A new generalized finite element method for two-scale simulations of propagating cohesive fractures in 3-D

Abstract

This paper presents a novel numerical framework based on the generalized finite element method with global-local enrichments (GFEM gl ) for two-scale simulations of propagating fractures in three dimensions. A non-linear cohesive law is adopted to capture objectively the dissipated energy during the process of material degradation without the need of adaptive remeshing at the macro scale or artificial regularization parameters. The cohesive crack is capable of propagating through the interior of finite elements in virtue of the partition of unity concept provided by the generalized/extended finite element method, and thus eliminating the need of interfacial surface elements to represent the geometry of discontinuities and the requirement of finite element meshes fitting the cohesive crack surface. The proposed method employs fine-scale solutions of non-linear local boundary-value problems extracted from the original global problem in order to not only construct scale-bridging enrichment functions but also to identify damaged states in the global problem, thus enabling accurate global solutions on coarse meshes. This is in contrast with the available GFEM gl in which the local solution field contributes only to the kinematic description of global solutions. The robustness, efficiency, and accuracy of this approach are demonstrated by results obtained from representative numerical examples. determination of an internal length scale for the correct scaling of the dissipated energy associated with the localization process is one of the main issues in the context of the finite element method (FEM) [11,12].Several strategies can be found in the literature to address those challenges in the FEM. One of the early representative approaches-the so-called cohesive zone model (CZM)-relies on the inclusion of interfacial surface elements (cohesive elements) between adjacent element boundaries [13][14][15][16]. The cohesive elements are then employed to capture the associated dissipated energy through a traction-separation law over a surface rather than stress-strain relationships in the bulk. This approach, however, inherently exhibits solutions that are dependent on the finite element mesh because of restricted paths of the discontinuity, thus potentially requiring adaptive refinement techniques at the macro scale [17][18][19][20].The difficulties with the direct incorporation of a strong discontinuity into the element interior have been tackled more recently by the generalized finite element method/extended finite element method (GFEM/XFEM), which is the method adopted in this study; see e.g., [21][22][23] for detailed reviews. This method relies on the construction of the so-called patch approximation spaces in a node-by-node manner and hierarchically enriches the standard FEM approximation space using the partition of unity (POU) framework [24][25][26][27]. A priori knowledge about solutions is typically employed to select enrichments. These nodal enrichments then allow the discontinuity to propagate independently of the ...

A non-intrusive global/local approach applied to phase-field modeling of brittle fracture

“…Non-intrusive global/local approaches have also been applied to a quite large number of situations: the computation of the propagation of cracks in a sound model using the extended finite element method (XFEM) [37], the computation of assembly of plates introducing realistic non-linear 3D modeling of connectors [38], the extension to non-linear domain decomposition methods [39] and to explicit dynamics [40,41] with an application to the prediction of delamination under impact using Abaqus [42]. Alternative strategies can be derived from the Partition of Unity Method [43,44].…”

A hierarchical non-intrusive algorithm for the generalized finite element method

“…The enrichment functions in the GFEM gl are the solution of local problems discretized with a GFEM enriched with analytically defined functions and a fine mesh around features of interest like fractures, material interfaces, etc. The GFEM gl has been formulated and applied to various classes of problems including transient heat transfer [22], linear fracture [23,24], cohesive fracture [25], local plasticity [26], material heterogeneity [27], and localized thermoplasticity [28]. A GFEM gl for linear elastic fracture mechanics problems and suitable for a non-intrusive implementation of the method is described next.…”

Delamination growth in composite laminates of variable stiffness

“…An alternative approach is to use the cohesive zone model, which contains the failure and the fracture criterion in a traction–separation law. Thus, adopting the cohesive formulation can eliminate the stress singularity at the delamination front, and also, the capability to capture the non‐linear responses at the crack tip is provided .…”