Direct estimation of generalized stress intensity factors using a three‐scale concurrent multigrid X‐FEM

A concurrent multigrid method is devised for the direct estimation of stress intensity factors and higher‐order coefficients of the elastic crack tip asymptotic field. The proposed method bridges three characteristic length scales that can be present in fracture mechanics: the structure, the crack and the singularity at the crack tip. For each of them, a relevant model is proposed. First, a truncated analytical reduced‐order model based on Williams' expansion is used to describe the singularity at the tip. Then, it is coupled with a standard extended finite element (FE) method model which is known to be suitable for the scale of the crack. A multigrid solver finally bridges the scale of the crack to that of the structure for which a standard FE model is often accurate enough. Dedicated coupling algorithms are presented and the effects of their parameters are discussed. The efficiency and accuracy of this new approach are exemplified using three benchmarks. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

Modeling arbitrary crack propagation in coupled shell/solid structures with X‐FEM

In this paper, the extended finite element method (X‐FEM) formulation for the modeling of arbitrary crack propagation in coupled shell/solid structures is developed based on the large deformation continuum‐based (CB) shell theory. The main features of the new method are as follows: (1) different kinematic equations are derived for different fibers in CB shell elements, including the fibers enriched by shifted jump function or crack tip functions and the fibers cut into two segments by the crack surface or connecting with solid elements. So the crack tip can locate inside the element, and the crack surface is not necessarily perpendicular to the middle surface. (2) The enhanced CB shell element is developed to realize the seamless transition of crack propagation between shell and solid structures. (3) A revised interaction integral is used to calculate the stress intensity factor (SIF) for shells, which avoids that the auxiliary fields for cracks in Mindlin–Reissner plates cannot satisfy exactly the equilibrium equations. Several numerical examples, including the calculation of SIF for the cracked plate under uniform bending and crack propagation between solid and shell structures are presented to demonstrate the performance of the developed method. Copyright © 2015 John Wiley & Sons, Ltd.     

An X‐FEM and level set computational approach for image‐based modelling: Application to homogenization

The advances in material characterization by means of imaging techniques require powerful computational methods for numerical analysis. The present contribution focuses on highlighting the advantages of coupling the extended finite elements method and the level sets method, applied to solve microstructures with complex geometries. The process of obtaining the level set data starting from a digital image of a material structure and its input into an extended finite element framework is presented. The coupled method is validated using reference examples and applied to obtain homogenized properties for heterogeneous structures. Although the computational applications presented here are mainly two‐dimensional, the method is equally applicable for three‐dimensional problems. Copyright © 2010 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.     

3D fracture mechanics investigation on surface fatigue crack propagation

Surface fatigue crack propagation is the typical failure mode of engineering structures. In this study, the experiment on surface fatigue crack propagation in 15MnVN steel plate is carried out, and the crack shape and propagation life are obtained. With the concept of ‘equivalent thickness’ brought into the latest three‐dimensional (3D) fracture mechanics theory, one closure model applicable to 3D fatigue crack is put forward. By using the above 3D crack‐closure model, the shape and propagation life of surface fatigue crack in 15MnVN plates are predicted. The simulative results show that the 3D fracture mechanics‐based closure model for 3D fatigue crack is effective.     

Crack face contact in X‐FEM using a segment‐to‐segment approach

In this study, we propose a segment‐to‐segment contact formulation (mortar‐based) that uses Lagrange's multipliers to establish the contact between crack faces when modeled with the extended finite element method (X‐FEM) in 2D problems. It is shown that, in general, inaccuracies arise when the contact is formulated following a point‐to‐point approach. This is due to the non‐linear character of the X‐FEM interpolation along the crack faces that leads to crack face interpenetration. However, the segment‐to‐segment approach optimizes the fulfilment of the contact constraints along the whole crack segment, and in practice the contact is modeled precisely. Convergence studies for mesh sequences have been performed, showing the advantages of the proposed methodology. Copyright © 2009 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.     

Prediction of 3D small fatigue crack propagation in shot-peened specimens

Shot peening has been widely applied in industrial design to improve fatigue durability of high loaded machine components. The compressive residual stress induced by shot peening is in general assumed to be responsible for the improvement of material fatigue strength. In the present work a cyclic cohesive zone model is extended to analyze three-dimensional fatigue crack growth in shot-peened specimens. Fatigue crack growth behaviors in both unpeened and peened specimens are investigated using 3D finite element analysis. The parameters of the cohesive zone model have been identified in 2D unpeened specimens and are applied to predict peened specimens directly. The results indicate that shot peening strongly affects crack initiation time and crack profiles, but has little effect on crack propagation rate. It implies that the shot peening will hardly change Paris’ law used for the damage tolerant design.     

