Dual boundary element method and finite element method for mixed‐mode crack propagation simulations in a cracked hollow shaft

Three‐dimensional mixed‐mode crack propagation simulations were performed by means of the dual boundary element method code BEASY and 2 finite element method‐based crack propagation codes: ZENCRACK (ZC) and CRACKTRACER3D (CT3D). The stress intensity factors (SIFs) along the front of an initial semielliptical crack, initiated from the external surface of a shaft, were calculated for 4 different load cases: bending, press fit, shear, and torsion. The methods used for the SIF assessment along the crack front were the J‐integral for BEASY and ZC and the quarter point element stress method for CT3D. Subsequently, crack propagation simulations were performed, with the crack growth rate evaluated by using Paris' law, calibrated for the material at stake (American Society for Testing and Materials A469 steel). The kink angles were evaluated by using the minimum strain energy density and maximum tangential stress criteria for BEASY, the maximum energy release rate and maximum tangential stress for ZC, and the maximum principal asymptotic stress for CT3D. The results obtained in terms of SIFs and crack propagation life show very good agreement among the 3 codes. Also, the shape of the propagated crack, which is significantly out‐of‐plane for the shear and torsion loading, matched very well.     

Three‐dimensional non‐planar crack growth by a coupled extended finite element and fast marching method

A numerical technique for non‐planar three‐dimensional linear elastic crack growth simulations is proposed. This technique couples the extended finite element method (X‐FEM) and the fast marching method (FMM). In crack modeling using X‐FEM, the framework of partition of unity is used to enrich the standard finite element approximation by a discontinuous function and the two‐dimensional asymptotic crack‐tip displacement fields. The initial crack geometry is represented by two level set functions, and subsequently signed distance functions are used to maintain the location of the crack and to compute the enrichment functions that appear in the displacement approximation. Crack modeling is performed without the need to mesh the crack, and crack propagation is simulated without remeshing. Crack growth is conducted using FMM; unlike a level set formulation for interface capturing, no iterations nor any time step restrictions are imposed in the FMM. Planar and non‐planar quasi‐static crack growth simulations are presented to demonstrate the robustness and versatility of the proposed technique. Copyright © 2008 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.     

Stress intensity factors for slanted through-wall cracks based on elastic finite element analyses

Based on detailed 3‐dimensional (3‐D) elastic finite element (FE) analyses, the present paper provides stress intensity factors (SIFs) for plates with slanted through‐wall crack (TWC) and cylinders with slanted circumferential TWC. Regarding loading conditions, axial tension was considered for the plates, whereas axial tension, global bending and internal pressure were considered for the cylinders. To cover a practical range, the geometric variables affecting the SIF were systematically varied. Based on FE results, SIFs along the crack front, including the inner and outer surface points, were provided. The present results can be used to evaluate the fatigue crack growth or stress corrosion cracking behaviour of a slanted TWC and furthermore to perform detailed Leak‐Before‐Break analysis considering a more realistic crack shape.     

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.     

Comparison of crack growth simulation by DBEM and FEM for SEN-specimens undergoing torsion or bending loading

The 3D fatigue crack growth behaviour in 3PB-specimens or in cantilever beam specimens under, respectively, bending or torsion loading, with inclined planes for the initial crack, is investigated by the dual boundary element method (DBEM). Mixed mode conditions along the crack edge are characterized: stress intensity factors (SIFs) are determined using the J-integral method, and the crack growth direction is computed by the minimum strain energy density criterion. The evolving crack shape (spatially twisted or twisted and warped crack faces), computed by the DBEM model, is successfully compared with experimental findings and FEM results.     

