高阶拉氏乘子法和弹性理论中更一般的广义变分原理

作者曾指出1],弹性理论的最小位能原理和最小余能原理都是有约束条件限制下的变分原理采用拉格朗日乘子法,我们可以把这些约束条件乘上待定的拉氏乘子,计入有关变分原理的泛函内,从而将这些有约束条件的极值变分原理,化为无条件的驻值变分原理.如果把这些待定拉氏乘子和原来的变量都看作是独立变量而进行变分,则从有关泛函的驻值条件就可以求得这些拉氏乘子用原有物理变量表示的表达式.把这些表达式代入待定的拉氏乘子中,即可求所谓广义变分原理的驻值变分泛函.但是某些情况下,待定的拉氏乘子在变分中证明恒等于零.这是一种临界的变分状态.在这种临界状态中,我们无法用待定拉氏乘子法把变分约束条件吸收入泛函,从而解除这个约束条件.从最小余能原理出发,利用待定拉氏乘子法,企图把应力应变关系这个约束条件吸收入有关泛函时,就发生这种临界状态,用拉氏乘子法,从余能原理只能导出Hellinger-Reissner变分原理2],3],这个原理中只有应力和位移两类独立变量,而应力应变关系则仍是变分约束条件,人们利用这个条件,从变分求得的应力中求应变.所以Hellinger-Reissner变分原理仍是一种有条件的变分原理.

Method of High-Order Lagrange Multiplier and Generalized Variational Principles of Elasticity with More General Forms of Functionals

It is known1] that the minimum principles of potential energy and complementary energy are the conditional variation principles under respective conditions of constraints. By means of the method of La-grange multipliers, we are able to reduce the functionals of conditional variation principles to new functionals of non-conditional variation principles. This method can be described as follows:Multiply undetermined Lagrange multipliers by various constraints, and add these products to the original functionals.Considering these undetermined Lagrange multipliers and the original variables in these new functionals as independent variables of variation,we can see that the stationary conditions of these functionals give these unceter -mined Lagrange multipliers in terms of original variables. The substitutions of Ihese results for Lagrange multipliers into the above functionals lead to the functionals of these non-conditional variation principles.However, in certain cases, some of the 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 cannot be eliminated by means of the simple Lagrange multiplier method. This is indeed the case when one tries to eliminate the constraint condition of stress-strain relation in the variational principle of minimum complementary energy by the me-thod of Lagrange multiplier. By means of Lagrange multiplier method one can only derive, from minimum complementary energy principle,the Hellinger-Reissner principle2,3], in which only two types of independent variables, stresses and displacements, exist in the new functional. The strain-stress relation remains to be a constraint,from which one derives the strain from the given stress. Thus the Hellinger-Reissner principle remains to be a conditional variation with one constraint uneliminated.In ordinary Lagrange multiplier method, only the linear terms of constraint conditions are taken into consideration. It is impossible to incorporate this condition of constraint into functional whenever the corresponding Lagrange multiplier turns out to be zero. Hence, we extend the Lagrange multiplier method by considering not only the linear term, but also the high-order terms,such as thequa-dratic terms of constraint in the Taylor's series expansion.We call this method the high order Lagrange multiplier method.With this method we find the more general form of functional of the generalized variational principle ever known to us from the Hellinger-Reissner principle. In particular, this more general form of functional can be all known functionals of existing generalized variational principles in elasticity. Similarly, we can also find the more general form of functional from He-Washizu principle4,5].It is also shown that there are equivalent theorem and related equivalent relation between these two general forms of functionals in elasticity.