Extraction of stress intensity factors from Irwin’s integral using high-order XFEM on triangular meshes

Gan Song, Haim Waisman, Mengyu Lan, Isaac Harari

Research output: Contribution to journalArticlepeer-review


This paper is dedicated to Professor Ted Belytscho’s 70th birthday. His seminal contributions to Fracture Mechanics and the extended finite element method have significantly inspired us in this work. The cur-rent paper extends our recent work on the extraction of stress intensity factors (SIFs) from Irwin’s integral, using a high-order extended FEM (XFEM) formulation. By matching leading r terms (r being the distance from the crack tip to any other material point) in XFEM with analytical expansion of Irwin’s Integral, a closed-form solution for SIFs is obtained. Hence, SIFs can directly be obtained upon solution of the XFEM discrete system, which were shown to be accurate on structured rectangular meshes. However, extension to unstructured quadrilateral elements may not be trivial. To this end, we extend the formulation to triangular elements, where significant simplification of the closed-form expression is obtained. Moreover, the formulation is directly applicable to unstructured meshes without extra work. Hence, this paper is considered a significant step toward automating and extending this method to more general applications. Numerical results show that accurate and consistent SIFs can be obtained on some pure mode and inclined crack benchmark problems, on structured and unstructured meshes. Examples of a crack approaching a hole and two cracks approaching each other are also investigated. The latter illustrates the advantage of this method over a J-integral class of methods, as SIFs can still be calculated when cracks are in proximity and no remeshing is required. Hence, potentially, this method can address crack coalescence, branching and proximity to boundaries more rigorously.

Original languageEnglish
Pages (from-to)528-550
Number of pages23
JournalInternational Journal for Numerical Methods in Engineering
Issue number3-4
StatePublished - Apr 2015


  • Irwin’s integral
  • XFEM
  • energy release rate
  • extended finite element method
  • high-order asymptotic functions
  • mixed-mode fracture
  • stress intensity factors

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • Engineering(all)
  • Applied Mathematics


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