非线性弹性理论变分原理的统一理论

本文旨在介绍和讨论非线性弹性理论的几个主要变分原理——古典的势能原理,余能原理以及目前争论甚多的另两个余能原理(Levinson原理和Fraeijs de Veubeke原理),同时给出了和这些原理相对应的广义变分原理.本文单一地从虚功原理出发,系统推导并严格论证了这些原理,并且指出了各原理间的内在联系.出发点是一个,采取不同的变量和Legendre变换就导致不同的原理.这样,各变分原理在统一的框架里构成一个有机的整体.文中未涉及的其它原理也同样可以纳入这框架.给出的关系图使读者更能看清各原理间的纵横关系.

Unified Theory of Variational Principles in Non-linear Theory of Elasticity

The purpose of this paper is to introduce and to discuss several main variational principles in nonlinear theory of elasticity,namely the classic potential energy principle, complementary energy principle, and other two complementary energy pri nciples(Levinson principle and Fraeijs de Veubeke principle), which are widely discussed in recent literatures.At the same time, the generalized variational principles are given also for all these principles. In this paper, Systematic derivation and rigorous proof are given to these variational principles on the unified bases of principle of virtual work, and the intrinsic relations between these principles are also indicated. It is shown that, these principles have unified bases, and their differences are solely due to the adoption of different variables and Legendre transformation. Thus, various variational principles constitute an organized totality in an unified frame. For those variational principles not discussed in this paper, the same frame can also be used.a diagram is given to illustrate the interrelationships between these principles.