An Application Of The Method Of Hamiltonian System For The Problem Of Special Cracks

Many members in industry can be treated as plane problems when bearing or deforming, such as beam, thus plane problem plays an important role in traditional research. Most of the investigations on plane problem employ semi-inverse method which is based on Lagrangian system and has several inevitable drawbacks. Zhong firstly introduced Hamiltonian system with two kinds of variables into elasticity, casting light on new methodology of plane problems. Compared with elementary solutions, Zhong’s process is more reasonable and general, providing a new direct solution. Based on Hamiltonian system, the stress intensity factors of special boundary conditions and special cracks are considered in this paper.Plane rectangular beam is investigated under Hamiltonian system. X-axis is simulated as time coordinate. By utilizing Hamiltonian variational principle and Legendre’s transformation, Hamiltonian canonical equations are obtained, together with side boundary conditions. A special method is introduced to transform arbitrary side loadings into inhomogeneous term of Hamiltonian canonical equations. Original problem is converted into determination of eigensolution coefficients with the help of separation of variables and symplectic expansion theorem and solved by virtue of orthogonal relations.As with stress concentrations caused by complex boundary condition of structures and stress intensity factor of special cracks, Hamiltonian system is introduced to reduce original problem into eigenvalues, eigensolutions and algebraic equations. The local stress concentration factor or stress intensity factor of structures can be directly expressed by the expansion coefficient of specific eigensolutions aided by orthogonal relations in Hamiltonian system, thus forming a numerical method and simulation technique. The results show that the local stress concentration factor is related with the irregular complex boundary of structures and external loads directly. Besides, the method proposed is simple and effective with fast convergence speed. It also provides a basis for the construction of singular element.