A contact algorithm for frictional crack propagation with the extended finite element method

We present an incremental quasi‐static contact algorithm for path‐dependent frictional crack propagation in the framework of the extended finite element (FE) method. The discrete formulation allows for the modeling of frictional contact independent of the FE mesh. Standard Coulomb plasticity model is introduced to model the frictional contact on the surface of discontinuity. The contact constraint is borrowed from non‐linear contact mechanics and embedded within a localized element by penalty method. Newton–Raphson iteration with consistent linearization is used to advance the solution. We show the superior convergence performance of the proposed iterative method compared with a previously published algorithm called ‘LATIN’ for frictional crack propagation. Numerical examples include simulation of crack initiation and propagation in 2D plane strain with and without bulk plasticity. In the presence of bulk plasticity, the problem is also solved using an augmented Lagrangian procedure to demonstrate the efficacy and adequacy of the standard penalty solution. Copyright © 2008 John Wiley & Sons, Ltd.     

Developing new enrichment functions for crack simulation in orthotropic media by the extended finite element method

New enrichment functions are proposed for crack modelling in orthotropic media using the extended finite element method (XFEM). In this method, Heaviside and near‐tip functions are utilized in the framework of the partition of unity method for modelling discontinuities in the classical finite element method. In this procedure, by using meshless based ideas, elements containing a crack are not required to conform to crack edges. Therefore, mesh generation is directly performed ignoring the existence of any crack while the method remains capable of extending the crack without any remeshing requirement. Furthermore, the type of elements around the crack‐tip remains the same as other parts of the finite element model and the number of nodes and consequently degrees of freedom are reduced considerably in comparison to the classical finite element method. Mixed‐mode stress intensity factors (SIFs) are evaluated to determine the fracture properties of domain and to compare the proposed approach with other available methods. In this paper, the interaction integral (M‐integral) is adopted, which is considered as one of the most accurate numerical methods for calculating stress intensity factors. Copyright © 2006 John Wiley & Sons, Ltd.     

A coupled molecular dynamics and extended finite element method for dynamic crack propagation

A multiscale method is presented which couples a molecular dynamics approach for describing fracture at the crack tip with an extended finite element method for discretizing the remainder of the domain. After recalling the basic equations of molecular dynamics and continuum mechanics, the discretization is discussed for the continuum subdomain where the partition‐of‐unity property of finite element shape functions is used, since in this fashion the crack in the wake of its tip is naturally modelled as a traction‐free discontinuity. Next, the zonal coupling method between the atomistic and continuum models is recapitulated. Finally, it is discussed how the stress has been computed in the atomic subdomain, and a two‐dimensional computation is presented of dynamic fracture using the coupled model. The result shows multiple branching, which is reminiscent of recent results from simulations on dynamic fracture using cohesive‐zone models. Copyright © 2009 John Wiley & Sons, Ltd.     

An extended finite element method with analytical enrichment for cohesive crack modeling

A recent approach to fracture modeling has combined the extended finite element method (XFEM) with cohesive zone models. Most studies have used simplified enrichment functions to represent the strong discontinuity but have lacked an analytical basis to represent the displacement gradients in the vicinity of the cohesive crack. In this study enrichment functions based upon an existing analytical investigation of the cohesive crack problem are proposed. These functions have the potential of representing displacement gradients in the vicinity of the cohesive crack and allow the crack to incrementally advance across each element. Key aspects of the corresponding numerical formulation and enrichment functions are discussed. A parameter study for a simple mode I model problem is presented to evaluate if quasi‐static crack propagation can be accurately followed with the proposed formulation. The effects of mesh refinement and mesh orientation are considered. Propagation of the cohesive zone tip and crack tip, time variation of the cohesive zone length, and crack profiles are examined. The analysis results indicate that the analytically based enrichment functions can accurately track the cohesive crack propagation of a mode I crack independent of mesh orientation. A mixed mode example further demonstrates the potential of the formulation. Copyright © 2008 John Wiley & Sons, Ltd.     