External surface cracks in shells under cyclic internal pressure

The stress analysis and fatigue crack growth behaviour of a part‐through‐cracked double‐curvature thin‐walled shell is examined. An external surface crack is assumed to lie in one of the principal curvature planes of the shell, and to present a semi‐elliptical shape. The stress intensity factors (SIFs) along the crack front for different elementary opening stresses acting on the crack faces are determined through a three‐dimensional finite element analysis. Then approximate values of SIF in the case of a cracked pressure vessel are computed by employing the above results together with the superposition principle and the power series expansion of the actual opening stress. Finally, a numerical simulation procedure is carried out to predict the crack growth under cyclic internal pressure. Some results are compared with those of other authors.     

Extended finite element method and fast marching method for three-dimensional fatigue crack propagation

A numerical technique for planar three-dimensional fatigue crack growth simulations is proposed. The new technique couples the extended finite element method (X-FEM) to the fast marching method (FMM). In the X-FEM, 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 modeled by finite elements with no explicit meshing of the crack surfaces. The initial crack geometry is represented by level set functions, and subsequently signed distance functions are used to compute the enrichment functions that appear in the displacement-based finite element approximation. The FMM in conjunction with the Paris crack growth law is used to advance the crack front. Stress intensity factors for planar three-dimensional cracks are computed, and fatigue crack growth simulations for planar cracks are presented. Good agreement between the numerical results and theory is realized.     

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.     

SGBEM analysis of crack-particle(s) interactions due to elastic constants mismatch

The effects of elastic constants mismatch on the interaction between a propagating crack and single or multiple inclusions in brittle matrix materials are investigated using numerical simulations. The simulations employ a quasi-static crack-growth prediction tool based upon the symmetric-Galerkin boundary element method (SGBEM) for multiregions, a modified quarter-point crack-tip element, the displacement correlation technique for evaluating stress intensity factors (SIFs), and the maximum principal stress criterion for crack-growth direction. It is shown that, even with this simple method for calculating SIF, the crack-growth prediction tool is both highly accurate and computationally effective. This is evidenced by results for the case of a single inclusion in an infinite plate, where the SGBEM results for the SIFs show excellent agreement with known analytical solutions. The simulation results for crack growth and stress intensity behaviors in particulate media are very stable. The crack-tip shielding and amplification behaviors, as seen in similar studies using other numerical approaches, can be clearly observed.     

Coupled FEM–DBEM approach on multiple crack growth in cryogenic magnet system of nuclear fusion experiment ‘Wendelstein 7‐X’

At the Max Planck Institute for plasma physics in Greifswald, Germany, the world's largest nuclear fusion experiment of modular stellarator type Wendelstein 7‐X has started plasma operation. The hot hydrogen plasma is confined in a plasma vessel by an electromagnetic field generated by 50 non‐planar and 20 planar superconducting coils. The superconducting coils are encased in cast stainless steel housings. The coils are bolted onto a central support ring and welded together by so called lateral support elements (LSEs). In this paper, a procedure, based on a global–local finite element method (FEM)–dual boundary element method (DBEM) approach, is developed to simulate the propagation of multiple cracks detected in LSEs and undergoing a fatigue load spectrum. The global stress analysis on the superconducting coils is performed by FEM whereas the sub‐modelling approach is adopted to solve the crack propagation in the DBEM environment. The boundary conditions applied on the DBEM submodel are the displacements calculated by the FEM global analysis, in correspondence of the cut surfaces (there are no body forces nor external loads applied on the submodel volume). Two cracks are simultaneously introduced, and a linear elastic fracture mechanics analysis is performed. Results in terms of cracks growth rates and evolving crack shapes are provided, and the residual life of the component is forecast.     

Sickle-shaped cracks in metallic round bars under cyclic eccentric axial loading

In the present paper, the fatigue propagation of an initial sickle-shaped surface crack in a metallic round bar under eccentric axial loading acting perpendicular to the crack plane is examined. Firstly, the stress-intensity factor (SIF) along the crack front is determined through a three-dimensional finite element analysis and the one-quarter point displacement method, for different values of the loading eccentricity. Then, the fatigue behaviour of the cracked bar is numerically analysed by a step-by-step procedure based on the Paris–Erdogan law. The results are plotted in terms of crack paths, intersection angle and crack depth evolution, by varying the loading eccentricity.     

