The numerical modelling of cracks and the analysis of cracked structures go back to the early 70’s. Until the middle of the 1990s, the focus of the research within the failure analysis field was based on the development of continuum me-chanics models, which have a strong dependence on the spatial discretization.
Since the mid 90’s, the models based on the concept of discontinuous displace-ment approximations have gained great popularity in the scientific community.
These models take into account discontinuities within the finite elements and can be categorized into an element-wise formulation Embedded Crack Models [Simo et al., 1993, Oliver, 1996a, Oliver, 1996b, Jirásek and Zimmermann, 2001a, Jirásek and Zimmermann, 2001b] and a node-wise formulationeXtended Finite El-ement Method [Moës et al., 1999]. Within the framework of the Embedded Crack Models, the discontinuities are usually incompatible at the element boundaries and, since the work of SIMO [Simo et al., 1993], these models are based on the En-hanced Assumed Strainconcept. This means that the jump in the displacement field is not modelled directly, and only included in the form of a discontinuous strain field. The eXtended Finite Element Method is based on the partition of unity con-cept [Melenk and Babuška, 1996], and offers the possibility to model the discon-tinuities by enriching the displacement field with discontinuous shape functions. In contrast to theEmbedded Crack Models, the discontinuities are compatible at the el-ement boundaries. A comparison of these two modelling philosophies can be found in [Jirásek and Belytschko, 2002].
A third possibility to simulate the crack propagation is the so-called remesh-ing. Within this approach, the finite element mesh changes as the crack grows, in order to match the new crack geometry. Even when this technique have been used successfully in many works, see [Nishioka, 1995, Bouchard et al., 2003, Réthoré et al., 2004, Shahani and Fasakhodi, 2009], among others, it presents three main disadvantages: the relatively high numerical effort, the loss of accuracy by mapping the information between meshes and the complexity of implementation in a finite element program.
The development of meshless methods, such as theElement-Free Galerkin Method [Belytschko et al., 1994], made the remeshing process in the simulation of discon-tinuities unnecessary. This fourth method of crack simulation uses only one set of nodes and corresponding shape functions for the discretization of the continuum.
Therefore, no remeshing is needed as the crack propagates. However, this advan-tage is offset by the very high computational cost of the method. For this reason, the coupling with the finite element method for non-damaged areas of the structures
was proposed in [Belytschko et al., 1995]. For the simulation of dynamic crack prop-agation using the element free GALERKINmethod see [Krysl and Belytschko, 1999]
and references therein. An overview on meshless methods can be found in [Belytschko et al., 1996] and in the more recent work [Nguyen et al., 2008].
The eXtended Finite Element Method starts with the pioneering work of BE
-LYTSCHKO and BLACK [Belytschko and Black, 1999], where a methodology to model crack growth with minimal remeshing is presented. Next, in the work of MOËS [Moës et al., 1999], the remeshing is completely eliminated by us-ing a HEAVISIDE function to model the displacement discontinuity on the crack, and a near-tip enrichment to take into account the singularity of the stress field at the crack tip. The doctoral thesis of DOLBOW [Dolbow, 1999]
resumes these concepts of the XFEM, including also the modelling of cracked plates [Dolbow et al., 2000b, Dolbow et al., 2000a], intersecting cracks, see also [Daux et al., 2000, Budyn et al., 2004]) and frictional contact cracks [Dolbow et al., 2001]. Next, SUKUMAR extended the method to three dimensional crack modelling [Sukumar et al., 2000b] and to the modelling of holes and inclusions by level sets in the XFEM [Sukumar et al., 2000a]. Non-planar three-dimensional crack growth was analysed in [Moës et al., 2002] using XFEM and level sets.
After the above mentioned contributions, the interest of the scientific commu-nity in XFEM increased rapidly. WELLS and SLUYS [Wells and Sluys, 2001] used only the HEAVISIDE function for modelling cohesive cracks using finite elements.
Within this approach, the crack tip is always located at the element boundaries.
