Stabilized global–local X‐FEM for 3D non‐planar frictional crack using relevant meshes

A stabilized global–local quasi‐static contact algorithm for 3D non‐planar frictional crack is presented in the X‐FEM/level set framework. A three‐field weak formulation is considered and allows an independent discretization of the bulk and the crack interface. Then, a fine discretization of the interface can be defined according to the possible complex contact state along the crack faces independently from the mesh in the bulk. Furthermore, an efficient stabilized non‐linear LATIN solver dedicated to contact and friction is proposed. It allows solving in a unified framework frictionless and frictional contact at the crack interface with a symmetric formulation, no iterations on the local stage (unilateral contact law with/without friction), no calculation of any global tangent operator, and improved convergence rate. 2D and 3D patch tests are presented to illustrate the relevance of the proposed model and an actual 3D frictional crack problem under cyclic fretting loading is modeled. Copyright © 2011 John Wiley & Sons, Ltd.     

An X‐FEM approach for large sliding contact along discontinuities

The extended finite element method (X‐FEM) has been developed to minimize requirements on the mesh design in a problem with a displacement discontinuity. This advantage, however, still remains limited to the small deformation hypothesis when considering sliding discontinuities. The approach presented in this paper proposes to couple X‐FEM with a Lagrangian large sliding frictionless contact algorithm. A new hybrid X‐FEM contact element was developed with a contact search algorithm allowing for an update of contacting surfaces pairing. The stability of the contact formulation is ensured by an algorithm for fulfilling Ladyzhenskaya‐Babuska‐Brezzi (LBB) condition. Several 2D simple examples are presented in this paper in order to prove its efficiency and stability. Copyright © 2008 John Wiley & Sons, Ltd.     

A stable X‐FEM in cohesive transition from closed to open crack

Ignoring crack tip effects, the stability of the X‐FEM discretizations is trivial for open cracks but remains a challenge if we constrain the crack to be closed (i.e., the bi‐material problem). Here, we develop a formulation for general cohesive interactions between crack faces within the X‐FEM framework. The stability of the new formulation is demonstrated for any cohesive crack stiffness (including the closed crack) and illustrated for a nonlinear cohesive softening law. A benchmark of the new model is carried out with simpler approaches for a closed crack (i.e., Lagrange multipliers) and for a cohesive crack (i.e., penalty approach). Due to the analogies between stable cohesive X‐FEM and Nitsche's methods, the new method simplifies the implementation and is attractive in dynamic explicit codes. Copyright © 2014 John Wiley & Sons, Ltd.     

Extension of the node‐to‐segment contact element for surface‐expansion‐dependent contact laws

A class of friction laws depending on the measure of contact surface expansion is defined in the paper within the continuum contact mechanics framework. The nominal and spatial forms of constitutive relations are discussed, including incremental penalty relations. Further, an extended node‐to‐segment element is derived which is capable of treating surface‐expansion‐dependent contact laws in a consistent way. The approach is suitable for any kind of node‐to‐segment contact elements. Finally, the computational efficiency of the extended element as well as other possible approaches are illustrated by numerical examples relevant to metal forming applications. Copyright © 2001 John Wiley & Sons, Ltd.     

Improved implementation and robustness study of the X‐FEM for stress analysis around cracks

Numerical crack propagation schemes were augmented in an elegant manner by the X‐FEM method. The use of special tip enrichment functions, as well as a discontinuous function along the sides of the crack allows one to do a complete crack analysis virtually without modifying the underlying mesh, which is of industrial interest, especially when a numerical model for crack propagation is desired. This paper improves the implementation of the X‐FEM method for stress analysis around cracks in three ways. First, the enrichment strategy is revisited. The conventional approach uses a ‘topological’ enrichment (only the elements touching the front are enriched). We suggest a ‘geometrical’ enrichment in which a given domain size is enriched. The improvements obtained with this enrichment are discussed. Second, the conditioning of the X‐FEM both for topological and geometrical enrichments is studied. A preconditioner is introduced so that ‘off the shelf’ iterative solver packages can be used and perform as well on X‐FEM matrices as on standard FEM matrices. The preconditioner uses a local (nodal) Cholesky based decomposition. Third, the numerical integration scheme to build the X‐FEM stiffness matrix is dramatically improved for tip enrichment functions by the use of an ad hoc integration scheme. A 2D benchmark problem is designed to show the improvements and the robustness. Copyright © 2005 John Wiley & Sons, Ltd.     

