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.     

Extended finite element fracture analysis of a cracked isotropic shell repaired by composite patch

An extended finite element method (XFEM) is developed to study fracture parameters of cracked metal plates and tubes that are repaired on top of the crack with a composite patch. A MATLAB® stand‐alone code is prepared to model such structures with eight‐noded doubly curved shell elements in the XFEM framework. Crack trajectory studies are performed for a diagonally cracked panel under fatigue loading. Verification studies are investigated on different shell type structures such as a cracked spherical shell and cracked cylindrical pipe with different crack orientations. The effects of using patch repairs with different fibre orientations on the reduction of stress intensity factors (SIFs) is also studied which can be useful for design purposes. XFEM is selected as any crack geometry can be embedded in the finite element mesh configuration with no need to coincide the crack geometry with meshed elements and so re‐meshing with fine mesh generation is not needed in the current method.     

Interaction between a cracked hole and a line crack under uniform heat flux

This article deals with the interaction between a cracked hole and a line crack under uniform heat flux. Using the principle of superposition, the original problem is converted into three particular cracked hole problems: the first one is the problem of the hole with an edge crack under uniform heat flux, the second and third ones are the problems of the hole under distributed temperature and edge dislocations, respectively, along the line crack surface. Singular integral equations satisfying adiabatic and traction free conditions on the crack surface are obtained for the solution of the second and third problems. The solution of the first problem, as well as the fundamental solutions of the second and third, is obtained by the complex variable method along with the rational mapping function approach. Stress intensity factors (SIFs) at all three crack tips are calculated. Interestingly, the results show that the interaction between the cracked hole and the line crack under uniform heat flux can lead to the vanishing of the SIFs at the hole edge crack tip. The fact has never been seen for the case of a cracked hole and a line crack under remote uniform tension.     

Numerical study on deformations in a cracked viscoelastic body with the extended finite element method

Deformations such as crack opening and sliding displacements in a cracked viscoelastic body are numerically investigated by the extended finite element method (XFEM). The solution is carried out directly in time domain with a mesh not conforming to the crack geometry. The generalized Heaviside function is used to reflect the displacement discontinuity across a crack surface while the basis functions extracted from the viscoelastic asymptotic fields are used to manifest the gradient singularity at a crack tip. With these treatments, the XFEM formulations are derived in an incremental form. In evaluating the stiffness matrix, a selective integration scheme is suggested for problems with high Poisson ratios often encountered in viscoelastic materials over different element types in the XFEM. Numerical examples show that the crack opening displacement and crack sliding displacement are satisfactory.     

A thermal-medium crack model

In analyzing the fracture behavior of a cracked thermoelastic material, of much importance are the effects of thermal loadings on the crack growth. Under the consideration of a medium in an opening crack, a thermal-medium crack model is proposed in this paper. The heat flux at the crack surfaces is assumed to depend on the jumps of the temperature and the elastic displacement across the crack. The thermally permeable and impermeable crack models are the limiting cases of a thermal-medium one. The proposed crack model is applied to solve the problem of a Griffith crack in a transversely isotropic material under thermal and mechanical loadings. Using two introduced displacement functions and the Fourier transform technique, the thermoelastic field and the elastic T-stress are determined in explicit forms by using elementary functions. Numerical results are presented to show the effects of the thermal conductivity inside a crack and applied mechanical loadings on the heat flux at the crack faces, the jumps of temperature across the crack and mode-II stress intensity factor in graphics respectively. The obtained results reveal that the mode-II stress intensity factor for a thermal-medium crack in a thermoelastic material depends not only on applied thermal loadings but also on applied mechanical ones.     

Inexact Schwarz‐algebraic multigrid preconditioners for crack problems modeled by extended finite element methods

Traditional algebraic multigrid (AMG) preconditioners are not well suited for crack problems modeled by extended finite element methods (XFEM). This is mainly because of the unique XFEM formulations, which embed discontinuous fields in the linear system by addition of special degrees of freedom. These degrees of freedom are not properly handled by the AMG coarsening process and lead to slow convergence. In this paper, we proposed a simple domain decomposition approach that retains the AMG advantages on well‐behaved domains by avoiding the coarsening of enriched degrees of freedom. The idea was to employ a multiplicative Schwarz preconditioner where the physical domain was partitioned into “healthy” (or unfractured) and “cracked” subdomains. First, the “healthy” subdomain containing only standard degrees of freedom, was solved approximately by one AMG V‐cycle, followed by concurrent direct solves of “cracked” subdomains. This strategy alleviated the need to redesign special AMG coarsening strategies that can handle XFEM discretizations. Numerical examples on various crack problems clearly illustrated the superior performance of this approach over a brute force AMG preconditioner applied to the linear system. Copyright © 2011 John Wiley & Sons, Ltd.     

