Issue 48

A. Takahashi et alii, Frattura ed Integrità Strutturale, 48 (2018) 473-480; DOI: 10.3221/IGF-ESIS.48.45 474 by the combination rule in the FFS code. The stress intensity factor calculation methods in the FFS code can then be applicable to the simple elliptical cracks. Therefore, the validity and reliability of the alignment and combination rules are of great interest to ensure the appropriate assessment of the structural integrity. In order to evaluate the validity and reliability of the alignment and combination rules, it is necessary to check the fatigue crack growth behavior and the interaction of the non-coplanar cracks. Owing to the development of computers and computer simulation techniques, computer simulation is now a powerful approach to complex crack growth problems. Kamaya et al. has applied the s-version finite element method (s-FEM) to fatigue crack growth simulations, and successfully simulated complex fatigue crack growth behavior [5]. Using the s-FEM, the cracks are modelled by local meshes. The local meshes are then superimposed onto a global mesh, which models the geometry and boundary conditions of the structure of interest. The global and local meshes can be separately modelled so that the complexity in the mesh generation of cracked structures can be drastically reduced. This remarkable property of s- FEM in the mesh generation process is a great advantage in the fatigue crack growth simulation, where the finite element meshes must be repeatedly remodeled in accordance with the updated crack shape. In this paper, the fatigue crack growth simulation of two non-coplanar embedded cracks using the s-FEM is presented, and the validity and reliability of the alignment rule for two non-coplanar cracks are evaluated. According to the previous numerical and experimental studies on two non-coplanar surface cracks [6], the simulated fatigue crack growth behavior is categorized into five patterns to discuss the criteria for the application of the alignment rule. Finally, the interaction of the cracks is evaluated by the stress intensity factor, and the categorization of the fatigue crack growth behavior is discussed in terms of the stress intensity factor. C OMPUTATIONAL METHOD n this study, the fatigue crack growth behavior of two non-coplanar cracks are simulated using the s-FEM. In the s- FEM, as shown in Fig. 1, the geometry and boundary conditions of structures are modeled with a global mesh. Cracks are modeled with local meshes separately from the global mesh, and can be inserted into the structures by superimposing the local meshes on the global mesh. The displacement functions of the global and local meshes are independently defined. As an example, if the structure has two cracks, the problem is modelled with a global mesh and two local meshes for each crack. The displacement function for the global mesh is defined as u i G ( x ) , and for the local meshes, u i L 1 ( x ) and u i L 2 ( x ) . By superimposing the displacement functions, the displacement field of the structure is defined as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 2 1 2 Ω Ω Ω Ω Ω Ω Ω Ω Ω G L L I G i G L G I i i i G L G I i i G L L I i i i x u x u x u x x u x u x u x x u x u x u x x   − + −  +   − =  +  −   + +   (1) where,  G is the entire volume of the structure,  L1 and  L2 are for the volumes defined by the local meshes, and  I is the volume overlapped with the local meshes. In order to preserve the continuity of the displacement function at the boundaries of local meshes, the displacement at the boundaries of the local meshes are fully constraint. Thus, the displacement can be equal to the displacement of global mesh. The displacement equation can then be calculated in accordance with the definition of displacement function given in Eqn. (1). The detailed information about the s-FEM can be found elsewhere [7]. Since the local meshes can be modelled separately from the global mesh, the complex shape of cracks can be easily modelled in the s-FEM. Thus, due to the remarkable advantage of the s-FEM in the modeling of cracks, the fatigue crack growth simulation, where the finite element mesh for cracks must be repeatedly updated, can be performed easily by the s-FEM. In our developed fatigue crack growth simulation system, the crack front shape is modelled with a number of segments, and the local meshes are automatically modeled using the segment data. The global mesh is prepared only at the beginning of the simulation, and is used repeatedly for the entire simulation. The energy release rate along the crack tip is calculated by the virtual crack closure method (VCCM) [8]. The energy release rate calculated by the VCCM is simply converted into the stress intensity factor. Then, the stress intensity factor range is calculated with the stress intensity factor and the stress ratio. Using the stress intensity factor range, the crack growth amount and direction are then determined by the Paris law [9] and I