Skip to main content
SpringerLink
Log in

Zur Variationsformulierung nichtlinearer Randwertprobleme

  • Published:
Ingenieur-Archiv Aims and scope Submit manuscript

Übersicht

Es werden verschiedene Bedingungen aufgestellt, die es erlauben, die durch die beiden (Systeme von) nichtlinearen DifferentialgleichungenA (u, σ) = q, B (u, σ) = ɛ und Randbedingungen zusammen mit den nichtlinearen algebraischen Relationenq = C(u, σ), ɛ = D(u, σ) beschriebene Aufgabe durch äquivalente Variationsprobleme zu ersetzen. Dabei zeigt sich ein enger Zusammenhang mit den in der Festkörpermechanik wohlbekannten Prinzipien der virtuellen Verschiebungen und der virtuellen Kräfte. Die auf systematischem Weg konstruierten Variationsfunktionale enthalten viele in der Physik bekannte Funktionale als Sonderfälle, insbesondere jene, die in der Elastomechanik nach Green, Castigliano, Hellinger, Reißner, Hu und Washizu benannt werden.

Summary

In this paper there are established various conditions which allow a variational formulation of the problem described by the two (systems of) nonlinear differential equationsA(u, σ) = q, B(u, σ) = ɛ and boundary conditions together with the nonlinear algebraic relationsq = C(u, σ), ɛ = D(u, σ). Besides a close relationship is revealed to the principles of virtual displacements and virtual forces which are wellknown in solid mechanics. The systematically constructed variational functional contain many functionals in physics as special cases, mainly those of Green, Castigliano, Hellinger, Reißner, Hu and Washizu in elastomechanics.

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

Literatur

  1. Noble, B.: Complementary variational principles for boundary value problems I, Basic principles. Univ. of Wisconsin, Math. Res. Center Report 473 (1964)

  2. Rall, L. B.: On complementary variational principles. J. Math. Anal. Appl. 14 (1966) S. 174

    Google Scholar 

  3. Sewell, M. J.: On dual approximation principles and optimization in continuum mechanics. Phil. Trans. Roy. Soc., London A 265 (1969) S. 319

    Google Scholar 

  4. Arthurs, A. M.: Complementary Variational Principles. Oxford University Press 1970

  5. Robinson, P. D.: Complementary variational principles, in [13], S. 507

  6. Noble, B.; Sewell, M. J.: On dual extremum principles in applied mathematics. J. Inst. Maths. Applies. 9 (1972) S. 123

    Google Scholar 

  7. Sewell, M. J.: On governing equations and extremum principles of elasticity and plasticity generated from a single functional. J. Struct. Mech. 2 (1) (1973) S. l und 2 (2) (1973) S. 135

    Google Scholar 

  8. Arthurs, A. M.: Dual extremum principles and error bounds for a class of boundary value problems. J. Math. Anal. Appl. 41 (1973) S. 781

    Google Scholar 

  9. Oden, J. T.; Reddy, J. N.: On dual-complementary variational principles in mathematical physics. Int. J. Engng. Sci. 12 (1974) S. 1

    Google Scholar 

  10. Bufler, H.: Variationsgleichungen und finite Elemente. Bayer. Akad. Wiss., math.-naturw. Klasse (1975) S. 155

  11. Tonti, E.: Variational formulation of nonlinear differential equations I and II. Bull, de l'Acad. roy. de Belgique (Classe des Sciences) 55 (1969) S. 137 und S. 262

    Google Scholar 

  12. Tonti, E.: A mathematical model for physical theories. Acc. Naz. Linc., Rend. Cl. Sc. Fis. Mat. Nat. Ser. 8 vol. 52 (1972) S. 175 und S. 350

    Google Scholar 

  13. Rall, L. B. (ed.): Nonlinear functional analysis and applications. New York, London 1971

  14. Nashed, M. Z.: Differentiability and related properties of nonlinear operators: Some aspects of the role of differentials in nonlinear functional analysis, in [13], S. 103

