对弹性理论中临界变分状态的一个注记

(t) 最近钱伟长教授指出[1],在某些情况下,用普通的拉氏乘子法,其待定的拉氏乘子在变分中恒等于零,这称为临界变分状态,在这种临界状态中,我们无法用待定拉氏乘子法把变分的约束条件吸收入泛函,从而解除这个约束条件.例如用拉氏乘子法,从最小余能原理只能导出Hellinger-Reissner变分原理,这个原理中只有应力和位移两类独立变量,而应力应变关系仍然是变分的约束条件.为了消除这个约束条件,钱伟长教授提出了高次拉氏乘子法,即在泛函中引入二次项
A_ifk1(e_ij-b_iimnσmn)(e_ki-b_k1pqσ_pq)
来消除应力应变这个约束条件. 本文目的是要证明,如果在泛函中引入如下二次项
A_ifk1(e_ij-b_iimnσ_mn)(e_ki-1/2u_k2-1/2u_1:k)
我们也可以用高次拉氏乘子法解除应力应变这个变分约束条件.用这种方法,我们不仅可以从Hel-linger-Reissner原理的基础上,找到更一般的广义变分原理.在特殊情况下,这个更一般的广义变分原理,可以还原为各种已知的弹性理论变分原理.同样,我们也可以从Hu-Washizu(胡海昌-鹫津久一郎)[4,5]变分原理,用高次拉氏乘子法,求得比该原理更一般的广义变分原理.

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].