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

Abstract

SUMMARYThis 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 current 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.

Direct evaluation of stress intensity factors for curved cracks using Irwin's integral and XFEM with high‐order enrichment functions

“…However, it is not yet clear how the VCCT can be employed together with XFEM, because cracks may be located anywhere within an element, and obtaining the crack‐tip force may not be trivial as in standard FEM. To this end, our recent work in this field reverted back to the original construct of Irwin integral, where XFEM with high‐order asymptotic enrichment functions was proposed in order to obtain highly accurate near‐tip stress fields . In this contribution, we further extend this framework to curved cracks.…”

“…However, and unlike the J‐integral, its extension to the XFEM is not trivial as the simple formula in terms of nodal forces and displacements may not hold for cracks located within enriched elements. To this end, our recent work demonstrated that the original definition of Irwin's crack closure integral, upon which the VCCT is based, is a more suited framework for computation of SIFs with the XFEM . In this approach, SIFs are obtained directly from a closed‐form evaluation of Irwin's integral once the XFEM discrete system is solved.…”

“…The objective of this contribution is to extend our approach that combines Irwin's integral and XFEM with high‐order enrichment functions to curved cracks. Unlike the previous work that assumes piecewise straight cracks, we consider the crack curvature within the elements by representing the crack geometry explicitly. Furthermore, we construct an efficient quadrature scheme using a generalized Duffy transformation and high‐order isoparametric mapping in order to integrate the singular fields in the element containing the curved crack tip.…”

A decomposition technique of generalized degrees of freedom for mixedmode crack problems

“…The aforementioned aspects are all implemented in the highorder NMM, so this study can be regarded as the development of the virtual crack extension technique and can also be seen as a prelude to an h-version high-order NMM.to extract the strain-energy release rates was provided by [18]. Under the context of XFEM, a direct analytical method to extract the mixed-mode strain-energy release rates from Irwin's integral was given by [19], and then this method was extended to high-order XFEM [20,21].Alternatively, the stiffness derivative technique (SDT) and virtual crack extension techniques (VCET) were proposed, respectively [22,23]. Whereafter, the VCET was applied to determine the SIFs of mode-I and mode-II by carrying out virtual crack extension along both the parallel and perpendicular directions to crack surface [24].…”

“…to extract the strain-energy release rates was provided by [18]. Under the context of XFEM, a direct analytical method to extract the mixed-mode strain-energy release rates from Irwin's integral was given by [19], and then this method was extended to high-order XFEM [20,21].…”