Note on the critical variational state in elasticity theory (Q1071540)

Recently \textit{W. Chien} [ibid. 4, 143-157 (1983; Zbl 0517.73018)] 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 constraints can not be eliminated by means of simple Lagrange multiplier method. Chien introduced a higher- order Lagrange multiplier method by adding the quadratic terms: \(A_{ijkl}(e_{ij}-b_{ijmn}\sigma_{mn})(e_{kl}- b_{klpq}\sigma_{pq})\) to the original functions. The purpose of this paper is to show that by adding \(A_{ijkl}(e_{ij}- b_{ijmn}\sigma_{mn})(e_{kl}-u_{k,l}-u_{i,k})\) to original functionals, one can also eliminate the constraint condition of strain- stress by the higher-order Lagrange multiplier method. We find a more general form of the functional of generalized variational principle ever known to us from the Hellinger-Reissner principle. In particular, this more general form of functional can be reduced to all known functions of existing generalized variational principles in elasticity. Similarly, we can also find a more general form of functional from the Hu-Washizu principle.