Symplectic Space Based Analytical Singular Finite Element And Its Applications In Fracture Mechanics

It’s well known that fracture problems arise frequently in many engineering structures, such as in aircrafts, rockets, ships, boilers, bridges, etc. The study on fracture mechanics including prevention and control has significant meaning since destroy accidents caused by fractures could always bring huge loses. Stress intensity factors are usually employed to judge the strength of stress and displacement fields around crack tips, and the corresponding strength criterion of instability propagation was proposed. Meanwhile, COD criterions are used in the crack problems based on cohesive models. Hence, calculating the parameters such as stress intensity factors and COD accurately and efficiently are very attractive research subjects in the strength analysis of engineering practicle structures.The basic theoretical study of fracture problems is well developed, but there are still some complex cases left unsolved, such as cracks with arbitrary tractions, anisotropic multi-material cracks, etc. The solving procedure for such complex problems in traditional methods is always too complex to achieve. With the presence of symplectic dual approach for applied mechanics, many complex problems previously considered impossible can be solved analytically, such as crack problems. In this study, some complex crack problems are considered in symplectic solving system, and analytical symplectic eigen solutions and special solutions are specified. Then, based on these analytical solutions and special solutions, a series of singular finite element are constructed for the numerical analysis of different crack problems. The main contents of the present study are expressed as follows:(1). The special solutions for arbitrary crack tractions acting on single material and bimaterial cracks under plane assumptions are specified in symplectic solving system. Crack tractions are expressed approximately in terms of polynomial expansion, and special solution of each expanding term is specified analytically. In this way, the final special solutions for arbitrary crack tractions can be obtained accordingly. Simultaneously, the symplectic solving system for antiplane crack problem is proposed, analytical symplectic eigen solution and special solution for antiplane cracks with arbitrary tractions are specified. The analysis procedure of multi-material antiplane crack problem is simplified using the coordinate transform technique. Some specific cases are considered as an example, and the corresponding eigen equation of stress singularity orders are given. Besides, multi-material anisotropic antiplane crack problem which couldn’t be solved by traditional sympectic approach is considered in this study. By combining subfield method and symplectic method, a new approximate approach for the stress singularity analyze of multi-material anisotropic antiplane crack is proposed, and this new approach is illustrated by numerical examples. (2). A series of singular finite element are constructed, respectively, based on analytical symplectic eigen solutions and special solutions for single material, bimaterial cracks under plane assumptions and those of antiplane cracks. Using the singular element around crack tips and conventional elements in the other area, cracked structures can be analyzed numerically with high efficiency. According to the relationship between expanding coefficients and stress intensity factors, stress intensity factors for mode Ⅰ,Ⅱ and Ⅲ can be obtained directly without any post process, hence the computational accuracy is ensured. Simultaneously, stiffness matrix of the singular finite elements are proved to be independent on element sizes, this feature is helpful to improve the stability of numerical calculation. As an attempt, the singular finite element for plane cracks is further applied on fatigue crack growth in order to get more precise growth path and more accurate estimated fatigue life. Numerical examples imply that the singular finite element which possesses good caltulation accuracy and stability is very effective for numerical analysis of cracked structures.(3). Based on analytical symplectic eigne solution and special solutions of plane cracks obtained using symplectic system, singular finite elements respectively for single material and bimaterial Ⅰ+Ⅱ mixed-mode Dugdale model based cracks are constructed. Using the singular element around crack tips and conventional elements in the other area, mixed-mode Dugdale crack problems can be solved numerically. Simultaneously, length of plastic zone, crack tip opening/sliding displacement, cohesive stress acting on virtual crack and other parameters in mixed-mode Dugdale model can all be specified by iteration. The stiffness matrix is proved to be independent on element size, and this feature of the singular finite element is helpful to ensure the stability of calculation. Furthermore, the singular element is applied on analyze of mode Ⅰ bimaterial Dugdale cracks with bridging tractions. Numerical examples imply that the present method possesses high calculation accuracy and fast iteration speed, etc. The singular element presented in this study has significant practical value, and it can be applied on the numerical analysis of cracks in elastro-plastic materials with high efficiency.A series of singular finite element are constructed, and the new method which has good stability of calculation has impoved the calculation accuracy and efficiency of numerical analysis of cracked structures. Also, transition elements are not required which has improved the commonality and compatibility of the present elements, hence, the singular finite elements can be integrated into most existing FEM softwares directly to make the most of its practical value in the analysis of engineering structures.