An XFEM model for cracked porous media: effects of fluid flow and heat transfer

In this work, a numerical model is developed to investigate the influence of fluid flow and heat transfer on the thermo-mechanical response of a cracked porous media. The fluid flow, governed by the Darcy’s law, is discretized with the nonconforming finite element method. Time splitting is used with the energy conservation equation to solve the fluid and the solid phases separately. A combination of Discontinuous Galerkin (DG) and multi-point flux approximation methods is used to solve the advection-diffusion heat transfer equation in the fluid phase. While the conductive heat transfers equation in the solid phase is solved using the eXtended finite element method (XFEM) to better handle the temperature discontinuities and singularities caused by the cracks. Further, the resulted temperature is used as body force to solve the thermo-mechanical problem using the XFEM. In the post processing stage, the thermal stress intensity factor is computed using the interaction integral technique at each time step and used to validate the obtained results. A good agreement was found when the results were compared with the existing ones in the literature.     

A thermomechanical finite element tool for Leak‐before‐Break analysis

A new finite element tool is presented, which utilises the extended FEM (XFEM) to model leaks through cracks. Heat flux and pressure boundary conditions are imposed on the crack in the form of jump terms. Enrichments are chosen to model either strong or weak discontinuities across the crack, as well as singularities at the crack tips. Excellent convergence rates are achieved for both the thermal and mechanical models, where errors are calculated relative to analytical solutions derived for this specific problem. A more general temperature approximation is also presented, which makes no assumptions about the continuity of temperature or heat flux across the crack. Results indicate that this is a robust way of modelling the temperature of a plate containing a crack with or without a leaking fluid. Thermomechanical simulations were then carried out to demonstrate the applicability of the FEM for analysing leak rates in nuclear reactor primary pipework. A two‐phase flow model based on the Henry–Fauske model is chosen for the fluid aspect, and this is coupled to the structure through a convection law. Crack closure is shown to reduce the leak rate by up to 40%. © 2014 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons, Ltd.     

Accuracy improvement of XFEM using Wilson's incompatible element technique

Based on the extended finite element method (XFEM) and Wilson incompatible element technique, an incompatible extended finite element method (IXFEM) is presented to deal with the cracked bending problems in this study. The additional displacement modes of Wilson's incompatible element are introduced into the approximation of the XFEM. The internal degrees of freedom induced by additional models are condensed to improve efficiency. Derivation of the equilibrium equations of the IXFEM is performed. Several examples are employed to validate the capabilities of the proposed method. The influence of bending degree on the performance of proposed method is studied by changing the crack length. The numerical results indicate that the stress and displacement solutions of the IXFEM show a slight oscillation. The IXFEM achieves better convergence rate and computational accuracy compared with the traditional XFEM for the bending problems.     

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.     

Element‐local level set method for three‐dimensional dynamic crack growth

An approximate level set method for three‐dimensional crack propagation is presented. In this method, the discontinuity surface in each cracked element is defined by element‐local level sets (ELLSs). The local level sets are generated by a fitting procedure that meets the fracture directionality and its continuity with the adjacent element crack surfaces in a least‐square sense. A simple iterative procedure is introduced to improve the consistency of the generated element crack surface with those of the adjacent cracked elements. The discrete discontinuity is treated by the phantom node method which is a simplified version of the extended finite element method (XFEM). The ELLS method and the phantom node technology are combined for the solution of dynamic fracture problems. Numerical examples for three‐dimensional dynamic crack propagation are provided to demonstrate the effectiveness and robustness of the proposed method. Copyright © 2009 John Wiley & Sons, Ltd.     

Stable 3D XFEM/vector level sets for non‐planar 3D crack propagation and comparison of enrichment schemes

We present a three‐dimensional vector level set method coupled to a recently developed stable extended finite element method (XFEM). We further investigate a new enrichment approach for XFEM adopting discontinuous linear enrichment functions in place of the asymptotic near‐tip functions. Through the vector level set method, level set values for propagating cracks are obtained via simple geometrical operations, eliminating the need for solution of differential evolution equations. The first XFEM variant ensures optimal convergence rates by means of geometrical enrichment, ie, the use of enriched elements in a fixed volume around the crack front, without giving rise to conditioning problems. The linear enrichment approach, significantly simplifies implementation and reduces the computational cost associated with numerical integration, while providing nonoptimal convergence rates similar to standard finite elements. The 2 dicretization schemes are tested for different benchmark problems, and their combination to the vector level set method is verified for nonplanar crack propagation problems.     

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.     

