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.     

Anomalous behavior of bilinear quadrilateral finite elements for modeling cohesive cracks with XFEM/GFEM

The performance of partition‐of‐unity based methods such as the generalized finite element method or the extended finite element method is studied for the simulation of cohesive cracking. The focus of investigation is on the performance of bilinear quadrilateral finite elements using these methods. In particular, the approximation of the displacement jump field, representing cohesive cracks, by extended finite element method/generalized finite element method and its effect on the overall behavior at element and structural level is investigated. A single element test is performed with two different integration schemes, namely the Newton‐Cotes/Lobatto and the Gauss integration schemes, for the cracked interface contribution. It was found that cohesive crack segments subjected to a nonuniform opening in unstructured meshes (or an inclined crack in a structured finite element mesh) result in an unrealistic crack opening. The reasons for such behavior and its effect on the response at element level are discussed. Furthermore, a mesh refinement study is performed to analyze the overall response of a cohesively cracked body in a finite element analysis. Copyright © 2013 John Wiley & Sons, Ltd.     

A finite element method for the computational modelling of cohesive cracks

The present contribution is concerned with the computational modelling of cohesive cracks in quasi‐brittle materials, whereby the discontinuity is not limited to interelement boundaries, but is allowed to propagate freely through the elements. In the elements, which are intersected by the discontinuity, additional displacement degrees of freedom are introduced at the existing nodes. Therefore, two independent copies of the standard basis functions are used. One set is put to zero on one side of the discontinuity, while it takes its usual values on the opposite side, and vice versa for the other set. To model inelastic material behaviour, a discrete damage‐type constitutive model is applied, formulated in terms of displacements and tractions at the surface. Some details on the numerical implementation are given, concerning the failure criterion, the determination of the direction of the discontinuity and the integration scheme. Finally, numerical examples show the performance of the method. Copyright © 2005 John Wiley & Sons, Ltd.     

A consistent partly cracked XFEM element for cohesive crack growth

Present extended finite element method (XFEM) elements for cohesive crack growth may often not be able to model equal stresses on both sides of the discontinuity when acting as a crack‐tip element. The authors have developed a new partly cracked XFEM element for cohesive crack growth with extra enrichments to the cracked elements. The extra enrichments are element side local and were developed by superposition of the standard nodal shape functions for the element and standard nodal shape functions for a sub‐triangle of the cracked element. With the extra enrichments, the crack‐tip element becomes capable of modelling variations in the discontinuous displacement field on both sides of the crack and hence also capable of modelling the case where equal stresses are present on each side of the crack. The enrichment was implemented for the 3‐node constant strain triangle (CST) and a standard algorithm was used to solve the non‐linear equations. The performance of the element is illustrated by modelling fracture mechanical benchmark tests. Investigations were carried out on the performance of the element for different crack lengths within one element. The results are compared with previously obtained XFEM results applying fully cracked XFEM elements, with computational results achieved using standard cohesive interface elements in a commercial code, and with experimental results. The suggested element performed well in the tests. Copyright © 2007 John Wiley & Sons, Ltd.     

New crack‐tip elements for XFEM and applications to cohesive cracks

An extended finite element method scheme for a static cohesive crack is developed with a new formulation for elements containing crack tips. This method can treat arbitrary cracks independent of the mesh and crack growth without remeshing. All cracked elements are enriched by the sign function so that no blending of the local partition of unity is required. This method is able to treat the entire crack with only one type of enrichment function, including the elements containing the crack tip. This scheme is applied to linear 3‐node triangular elements and quadratic 6‐node triangular elements. To ensure smooth crack closing of the cohesive crack, the stress projection normal to the crack tip is imposed to be equal to the material strength. The equilibrium equation and the traction condition are solved by the Newton–Raphson method to obtain the nodal displacements and the external load simultaneously. The results obtained by the new extended finite element method are compared to reference solutions and show excellent agreement. Copyright © 2003 John Wiley & Sons, Ltd.     