A local multigrid X‐FEM strategy for 3‐D crack propagation

A multiscale method for 3‐D crack propagation simulation in large structures is proposed. The method is based on the extended finite element method (X‐FEM). The asymptotic behavior of the crack front is accurately modeled using enriched elements and no remeshing is required during crack propagation. However, the different scales involved in fracture mechanics problems can differ by several orders of magnitude and industrial meshes are usually not designed to account for small cracks. Enrichments are therefore useless if the crack is too small compared with the element size. To overcome this drawback, a project combining different numerical techniques was started. The first step was the implementation of a global multigrid algorithm within the X‐FEM framework and was presented in a previous paper (Eur. J. Comput. Mech. 2007; 16 :161–182). This work emphasized the high efficiency in cpu time but highlighted that mesh refinement is required on localized areas only (cracks, inclusions, steep gradient zones). This paper aims at linking the different scales by using a local multigrid approach. The coupling of this technique with the X‐FEM is described and computational aspects dealing with intergrid operators, optimal multiscale enrichment strategy and level sets are pointed out. Examples illustrating the accuracy and efficiency of the method are given. Copyright © 2008 John Wiley & Sons, Ltd.     

Fatigue analysis of railway wheel using a multiaxial strain‐based critical‐plane index

A fatigue damage model to assess the development of subsurface fatigue cracks in railway wheels is presented in this paper. A 3‐dimensional finite element model (FEM) is constructed to simulate repeated cycles of contact loading between a railway wheel and a rail. The computational approach includes a hard‐contact over‐closure relationship and an elastoplastic material model with isotropic and kinematic hardening. Results from the simulation are used in a multiaxial critical‐plane fatigue damage analysis. The employed strain‐based critical‐plane fatigue damage approach is based on Fatemi‐Socie fatigue index that takes into account the non‐proportional and out‐of‐phase nature of the multiaxial state of stress occurs when a railway wheel rolls on a rail. It predicts fatigue‐induced micro‐crack nucleation at a depth of about 3.7 mm beneath the wheel tread, as well as the crack plane growth orientation which indicates the possible failure pattern. Additionally, the influence of various factors such as contribution of normal stresses, higher wheel load, and material model have been investigated.     

Three‐dimensional simulation of crack with curved front with direct estimation of stress intensity factors

This paper consists of an extension of simulation with direct estimation of stress intensity factors to the three‐dimensional case. Here, it combines X‐FEM with localized multigrids and direct estimation of quantities of interest along the crack front (SIF, T‐stress, etc.) based on crack tip asymptotic series expansion. In practice, a three‐dimensional patch is introduced locally with a truncated basis of Williams series expansion and is linked in a weak sense with the X‐FEM localized multigrids. Some examples (with available analytical solutions) illustrate the efficiency and the robustness of the method. These examples consider planar cracks with curved front, but the proposed method aims to apply to any continuously curved crack. Copyright © 2014 John Wiley & Sons, Ltd.     

A quasi‐static crack growth simulation based on the singular ES‐FEM

In the edge‐based smoothed finite element method (ES‐FEM), one needs only the assumed displacement values (not the derivatives) on the boundary of the edge‐based smoothing domains to compute the stiffness matrix of the system. Adopting this important feature, a five‐node crack‐tip element is employed in this paper to produce a proper stress singularity near the crack tip based on a basic mesh of linear triangular elements that can be generated automatically for problems with complicated geometries. The singular ES‐FEM is then formulated and used to simulate the crack propagation in various settings, using a largely coarse mesh with a few layers of fine mesh near the crack tip. The results demonstrate that the singular ES‐FEM is much more accurate than X‐FEM and the existing FEM. Moreover, the excellent agreement between numerical results and the reference observations shows that the singular ES‐FEM offers an efficient and high‐quality solution for crack propagation problems. Copyright © 2011 John Wiley & Sons, Ltd.     

