PSI - Issue 13

Kota Kishi et al. / Procedia Structural Integrity 13 (2018) 1111–1116 Author name / Structural Integrity Procedia 00 (2018) 000 – 000

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to analyze whole of brittle crack propagation and arrest behavior, it is desirable to evaluate the local tensile stresses in finite element analyses. However, because used meshes need to be fine enough to evaluate local tensile stress in the vicinity of the crack tip, the mesh size near the crack tip should be about 0.05~0.1 mm. Furthermore, because explicit analysis is not suitable for accurate local stress evaluation, implicit analysis has to be employed. This means that the analyses considering the local tensile stress need a lot of computational cost even in 2D, and therefore it is nearly impossible to conduct elasto-plastic 3D analyses. In fact, past studies evaluating local tensile stress basically employed 2D finite element models. Therefore, considering above background, we focused on mesh superposition method to evaluate local tensile stress accurately in realistic computational cost. This study carried out first application of mesh superposition method to local tensile stress evaluation of rapid crack propagation and investigated its accuracy. In this research, stationary crack analysis is carried out using mesh superposition method at first in order to prove that mesh superposition method is valid for crack analyses. Minute verification is made using relative L 2 norm. Then dynamic crack analysis is made in order to establish the basic theory on the local fracture stress criterion. 2. Preparation of experiments 2.1. Mesh superposition method for static crack analysis Mesh superposition method is one of the FEM analysis methods presented by Fish (Fish, 1992). Fig. 1 shows the concept of mesh superposition method. In mesh superposition method, all the analysis area is covered with global mesh and the area which needs to be searched minutely is covered with local mesh. The area covered by global mesh and local mesh is defined as the global area and local area respectively. By using mesh superposition method, computational cost can be decreased because only local mesh needs to be fine when local tensile stress is evaluated.

Global mesh

Local mesh

Fig. 1 Schematic illustration of mesh superposition method This is formulation of mesh superposition method. In formulas, the subscripts “G” and “L” mean global mesh and local mesh respectively. Considering the overall area Ω , the local area is defined as Ω L . In Ω and Ω L , displacement field is defined as G and L in the global and local area respectively and displacement in Ω L is defined as the sum of displacement in both meshes, as shown in Eq. (1). = G + L in Ω L (1) In order to ensure the continuity of displacement for Γ GL , the boundary between both areas, L = 0 is satisfied. By discretization of each displacement field and principle of virtual work, Eq. (2) is obtained, [ G GL GL T L ] { G L } = { G L } (2) G and L are usual stiffness matrices for the global area and local area respectively and GL is a stiffness matrix representing the relationship between the both meshes. G and L are external force vectors in both meshes.