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On the foundation of the stability theory of elastic structures

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Sommario

Vengono esaminate alcune difficoltà e problemi ancora aperti nella teoria della stabilità dei sistemi continui. Dopo aver approfondito il significato e l'importanza della scelta della “distanza” tra due configurazioni di un sistema continuo, il problema della stabilità dell'equilibrio delle strutture elastiche viene formulato in un opportuno spazio funzionale, nel quale è possibile dimostrare la validità del criterio dell'energia come condizione sufficiente di stabilità nel senso di Lyapunov.

Si puntualizzano infine le questioni ancora aperte sul criterio di stabilità basato sulla variazione seconda e del quale si fornisce una parziale giustificazione.

Summary

Some difficulties and unsolved problems of the stability theory of continuous systems will be investigated. After having closely examined the meaning and importance of the choices of the “distance” between two configuration of a continuous system, the problem of the equilibrium stability of elastic structures will be formulated in a proper functional space, where it is possible to demonstrate the validity of the energy criterion as sufficient condition of stability in Lyapunov's sense.

Lastly, are pointed out the still open problems on the stability criterion based on the second variation of the potential energy of which a partial proof is given.

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References

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    Knops R. J., Wilkes E. W., “Theory of elastic stability,”Handbuch der Physik, Band VIa/3, Springer-Verlag, 1973.

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    Koiter W. T., “The energy criterion of stability for continuous elastic bodies,” Proceedings of Kon. Ned. Akad. Wetensch, Amsterdam. Ser. B, vol. 68, N. 4, pp. 178–189 (I) and 190–202 (II), 1965.

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    Naghdi P. M., Trapp J. A., “On the general theory of stability for elastic bodies,”Arch. Rat. Mech., vol. 51, 1973.

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    Como M., Grimaldi A., “Stability, buckling and postbuckling of elastic structures,” Rep. n. 1-2-3, Dipartimento di Strutture, Università degli Studi della Calabria, 1974.

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    Mikhlin S. G.,The problem of the minimum of a quadratic functional, Holden-Day, 1965.

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    Mikhlin S. G.,Mathematical physics, an advanced course, North-Holland Publ. Comp., 1970.

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Como, M., Grimaldi, A. On the foundation of the stability theory of elastic structures. Meccanica 10, 171–177 (1975). https://doi.org/10.1007/BF02149029

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Keywords

  • Mechanical Engineer
  • Potential Energy
  • Civil Engineer
  • Open Problem
  • Stability Criterion