Determination of crack-tip stress intensity factors in complex geometries by the composition of constituent weight function solutions

Linear elastic fracture mechanics (LEFM) is the science frequently used to understand the stable and progressive fatigue crack growth that often occurs in engineering components under varying applied stress. The stress intensity factor (SIF) is its basis and describes the stress state at the crack tip. This can be used with the appropriate material properties to calculate the rate at which the crack will propagate in a linear elastic manner. Unfortunately, the SIF is difficult to compute or measure, particularly if the crack is situated in a complex three‐dimensional geometry or subjected to a non‐simple stress state. This is because the SIF is not only a function of the crack and component geometry but is also dependent on the applied stress field. In the last 20 years, the SIF weight function has gained prominence as a method for calculating and presenting SIFs independent of applied stress. This paper demonstrates that the real promise of the SIF weight Function lies in its use to rapidly generate SIF solutions for cracks in complex geometries by simple composition of geometric influences from reference constituent solutions.     

Assessing the accuracy of fatigue life in surface‐cracked straight pipes

The demonstration of leak before brake (LBB) based on fracture mechanics requires information on the initial size of a defect, initiation of crack growth from the inherent defect and subsequent crack growth rates. In the present paper the prediction methodologies have been tested for three different full scale pipes geometry experimentally tested data. The prediction accuracy of two SIF solutions available in the literature has also been judged. The effect of fatigue crack closure and corrections needed in the numerical prediction methodology using FEM have also been included. The results showed that the FEM could fairly predict the fatigue crack initiation and crack growth life of full‐scale piping components having a constant depth crack profile.     

AN IMPROVED NUMERICAL TECHNIQUE FOR SIMULATING THE GROWTH OF PLANAR FATIGUE CRACKS

Abstract— This paper describes a versatile technique for simulating the fatigue growth of a wide range of planar cracks of practical significance. Crack growth is predicted on a step-by-step basis from the Paris law using stress intensity factors calculated by the finite element method. The crack front is defined by a cubic spline curve from a set of nodes. Both the 1/4-node crack opening displacement and the three-dimensional J -integral (energy release rate) methods are used to calculate the stress intensity factors. Automatic remeshing of the finite element model to a new position which defines the new crack front enables the crack propagation to be followed. The accuracy and capability of this finite element simulation technique are demonstrated in this paper by the investigation of various problems of both theoretical and practical interest. These include the shape growth trend of an embedded initially penny-shaped defect and an embedded initially elliptical defect in an infinite body, the growth of a semi-elliptical surface crack in a finite thickness plate under tension and bending, the propagation of an internal crack in a round bar and the shape change of an external surface crack in a pressure vessel.     

Improvements of explicit crack surface representation and update within the generalized finite element method with application to three‐dimensional crack coalescence

This paper presents improvements to three‐dimensional crack propagation simulation capabilities of the generalized finite element method. In particular, it presents new update algorithms suitable for explicit crack surface representations and simulations in which the initial crack surfaces grow significantly in size (one order of magnitude or more). These simulations pose problems in regard to robust crack surface/front representation throughout the propagation analysis. The proposed techniques are appropriate for propagation of highly non‐convex crack fronts and simulations involving significantly different crack front speeds. Furthermore, the algorithms are able to handle computational difficulties arising from the coalescence of non‐planar crack surfaces and their interactions with domain boundaries. An approach based on moving least squares approximations is developed to handle highly non‐convex crack fronts after crack surface coalescence. Several numerical examples are provided, which illustrate the robustness and capabilities of the proposed approaches and some of its potential engineering applications. Copyright © 2013 John Wiley & Sons, Ltd.     