A contact algorithm for cohesive cracks in the extended finite element method

Cracks with quasibrittle behavior are extremely common in engineering structures. The modeling of cohesive cracks involves strong nonlinearity in the contact, material, and complex transition between contact and cohesive forces. In this article, we propose a novel contact algorithm for cohesive cracks in the framework of the extended finite element method. A cohesive-contact constitutive model is introduced to characterize the complex mechanical behavior of the fracture process zone. To avoid the stress oscillations and ill-conditioned system matrix that often occur in the conventional contact approach, the proposed algorithm employs a special dual Lagrange multiplier to impose the contact constraint. This Lagrange multiplier is constructed by means of the area-weighted average and biorthogonality conditions at the element level. The system matrix can be condensed into a positive definite matrix with an unchanged size at a very low computational cost. In addition, we illustrate solving the cohesive crack contact problem using a novel iteration strategy. Several numerical experiments are performed to illustrate the efficiency and high-quality results of our method in contact analysis of cohesive cracks.     

Cracking node method for dynamic fracture with finite elements

A new method for modeling discrete cracks based on the extended finite element method is described. In the method, the growth of the actual crack is tracked and approximated with contiguous discrete crack segments that lie on finite element nodes and span only two adjacent elements. The method can deal with complicated fracture patterns because it needs no explicit representation of the topology of the actual crack path. A set of effective rules for injection of crack segments is presented so that fracture behavior beginning from arbitrary crack nucleations to macroscopic crack propagation is seamlessly modeled. The effectiveness of the method is demonstrated with several dynamic fracture problems that involve complicated crack patterns such as fragmentation and crack branching. Copyright © 2008 John Wiley & Sons, Ltd.     

Theory and numerics for finite deformation fracture modelling using strong discontinuities

A general finite element approach for the modelling of fracture is presented for the geometrically non‐linear case. The kinematical representation is based on a strong discontinuity formulation in line with the concept of partition of unity for finite elements. Thus, the deformation map is defined in terms of one continuous and one discontinuous portion, considered as mutually independent, giving rise to a weak formulation of the equilibrium consisting of two coupled equations. In addition, two different fracture criteria are considered. Firstly, a principle stress criterion in terms of the material Mandel stress in conjunction with a material cohesive zone law, relating the cohesive Mandel traction to a material displacement ‘jump’ associated with the direct discontinuity. Secondly, a criterion of Griffith type is formulated in terms of the material‐crack‐driving force (MCDF) with the crack propagation direction determined by the direction of the force, corresponding to the direction of maximum energy release. Apart from the material modelling, the numerical treatment and aspects of computational implementation of the proposed approach is also thoroughly discussed and the paper is concluded with a few numerical examples illustrating the capabilities of the proposed approach and the connection between the two fracture criteria. Copyright © 2005 John Wiley & Sons, Ltd.     

A discrete cohesive model for fractal cracks

The fractal crack model described here incorporates the essential features of the fractal view of fracture, the basic concepts of the LEFM model, the concepts contained within the Barenblatt-Dugdale cohesive crack model and the quantized (discrete or finite) fracture mechanics assumptions proposed by Pugno and Ruoff [Pugno N, Ruoff RS. Quantized fracture mechanics. Philos Mag 2004;84(27):2829-45] and extended by Wnuk and Yavari [Wnuk MP, Yavari A. Discrete fractal fracture mechanics. Engng Fract Mech 2008;75(5):1127-42]. The well-known entities such as the stress intensity factor and the Barenblatt cohesion modulus, which is a measure of material toughness, have been re-defined to accommodate the fractal view of fracture.For very small cracks or as the degree of fractality increases, the characteristic length constant, related to the size of the cohesive zone is shown to substantially increase compared to the conventional solutions obtained from the cohesive crack model. In order to understand fracture occurring in real materials, whether brittle or ductile, it seems necessary to account for the enhancement of fracture energy, and therefore of material toughness, due to fractal and discrete nature of crack growth. These two features of any real material appear to be inherent defense mechanisms provided by Nature.     

Elastic crack growth in finite elements with minimal remeshing

A minimal remeshing finite element method for crack growth is presented. Discontinuous enrichment functions are added to the finite element approximation to account for the presence of the crack. This method allows the crack to be arbitrarily aligned within the mesh. For severely curved cracks, remeshing may be needed but only away from the crack tip where remeshing is much easier. Results are presented for a wide range of two‐dimensional crack problems showing excellent accuracy. Copyright © 1999 John Wiley & Sons, Ltd.     

