三维裂纹分析的主值型面力边界积分方程法

A PRINCIPAL TYPED TRACTION BIEM FOR 3D CRACK PROBLEMS 1)

给出了一组只包含Ｃａｕｃｈｙ主值积分、不含有强奇异积分的三维静动力边界积分方程及其应用于裂纹问题的具体列式，并给出了几何轴对称问题的相应半解析边界元求解方法，将三维问题降阶为一维数值问题．文中分析了无限、半无限介质中圆裂纹、平行圆裂纹系、球面裂纹等在静载及应力波作用下的静力或瞬态动力响应问题，求得了相应的应力强度因子．

This paper presents a principal typed traction boundary integral equation method for three dimensional static and dynamic crack problems. Based on a set of traction boundary integral equations for elasto statics/dynamics derived from conservation laws of elasticity, which contain no hyper singular integral, corresponding equations for crack(s) embedded in an infinite medium and in a free surfaced finite body are formulated. Because no hyper singular integral encountered, there are no distinct difficulties to integrate the kernel functions by numerical methods. Furthermore, by subdividing the boundaries(including the crack's upper face and the body's surfaces) and the total time concerned into time space elements and by choosing suitable trial functions for displacements and tractions in each element, we can reduce the principal integrals into weak singular or regular ones by integrating by parts, which simplifies the integrals even more, we obtain linear algebra equations relevant to unknown displacements, CODs and tractions. In addition, for axis symmetric problems, we can subdivide the boundaries along their meridians into ring elements. The corresponding trial functions and integral formulae are presented in the paper. Thus, a semi analytical scheme is set up for the analysis of complicated elasto dynamic 3D crack problems. It can greatly reduce the number of elements, that of DOFs and the computational work. Test examples about the interaction between a single penny shaped crack or spherical crack embedded in an infinite body and stress waves are analyzed, which shows good agreement with the results in literature. Then the static and dynamic interaction between two axial penny shaped cracks and the problem about a penny shaped crack embedded in a free surfaced half space under impact are analyzed and discussed. These examples demonstrate the effectiveness and accuracy of the present method as well as offering useful results to theoretical research and engineering practice.