Stress intensity factors for surface fatigue crack in a round bar under cyclic axial loading

In this paper, the surface fatigue crack growth shape for an initial straight-fronted edge crack in an elastic bar of circular cross-section is determined through experiments under pure fatigue axial loading. Three different initial notch depths are discussed. The relations of the aspect ratio (b/c) and relative crack depth (b/D) are obtained, and it is shown that there is a great difference in the growth of cracks with different initial front shapes and crack depths. Further, using the three-dimensional finite element method, the stress intensity factors (SIFs) are determined under remote uniform tension loading. Since the relationship of b/c and b/D changes during the fatigue crack growth, the SIFs are determined for different surface crack configurations.     

Effects of CFRP bond locations on the Mode I stress intensity factor of centre‐cracked tensile steel plates

The fatigue life of cracked steel members can be greatly extended by externally attached carbon fibre reinforced plastics (CFRP), which reduces the stress intensity factors (SIFs) at the crack tip. Access to cracks is sometimes limited and the CFRP has to be attached away from the cracks. There is a lack of knowledge on SIFs for such strengthening scheme. This paper presents the effects of CFRP bond locations on the Mode I SIF of centre‐cracked tensile (CCT) steel plate. The Mode I SIF at the crack tip is calculated using the finite element (FE) models. A correction factor is introduced as a function of CFRP bond location and crack length. The FE results are compared and agree well with experimental tests conducted by the authors. By combining with another two factors (one considering CFRP mechanical properties and the other considering CFRP bond width) derived previously by the authors, SIF formulae are proposed for CFRP reinforced CCT steel plates.     

An analytical expression for the J2‐integral of an interfacial crack in orthotropic bimaterials

This paper presents a new analytical expression relating the J₂‐integral and stress intensity factors (SIF) in an in‐plane traction‐free crack between two orthotropic elastic solids using the complex function method. The singular oscillatory near tip field of a bimaterial interfacial crack is usually characterized by a pair of SIFs. In linear elastic interfacial fracture mechanics, the majority of numerical and experimental methods rely on the analytical equations relating J_k‐integrals and SIFs. Although an analytical equation relating J₁‐integral or strain energy release rate and SIFs is available, a similar relation for J₂‐integral in debonded anisotropic solids is non‐existent. Using this new analytical expression, in conjunction with the values of J_k, the SIFs can be computed without the need for an auxiliary relation. An example with known analytical solutions for SIFs is presented to show the variation of the J₂‐integral near the crack tip of a bimaterial orthotropic plate. Different bimaterial combinations are considered, and the effect of material mismatch on J_k is demonstrated.     

SGBEM–FEM coupling for analysis of cracks in 3D anisotropic media

A weakly singular symmetric Galerkin boundary element method (SGBEM) is coupled with the standard finite element method (FEM) in order to establish an accurate and efficient numerical technique for analysis of fractures in three‐dimensional, anisotropic, linearly elastic media. In the strategy, the weakly singular SGBEM developed by Rungamornrat and Mear (textitInt. J. Solids Struct. 2008; 45 :1283–1301; Comput. Methods Appl. Mech. Engrg 2008; 197 :4319–4332) is utilized to model a small‐scale region containing the crack while the (possibly large and complex) compliment region is treated by the FEM. The coupled technique exploits the positive features of both methods; the SGBEM proves to be a convenient and highly accurate method for obtaining mixed‐mode stress intensity factors along the crack front, whereas the FEM is very efficient for modeling large‐scale problems in the absence of cracks. An important aspect of the formulation and implementation of the technique is that continuity of displacement and traction across the interface between the SGBEM and FEM regions is enforced in a weak sense. This allows the two regions (one modeled by the SGBEM and the other by the FEM) to be discretized independently without the need for the resulting meshes to conform on the interface separating the regions, and this flexibility in the discretization process leads to a significant reduction in the modeling effort. To demonstrate the utility and accuracy of the technique, several boundary value problems involving both embedded and surface breaking cracks are treated, and it is shown that the coupled technique yields highly accurate stress intensity factors that exhibit only a slight dependence upon mesh refinement. Copyright © 2010 John Wiley & Sons, Ltd.