Applied Mathematics and Mechanics

, Volume 5, Issue 6, pp 1875–1881 | Cite as

Note on the critical variational state in elasticity theory

  • Liu Cheng-qun


Recently Prof. Chien Wei-zang[1] pointed out that in certain cases, by means of ordinary Lagrange multiplier method, some of undetermined Lagrange multipliers may turn out to be zero during variation. This is a critical state of variation. In this critical state, the corresponding variational constraint can not be eliminated by means of simple Lagrange multiplier method. This is indeed the case when one tries to eliminate the constraint condition of strain-stress relation in variational principle of minimum complementary energy by the method of Lagrange multiplier. By means of Lagrange multiplier method, one can only derive, from minimum complementary energy principle, the Hellinger-Reissner principle[2,3], in which only two type of independent variables, stresses and displacements, exist in the new functional. Hence Prof. Chien introduced the high-order Lagrang multiplier method bu adding the quadratic terms.
$$A_{ijki} (e_{ij} - b_{ijmn} \sigma _{mn} )(e_{ki} - b_{kipq} \sigma _{pq} )$$

to original functions.

The purpose of this paper is to show that by, adding
$$A_{ijki} \left( {e_{ij} - b_{ijmn} \sigma _{mn} } \right)\left( {e_{kl} - \frac{1}{2}u_{k,l} - \frac{1}{2}u_{l,k} } \right)$$

to original functionals one can also eliminate the constraint condation of strain-stress by the high-order Lagrange multiplier method. With this methods, we find more general form of functional of generalized variational principle ever known to us from Hellinger-Reissner principle. In particular, this more general form of functional can be reduced into all known functions of existing generalized variational principles in elasticity. Similarly, we can also find more general form of functional from H. C. Hu-Washizu principle[4,5].


Lagrange Multiplier Critical State Variational Principle Elasticity Theory Constraint Condition 
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  1. (1).
    Chien Wei-zany, Method of high-order Lagrange multiplier and generalized variational principles of clasticity with, more genoral forme of functionals Applied Mathematics and Mechanics, Vol. 4, No. 2 (1983).Google Scholar
  2. (2).
    Hellinger, E., Der Allgemeine Ansats der Meshanik der Kontirua, clopadia der Mathematishem Wissenshaften, Vol. 4, IV (1914), 602–604.Google Scholar
  3. (3).
    Keissner, E., On a variational theorem in elasticity, Journal of Matimulatics and Physics, 29, 2(1950), 90–95.Google Scholar
  4. (4).
    Hu Hai-Chang, On some variational, principles in the theory of clasticity and the theory of plasticity. Scientia Sinica, 4, 1(1955), 33. (in Chinese)Google Scholar
  5. (5).
    Washizu, K., On the variational principles of elasticity and plasticity Aeroclastic and Structures Research Laboratory, Masschusetts Institute Technology, Technical Report, 25–18, March (1955).Google Scholar

Copyright information

© HUST Press 1984

Authors and Affiliations

  • Liu Cheng-qun
    • 1
  1. 1.Chongqing UnivesityChongqing

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