Note on the Critical Variational State in Elasticity Theory

Recently Prof, Chien Wei-zang^[1] pointed out that 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 stress-strain 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 types of independent variables, stresses and displacements, exist in the new functional.Hence Prof, Chien Wei-zang introduced the high-order Laaranae multiplier method by addine the quadratic terms
A_ifk1(e_ij-b_iimnσmn)(e_ki-b_k1pqσ_pq)
to the original functionals.The purpose of this paper is to show that by adding the quadratic terms
A_ifk1(e_ij-b_iimnσ_mn)(e_ki-1/2u_k2-1/2u_1:k)
to original functionals one can also eliminate the constraint condition of strain-stress by the high-order Lagrange multiplier method. With this method, we find more general form of generalized variational principle ever known to us from Helliager-Reissner principle, In particular, this more general form of functional can be, reduced into all known functionals of eaisting generalized variational principles in elasticity. Similarly, we can also find snore general form of functional by Hu-Washizu principle^[4,5].