Applied Mathematics and Mechanics

, Volume 1, Issue 1, pp 1–22 | Cite as

Unified theory of variation principles in non-linear theory of elasticity

  • Guo Zhong-heng


The purpose of this paper is to introduce and to discuss several main variation principles in nonlinear theory of elasticity — namely the classic potential energy principle, complementary energy principle, and other two complementary energy principles (Levinson principle and Fraeijs de Veubeke principle) which are widely discussed in recent literatures. At the same time, the generalized variational principles are given also for all these principles. In this paper, systematic derivation and rigorous proof are given to these variational principles on the unified bases of principle of virtual work, and the intrinsic relations between these principles are also indicated. It is shown that, these principles have unified bases, and their differences are solely due to the adoption of different variables and Legendre tarnsformation. Thus, various variational principles constitute an organized totality in an unified frame. For those variational principles not discussed in this paper, the same frame can also be used, a diagram is given to illustrate the interrelationships between these principles.


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  1. 1.
    Courant, R., Hilbert, D., Methode der Mathematischen Physik I. 3. Auflage, Springer, 1968, 201–207.Google Scholar
  2. 2.
    Sewell, M. J., On dual approximation principles and optimization in continuum mechanics,Philos. Trans. Roy. Soc. London, Ser. A265 (1969), 319–351.Google Scholar
  3. 3.
    Reissner, E., On a variational theorem for finite elastic deformations,J. Math. Phys., 32 (1953), 129–135.Google Scholar
  4. 4.
    Truesdell, C., Noll, W., Non-linear Field Theories of Mechanics, Handbuch der Physik Bd. III/3, Springer, (1965).Google Scholar
  5. 5.
    Nemat-Nasser, S., General variational principles in nonlinear and linear elasticity with applications, Mechanics Today vol. 1, Pergamon, (1972).Google Scholar
  6. 6.
    Washizu, K., Variational Methods in Elasticity and Plasticity, ed. II, Oxford, (1975).Google Scholar
  7. 7.
    Washizu, K., Complementary Variational Principles in Elasticity and Plasticity, Lecture at the Conference on “Duality and complementary in Mechanics of Deformable Solids”, Jablonna Poland. (1977).Google Scholar
  8. 8.
    Levinson, M., The complementary energy theorem in finite elasicity, Trans. ASME, Ser. E, J. Appl. Mech., 87 (1965), 826–828.Google Scholar
  9. 9.
    Zubov, L. M., The stationary principle of complementary work in nonlinear theory of elasticity, Prikl, Mat. Meh., 34 (1970), 228–232.Google Scholar
  10. 10.
    Koiter, W. T., On the principle of stationary complementary energy in the nonlinear theory of elasticity, SIAM, J. Appl. Math., 25 (1973), 424–434.Google Scholar
  11. 11.
    Koiter, W. T., On the complementary energy theorem in nonlinear elasticity theory, Trends in Appl. of pure Math. to Mech. ed. G. Fichera, Pitman, (1976).Google Scholar
  12. 12.
    Fraeijs de Veubeke, B., A new variational principle for finite elastic displacements,Int. J. Engng. Sci., 10 (1972), 745–763.Google Scholar
  13. 13.
    Christoffersen, J., On Zubov's principle of stationary complementary ebergy and a related principle, Rep. No. 44, Danish Center for Appl. Math. and Mech., April, (1973).Google Scholar
  14. 14.
    Odgen, R. W., A note on variational theorems in non-linear elastostatics, Math. Proc. Cams. Phil. Soc., 77 (1975), 609–615.Google Scholar
  15. 15.
    Dill, E. H., The complementary energy principle in nonlinear elasticity,Lett. Appl. and Engng. Sci., 5 (1977), 95–106.Google Scholar
  16. 16.
    Ogden, R. W., Inequalities associated with the inversion of elastic stress-deformation relations and their implications,Math. Proc. Camb. Phil. Soc., 81 (1977), 313–324.Google Scholar
  17. 17.
    Ogden, R. W., Extremum principles in non-linear elasticity and their application to composite-I, Int. J. Solids Struct., 14 (1978), 265–282.Google Scholar
  18. 18.
    Truesdell, C., Toupin, R., The Classical Field Theories, Handbuch der Physik Bd. II/1, Springer, (1960).Google Scholar
  19. 19.
    Guo Zhong-heng, Theorem of nonlinear elasticity, Scientific Publishing House (1980)Google Scholar
  20. 20.
    Macvean, D. B., Die Elementararbeit in einem Kontinuum und die Zuordnung von Spannungs- und Verzerrungstensoren,ZAMP, 19 (1968), 157–185.Google Scholar
  21. 21.
    Hill, R., On constitutive inequalities for simple materials-I,J. Mech. Phys. Solids, 16 (1968), 229–242.Google Scholar
  22. 22.
    Novozhilov, V. V., Theory of Elasticity, Pergamon, (1961).Google Scholar
  23. 23.
    Hu Hai-chang, Some variation principles in the elasto-plastic theorems (in Chinese), Chinese Science, 4 (1955). 33–54.Google Scholar
  24. 24.
    Chien Wei-zang, The Study of the generalized variation principle and its application in the finite element computation (in Chinese), Symposium of lecturers on science of Tsing Hua University (1975).Google Scholar

Copyright information

© Techmodern Business Promotion Centre 1980

Authors and Affiliations

  • Guo Zhong-heng
    • 1
  1. 1.Department of MathematicsPeking UniversityPekingChina

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