The Boundary Element Method Based On The Parametric Space And Its Application In Fracture Mechanics

Fracture is one of the main failure forms for structures in civil engineering, and the strength of materials may be the most important influence factor. Due to the complicated geometry and materials in actual engineering problems, analytical solutions only exist for a few simple fracture mechanics problems, and almost all of them should be solved by means of numerical methods. Conventional numerical methods may be powerless as to the existence of the singular stress field around the crack tips, and some special techniques should be employed according to the features of fracture mechanics. However, the singular nature of the fundamental solution used as the weight function in the boundary element method (BEM) results in that the diagonal and near diagonal elements in the final algebraic equations are much larger than the others. This feature makes the BEM very suitable for crack problems with great variation of gradient field. Moreover, only the boundary needs to be discretized in BEM, and the dimension reduction property decreases the pre-processing information and matrix scale.Three aspects of research work related to fracture mechanics have been carried out based on the BEM as follows:(1) how to modelling the crack geometry accurately;(2) how to evaluate the singular stress field around crack tips accurately;(3) how to improve the accuracy and efficiency.Firstly, in order to model the crack geometry accurately, the BEM based on the parametric space have been developed to eliminate the geometry error caused by discretization. The boundary representation (B-rep) data structure in CAD packages is introduced in the conventional BEM. The parametric face in B-rep data structure is seen as a large isoparametric element, and divided into a series of boundary elements in parametric space. These elements is only used for variable interpolation and numerical integration, and the geometry approximate is implemented by parametric mapping, which avoid the geometry error effectively.Then, in order to evaluate the nearly weak and strong singular integrals accurately, two types of nonlinear transformations have been investigated: the distance transformation and the sinh transformation. When calculate the singular stress field around the crack tips, the source point may be very close to the field point and nearly singular integrals appear. Regular Gauss quadrature cannot calculate these integrals effectively. By introducing the distance function and the nonlinear transformations, the Gauss points move close to the singular point and the nearly singular integrals can be evaluated accurately.Finally, in order to improve the accuracy and efficiency, a direct traction boundary integral equation method (DTBIEM) applicable for crack problems have been proposed. According to the traction equilibrium assumed on the crack surfaces and the properties of fundamental solutions, the crack opening displacements are chosen as unknowns on either side of crack surfaces. The traction boundary integral equation is collocated on both the external boundary and either side of the crack surfaces.The study shows that the BEM based on the parametric space can model the geometry accurately, and keeps high accuracy and convergence for two and three dimensional elasticity problems. Besides, it is also applicable for complicated geometry models. The proposed direct traction boundary integral equation method for crack problems only considers either crack surface and keeps high accuracy and efficiency, which can be applied into fracture problems in practical engineering.