A method to compute mixed‐mode stress intensity factors for nonplanar cracks in three dimensions

Abstract

Summary 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.

Comments on the paper by B.E. Grossman‐Ponemon, L.M. Keer, and A.J. Lew “A method to compute mixed‐mode stress intensity factors for nonplanar cracks in three dimensions” (Int. J. Numer. Methods Eng., 2020)

“…The above leads to an exact solution that is constant in the x 3 direction and given by where , with , the well‐known angular variation of the asymptotic fields, and e i a base vector of the Cartesian system. Function can be found in, for example, Reference 6 and appendix B1 of Reference 1.…”

“…Solution u U (x) is the same as equation (30) of Reference 1 with . It is also noted that 1 adopts and periodic boundary conditions on boundary faces with | x 3 | = 0.5.…”

“…Grossman‐Ponemon et al 1 use interaction integrals 2 and the Displacement Correlation Method (DCM) 3 to extract stress intensity factors (SIFs) along three‐dimensional crack fronts. Finite element approximations defined on unstructured meshes of trilinear hexahedral and linear tetrahedral elements are adopted by Grossman‐Ponemon et al 1 They show that SIFs extracted with the DCM fail to converge with mesh refinement. The DCM is denoted in that work as “Gupta, Duarte, and Dhankhar's displacement correlation method (GDD‐DCM)”.…”

“…Gupta et al 4 study the performance of the DCM when applied to Generalized/eXtended Finite Element (G/XFEM) approximations – not the finite element method (FEM) as in Grossman‐Ponemon et al 1 Numerical studies presented in Gupta et al 4 show that unless a proper G/XFEM enrichment scheme is adopted , SIFs extracted with the DCM are of low accuracy and sensitive to extraction parameters. Since the FEM is the same as the G/XFEM without any enrichment, it is not surprising that the DCM is not convergent for the problems solved by Grossman‐Ponemon et al 1 …”

“…That work was not the first to use the DCM with G/XFEM approximations, as also discussed in Reference 4. Therefore, using the DCM with standard FEM discretizations while referring to it as GDD‐DCM is a misrepresentation of the work presented in Gupta et al 4 One problem adapted from 1 is solved next. It confirms the findings of Gupta et al 4 …”