Abstract
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.
References
Courant, R., Hilbert, D., Methode der Mathematischen Physik I. 3. Auflage, Springer, 1968, 201–207.
Sewell, M. J., On dual approximation principles and optimization in continuum mechanics,Philos. Trans. Roy. Soc. London, Ser. A265 (1969), 319–351.
Reissner, E., On a variational theorem for finite elastic deformations,J. Math. Phys., 32 (1953), 129–135.
Truesdell, C., Noll, W., Non-linear Field Theories of Mechanics, Handbuch der Physik Bd. III/3, Springer, (1965).
Nemat-Nasser, S., General variational principles in nonlinear and linear elasticity with applications, Mechanics Today vol. 1, Pergamon, (1972).
Washizu, K., Variational Methods in Elasticity and Plasticity, ed. II, Oxford, (1975).
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).
Levinson, M., The complementary energy theorem in finite elasicity, Trans. ASME, Ser. E, J. Appl. Mech., 87 (1965), 826–828.
Zubov, L. M., The stationary principle of complementary work in nonlinear theory of elasticity, Prikl, Mat. Meh., 34 (1970), 228–232.
Koiter, W. T., On the principle of stationary complementary energy in the nonlinear theory of elasticity, SIAM, J. Appl. Math., 25 (1973), 424–434.
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).
Fraeijs de Veubeke, B., A new variational principle for finite elastic displacements,Int. J. Engng. Sci., 10 (1972), 745–763.
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).
Odgen, R. W., A note on variational theorems in non-linear elastostatics, Math. Proc. Cams. Phil. Soc., 77 (1975), 609–615.
Dill, E. H., The complementary energy principle in nonlinear elasticity,Lett. Appl. and Engng. Sci., 5 (1977), 95–106.
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.
Ogden, R. W., Extremum principles in non-linear elasticity and their application to composite-I, Int. J. Solids Struct., 14 (1978), 265–282.
Truesdell, C., Toupin, R., The Classical Field Theories, Handbuch der Physik Bd. II/1, Springer, (1960).
Guo Zhong-heng, Theorem of nonlinear elasticity, Scientific Publishing House (1980)
Macvean, D. B., Die Elementararbeit in einem Kontinuum und die Zuordnung von Spannungs- und Verzerrungstensoren,ZAMP, 19 (1968), 157–185.
Hill, R., On constitutive inequalities for simple materials-I,J. Mech. Phys. Solids, 16 (1968), 229–242.
Novozhilov, V. V., Theory of Elasticity, Pergamon, (1961).
Hu Hai-chang, Some variation principles in the elasto-plastic theorems (in Chinese), Chinese Science, 4 (1955). 33–54.
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).
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Zhong-heng, G. Unified theory of variation principles in non-linear theory of elasticity. Appl Math Mech 1, 1–22 (1980). https://doi.org/10.1007/BF01872624
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DOI: https://doi.org/10.1007/BF01872624