Finite strain fracture analysis using the extended finite element method with new set of enrichment functions

Nonlinear fracture analysis of rubber‐like materials is computationally challenging due to a number of complicated numerical problems. The aim of this paper is to study finite strain fracture problems based on appropriate enrichment functions within the extended finite element method. Two‐dimensional static and quasi‐static crack propagation problems are solved to demonstrate the efficiency of the proposed method. Complex mixed‐mode problems under extreme large deformation regimes are solved to evaluate the performance of the proposed extended finite element analysis based on different tip enrichment functions. Finally, it is demonstrated that the logarithmic set of enrichment functions provides the most accurate and efficient solution for finite strain fracture analysis. Copyright © 2015 John Wiley & Sons, Ltd.     

The improvement of crack propagation modelling in triangular 2D structures using the extended finite element method

In this paper, a novel geometric method combined with the piecewise linear function method is introduced into the extended finite element method (XFEM) to determine the crack tip element and crack surface element. Then, by combining with the advanced mesh technique, a novel method is proposed to improve the modelling of crack propagation in triangular 2D structure with the XFEM. The numerical tests show that the accuracy, the convergence, and the stability of the XFEM can be improved using the proposed method. Moreover, the applicability of the conventional multiple enrichment scheme is discussed. Compared with the proposed method, the conventional multiple enrichment scheme has deficiency in mixed mode I and II crack. Finally, a comparative study shows that the performance of the XFEM by using the proposed method to model the crack propagation can be greatly improved.     

The extended/generalized finite element method: An overview of the method and its applications

An overview of the extended/generalized finite element method (GEFM/XFEM) with emphasis on methodological issues is presented. This method enables the accurate approximation of solutions that involve jumps, kinks, singularities, and other locally non‐smooth features within elements. This is achieved by enriching the polynomial approximation space of the classical finite element method. The GEFM/XFEM has shown its potential in a variety of applications that involve non‐smooth solutions near interfaces: Among them are the simulation of cracks, shear bands, dislocations, solidification, and multi‐field problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Numerical simulation of elastic plastic fatigue crack growth in functionally graded material using the extended finite element method

In the present work, the extended finite element method has been used to simulate the fatigue crack growth problems in functionally graded material in the presence of holes, inclusions, and minor cracks under plastic and plane stress conditions for both edge and center cracks. Both soft and hard inclusions have been implemented in the problems. The validity of linear elastic fracture mechanics theory is limited to the brittle materials. Therefore, the elastic plastic fracture mechanics theory needs to be utilized to characterize the plastic behavior of the material. A generalized Ramberg-Osgood material model has been used for modeling purposes.     

Modelling crack propagation in reinforced concrete using a hybrid finite element–scaled boundary finite element method

A previously developed hybrid finite element–scaled boundary finite element method (FEM–SBFEM) is extended to model multiple cohesive crack propagation in reinforced concrete. This hybrid method can efficiently extract accurate stress intensity factors from the semi-analytical solutions of SBFEM and is also flexible in remeshing multiple cracks. Crack propagation in the concrete bulk is modelled by automatically inserted cohesive interface elements with nonlinear softening laws. The concrete–reinforcement interaction is also modelled by cohesive interface elements. The bond shear stress–slip relation of CEB-FIP Model Code 90 and an empirical confining stress–crack opening relation are used to characterise slip and split failure at the concrete–reinforcement interface, respectively. Three RC beams were simulated. The numerical results agreed well with both experimental and numerical results available in the literature. Parametric studies demonstrated the importance of modelling both slip and split failure mechanisms at the concrete–reinforcement interface.     

Delamination analysis of composites by new orthotropic bimaterial extended finite element method

The extended finite element method (XFEM) is further improved for fracture analysis of composite laminates containing interlaminar delaminations. New set of bimaterial orthotropic enrichment functions are developed and utilized in XFEM analysis of linear‐elastic fracture mechanics of layered composites. Interlaminar crack‐tip enrichment functions are derived from analytical asymptotic displacement fields around a traction‐free interfacial crack. Also, heaviside and weak discontinuity enrichment functions are utilized in modeling discontinuous fields across interface cracks and bimaterial weak discontinuities, respectively. In this procedure, elements containing a crack‐tip or strong/weak discontinuities are not required to conform to those geometries. In addition, the same mesh can be used to analyze different interlaminar cracks or delamination propagation. The domain interaction integral approach is also adopted in order to numerically evaluate the mixed‐mode stress intensity factors. A number of benchmark tests are simulated to assess the performance of the proposed approach and the results are compared with available reference results. Copyright © 2011 John Wiley & Sons, Ltd.     

