Int J Fract (2010) 162:33–49 DOI 10.1007/s10704-009-9405-9 ORIGINAL PAPER Time dependent crack tip enrichment for dynamic crack propagation Thomas Menouillard · Jeong-Hoon Song · Qinglin Duan · Ted Belytschko Received: 8 April 2009 / Accepted: 2 September 2009 / Published online: 29 September 2009 © Springer Science+Business Media B.V. 2009 Abstract We study several enrichment strategies for dynamic crack propagation in the context of the extended finite element method and the effect of different directional criteria on the crack path. A new enrichment method with a time dependent enrich- ment function is proposed. In contrast to previous approaches, it entails only one crack tip enrichment function. Results for stress intensity factors and crack paths for different enrichments and direction criteria are given. Keywords Dynamic · Fracture · XFEM 1 Introduction Classical finite element strategies for dynamic crack propagation simulation are limited because of the evolution of the topology of the crack. They require remeshing after each time step, and also a projection T. Menouillard (B ) · J.-H. Song · Q. Duan · T. Belytschko Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL, 60208-3111, USA e-mail: t-menouillard@northwestern.edu J.-H. Song e-mail: jeonghoonsong2008@u.northwestern.edu Q. Duan e-mail: q-duan@northwestern.edu T. Belytschko e-mail: tedbelytschko@northwestern.edu of the variables on the new discretization. This manda- tory process of updating the model is quite costly and awkward. Meshless methods are able to handle such a discontinuity evolution in the structure a lot more easily. Belytschko et al. (1996) proposed meshless approximations based on moving least-squares. To adapt the standard finite element method to frac- ture computations, the extended finite element method (XFEM) has been developed, which completely avoids remeshing (Black and Belytschko 1999; Moës et al. 1999). Also see Belytschko et al. (2001) and Stolarska et al. (2001) where it was combined with level sets. This XFEM method is based on the partition of unity pioneered by Melenk and Babuška (1996), whereby specific functions are used to describe the physical behavior in subdomains of the problem. Thus in Moës et al. (1999), the discontinuous enrichment function was used along the crack in order to describe a discon- tinuous displacement. Belytschko and Chen (2004) presented a singular tip enrichment for elastodynamic crack propagation. For plasticity, Elguedj et al. (2006) developed partic- ular functions for elastic-plastic materials for static case. These tip enrichment functions are related to the asymptotic displacement field near the crack tip for such a material law. Belytschko et al. (2003) developed a method for dynamic crack propagation with loss of hyperbolici- ty as a propagation criterion. They developed a spe- cial element with linear crack opening at the tip. Moës and Belytschko (2002) adapted the cohesive law for 123