We consider the elastic stress near a hole with corners in an infinite plate under biaxial stress. The elasticity problem is formulated using complex Goursat functions, resulting in a set of singular integro-differential equations on the boundary. The resulting boundary integral equations are solved numerically using a Chebyshev collocation method which is augmented by a fractional power term, derived by asymptotic analysis of the corner region, to resolve stress singularities at corners of the hole. We apply our numerical method to the test case of the hole formed by two partially overlapping circles, which can include either a corner pointing into the solid or a corner pointing out of the solid. Our numerical results recover the exact stress on the boundary to within relative error \(10^{-3}\) for modest computational effort.

我们考虑了在双轴应力下无限板中带有拐角的孔附近的弹性应力。弹性问题是使用复杂的Goursat函数公式化的，从而在边界上产生了一组奇异的积分微分方程。使用切比雪夫搭配方法对所得的边界积分方程进行数值求解，该方法通过对角区域进行渐近分析得出的分数次幂项进行扩展，以解决孔角处的应力奇异性。我们将数值方法应用于由两个部分重叠的圆形成的孔的测试案例，该圆可以包括指向实体的角或指向实体之外的角。我们的数值结果将边界上的精确应力恢复到相对误差\（10 ^ {-3} \）内 进行适度的计算。