Dual Boundary Integral Equations Method For Linear Elastic Fracture Problems

Fracture is one of the most dangerous failure modes for structures and materials, which often cause serious disasters. From this point of view, fracture mechanics is a very important and urgent research area, and many scholars have made great efforts in this field. By now, the currently used methods for solving fracture problems include:theoretical analysis, experimental technique and numerical simulation. With the development of computational science, numerical methods are becoming more and more popular in scholars and engineers. Boundary element method (BEM), as one kind of numerical methods, is established based on singular fundamental solutions. It has the advantages of high precision and mesh reduction characteristic. BEM has been proved to be very suitable for solving fracture problems. However, when solving crack problems using BEM, the traditional displacement boundary integral equation is ill-posed on crack surfaces. In this paper, the traction boundary integral equation is introduced, and the dual boundary integral equation method is derived for solving the fracture problems.In this paper, the dual boundary integral equations for elasticity problem are presented firstly, then the corresponding numerical methods for evaluating different kinds of singular boundary integrals in BEM is described. Especially for evaluating the strongly singular and hypersingular integrals involved in the traction boundary integral equation, a projection line? plane power series expansion method is adopted in this paper to evaluate these singular integrals directly and precisely. Several examples are given to validate the correctness of the proposed dual boundary integral equation method in this paper. Numerical examples are also given to verify the correctness of the adopted singular integration method, which solids the foundation of applying the dual integral equation method to solve fracture mechanics problems.Then, the dual boundary integral equation method for linear elastic fracture problems is presented by taking the crack opening displacements(CODs) as unknown quantities. Only one side of the crack surface need to be collocated points. On the one hand, this method reduces the amount of computational cost, and on the other hand, the computed CODs can be directly used to calculate the stress intensity factors. In order to capture the singular characteristic of the displacement field near the crack tip, two kinds of special crack tip elements are constructed in this paper, in which (?) is introduced in the interpolation functions of CODs. Several examples are given to verify the efficiency of the presented method for 2-dimensional, 3-dimensional, infinite domain, finite domain, straight or curved lines, plane or curved surface crack problems.Finally, the nonhomogeneous and multiple media material fracture problems are discussed, and the corresponding dual boundary integral equations are established in this paper, which providing the direction for the further development of this method.