对合变换和薄板弯曲问题的多变量变分原理

本文利用拉氏乘子法把薄板弯曲问题的最小位能原理和最小余能原理的变分约束条件解除.从而导出了常见的广义变分原理.为了降低泛函中变量导数的阶次.我们用对合变换引进新的正则变量.于是,我们可以进一步利用拉氏乘子法,把这些对合变换当作变分约束而予以消除,从而导出了各种多变量的薄板弯曲广义变分原理.事实证明,使用上述拉氏乘子法,并不能消除一切变分约束;为此,我们进一步引用高阶拉氏乘子法消除这些剩下来的约束条件,从而导得了薄板弯曲问题的更一般的广义变分原理.

Involutory Transformations and Variational Principles with Multi-Variables in Thin Plate Bending Problems

In this paper, the generalizd variational principles of plate bending, problems are established from their minimum potential energy principle and minimum complementary energy principle through the elimination of their constraints by means of the method of Lagrange multipliers. The involutory transformations are also introduced in order to reduce the order of differentiations for the variables in the variation. Funhermore, these involutory transformations become infacl the additional constraints in the varialion, and additional Lagrange multipliers may be used in order to remove these additional constraints. Thus, various multi-variable variational principles are obtained for the plate bending problems. However, it is observed that, not all the constrainls of variaticn can be removed simply by the ordinary method of linear Lagrange multipliers. In such cases, the method of high-order Lagrange multipliers are used to remove those constrainls left over by ordinary linear multiplier method. And consequently, some functionals of more general forms are oblained for the generaleed variational principles of plate bending problems.