Modeling of cracked structures containing voids subjected to fatigue and dynamic loads using XFEM KIRED Mohamed Riad1, HACHI Brahim Elkhalil1,*, GUESMI Mohamed1, RECHAK Said 2, BADAOUI Mohamed1. 1 Laboratoire de Développement en Mécanique et Matériaux, University of Djelfa, PB 3117, Djelfa, Algeria. 2 Laboratoire de Génie Mécanique et Développement, ENP, Algiers, Algeria. *br_khalil@yahoo.fr Abstract. In this paper, we present a modeling of planar structures under dynamic loading containing stationary cracks in order to determine the dynamic stress intensity factor (DSIF). This parameter will be evaluated by using the eXtended Finite Element Method (XFEM) coupled with the interaction integral technique. Some examples of validation of the computer code developed in this work were tested. The good correlation of the obtained results in fatigue with the literature proves the effectiveness of the method as well as the developed computer code. In the dynamic case, a parametric study on the presence, position and size of the void with respect to the crack and also on the crack type (crack edge and central crack) was conducted for some practical applications. Keywords: Stress Intensity Factor, Extended Finite Element Method, Dynamic Loads, Fatigue, Void. 1. Introduction and stat of the art In the cracking of fragile and quasi-fragile structures containing voids (holes) and subjected to quasi-static and dynamics loadings, the characterized parameter is the Stress Intensity Factor (SIF). Many techniques have been used in literature to evaluate this parameter. Among which we mention the finite element method FEM [1], the boundary element method BEM [2], the finite difference method FDM [3], and the symmetric-Galerkin boundary element method SGBEM [4]. We note that the FEM is the most popular for its flexibility and efficiency. However, it requires a special treatment of discontinuities and singularities of fields due to the presence of the crack. For this purpose, a new FEM approach has been developed by Belytschko and Black [5] named eXtended Finite Element Method (XFEM). It consists to take into account the discontinuity at the crack edges and the singularity at the crack tip by enrichment of neighboring nodes with new degrees of freedom via the new shape functions associated with elements containing these nodes. Among the first who addressed the problem of voids by using XFEM in static are Sukumar and Chopp [9], by introducing a new enrichments for voids. Recently, J M Pais [10] has treated voids problem using XFEM but limited on static and quasi-static loadings. In this context, this work seeks to model the behavior of structures containing simultaneously voids and stationary cracks and subjected to different types of loads (fatigue loads and dynamic Heaviside step loading). The SIF will be evaluated using a global approach; based on the J integral. Also, in this work, we will test the effect of size and position of the void. The obtained results will be compared with other works in literature. 2. XEFM formulation The XFEM introduces in the approximation of the displacement field three types of enrichments [5]: -A discontinuous function H (Heaviside function) that enriches the split nodes (Fig. 1): ( ) 0 1 ( ) 0 1 ( ) if x if x H x . (1) Where is the level set function that determines the normal position of node (x) from the crack. -Four (04) singular functions for each tip node (Fig. 1): (2)
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