Issue 17

E. Benvenuti et alii, Frattura ed Integrità Strutturale, 17 (2011) 23-31; DOI: 10.3221/IGF-ESIS.17.03 24 cohesive crack approach can be extended to rounded notches and makes it possible to deal with non-linear materials. A limitation of the cohesive-crack models is that the crack path has to be placed along the element boundaries. Therefore, the crack geometry depends on the mesh topology. This drawback can be overcome by adopting the eXtended Finite Element method (XFEM) method which has been developed by Belytschko and coworkers [12]. XFEM makes it possible to consider crack paths, which are embedded within the finite elements. XFEM is based on the partition of unity property of the polynomial shape functions, which are used in standard finite element analyses. For instance, in the case of crack problems, where a displacement discontinuity has to be simulated, the standard piecewise continuous approximation of the displacement field is enriched by additional terms, which contain the Heaviside function. More recently, enhanced XFEM approaches have been proposed were the truncated series of William’s expansion of the displacement field are considered as enrichment functions [13-15]. In quasi-brittle materials, experimental evidence shows that macro-cracks stem from the interaction if micro-cracks distributed within a zone of finite width, called fracture process zone (e.g. [16]). Benvenuti et al. [17-19] have proposed a modified XFEM approach which is based on the use of regularized enrichment functions. In particular, the discontinuous step function is replaced by a regularized Heaviside function where a regularization length is introduced. A cohesive-like stress field emerges along the crack path which is always non singular and smooth. The smoothing effect on the stress field increases for increasing values of the regularization length. In the present study, a static failure criterion based on the regularized XFEM approach is applied to the analysis of the tensile PMMA specimens previously proposed by Seweryn [2]. The pros and cons of the present approach are discussed. B ASICS OF THE PROPOSED E X TENDED F INITE E LEMENT FORMULATION et us consider a tensile specimen with a sharp notch. The displacement field is supposed to exhibit a jump    u u across a line d S starting from the notch tip. The vector field n normal to d S is introduced. Let us discretize the geometry through e N finite elements with n N nodes, and introduce the regularized Heaviside function   ( ) / ( ( ))(1 )    s x ρ H x sign s x exp (1) where ( ) s x denotes the signed distance function from the displacement discontinuity defined as ( ): s   x x x if ( ) 0    x x n x and ( ) s    x x x if ( ) 0    x x n x . A picture of the regularized Heaviside function is given in Fig. 2 for various  . Figure 1 : Tensile element with a notch. According to the regularized XFEM formulation recently proposed by Benvenuti et al. [17-19], the displacement field can be expressed as