Unified theory of variation principles in non-linear theory of elasticity

Abstract

The purpose of this paper is to introduce and to discuss several main variation principles in nonlinear theory of elasticity ~ namely the classic potential energy principle, complementary energy principle, and other two complementary energy principles (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 variat)onal 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 anrT Legendre tarnsformation. Thus, various variational principles constitute an organized totality in at" 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.