We derive formulas extracting stress intensity factors of the biharmonic equations on cracked domains with clamped (or simply supported or free) boundary conditions along the crack faces. Each of these formulas can be written in terms of the integral of the given source function multiplied by the cut-off dual singularity and the integral of the unknown true solution multiplied by the cut-off dual singularity over the unit disk. The unknown true solution in the extraction formulas is calculated by either the Implicitly Enriched Galerkin Method or the Iteration Methods. The former was developed by the authors (Kim, Oh, Palta, Kim) and the latter proposed in this paper is the sum of the solution of a regular biharmonic equation and singular functions with iteratively estimated stress intensity factors as coefficients. We show the Iteration Methods quickly converge and the proposed Enrichment Method yields highly accurate stress intensity factors. We also demonstrate that for a known true solution, the extraction formulas yield exact stress intensity factors.

我们推导了公式，该公式提取了沿裂纹面具有约束（或简单支撑或自由）边界条件的裂纹域上双调和方程的应力强度因子。这些公式中的每一个都可以用给定源函数的积分乘以截止对偶奇点和未知真实解的积分乘以截止对偶奇点来写。提取公式中的未知真实解是通过隐式丰富Galerkin方法或迭代方法计算的。前者是由作者（Kim，Oh，Palta，Kim）开发的，而后者是由规则的双调和方程和奇异函数（以迭代估计的应力强度因子作为系数）的解的总和。我们展示了迭代方法快速收敛，并且所提出的富集方法产生了高度准确的应力强度因子。我们还证明，对于已知的真实解，提取公式可得出精确的应力强度因子。