A Method Of Hamiltonian System For Reinforcement Of Cracked Structures

The security of equipments and structures has been drawn more attention in engineering. Specially, the reinforcement problem of cracked structures has relation to security of structures, extending of service life of structures and resource saving. However, structures may appear sometimes spallations on the interfaces after they are reinforced. Thus the mechanism of layer crack problem need to be researched and the reinforcement craft is demanded to be improved with technology content. So far, the finite element method is used almost for analyzing the problem. Nevertheless, for computing stress intensity factors, the mesh density of the elements and the path of limit affect directly precision of the factors. For the problem, in this degree thesis, an impactful method is presented to analyze cracked structures, to reveal the cracking mechanism and to study the key scientific problem for the reinforcement of cracked structures. In this dissertation, the reinforcement problem of cracked structures is discussed as a background and application, and the method of Hamiltonian system for structures and the model of symplectic singular element are researched and analyzed systemically. Some main results are obtained and listed below:The Hamiltonian system is generalized to spatial fundamental problem of anisotropic elasticity. In the system, one spatial coordinate is simulated to the time and the dual variable is obtained from the elastic potential energy. The dual equations, in Hamiltonian system, are shown with the aid of the Hamiltonian principle. Thus, the fundamental problem is reduced to symplectic eigenvalues and eigensolutions in symplectic space. Since sub-symplectic system is introduced again, a direct method is presented for solving symplectic eigenvalues and eigensolutions. Results show that Zero eigenvalue solutions belong to Saint Venant solutions and non-zero-eigenvalue solutions corresponding to the solutions that are coverd by the Saint Venant principle. In the complete space of eigensolutions, a new symplectic adjoint relationship of biorthogonality is presented among the eigensolutions. Thus, complete and perfect system for solving solutions is founded. In further research, analyzing singularity for cracked bi-materials is discussed as a breach. A method, which shows Hamiltonian governing equations and eigensolutions divided regions, is presented. Therefore, eigensolutions and general solutions can be expressed uniformly. It is taken notice of that non-zero-eigenvalue solutions have local character. Especially,1/2eigenvalue solutions reveals the singularity of stresses. Namely, the stress intensity factors can be identified directly to be the coefficients of the series of certain eigenvalue solutions. After the functional of subregional integration is introduced, a new and idiographic symplectic relationship of adjoint orthogonality is presented. Based on the key technique, the relationship, the coefficients of the series, which is expanded by symplectic eigenvalue solutions, are determined from the boundary conditions. Thus, the stress intensity factors can be given directly an expression. Based on the results, a model of symplectic singular element is presented. Corresponding relationships between the coefficients of the series symplectic eigenvalue solutions and node displacements are obtained aid the close combination of symplectic functions of eigensolutions and shape functions of elements. Farther, the element stiffness matrix is given directly. The stress intensity factors have clear expressions in the system. The method of symplectic singular element combining with finite element software gives a numerical method for analyzing cracked structures and improves the accuracy of stress intensity factors. As an application of symplectic singular element, it combines with finite element software to analyze reinforcement problem of cracked structures. By analyzing and discussing, a series of results and conclusions is obtained. Results show that crack of the structure and bond strength can be in safety range under reasonable bond and design based on the length of crack in structure, material of cracked structure, constants of reinforced material, geometry size and strength of bond. The optimum reinforcement scheme is to reduce stress intensity factors at the tip of the crack on the interface of two-materials, as well as the cracked structure. The research provides reliable basis on reinforcement technology for the cracked structure.