Arbitrary branched and intersecting cracks with the extended finite element method

Extensions of a new technique for the finite element modelling of cracks with multiple branches, multiple holes and cracks emanating from holes are presented. This extended finite element method (X‐FEM) allows the representation of crack discontinuities and voids independently of the mesh. A standard displacement‐based approximation is enriched by incorporating discontinuous fields through a partition of unity method. A methodology that constructs the enriched approximation based on the interaction of the discontinuous geometric features with the mesh is developed. Computation of the stress intensity factors (SIF) in different examples involving branched and intersecting cracks as well as cracks emanating from holes are presented to demonstrate the accuracy and the robustness of the proposed technique. Copyright © 2000 John Wiley & Sons, Ltd.     

Cracking particles: a simplified meshfree method for arbitrary evolving cracks

A new approach for modelling discrete cracks in meshfree methods is described. In this method, the crack can be arbitrarily oriented, but its growth is represented discretely by activation of crack surfaces at individual particles, so no representation of the crack's topology is needed. The crack is modelled by a local enrichment of the test and trial functions with a sign function (a variant of the Heaviside step function), so that the discontinuities are along the direction of the crack. The discontinuity consists of cylindrical planes centred at the particles in three dimensions, lines centred at the particles in two dimensions. The model is applied to several 2D problems and compared to experimental data. Copyright © 2004 John Wiley & Sons, Ltd.     

Fully-automatic modelling of cohesive crack growth using a finite element-scaled boundary finite element coupled method

This study develops a method coupling the finite element method (FEM) and the scaled boundary finite element method (SBFEM) for fully-automatic modelling of cohesive crack growth in quasi-brittle materials. The simple linear elastic fracture mechanics (LEFM)-based remeshing procedure developed previously is augmented by inserting nonlinear interface finite elements automatically. The constitutive law of these elements is modelled by the cohesive/fictitious crack model to simulate the fracture process zone, while the elastic bulk material is modelled by the SBFEM. The resultant nonlinear equation system is solved by a local arc-length controlled solver. The crack is assumed to grow when the mode-I stress intensity factor K_I vanishes in the direction determined by LEFM criteria. Other salient algorithms associated with the SBFEM, such as mapping state variables after remeshing and calculating K_I using a “shadow subdomain”, are also described. Two concrete beams subjected to mode-I and mixed-mode fracture respectively are modelled to validate the new method. The results show that this SBFEM-FEM coupled method is capable of fully-automatically predicting both satisfactory crack trajectories and accurate load-displacement relations with a small number of degrees of freedom, even for problems with strong snap-back. Parametric studies were carried out on the crack incremental length, the concrete tensile strength, and the mode-I and mode-II fracture energies. It is found that the K_I ? 0 criterion is objective with respect to the crack incremental length.     

Automated dynamic fracture procedure for modelling mixed-mode crack propagation using explicit time integration brick finite elements

Several fracture codes have been developed in recent years to perform analyses of dynamic crack propagation in arbitrary directions. However, general-purpose, commercial finite-element software which have capabilities to do fracture analyses are still limited in their use to stationary cracks and crack propagation along trajectories known a priori . In this paper, we present an automated fracture procedure implemented in the large-scale, nonlinear, explicit, finite-element code DYNA3D which can be used to simulate dynamic crack propagation in arbitrary directions. The model can be used to perform both generation- and application-phase simulations of self-similar as well as non-self-similar dynamic crack propagation in linear elastic structures without user intervention. It is developed based on dynamic fracture mechanics concepts and implemented for three-dimensional solid elements. Energy approach is used in the model to check for crack initiation/propagation. Dynamic energy release rate and stress intensity factors are determined from far-field finite-element field solutions using finite-domain integrals. Fracture toughness is input as a function of crack-tip velocity, and when the criterion for crack growth is satisfied, an element deletion-and-replacement re-meshing procedure is used along with a gradual nodal release technique to update the crack geometry and model the crack propagation. Direction of crack propagation is determined using the maximum circumferential stress criterion. Numerical simulations of experiments involving non-self-similar crack propagation are performed, and results are presented as verification examples.     