This technique was improved in [Zi and Belytschko, 2003], where a new crack-tip element for XFEM and its application to cohesive cracks was proposed. The method was again based only on the HEAVISIDE function, but the crack tip can be located anywhere within an element. Higher-order displacement discontinuity in a standard finite element model was introduced using the XFEM approach in [Mariani and Perego, 2003] for the simulation of quasi-static cohesive crack propa-gation. An energy-based modelling of cohesive and cohesionless cracks via XFEM was proposed in [Meschke and Dumstorff, 2007]. The method is formulated within an XFEM based analysis model, leading to a variational formulation in terms of dis-placements, crack lengths and crack kink angles. The influence of both the numerical integration and the cohesive law on the energy distribution and the crack path was studied in the subsequent work [Meschke et al., 2006].
A slightly different approach to XFEM is the HANSBO&HANSBO method [Hansbo, 2002, Hansbo, 2004]. Within this framework, the discontinuous approx-imation is accomplished by doubling the degrees of freedom in the discontin-uous elements. Inspired in this work, a geometrically non-linear finite element framework for modelling three-dimensional crack propagation was presented in [Mergheim et al., 2006].
The convergence properties of the XFEM was studied in [Bechet et al., 2005, Laborde et al., 2005, Chahine et al., 2006] inspired in the previous work of [Strang and Fix, 1973] in the context of the finite element method. In [Laborde et al., 2005], some improvements of the XFEM were proposed in or-der to obtain optimal convergence properties for high-oror-der elements. Furthermore,
thealmost polar integrationfor the elements containing the crack tip was introduced.
This integration issue was also studied in [Bechet et al., 2005], along with a special-ized preconditioner for enriched finite elements. Next, the use of a cut-off function in a zone independent of the mesh size was studied in [Chahine et al., 2006], where also a quasi-optimal error estimate for first order LAGRANGE finite elements on a regular triangular mesh was obtained. The influence of the partially enriched elements (blending elements) on the performance of local partition of unity en-richments was studied in [Chessa et al., 2003]. In this paper, an enhanced strain formulation, capable of achieving the optimal rate of convergence, was developed.
In [Liu et al., 2004] the XFEM was improved to directly evaluate the stress intensity factors without post-processing, considering not only homogeneous materials but also bimaterials. A statically admissible stress recovery scheme was proposed in [Xiao and Karihaloo, 2006b] in order to improve the accuracy of crack tip fields.
Within the framework of dynamic crack propagation, WELLS [Wells, 2001] pro-posed the modelling of cohesive cracks with XFEM using only the HEAVISIDE
function, again with the crack tip located always at the element boundary. A similar method was presented in [Zi et al., 2005], coupling XFEM with level sets to represent the crack. BELYTSCHKO [Belytschko et al., 2003] proposed a model based on the loss of hyperbolicity and a new discontinuous enrichment without tip enrichment, but allowing the crack tip to be located within the elements. The crack tip behaviour was modelled using a cohesive zone model. For brittle ma-terials, RÉTHORÉ [Réthoré et al., 2005b] proposed an energy conserving scheme for modelling dynamic crack growth using XFEM, an latter developed a combined space-time extended finite element method STX-FEM [Réthoré et al., 2005a] in order to improve the quality of the numerical simulation when time discontinu-ities are present. Space-time discontinuous finite element have been also anal-ysed in [Chessa and Belytschko, 2004, Chessa and Belytschko, 2006]. A singular enrichment finite element method was proposed in [Belytschko and Chen, 2004], the method uses an enriched basis that spans the first terms of the asymptotic dynamic solution, where the stress intensity factorsKI andKII are taken as ad-ditional unknowns. XFEM simulations and mixed-mode loading experiments have been compared in [Grégoire et al., 2007].