  15. Vainberg, M. M.: Variational methods for the study of nonlinear operators, Holden-Day, San Francisco, London, Amsterdam 1964

    Google Scholar 

  16. Vainberg, M. M.: Variational method and method of monotone operators in the theory of nonlinear equations, New York, Toronto 1973 (translation of the Russian edition, Moscow 1972)

  17. Bufler, H.: Erweiterung des Prinzips der virtuellen Verschiebungen und des Prinzips der virtuellen Kräfte. Z. Angew. Math. Mech. 50 (1970) S. T 104

    Google Scholar 

  18. Courant, R.; Hilbert, D.: Methoden der mathematischen Physik I, 3. Aufl., Berlin, Heidelberg, New York 1968

  19. Hu, Hai-Chang: On some variational principles in the theory of elasticity and the theory of plasticity. Scientia Sinica 4 (1955) S. 33

    Google Scholar 

  20. Washizu, K.: On the variational principles of elasticity and plasticity. Techn. Rep. 25-18 of the Aeroel. and Struct. Res. Lab. of the MIT 1955

  21. Washizu, K.: Variational Methods in Elasticity and Plasticity. London, Edinburgh 1968

  22. Hellinger, E.: Die allgemeinen Ansätze der Mechanik der Kontinua. Enzykl. math. Wiss. 4 (1914) S. 601

    Google Scholar 

  23. Reißner, E.: On a variational theorem for finite elastic deformations. J. Math. Phys. 32 (1953) S. 129

    Google Scholar 

  24. Koppe, E.: Die Ableitung der Minimalprinzipien der nichtlinearen Elastizitätstheorie mittels kanonischer Transformation. Nachr. Akad. d. Wiss. Göttingen, math.-phys. Kl. (1956) S. 259

  25. Born, M.: Untersuchungen über die Stabilität der elastischen Linie in Ebene und Raum unter verschiedenen Grenzbedingungen, Dissertation, Göttingen 1906

  26. Prange, G.: Das Extremum der Formänderungsarbeit, Habilitationsschrift, Hannover 1916

  27. Hlaváček, I.: Variational principles in the linear theory of elasticity for general boundary conditions. Aplikace Matematiky 12 (1967) S. 15

    Google Scholar 

  28. Friedrichs, K.: Ein Verfahren der Variationsrechnung, das Minimum eines Integrals als das Maximum eines anderen Ausdrucks darzustellen. Nachr. Ges. Wiss. Göttingen (1929) S. 13

    Google Scholar 

  29. Koiter, W. T.: On the principle of stationary complementary energy in the nonlinear theory of elasticity. SIAM J. Appl. Math. 25 (1973) S. 424

    Google Scholar 

  30. Stumpf, H.: Dual extremum principles and error bounds in the theory of plates with large deflections. Arch. Mech. 27 (1975) S. 485

    Google Scholar 

  31. Krawietz, A.: A comprehensive constitutive inequality in finite elastic strain. Arch. Rat. Mech. 58 (1975) S. 127

    Google Scholar 

  32. Prager, W.: Variational principles for elastic plates with relaxed continuity requirements. Int. J. Solids Struct. 4 (1968) S. 837

    Google Scholar 

  33. Nemat-Nasser, S.: On variational methods in finite and incremental elastic deformation problems with discontinuous fields. Quart. Appl. Math. 30 (1972) S. 143

    Google Scholar 

  34. Tonti, E.: A systematic approach to the search for variational principles. Dep. Civ. Eng., Southampton Univ., England 1972

    Google Scholar 

  35. Magri, F.: Variational formulation for everylinear equation. Int. J. Engng. Sci. 12 (1974) S. 537

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mechanik (Bauwesen), Universität Stuttgart, Keplerstraße 11, D-7 Stuttgart 1

    H. Bufler

Authors
  1. H. Bufler
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bufler, H. Zur Variationsformulierung nichtlinearer Randwertprobleme. Ing. arch 45, 17–39 (1976). https://doi.org/10.1007/BF00534244

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00534244

Search

Navigation