Convergence of domain integrals for stress intensity factor extraction in 2‐D curved cracks problems with the extended finite element method

Abstract

WileyGonzález Albuixech, VF.; Giner Maravilla, E.; Tarancón Caro, JE.; Fuenmayor Fernández, FJ.; Gravouil, A. (2013). Convergence of domain integrals for stress intensity factor extraction in 2-D curved cracks problems with the extended finite element method. International Journal for Numerical Methods in Engineering. 94(8):740-757. doi:10.1002/nme.4478. This is the peer reviewed version of the following article: Int. J. Numer. Meth. Engng 2013; 94:740-757, which has been published in final form at Wiley Online Library (wileyonlinelibrary.com SUMMARYThe aim of this study is the analysis of the convergence rates achieved with domain energy integrals for the computation of the stress intensity factors (SIF) when solving 2-D curved crack problems with the extended finite element method (XFEM). Domain integrals, specially the J-integral and the interaction integral, are widely used for SIF extraction and provide high accurate estimations with finite element methods. The crack description in XFEM is usually realized using level sets. This allows to define a local basis associated with the crack geometry. In this work the effect of the level set local basis definition on the domain integral has been studied. The usual definition of the interaction integral involves hypotheses that are not fulfilled in generic curved crack problems and we introduce some modifications to improve the behavior in curved crack analyses. Despite the good accuracy of domain integrals, convergence rates are not always optimal and convergence to the exact solution cannot be assured for curved cracks. The lack of convergence is associated with the effect of the curvature on the definition of the auxiliary extraction fields. With our modified integral proposal, the optimal convergence rate is achieved by controlling the q-function and the size of the extraction domain.

Computing stress intensity factors for curvilinear cracks

“…For β aux = β TF , we begin by taking advantage of (27) and the linearity of λ in the second argument to write λ β, β TF = λ β, β DFC + λ (β, β S ) .…”

“…In [27,26], González-Albuixech et al studied the properties of the aforementioned methods in two-and three-dimensions, respectively, but not with all terms arising from the derivation of the interaction integral. Such omission of terms may have contributed to the observed slow convergence and occasional divergence.…”

Domain integral formulation for 3-D curved and non-planar cracks with the extended finite element method

“…Some minor modifications are introduced in the interaction integral formulation to minimize the possible error associated with the use of straight crack fields. These improvements have been studied in [13] using convergence rates in 2D. The modifications do not depend on the dimensionality of the problem, and therefore the introduced formulation can be implemented in three dimensional cases.…”

“…In this work, we will enforce the orthogonality of the level sets locally for all the numerical calculations, since it has been shown in [13] that it is necessary to guarantee convergence [3].…”

“…The solution fields for curved cracks are not the same as the fields for a straight crack and, to the authors' knowledge, a generic expression is still not available. Consequently, the stress intensity factor extraction for curved crack needs a careful approach as shown in [12,13] for 2-D problems. Gosz introduces corrections to the interaction integral in the study of 3-D curved cracks using the finite element method [9,10] achieving an improvement of the results.…”