横观各向同性材料的三维断裂力学问题

从三维横观各向同性材料弹性力学理论出发, 使用Hadamard有限部积分概念, 导出了三维状态下单位位移间断(位错)集度的基 本解. 在此基础上, 进一步运用极限理论, 将任意载荷作用下, 三维无限大横观各向 同性材料弹性体中, 含有一个位于弹性对称面内的任意形状的片状裂纹问题, 归结为求 解一组超奇异积分方程的问题. 通过二维超奇异积分的主部分析方法, 精确地求得了裂纹前沿光滑点附近的应力奇异指数和奇异应力场, 从而找到了以裂纹表面位移间断表示的应力强度因子表达式及裂纹局部扩展所提供 的能量释放率. 作为以上理论的实际应用，最后给出了一个圆形片状裂纹问题 的精确解例和一个正方形片状裂纹问题的数值解例. 对受轴对称法向均布载荷作用下圆形片状裂纹问题, 讨论了超奇异积分方程的精确求解方法, 并获得了位移间断和应力强度因子的封闭解, 此结果与现有理论解完全一致.

Three-dimensional fracture mechanics for transversely isotropic materials

In this paper,based on three-dimensional elastic theory for transversely isotropic materials,the fundamental solutions for a displacement-jump(dislocation)were derived with Hadamard's finite-part concepts. Subsequently,the limit theory was used to reduce the three-dimension problems for arbitrary-shaped planar cracks in an infinite transversely isotropic solid to the solutions of a set of hypersingular integral equations.With a dominant-part analysis of two-dimensional hypersingular integrals,the stress singular indices and singular stress fields near smooth periphery of the crack front were exactly derived and thus the stress intensity factors and the energy release rate for local extension of the crack were expressed in the form of the displacement jumps on the crack surfaces.As applications of the results obtained,two examples are given,an exact solution for a penny-shaped crack problem and a numerical solution for a square crack.The exact analytical method for hypersingular integral equations under axially uniform extension loading for the penny-shaped crack was discussed.The closed solutions of the displacement jumps and the stress intensity factors were obtained.These solutions agree with available results.