Non‐planar 3D crack growth by the extended finite element and level sets—Part I: Mechanical model

Abstract

SUMMARYA methodology for solving three-dimensional crack problems with geometries that are independent of the mesh is described. The method is based on the extended ÿnite element method, in which the crack discontinuity is introduced as a Heaviside step function via a partition of unity. In addition, branch functions are introduced for all elements containing the crack front. The branch functions include asymptotic near-tip ÿelds that improve the accuracy of the method. The crack geometry is described by two signed distance functions, which in turn can be deÿned by nodal values. Consequently, no explicit representation of the crack is needed. Examples for three-dimensional elastostatic problems are given and compared to analytic and benchmark solutions. The method is readily extendable to inelastic fracture problems.

Direct estimation of generalized stress intensity factors using a three‐scale concurrent multigrid X‐FEM

“…The eXtended Finite Element Method (X-FEM [23,24,11]), was initially developed for the simulation of failure in solids. It is able to account for the displacement discontinuity across the crack and the asymptotic displacement at the crack tip by the introduction of additional enrichment functions into the classical finite element space, thanks to the Partition of the Unity (PUM) [22].…”

Stability of incompressible formulations enriched with X-FEM

“…The resolution of compressible fracture mechanics problems has been extensively studied in the context of the X-FEM for both 2D [39,18,40,32,24] and 3D fracture mechanics [21,22]. The most common enrichment strategy consists in using the asymptotic displacement field as an enrichment for the displacement finite element approximation.…”

“…Proper enrichment of the finite element basis makes it possible to model crack, material inclusions and holes with non-conforming meshes. The X-FEM method has been used for the simulation of a wide variety of problems such as fracture mechanics problems (2D [18][19][20], 3D [21][22][23], plates [24,25], cohesive zone modeling [26,27], dynamic fracture [28], nonlinear fracture mechanics [29][30][31]), holes [32,33], but also material inclusions [33,34] or multiple phase flows [35]. Here, we focus on the application of this method to mixed formulations for the treatment of holes, material inclusions and cracks in the incompressible limit.…”

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

“…However, the study of a generic 3-D problem using XFEM may still need the construction of a local refined mesh, although the element size should not be so refined as the adequate mesh for the standard finite element approach and the element topology does not need to match the crack geometry [3]. The XFEM and level set methods [4,5] also simplify the analysis and description of curved and/or non-planar cracks in three dimensions [6][7][8], as they provide the appropriate tools to build a local coordinate system natural to the crack geometry as shown in Fig. 1, where direction 1 is the normal direction to the crack front, contained in the crack plane, direction 2 is the normal direction to the crack surface and direction 3 is the tangential direction to the crack front.…”