Extended FEM modeling of crack paths near inclusions

The extended FEM is applied to model crack growth near inclusions. A procedure to handle different propagation rates at different crack tips is presented. The examples considered investigate uniform tension as well as equibiaxial tension under plane strain conditions. A parameter study analyzes the effects on the crack path when changing the relative stiffness between inclusion and matrix material, the relative distance between initial crack and inclusion, and the size of the inclusion. Both edge cracks and internal cracks are studied. An example with an internal crack near an inclusion is presented, where both crack tips propagate at different growth rates until one crack tip eventually stops growing, as the related energy release rate drops below the critical value. In another example, only one crack tip propagates initially, but eventually, the energy release rate of the second crack tip becomes critical, and both crack tips propagate. Finally, an example of two cracks near an inclusion is presented in which up to four crack tips propagate simultaneously. Copyright © 2011 John Wiley & Sons, Ltd.     

Nonlinear fatigue crack propagation in a baffle module of Wendelstein 7‐X under cyclic bending loads

Simulation of the fatigue crack propagation in a Wendelstein 7‐X baffle module is performed in this study using both a finite element method‐based software and the UniGrow nonlinear model for small‐scale yielding (SSY) conditions. Some experimental fatigue tests of several cracked baffle modules have been performed through a servo‐hydraulic machine. One of these experimental tests has been considered to simulate fatigue crack propagation in the baffle module. Before starting the experimental test, a first crack partly contained in the welding seam and partly in the steel pipe is found. Subsequently, owing to the applied load, the crack propagated both into the welding seam and into the steel pipe until the plastic zone in the near field attains SSY conditions. Finally, owing to the increase in the extension of the plastic zone, SSY conditions are not more valid, and the breakage of the steel pipe is produced by plastic collapse.     

3D Nanoprinted Plastic Kinoform X‐Ray Optics

High‐performance focusing of X‐rays requires the realization of very challenging 3D geometries with nanoscale features, sub‐millimeter‐scale apertures, and high aspect ratios. A particularly difficult structure is the profile of an ideal zone plate called a kinoform, which is manufactured in nonideal approximated patterns, nonetheless requires complicated multistep fabrication processes. Here, 3D fabrication of high‐performance kinoforms with unprecedented aspect ratios out of low‐loss plastics using femtosecond two‐photon 3D nanoprinting is presented. A thorough characterization of the 3D‐printed kinoforms using direct soft X‐ray imaging and ptychography demonstrates superior performance with an efficiency reaching up to 20%. An extended concept is proposed for on‐chip integration of various X‐ray optics toward high‐fidelity control of X‐ray wavefronts and ultimate efficiencies even for harder X‐rays. Initial results establish new, advanced focusing optics for both synchrotron and laboratory sources for a large variety of X‐ray techniques and applications ranging from materials science to medicine.     

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.     

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.     

X‐FEM in isogeometric analysis for linear fracture mechanics

The extended finite element method (X‐FEM) has proven to be an accurate, robust method for solving problems in fracture mechanics. X‐FEM has typically been used with elements using linear basis functions, although some work has been performed using quadratics. In the current work, the X‐FEM formulation is incorporated into isogeometric analysis to obtain solutions with higher order convergence rates for problems in linear fracture mechanics. In comparison with X‐FEM with conventional finite elements of equal degree, the NURBS‐based isogeometric analysis gives equal asymptotic convergence rates and equal accuracy with fewer degrees of freedom (DOF). Results for linear through quartic NURBS basis functions are presented for a multiplicity of one or a multiplicity equal the degree. Copyright © 2011 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.     

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.     

Precise 3D crack growth simulations

The continuous growth of 3D cracks under cyclic loading conditions is considered within a discrete simulation procedure. It is performed within the framework of linear elastic fracture mechanics. An incremental procedure is applied to consider the non-linear behavior of crack growth within the simulation. In each increment the direction and magnitude of the crack propagation for each point along the crack front are needed to define the new crack front. Within the present context the crack deflection results from the maximum tangential stress criterion and the crack extension is obtained by the evaluation of a crack propagation rate. To simulate the crack propagation as exactly as possible the evolution of the stress field between two consecutive crack fronts is taken into account. The analysis of the changing stress field is utilized for optimization of the predicted crack fronts. The whole procedure is realized in terms of a predictor–corrector scheme. Numerical examples are presented to demonstrate the benefits of this concept.