Combined interior‐point method and semismooth Newton method for frictionless contact problems

In the present paper, a solution scheme is proposed for frictionless contact problems of linear elastic bodies, which are discretized using the finite element method with lower order elements. An approach combining the interior‐point method and the semismooth Newton method is proposed. In this method, an initial active set for the semismooth Newton method is obtained from the approximate optimal solution by the interior‐point method. The simplest node‐to‐node contact model is considered in the present paper, that is, pairs of matching nodes exist on the contact surfaces. However, the discussions can be easily extended to a node‐to‐segment or segment‐to‐segment contact model. In order to evaluate the proposed method, a number of illustrative examples of the frictionless contact problem are shown. The proposed combined method is compared with the interior‐point method and the semismooth Newton method. Two numerical examples that are difficult to solve using the semismooth Newton method are solved effectively using the proposed combined method. It is shown that the proposed method converges within far fewer iterations than the semismooth Newton methods or the interior‐point method. Copyright © 2009 John Wiley & Sons, Ltd.     

Non‐planar mixed‐mode growth of initially straight‐fronted surface cracks,in cylindrical bars under tension,torsion and bending,using the symmetric Galerkin boundary element method‐finite element method alternating method

In this paper, the stress intensity factor (SIF) variations along an arbitrarily developing crack front, the non‐planar fatigue‐crack growth patterns, and the fatigue life of a round bar with an initially straight‐fronted surface crack, are studied by employing the 3D symmetric Galerkin boundary element method‐finite element method (SGBEM‐FEM) alternating method. Different loading cases, involving tension, bending and torsion of the bar, with different initial crack depths and different stress ratios in fatigue, are considered. By using the SGBEM‐FEM alternating method, the SIF variations along the evolving crack front are computed; the fatigue growth rates and directions of the non‐planar growths of the crack surface are predicted; the evolving fatigue‐crack growth patterns are simulated, and thus, the fatigue life estimations of the cracked round bar are made. The accuracy and reliability of the SGBEM‐FEM alternating method are verified by comparing the presently computed results to the empirical solutions of SIFs, as well as experimental data of fatigue crack growth, available in the open literature. It is shown that the current approach gives very accurate solutions of SIFs and simulations of fatigue crack growth during the entire crack propagation, with very little computational burden and human–labour cost. The characteristics of fatigue growth patterns of initially simple‐shaped cracks in the cylindrical bar under different Modes I, III and mixed‐mode types of loads are also discussed in detail.     

Three‐dimensional brittle fracture: configurational‐force‐driven crack propagation

This paper presents a computational framework for quasi‐static brittle fracture in three‐dimensional solids. The paper sets out the theoretical basis for determining the initiation and direction of propagating cracks based on the concept of configurational mechanics, consistent with Griffith's theory. Resolution of the propagating crack by the FEM is achieved by restricting cracks to element faces and adapting the mesh to align it with the predicted crack direction. A local mesh improvement procedure is developed to maximise mesh quality in order to improve both accuracy and solution robustness and to remove the influence of the initial mesh on the direction of propagating cracks. An arc‐length control technique is derived to enable the dissipative load path to be traced. A hierarchical hp‐refinement strategy is implemented in order to improve both the approximation of displacements and crack geometry. The performance of this modelling approach is demonstrated on two numerical examples that qualitatively illustrate its ability to predict complex crack paths. All problems are three‐dimensional, including a torsion problem that results in the accurate prediction of a doubly‐curved crack. Copyright © 2013 John Wiley & Sons, Ltd.     

