Equivalent theorem of Hellinger-Reissner and Hu-Washizu variational principles
- 431 Downloads
The paper has proved that Hellinger-Reissner and Hu-Washizu variational principles are but equivalent principles in elasticity by following three ways: 1) Lagrange multiplier method. The paper points out that only a new independent variable can be introduced when one constraint equation has been eliminated by one Lagrange multiplier, which must be expressed as a function of the original variable(s) and/or the new introduced variable after identification. In using Lagrange multiplier method to deduce Hu-Washizu principle from the minimum potential energy principle, which has only one kind of independent variable namely displacement, by eliminating the constraint equations of stress-displacement relations, one can only obtain a principle with two kinds of variables namely displacement and stress; 2) involutory transformation, with such method Hu-Washizu variational principle can be deduce directly from the Hellinger-Reissner variational principle under the same variational constraints of stress-strain relation, and vice verse; 3)semi-inverse method, by which both of the above variational principles can be deduced from the minimum potential energy principle with the same variational constraints. So the three kinds of variational functions in Hu-Washizu variational principle are not independent to each other, the stress-strain relationships are still its constraint conditions.
Keywordsvariational principles in elasticity Hellinger-Reissner variational principle Hu-Washizu variational principle semi-inverse method trial-functional
Unable to display preview. Download preview PDF.
- 1.Hu, H.C. Some Variational Principles in Elasticity and Plasticity, Acta Physica Sinica, 10, 3 259–290 (1954)Google Scholar
- 2.Chien, W.Z. On Generalized Variational Principles of Elasticity and Its Application to Plate and Shell Problems, Selected Works of Wei-Zang Chien, 419–444 Fujian Education Press 1989 (in Chinese)Google Scholar
- 4.Chien, W.Z. Further Study on Generalized Variational Principles in Elasticity — Discussion with Mr. Hu Hai-Chang on the Problem of Equivalent Theorem, Acta Mechanica Sinica, July 1983, 313–323 (in Chinese)Google Scholar
- 6.Liu, G.L. A Systematic Approach to the Research and Transformation for Variational Principles in Fluid Mechanics with Emphasis on Inverse and Hybrid Problems, Proc. of 1st Int. Symp. Aerothermo-Dynamics of Internal Flow, 1990, 128–135Google Scholar
- 7.He, J.H. On Variational Crisis in Fluid Mechanics and Their Removal, 4th Congress of China Industry & Applied Mathematics. Shanghai 1996 (in Chinese)Google Scholar
- 8.He, J.H. Semi-Inverse Method of Establishing Generalized Variational Principles for Fluid Mechanics With Emphasis on Turbomachinery Aerodynamics, International J. of Turbo & Jet-Engines, 14(1997).Google Scholar
- 9.He, J.H. Semi-Inverse Method — A New Approach to Establishing Variational Principles for Fluid Mechanics, Chinese Annual Congress on Engineering Thermophysics, Beijing 1996 (in Chinese)Google Scholar
- 11.Washizu, K. On the variational principles of elasticity and plasticity, Aeroelastic and Structures Research Laboratory, Massachuetts Institute of Technology, Technical Report, 25–18, March 1955Google Scholar
- 12.Hu, H.C. On Hellinger-Reissner Pinciple and Hu-Washizu Principle (in Chinese), Acta Mechanica Sinica, 3, 1983.Google Scholar
- 13.Hu, H.C. On Generalized Variational Principle and Nonconditional Variational Principle, Journal of Solid Mechanics, 3 462–463 (1984), (in Chinese)Google Scholar
- 14.Chien, W. Z. Also on Generalized Variational Principle and Nonconditional Variational Principle, Journal of Solid Mechanics, 3 451–468 (1984) (in Chinese)Google Scholar