Note on the critical variational state in elasticity theory

Abstract

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].