Hamiltonian Formalisms Applied to Continuum Mechanics: Potential Use for Fracture Mechanics

Abstract

The first part of this chapter deals with several Hamiltonian formalisms in elasticity. The formalisms of Zhong ((1995) Dalian Science & Technology University Press, Liaoning, China) and Bui ((1993) Introduction aux problèmes inverses en mécaniques des matériaux, Editions Eyrolles, Paris), which resolve respectively the two-end problem and the Cauchy problem in elasticity, are presented briefly. Then we propose a new Hamiltonian formalism, which resolves simultaneously the two problems mentioned above and shows the link between the two formalisms. The potential use for fracture mechanics purposes is then mentioned. In fact, when traditional theories in fracture mechanics are used, asymptotic analyses are often carried out by using high-order differential equations governing the stress field near the crack tip. The solution of the high-order differential equations becomes difficult when one deals with anisotropic or multilayer media etc. The key of our idea was to introduce the Hamiltonian system, usually studied in rational mechanics, into continuum mechanics. By this way, one can obtain a system of first-order differential equations, instead of the high-order differential equation. This method is very efficient and quite simple to obtain a solution of the governing equations of this class of problems. It allows dealing with a large range of problems, which may be difficult to resolve by using traditional methods. Also, recently we developed another new way to resolve fracture mechanics problems with the use of ordinary differential equations (ODEs) with respect to the circumferential coordinate θ around the crack (or notch) tip. This method presents the opportunity to be coupled with finite element analysis and then allows resolving more complicated geometries.