A simple and efficient XFEM approach for 3-D cracks simulations

In this work, a simple and efficient XFEM approach has been presented to solve 3-D crack problems in linear elastic materials. In XFEM, displacement approximation is enriched by additional functions using the concept of partition of unity. In the proposed approach, a crack front is divided into a number of piecewise curve segments to avoid an iterative solution. A nearest point on the crack front from an arbitrary (Gauss) point is obtained for each crack segment. In crack front elements, the level set functions are approximated by higher order shape functions which assure the accurate modeling of the crack front. The values of stress intensity factors are obtained from XFEM solution by domain based interaction integral approach. Many benchmark crack problems are solved by the proposed XFEM approach. A convergence study has been conducted for few test problems. The results obtained by proposed XFEM approach are compared with the analytical/reference solutions.     

XFEM analysis of the effects of voids,inclusions and other cracks on the dynamic stress intensity factor of a major crack

This paper is devoted to the extraction of the dynamic stress intensity factor (DSIF) for structures containing multiple discontinuities (cracks, voids and inclusions) by developing the extended finite element method (XFEM). In this method, four types of enrichment functions are used in the framework of the partition of unity to model interface discontinuity within the classical finite element method. In this procedure, elements that include a crack segment, the boundary of a void or the boundary of an inclusion are not required to conform to discontinuous edges. The DSIF is evaluated by the interaction integral. After the effectiveness of the implemented XFEM program is verified, the effects of voids, inclusions and other cracks on the DSIF of a stationary major crack are investigated by using XFEM. The results show that the dynamic effects have an influence on the path independence of the interaction integral, and these voids, inclusions and other cracks have a significant effect on the DSIF of the major crack.     

The penny-shaped crack problem in micropolar thermoelasticity

The axisymmetry problem of a penny-shaped crack opened out by thermal loads is studied. The linear theory of micropolar elasticity is employed. Two types of thermal loads are considered—prescribed temperature on the crack faces and prescribed heat flux across the faces. It is shown that, in both the cases, the problem is equivalent to the isothermal problem of the crack opened out by suitable normal tractions on the crack faces. The stress intensity factors are found to depend on, in addition to the usual parameters, two parameters N and M; N is a number characterising the coupling of the displacement field with the microtation field and M is the ratio N/τ where τ is a non-dimensional material characteristic length. The classical values of the stress intensity factors are recovered as a limiting case. Numerical results are presented for the case of constant heat flux across the crack faces. These results show that the presence of couple stresses elevates the values of the stress intensity factors.     

A versatile interface model for thermal conduction phenomena and its numerical implementation by XFEM

A general interface model is presented for thermal conduction and characterized by two jump relations. The first one expresses that the temperature jump across an interface is proportional to the interfacial average of the normal heat flux while the second one states that the normal heat flux jump is proportional to the surface Laplacian of the interfacial average of the temperature. By varying the two scalar proportionality parameters, not only the Kapitza resistance and highly conducting interface models can be retrieved but also all the intermediate cases can be covered. The general interface model is numerically implemented by constructing its weak form and by using the level-set method and XFEM. The resulting numerical procedure, whose accuracy and robustness are thoroughly tested and discussed with the help of a benchmark problem, is shown to be efficient for solving the problem of thermal conduction in particulate composites with various imperfect interfaces.     

Crack tip enrichment in the XFEM using a cutoff function

We consider a variant of the eXtended Finite Element Method (XFEM) in which a cutoff function is used to localize the singular enrichment surface. The goal of this variant is to obtain numerically an optimal convergence rate while reducing the computational cost of the classical XFEM with a fixed enrichment area. We give a mathematical result of quasi‐optimal error estimate. One of the key points of this paper is to prove the optimality of the coupling between the singular and the discontinuous enrichments. Finally, we present some numerical computations validating the theoretical result. These computations are compared with those of the classical XFEM and a non‐enriched method. Copyright © 2008 John Wiley & Sons, Ltd.     

Transient thermal contact problem for axial crack in a hollow circular cylinder

The transient behavior of an axial-cracked hollow circular cylinder subjected to a sudden heating is investigated. It is shown that surface heating may induce compressive thermal stress near the inner surface of the cylinder which in turn may force the cracked surfaces to close together. Assuming that the existence of the crack does not alter the temperature distribution, this problem can be divided into two parts and solved by the principle of superposition. First, the temperature and transient thermal stress distributions along the axisymmetric surface of the imaginary cylinder without a crack are obtained by finite element/implicit time integration method. The calculated temperature and thermal stress distributions are in good agreement with the values predicted by the analytical method. Secondly, the opposite senses of the stress distributions along the cracked surfaces, which are obtained previously, are treated as the traction boundary conditions, and the contact length and contact pressure of the real cracked cylinder are obtained by a modified elimination finite element scheme. In this scheme, the concepts of contact-node-pairs' penetration, contact-double-forces and compliance matrix are introduced. The calculated results indicate that the contact length ratio becomes smaller when the crack length ratio increases, and becomes larger as the radius ratio increases. Finally, the normalized stress intensity factor for the crack tip of the cylinder is obtained. It is shown that the larger the crack length ratio the higher the stress intensity factor.