. [17]
    Neuber, H.: Über Probleme der Spannungskonzentration im Cosserat-Körper. Acta Mech. (1966).Google Scholar
  18. [18]
    Neuber, H.: Das statische und kinetische Torsionsproblem für den Cosserat-Kreiszylinder. Z. angew. Math. Mech. (1966).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1966

Authors and Affiliations

  • H. Neuber
    • 1
  1. 1.Technical University of MunichGermany

Personalised recommendations