A novel singular node‐based smoothed finite element method (NS‐FEM) for upper bound solutions of fracture problems

It is well known that the lower bound to exact solutions in linear fracture problems can be easily obtained by the displacement compatible finite element method (FEM) together with the singular crack tip elements. It is, however, much more difficult to obtain the upper bound solutions for these problems. This paper aims to formulate a novel singular node‐based smoothed finite element method (NS‐FEM) to obtain the upper bound solutions for fracture problems. In the present singular NS‐FEM, the calculation of the system stiffness matrix is performed using the strain smoothing technique over the smoothing domains (SDs) associated with nodes, which leads to the line integrations using only the shape function values along the boundaries of the SDs. A five‐node singular crack tip element is used within the framework of NS‐FEM to construct singular shape functions via direct point interpolation with proper order of fractional basis. The mix‐mode stress intensity factors are evaluated using the domain forms of the interaction integrals. The upper bound solutions of the present singular NS‐FEM are demonstrated via benchmark examples for a wide range of material combinations and boundary conditions. Copyright © 2010 John Wiley & Sons, Ltd.     

Modelling of reinforced‐concrete structures providing crack‐spacing based on X‐FEM,ED‐FEM and novel operator split solution procedure

In this work, we present a novel approach to the finite element modelling of reinforced‐concrete (RC) structures that provides the details of the constitutive behavior of each constituent (concrete, steel and bond‐slip), while keeping formally the same appearance as the classical finite element model. Each component constitutive behavior can be brought to fully non‐linear range, where we can consider cracking (or localized failure) of concrete, the plastic yielding and failure of steel bars and bond‐slip at concrete steel interface accounting for confining pressure effects. The standard finite element code architecture is preserved by using embedded discontinuity (ED‐FEM) and extended (X‐FEM) finite element strain representation for concrete and slip, respectively, along with the operator split solution method that separates the problem into computing the deformations of RC (with frozen slip) and the current value of the bond‐slip. Several numerical examples are presented in order to illustrate very satisfying performance of the proposed methodology. Copyright © 2010 John Wiley & Sons, Ltd.     

K‐dominance of static crack tip in functionally gradient materials

K‐dominance of static crack tip in functionally gradient materials (FGMs) with a crack oriented along the direction of the elastic gradient is studied through coherent gradient sensing (CGS), digital speckle correlation method (DSCM) and finite element method (FEM). In the direction of crack propagation, the shear modulus has a linear variation with constant mass density and Poisson's ratio. First, the CGS and DSCM governing equations related to the measurements and the elastic solutions at mode I crack in FGMs are obtained in terms of the stress intensity factor, material constants and graded index. Secondly, two kinds of FGMs specimens and one homogenous specimen are prepared to observe the influences of the property variation on the K‐dominance. Then, CGS and DSCM experiments using three‐point‐bending of FGMs and homogenous beams are performed. Thirdly, based on the results of the experiments, the stress intensity factors of three kinds of specimens are calculated by CGS and DSCM. Meanwhile, the stress intensity factors are obtained by FEM. Finally, comparing the results from CGS, DSCM and FEM, the K‐dominance of mode‐I static crack tip in FGMs is discussed in detail.     

Analysis of edge crack behaviour influenced by non‐symmetrical contact tractions and bulk tension

This paper analyses a crack growth behaviour, which is initiated from the contact edge between a square punch with rounded edges and a half plane. Investigated are the influences of the contact profile, magnitude of the bulk tension and, crack obliquity, in particular, misalignment between the punch and half plane on the variation of the stress intensity factors K_I and K_II during the crack growth. The misalignment is simulated by a tilting of the punch. A partial slip regime is considered for the contact shear force to accommodate a general fretting fatigue condition. It was found that a crack closure occurs if only the contact forces are applied. The crack grows longer before it is closed if the punch is tilted (clockwise, in this paper) such that it initiates at the opposite site with respect to the direction of tilting. The closure phenomenon disappears when the bulk tension is added and exceeds a certain magnitude, which significantly depends on not only the contact profile but also the degree and direction of tilting. Provided are the lowest values of the bulk tensile stress due to a fatigue load necessary to extend the crack without a closure for each condition of the contact profile and misalignment. This may be used as a design guideline to restrain the contact‐induced failure.     