An embedded cohesive crack model for finite element analysis of mixed mode fracture of concrete*

An embedded cohesive crack model is proposed for the analysis of the mixed mode fracture of concrete in the framework of the Finite Element Method. Different models, based on the strong discontinuity approach, have been proposed in the last decade to simulate the fracture of concrete and other quasi‐brittle materials. This paper presents a simple embedded crack model based on the cohesive crack approach. The predominant local mode I crack growth of the cohesive materials is utilized and the cohesive softening curve (stress vs. crack opening) is implemented by means of a central force traction vector. The model only requires the elastic constants and the mode I softening curve. The need for a tracking algorithm is avoided using a consistent procedure for the selection of the separated nodes. Numerical simulations of well‐known experiments are presented to show the ability of the proposed model to simulate the mixed mode fracture of concrete.     

Stress analysis around crack tips in finite strain problems using the eXtended finite element method

Fracture of rubber‐like materials is still an open problem. Indeed, it deals with modelling issues (crack growth law, bulk behaviour) and computational issues (robust crack growth in 2D and 3D, incompressibility). The present study focuses on the application of the eXtended Finite Element Method (X‐FEM) to large strain fracture mechanics for plane stress problems. Two important issues are investigated: the choice of the formulation used to solve the problem and the determination of suitable enrichment functions. It is demonstrated that the results obtained with the method are in good agreement with previously published works. Copyright © 2005 John Wiley & Sons, Ltd.     

Extended meshfree methods without branch enrichment for cohesive cracks

An extended meshless method for both static and dynamic cohesive cracks is proposed. This new method does not need any crack tip enrichment to guarantee that the crack closes at the tip. All cracked domains of influence are enriched by only the sign function. The domain of influence which includes a crack tip is modified so that the crack tip is always positioned at its edge. The modification is only applied for the discontinuous displacement field and the continuous field is kept unchanged. In addition to the new method, the use of Lagrange multiplier is explored to achieve the same goal. The crack is extended beyond the actual crack tip so that the domains of influence containing the crack tip are completely cut. It is enforced that the crack opening displacement vanishes along the extension of the crack. These methods are successfully applied to several well-known static and dynamic problems.     

A partition of unity‐based multiscale approach for modelling fracture in piezoelectric ceramics

The development of models for a priori assessment of the reliability of micro electromechanical systems is of crucial importance for the further development of such devices. In this contribution a partition of unity‐based cohesive zone finite element model is employed to mimic crack nucleation and propagation in a piezoelectric continuum. A multiscale framework to appropriately represent the influence of the microstructure on the response of a miniaturized component is proposed. It is illustrated that using the proposed multiscale method a representative volume element exists. Numerical simulations are performed to demonstrate the constitutive homogenization framework. Copyright © 2009 John Wiley & Sons, Ltd.     

Extended finite element method for three‐dimensional crack modelling

An extended finite element method (X‐FEM) for three‐dimensional crack modelling is described. A discontinuous function and the two‐dimensional asymptotic crack‐tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity. This enables the domain to be modelled by finite elements with no explicit meshing of the crack surfaces. Computational geometry issues associated with the representation of the crack and the enrichment of the finite element approximation are discussed. Stress intensity factors (SIFs) for planar three‐dimensional cracks are presented, which are found to be in good agreement with benchmark solutions. Copyright © 2000 John Wiley & Sons, Ltd.     

Harmonic enrichment functions: A unified treatment of multiple,intersecting and branched cracks in the extended finite element method

A unifying procedure to numerically compute enrichment functions for elastic fracture problems with the extended finite element method is presented. Within each element that is intersected by a crack, the enrichment function for the crack is obtained via the solution of the Laplace equation with Dirichlet and vanishing Neumann boundary conditions. A single algorithm emanates for the enrichment field for multiple cracks as well as intersecting and branched cracks, without recourse to special cases, which provides flexibility over the existing approaches in which each case is treated separately. Numerical integration is rendered to be simple—there is no need for partitioning of the finite elements into conforming subdivisions for the integration of discontinuous or weakly singular kernels. Stress intensity factor computations for different crack configurations are presented to demonstrate the accuracy and versatility of the proposed technique. Copyright © 2010 John Wiley & Sons, Ltd.