In the context of explicit dynamics, dynamic crack and shear band propagation was analysed in [Song et al., 2006] using phantom nodes, inspired in the work of HANSBO&HANSBO. The main problem of the method is that the critical time step tends to zero when the crack passes very close to a node. To alleviate this problem, mass lumping techniques for the XFEM have been analysed by MENOUILLARD [Menouillard et al., 2006, Menouillard et al., 2008] and also in the works [Rozycki et al., 2008, Elguedj et al., 2009, Gravouil et al., 2009]. The use of basis functions which are non-zero only in the elements containing the crack was introduced in [Svahn et al., 2007], together with a rate-dependent cohesive crack model based on anisotropic damage coupled to plasticity. Time integration schemes in the XFEM for moving discontinuities were studied in [Fries and Zilian, 2009], with special emphasis on the advection-difussion equation. Recently, MENOUILLARDand co-workers [Menouillard et al., 2010] proposed a time dependent crack tip enrich-ment for dynamic crack propagation.
A comparative study of several finite element methods for dynamic fracture can be found in [Song et al., 2007]. For a state of the art review on the XFEM the interested reader is referred to the articles [Karihaloo and Xiao, 2003, Belytschko et al., 2009]
and the XFEM book [Mohammadi, 2008].
After the problem is discretized in the spatial domain, one arrives to a semi-discrete equation of motion, that is, a set of differential equations in time. The most general approach to solve the problem is the direct integration of the equation of motion.
Several time integration schemes have been developed for structural dynamics ap-plications and one of the most famous methods is the NEWMARKmethod proposed in 1959 [Newmark, 1959]. The method is a one-step procedure, which means that the solution at timetn+1depends only on the solution at timetn, and can be uncondi-tionally stable for the right choice of the parameters involved in the method. Second order accuracy can be achieved forγ= 1/2, but this value ofγleads to no numerical dissipation. The WILSON-θmethod, see [Bathe and Wilson, 1973], solves this prob-lem achieving second order accuracy together with numerical dissipation. Neverthe-less, the method overshoots significantly the exact solution in the first time steps [Goudreau and Taylor, 1972]. A further improvement of the NEWMARKmethod was introduced in [Hilber et al., 1977a] and calledαmethod, also known as the HHT-α method, which shows a better behaviour with respect to overshooting compared with the WILSON-θmethod, see [Hilber and Hughes, 1978]. Several years later CHUNG
& HULBERT proposed theGeneralized-αmethod [Chung and Hulbert, 1993]. The method offers the possibility of controlling the numerical dissipation, achieving high frequency dissipation while minimizing unwanted low frequency dissipation. The method contains the HHT-α [Hilber et al., 1977b] and WBZ-α [Wood et al., 1981]
methods.
As a consequence of the DAHLQUISTtheorem [Dahlquist, 1963], the order of ac-curacy of the methods of the NEWMARKfamily described before, and in general of any A-stable linear multistep method, can not exceed two. Due to this fact, contin-uous and discontincontin-uous GALERKINmethods were considered in structural dynam-ics. These methods can achieve an arbitrary accuracy order by incrementing the polynomial degree of the finite element basis. The first ideas of time finite element methods were presented in [Argyris and Scharpf, 1969, Fried, 1969, Oden, 1969], several years later space finite element methods were introduced, see for exam-ple [Turner et al., 1956, Argyris, 1960]. Continuous GALERKINtime integration meth-ods were proposed in [Hulme, 1972, Argyris et al., 1977] to solve first order ini-tial value problems, while in [French, 1993] the method was used to solve the Wave Equation. The time discontinuous GALERKIN method followed the same path as its continuous counterpart, and was developed after the space discon-tinuous GALERKIN method [Reed and Hill, 1973, Le Saint and Raviart, 1974]. In [Jamet, 1978] a discontinuous GALERKIN method was used to solve parabolic initial value problems. The numerical properties of the method were anal-ysed in the latter works [Johnson et al., 1984, Eriksson et al., 1985]. In the con-text of linear elastodynamics, time discontinuous GALERKIN methods were de-veloped and analysed in the works [Gellert, 1978, Hughes and Hulbert, 1988, Hulbert, 1989, Hulbert and Hughes, 1990, Hulbert, 1992, Borri and Bottasso, 1975, Li, 1996, Li and Wiberg, 1996, Li and Wiberg, 1998, Johnson, 1993, Maute, 2001,
Steeb et al., 2003, Wiberg and Li, 1999].