Methods to compute the stress intensity factors along a three‐dimensional (3D) crack front often display a tenuous rate of convergence under mesh refinement or, worse, do not converge, particularly when applied on unstructured meshes. In this work, we propose an alternative formulation of the interaction integral functional and a method to compute stress intensity factors along the crack front which can be shown to converge. The novelty of our method is the decoupling of the two discretizations: the bulk mesh for the finite element solution and the mesh along the crack front for the numerical stress intensity factors, and hence we term it the multiple mesh interaction integral (MMII) method. Through analysis of the convergence of the functional and method, we find scalings of these two mesh sizes to guarantee convergence of the computed stress intensity factors in a variety of norms, including maximum pointwise error and total variation. We demonstrate the MMII on four examples: a semiinfinite straight crack with the asymptotic displacement fields, the same geometry with a nonuniform stress intensity factor along the crack front, a spherical cap crack in a cylinder under tension, and the elliptical crack under far‐field tension and shear.

中文翻译：

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计算三维非平面裂纹混合模应力强度因子的一种方法

沿三维（3D）裂纹前沿计算应力强度因子的方法通常在网格细化下显示出收敛速度较慢，或者更糟的是，它们不收敛，特别是应用于非结构化网格时。在这项工作中，我们提出了相互作用积分函数的一种替代形式，以及一种计算沿裂纹前沿的应力强度因子的方法，该方法可以证明是收敛的。我们方法的新颖之处在于两个离散化的解耦：有限元解的体网格和数值应力强度因子沿裂纹前沿的网格，因此我们将其称为多重网格相互作用积分（MMII）方法。通过分析功能和方法的融合，我们找到了这两种网格尺寸的缩放比例，以确保在各种规范（包括最大点误差和总变化）中所计算的应力强度因子的收敛。我们在四个示例上演示MMII：具有渐近位移场的半无限直裂纹，沿裂纹前沿的应力强度因子不均匀的相同几何形状，在拉伸状态下圆柱体中的球形帽裂纹以及在远场下的椭圆形裂纹张力和剪切力。