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Variational principles in finite elastostatics

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Summary

A variational principle of the complementary energy type is derived for an equilibrium state of a finitely deformed elastic body. Trial functions for the actual deformation gradient are used in the formulation of the principle in order to avoid any difficulties arising in the inversion of the stressdeformation gradient relationship. For some equilibrium states the principle becomes a local extremum principle and it can then be used in conjunction with the potential energy principle to provide bounds on the total potential energy. Corresponding principles for elastic materials subject to kinematical constraints are presented. The same approach provides variational principles for infinitesimal deformations superposed on a known finite deformation of an elastic body.

Zusammenfassung

Es wird für den Gleichgewichtszustand eines endlich deformierten elastischen Körpers ein Variationsprinzip des Komplementärenergie-Typs entwickelt. Bei seiner Formulierung werden Versuchsfunktionen für den wirklichen Deformationsgradienten verwendet, um die bei der Umkehrung der Beziehung zwischen der Spannung und dem Deformationsgradienten entstehenden Schwierigkeiten zu umgehen. Für einige Gleichgewichtszustände wird das Prinzip zum lokalen Extremalprinzip, und es kann dann zusammen mit dem Prinzip der potentiellen Energie zur Gewinnung von Schranken für die potentielle Gesamtenergie verwendet werden. Es werden entsprechende Prinzipien für elastische Körper mit kinematischen Bindungen aufgestellt. Das gleiche Verfahren liefert Variationsprinzipien für infinitesimale Verformungen, die der bekannten endlichen Deformation eines elastischen Körpers überlagert sind.

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References

  1. W. T. Koiter, ‘On the complementary Energy Theorem in Non-linear Elasticity’,Trends in Applications of Pure Mathematics, Ed. G. Fichera, Pitman Publ., London 1976, pp. 207–232.

    Google Scholar 

  2. M. Levinson,The Complementary Energy Theorem in Finite Elasticity, J. Appl. Mech.,32, 826–828 (1965).

    Google Scholar 

  3. B. Fraeys de Veubeke,A New Variational Principle for Finite Elastic Displacements, Int. J. Engng. Sci.,10, 745–763 (1972).

    Google Scholar 

  4. J. Christofferson,On Zubov's Principle of Stationary Complementary Energy and a Related Principle, Report 44, Danish Center for Applied Mathematics and Mechanics, 1973.

  5. C. A. Truesdell, andW. Noll, ‘The Non-linear Field Theories of Mechanics’,Handbuch der Physik, Vol. III/3, Springer-Verlag, Berlin (1965).

    Google Scholar 

  6. R. T. Shield,On the Stability of Linear Continuous Systems, Z. angew. Math. Phys.,16, 649–686 (1965).

    Google Scholar 

  7. T. K. Caughey, andR. T. Shield,Instability and the Energy Criterion, Z. angew. Math. Phys.,19, 485–492 (1968).

    Google Scholar 

  8. S. J. Lee,Variational Principles in Finite Elasticity with Applications, Ph.D. Thesis, University of Illinois at Urbana-Champaign, December 1979. T. & A.M. Report No. 437.

  9. A. E. Green, andJ. E. Adkins,Large Elastic Deformations, Oxford University (1960).

  10. A. E. Green, R. S. Rivlin, andR. T. Shield,General Theory of Small Elastic Deformations Superposed on Finite Elastic Deformations, Proc. Roy. Soc. London A.,211, 128–154 (1952).

    Google Scholar 

  11. R. L. Fosdick, andR. T. Shield, ‘Extremum Principles in the Theory of Small Elastic Deformations Superposed on Large Elastic Deformations’,Progress in Applied Mechanics, Prager Anniv. Vol., MacMillan, London, 1963, pp. 107–125.

    Google Scholar 

  12. M. E. Gurtin, ‘The Linear Theory of Elasticity,’,Handbuch der Physik, Vol. VIa/2, Springer-Verlag, Berlin (1972).

    Google Scholar 

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Author notes

  1. Sang J. Lee

    Present address: Mead Office Systems, 75081, Richardson, Texas, USA

Authors and Affiliations

  1. Dept of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA

    Sang J. Lee & Richard T. Shield

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  1. Sang J. Lee
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  2. Richard T. Shield
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Lee, S.J., Shield, R.T. Variational principles in finite elastostatics. Journal of Applied Mathematics and Physics (ZAMP) 31, 437–453 (1980). https://doi.org/10.1007/BF01590856

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  • DOI: https://doi.org/10.1007/BF01590856

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