建立广义变分原理的一种方法
A Method for Establishing Generalized Variational Principle
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摘要: 本文建议了一种根据问题的力学意义来建立广义变分原理的方法,本方法对于那些尚未建立起与之相应的变分原理的问题建立其相应的变分原理是有用的.文中不从最小势能原理的推广出发而从力学意义出发导出了弹性力学中的Hu-Washizu广义变分原理和胡海昌广义余能原理,给出了这两个广义变分原理的正确证明.本文并证明了,如果根据Hu-Washizu广义变分原理及胡海昌广义余能原理中含有σij,eij和ui三类变量,就认为这三类变量相互独立,就会导致错误.文中并阐明了这两个广义变分原理正确运用的条件.Abstract: A method for establishing generalized variational principle is proposed in this paper. It is based on the analysis of mechanical meaning and it can be applied to problems in which the variational principles are needed but no corresponding variational principle is available. In this paper, the Hu-Washizu's generalized variational principle and the Hu's generalized principle of complementary energy are derived from the mechanical meaning instead of from the generalization of the principle of minimum potenlial energy and the correct proofs of these two generaleed variational principles are given. It is also proved that this is wrong if one beleives that σij, eij and ui are independent variables each other based on the reason that these three kinds of variables are all contained in these two generalized variational principles. The condition of using these two variational principles in a correct manner is also explained.
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[1] 胡海昌,论弹性体力学与受范性体力学中的一般变分原理.物理学报,10, 3 (1954),259. [2] 钱伟长,再论弹性力学中的广义变分原理—就等价定理问题和胡海昌先生商榷,力学学报,4 (1983),325. [3] 钱伟长,关于弹性力学中的广义变分原理及其在板壳问题上的应用,未发表,见力学学报编辑部64-057号来信,1964年10月6日. [4] 钱伟长,弹性理论中广义变分原理的研究及其在有限元计算中的应用,机械工程学报,15,2 (1979),1-23, [5] 钱伟长,《变分法和有限元》,科学出版社,北京(1980). [6] 钱伟长,高阶拉氏乘子法和弹性理论中更一般的广义变分原理,应用数学和力学,4, 2(1983),137-150. [7] 胡海昌,《弹性力学的变分原理及其应用》,科学出版社,北京(1981), 387. [8] 胡海昌,略论Hellinger-Reissner和胡海昌一鹜津久一郎两种广义变分原理的联系,力学学报,3 (1983),301. -
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