A computational strategy for damage‐tolerant design of hollow shafts under mixed‐mode loading condition

Three‐dimensional numerical analyses, using the finite element method (FEM), have been adopted to simulate fatigue crack propagation in a hollow cylindrical specimen, under pure axial or combined axial‐torsion loading conditions. Specimens, made of Al alloys B95AT and D16T, have been experimentally tested under pure axial load and combined in‐phase constant amplitude axial and torsional loadings. The stress intensity factors (SIFs) have been calculated, according to the J‐integral approach, along the front of a part through crack, initiated in correspondence of the outer surface of a hollow cylindrical specimen. The crack path is evaluated by using the maximum energy release rate (MERR) criterion, whereas the Paris law is used to calculate crack growth rates. A numerical and experimental comparison of the results is presented, showing a good agreement in terms of crack growth rates and paths.     

Estimation and consequences of the crack thickness parameter in the assessment of crack growth behaviour of “squat” type cracks in the rail-wheel contact zone

The paper presents the results obtained from investigations of the “squat” type crack serving as an example of the rolling contact fatigue crack that appears in rail heads. Some geometrical parameters of the crack can be defined. The work is devoted mainly to the problem of crack thickness, i.e. the gap between crack faces. The observation of crack faces allowed us to formulate a concept of how this gap is forming. The crack thickness distribution over the whole crack face was measured. The results obtained were introduced into 3D FEM models. The paper also describes the changes in crack thickness distribution in the course of wheel rolling over the crack and the resulting crack volume changes.     

New development in extended finite element modeling of large elasto‐plastic deformations

This paper presents new achievements in the extended finite element modeling of large elasto‐plastic deformation in solid problems. The computational technique is presented based on the extended finite element method (X‐FEM) coupled with the Lagrangian formulation in order to model arbitrary interfaces in large deformations. In X‐FEM, the material interfaces are represented independently of element boundaries, and the process is accomplished by partitioning the domain with some triangular sub‐elements whose Gauss points are used for integration of the domain of elements. The large elasto‐plastic deformation formulation is employed within the X‐FEM framework to simulate the non‐linear behavior of materials. The interface between two bodies is modeled by using the X‐FEM technique and applying the Heaviside‐ and level‐set‐based enrichment functions. Finally, several numerical examples are analyzed, including arbitrary material interfaces, to demonstrate the efficiency of the X‐FEM technique in large plasticity deformations. Copyright © 2008 John Wiley & Sons, Ltd.     

Contact with friction in multi‐material arbitrary Lagrangian‐Eulerian formulations using X‐FEM

A contact method with friction for the multi‐dimensional Lagrangian step in multi‐material arbitrary Lagrangian–Eulerian (ALE) formulations is presented. In our previous research, the extended finite element method (X‐FEM) was used to create independent fields (i.e. velocity, strain rate, force, mass, etc.) for each material in the problem to model contact without friction. The research presented here includes the extension to friction and improvements to the accuracy and robustness of our previous study. The accelerations of the multi‐material nodes are obtained by coupling the material force and mass fields as a function of the prescribed contact; similarly, the velocities of the multi‐material nodes are recalculated using the conservation of momentum when the prescribed contact requires it. The coupling procedures impose the same nodal velocity on the coupled materials in the direction normal to their interface during the time step update. As a result, the overlap of materials is prevented and unwanted separation does not occur. Three different types of contacts are treated: perfectly bonded, frictionless slip, and slip with friction. Example impact problems are solved and the numerical solutions are presented. Copyright © 2008 John Wiley & Sons, Ltd.