Ductile failure analysis and crack behavior of X65 buried pipes using extended finite element method

This paper aims to study the ductile fracture mechanism of API X65 buried pipes including crack initiation and propagation using the extended finite element method (XFEM). First, the crack evolution histories of X65 specimens with initial crack-like flaws during tensile and three-point bending tests are illustrated, and the numerical results are compared with experimental data. In addition, effects of different crack configurations, damage initiation and evolution criteria are investigated. Second, the burst processes of straight pipes with initial gouge flaws are presented, and the FE results are compared with assessment in related standards and experiments. Finally, the crack onset and growth of buried pipes due to deflection arising from landslide movements are predicted, and the numerical results are compared with previous study. Particularly, the internal pressure, wall thickness, and soil properties on crack behavior and limit load-bearing ability are investigated. This paper provides a fundamental support for the integrity assessment and safety evaluation of buried pipes.     

Finite deformation formulation for embedded frictional crack with the extended finite element method

We reformulate an extended finite element (FE) framework for embedded frictional cracks in elastoplastic solids to accommodate finite deformation, including finite stretching and rotation. For the FE representation, we consider a Galerkin approximation in which both the trial and weighting functions adapt to the current contact configuration. Contact and frictional constraints employ two Kuhn–Tucker conditions, a contact/separation constraint nesting over a stick/slip constraint for the case when the crack faces are in frictional sliding mode. We integrate finite deformation bulk plasticity into the formulation using the multiplicative decomposition technique of nonlinear continuum mechanics. We then present plane strain simulations demonstrating various aspects of the extended FE solutions. The mechanisms considered include combined opening and frictional sliding in initially straight, curved, and S‐shaped cracks, with and without bulk plasticity. To gain further insight into the extended FE solutions, we perform mesh convergence studies focusing on both the global and the local responses of structures with cracks, including the distribution of the normal component of traction on the crack faces. Copyright © 2009 John Wiley & Sons, Ltd.     

On enrichment functions in the extended finite element method

This paper presents mathematical derivation of enrichment functions in the extended finite element method for numerical modeling of strong and weak discontinuities. The proposed approach consists in combining the level set method with characteristic functions as well as domain decomposition and reproduction technique. We start with the simple case of a triangular linear element cut by one interface across which displacement field suffers a jump. The main steps towards the derivation of enrichment functions are as follows: (1) extension of the subfields separated by the interface to the whole element domain and definition of complementary nodal variables; (2) construction of characteristic functions for describing the geometry and physical field; (3) determination of the sets of basic nodal variables; (4) domain decompositions according to Step 3 and then reproduction of the physical field in terms of characteristic functions and nodal variables; and (5) comparison of the piecewise interpolations formulated at Steps 3 and 4 with the standard extended finite element method form, which yields enrichment functions. In this process, the physical meanings of both the basic and complementary nodal variables are clarified, which helps to impose Dirichlet boundary conditions. Enrichment functions for weak discontinuities are constructed from deeper insights into the structure of the functions for strong discontinuities. Relationships between the two classes of functions are naturally established. Improvements upon basic enrichment functions for weak discontinuities are performed so as to achieve satisfactory convergence and accuracy. From numerical viewpoints, a simple and efficient treatment on the issue of blending elements is also proposed with implementation details. For validation purposes, applications of the derived functions to heterogeneous problems with imperfect interfaces are presented. Copyright © 2012 John Wiley & Sons, Ltd.     

Analysis of three‐dimensional crack initiation and propagation using the extended finite element method

An Erratum has been published for this article in International Journal for Numerical Methods in Engineering 2005, 63(8): 1228. We present a new formulation and a numerical procedure for the quasi‐static analysis of three‐dimensional crack propagation in brittle and quasi‐brittle solids. The extended finite element method (XFEM) is combined with linear tetrahedral elements. A viscosity‐regularized continuum damage constitutive model is used and coupled with the XFEM formulation resulting in a regularized ‘crack‐band’ version of XFEM. The evolving discontinuity surface is discretized through a C⁰ surface formed by the union of the triangles and quadrilaterals that separate each cracked element in two. The element's properties allow a closed form integration and a particularly efficient implementation allowing large‐scale 3D problems to be studied. Several examples of crack propagation are shown, illustrating the good results that can be achieved. Copyright © 2005 John Wiley & Sons, Ltd.     

