弹性理论中各种变分原理的分类

本文按弹性理论中各种变分原理的约束条件的不同,对所有变分原理进行分类.我们在前文中业已指出,应力应变关系这样的约束条件是不能用拉氏乘子法解除的.剩下的可能约束条件共有四种:(1)平衡方程,(2)应变位移关系,(3)边界外力已知的边界条件,和(4)边界位移已知的边界条件.弹性理论的各种变分原理中,有的只有一种约束条件,有的有两种或三种,最多只能有四种约束条件.这样一共可能有15种变分原理,但是每种变分原理既可以用应变能A表示,又可以用余能B表示.这样,我们一共应有30种形式完全不同的变分原理,我们全部列出了这三十种形式的变分原理.

Classification of Variational Principles in Elasticity

In this paper, variational principles in elasticity are classified accordiag to the differences is the constraints used in these principles, It is shown in a previous paper[4] that the stress-strain relations are the constraint conditions in all these variational principles, and can not be removed by the method of liaear Lagrange multiplier.The other possible constraats are four of them:(1)equations of equilibrium,(2)Strain-displacement relations,(3)boundary conditions of given external forces and boundary conditions of given boundary displacements. In variational principles of elasticity, some of them have only one kind of such constraints, some have two kinds or three kinds of constraints and at the most four kinds of constraints. Thus, we have altogether 15 kinds of possible variational principles, However, for every possible variatioaal prineiple, either the strain energy density,or the complementary energy density may be used, Hence, there are altogether 3p classes of functionals of variational principles in elasticity. In this paper, all these functionals are tabulated in detail.