Ingenieur-Archiv

, Volume 43, Issue 1, pp 44–57 | Cite as

Inadaptation theorems in the dynamics of elastic-work hardening structures

  • L. Corradi
  • G. Maier
Article
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Summary

Some broad classes of discrete structural models and piecewise linear yield loci and hardening rules are considered. The dynamic elastoplastic response to a given history of rapidly variable loads and imposed strains and displacements (“dislocations”) is studied on the basis of a suitable matrix description. Inadaptation means that the plastic work and, hence, some plastic deformations increase unlimitedly in time, i.e. the structure does not shakedown. Sufficient and necessary conditions for this occurrence are established and formulated in a theorem, which represents the extension to the dynamic range and to work-hardening structures of the second (Koiter's) shakedown theorem of classical plasticity.

Keywords

Neural Network Plastic Deformation Information Theory Nonlinear Dynamics Broad Class 
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.

Übersicht

Es werden einige Klassen diskreter Strukturmodelle und stückweise linearer Fließ- und Verfestigungsregeln betrachtet. Dabei wird die elasto-plastische Reaktion eines Bauteiles auf eine vorgegebene Belastung durch schnell veränderliche Lasten sowie aufgeprägte Verzerrungen oder Verschiebungen mit Hilfe einer geeigneten Matrizen-Darstellung untersucht. „Inadaptation” bedeutet, daß die plastische Verformungsarbeit und folglich auch einige plastische Deformationen mit der Zeit unbegrenzt anwachsen, so daß kein Abfangen bei endlichen Verformungen („adaptation”) stattfindet. Es werden notwendige und hinreichende Bedingungen für ein derartiges Verhalten aufgestellt und in einem Satz formuliert, der als eine Verallgemeinerung des zweiten (des Koiterschen) Satzes der klassischen Plastizitätstheorie aufgefaßt werden kann.

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References

  1. 1.
    Melan, E.: Zur Plastizität des räumlichen Kontinuums. Ing.-Arch. 9 (1938) pp. 116–126.Google Scholar
  2. 2.
    Koiter, W. T.: General theorems for elastic-plastic solids. In: Progress in Solid Mechanics, Vol. 1, Amsterdam 1960, p. 167Google Scholar
  3. 3.
    Koiter, W. T.: A new general theorem on shakedown of elastic-plastic structures. Proc. Kon. Nederl. Akad. Wet., B 59, 4 (1956) pp. 24Google Scholar
  4. 4.
    Maier, G.: A shakedown matrix theory allowing for workhardening and second order geometric effects. Proc. Symp. Foundations of Plasticity, Warsaw, 1972, pp. 250–260Google Scholar
  5. 5.
    König, J. A.: A shakedown theorem for temperature dependent elastic moduli. Bull. Acad. Pol. Sci., Ser. Sci. Techn. 17 (1969) pp. 161–165Google Scholar
  6. 6.
    Maier, G.: Shakedown theory in perfect elastoplasticity with associated and non associated flow laws: a finite element, linear programming approach. Meccanica 3 (1969) pp. 250–260Google Scholar
  7. 7.
    Cerádini, G.: Sull'adattamento dei corpi elastoplastici soggetti ad azioni dinamiche. Giorn. Genio Civile, May 1969, pp. 239–250Google Scholar
  8. 8.
    Maier, G.: A matrix structural theory of piecewise linear elastoplasticity with interacting yield planes. Meccanica 5 (1970) pp. 54–66Google Scholar
  9. 9.
    Maier, G.: Shakedown of plastic structures with unstable parts. Techn. Report ISTC, Milan, June 1971, and J. Eng. Mech. Div., Proc. ASCE, Oct. 1972, pp. 1322–1327,Google Scholar
  10. 10.
    De Donato, O.: Second shakedown theorem allowing for cycles of both loads and temperatures. Rendic. 1st. Lombardo, Cl. Sci. A, 104 (1970) pp. 265–277Google Scholar
  11. 11.
    Maier, G.; Vitiello, E.: Bounds on plastic strains and displacements in dynamic shakedown of hardening structures. Techn. Report, ISTC Politecnico Milano, 1972Google Scholar
  12. 12.
    Corradi, L.; Maier, G.: Dynamic inadaptation theorem for elastic perfectly plastic continua. Techn. Report, ISTC Politecnico Milano, Jan. 1973, to appear in J. Mech. Physics of SolidsGoogle Scholar
  13. 13.
    Sawczuk, A.: Evaluation of upper bounds to shakedown loads for shells. J. Mech. Phys. Solids 4 (1969) pp. 291–301Google Scholar
  14. 14.
    Argyris, J. H.: Continua and discontinua. Proc. Conf. Matrix Methods in Struct. Mech., Wright-Patterson A. F. B., Ohio, 1967, p. 12Google Scholar
  15. 15.
    Zienkiewicz, O. C.: The finite element method in engineering sciences. New York, 1971Google Scholar
  16. 16.
    Maier, G.: A quadratic programming approach for certain classes of nonlinear structural problems. Meccanica 3 (1968) pp. 121–130Google Scholar
  17. 17.
    Hodge, P. G.: Plastic analysis of structures. New York, 1959Google Scholar
  18. 18.
    Zangwill, W. J.: Nonlinear programming, a unified approach. Englewood Cliffs, N. J., 1969Google Scholar

Copyright information

© Springer-Verlag 1973

Authors and Affiliations

  • L. Corradi
    • 1
  • G. Maier
    • 2
  1. 1.Istituto di Scienza e Tecnica delle CostruzioniPolitecnico di MilanoMilanoItaly
  2. 2.Istituto di Scienza e Tecnica delle CostruzioniPolitecnico di MilanoMilanoItaly

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