PSI - Issue 13

3

Kazuki Shibanuma et al. / Procedia Structural Integrity 13 (2018) 1238–1243 Author name / Structural Integrity Procedia 00 (2018) 000–000

1240

Crack

: Nodes in set J : Nodes in set C

Finite element mesh

Crack shape

Polycrystal

Cleavage plane formation

Enrichment

Fig. 2 Relationship among the finite element mesh, the crack shape, and the polycrystal

where the first term is the ordinary finite element approximation. The second and third terms are the enrichment terms to express the crack discontinuity and the asymptotic field near the crack front, respectively. � � � is the standard interpolation functions related to node . � � is the Heaviside function to describe crack discontinuity. � � � is the set of the enrichment functions called as the branch functions to describe the asymptotic field near the crack front. These enrichment functions are the same ones in the original formulation proposed by Moës et al. (2002), so that the details of the definitions are found in their work. � , � and �� are nodal degrees of freedom. is the set of all nodes. is the set of nodes related to the branch enrichment defined as the all nodes of the elements where the crack front is included. is the set of nodes related to the Heaviside enrichment defined as the all nodes of the elements where the crack surface is included except the nodes belonging to . A schematic of the crack and relating sets of nodes is shown in Fig. 3.

: Nodes in set J : Nodes in set C

Fig. 3 Sets of nodes for XFEM approximation corresponding to the crack shape