Three‐dimensional fatigue crack growth modelling in a helical gear using extended finite element method

In this paper, the fatigue crack growth in helical gear tooth root has been simulated using linear elastic fracture mechanics. The extended finite element method has been used to simulate 3D fatigue crack growth and obtain growth path. Paris equation has been used to calculate the fatigue life of the gear. The modelling time has reduced considerably compared to previous works carried out on 3D crack growth in gears. Some verifications have been carried out to ensure the reliability of the results.     

Modelling crack growth by level sets in the extended finite element method

An algorithm which couples the level set method (LSM) with the extended finite element method (X‐FEM) to model crack growth is described. The level set method is used to represent the crack location, including the location of crack tips. The extended finite element method is used to compute the stress and displacement fields necessary for determining the rate of crack growth. This combined method requires no remeshing as the crack progresses, making the algorithm very efficient. The combination of these methods has a tremendous potential for a wide range of applications. Numerical examples are presented to demonstrate the accuracy of the combined methods. Copyright © 2001 John Wiley & Sons, Ltd.     

Modelling dynamic crack propagation using the scaled boundary finite element method

This study presents a novel application of the scaled boundary finite element method (SBFEM) to model dynamic crack propagation problems. Accurate dynamic stress intensity factors are extracted directly from the semi‐analytical solutions of SBFEM. They are then used in the dynamic fracture criteria to determine the crack‐tip position, velocity and propagation direction. A simple, yet flexible remeshing algorithm is used to accommodate crack propagation. Three dynamic crack propagation problems that include mode‐I and mix‐mode fracture are modelled. The results show good agreement with experimental and numerical results available in the literature. It is found that the developed method offers some advantages over conventional FEM in terms of accuracy, efficiency and ease of implementation. Copyright © 2011 John Wiley & Sons, Ltd.     

A spline‐based enrichment function for arbitrary inclusions in extended finite element method with applications to finite deformations

A novel enrichment function, which can model arbitrarily shaped inclusions within the framework of the extended finite element method, is proposed. The internal boundary of an arbitrary‐shaped inclusion is first discretized, and a numerical enrichment function is constructed ‘on the fly’ using spline interpolation. We consider a piecewise cubic spline which is constructed from seven localized discrete boundary points. The enrichment function is then determined by solving numerically a nonlinear equation which determines the distance from any point to the spline curve. Parametric convergence studies are carried out to show the accuracy of this approach compared with pointwise and linear segmentation of points for the construction of the enrichment function in the case of simple inclusions and arbitrarily shaped inclusions in linear elasticity. Moreover, the viability of this approach is illustrated on a neo‐Hookean hyperelastic material with a hole undergoing large deformation. In this case, the enrichment is able to adapt to the deformation and effectively capture the correct response without remeshing. Copyright © 2013 John Wiley & Sons, Ltd.     

Combined extended and superimposed finite element method for cracks

A combination of the extended finite element method (XFEM) and the mesh superposition method (s‐version FEM) for modelling of stationary and growing cracks is presented. The near‐tip field is modelled by superimposed quarter point elements on an overlaid mesh and the rest of the discontinuity is implicitly described by a step function on partition of unity. The two displacement fields are matched through a transition region. The method can robustly deal with stationary crack and crack growth. It simplifies the numerical integration of the weak form in the Galerkin method as compared to the s‐version FEM. Numerical experiments are provided to demonstrate the effectiveness and robustness of the proposed method. Copyright © 2004 John Wiley & Sons, Ltd.     

An adaptive multiscale method for crack propagation and crack coalescence

This work presents a new multiscale technique for the efficient simulation of crack propagation and crack coalescence of macrocracks and microcracks. The fully adaptive multiscale method is able to capture localization effect mesh independently. By modeling macrocracks and microcracks, the extended finite element method offers an accurate solution and captures cracks and their propagation without changing the mesh topology. Propagating and coaliting cracks of different length scales can therefore be easily modeled and updated during the computation process. Hence, the presented method is an efficient and accurate option for modeling cracks of different length scales. This is demonstrated in several numerical examples showing the interaction of microcracks and macrocracks. Copyright © 2012 John Wiley & Sons, Ltd.