Three-dimensional effects on fatigue crack closure in the small-scale yielding regime – a finite element study

Plasticity induced closure often strongly influences the behaviour of fatigue cracks at engineering scales in metallic materials. Current predictive models generally adopt the effective stress‐intensity factor (ΔΚ_eff = Κ_max–Κ_op) in a Paris law type relationship to quantify crack growth rates. This work describes a 3D finite element study of mode I fatigue crack growth in the small‐scale yielding (SSY) regime under a constant amplitude cyclic loading with zero T‐stress and a ratio Κ_min/Κ_max = 0 . The material behaviour follows a purely kinematic hardening constitutive model with constant hardening modulus. Dimensional analysis suggests, and the computational results confirm, that the normalized remote opening load value, Κ_op/Κ_max, at each location along the crack front remains unchanged when the peak load (Κ_max), thickness (B) and material flow stress (σ₀) all vary to maintain a fixed value of . Through parametric computations at various K levels, the results illustrate the effects of normalized peak loads on the through‐thickness opening–closing behaviour and the effects of σ₀/E, where E denotes material elastic modulus. The examination of deformation fields along the fatigue crack front provides additional insight into the 3D closure process.     

Bauschinger effect of alloys and plasticity-induced crack closure: a finite element analysis

The effect of overloads, underloads and stress ratio on plasticity-induced crack opening level is examined for different 'model' materials. This study is focused on the consequences of the Bauschinger effect on the crack opening level. Various finite element analyses were conducted using ABAQUS to test these effects, involving the Chaboche constitutive equations that take into account both the Bauschinger effect of the material and its cyclic hardening or softening. The cyclic plastic behaviour of the material is found to strongly affect the crack behaviour after an overload or an underload. The experimental data obtained on a 0.4% carbon mild steel confirm the numerical results.     

Non‐linear seismic behaviour of concrete gravity dams using coupled finite element–boundary element technique

In this paper, the seismic response of concrete gravity dams is presented using the concept of Continuum Damage Mechanics (CDM) and adopting the hybrid Finite Element–Boundary Element technique (FE–BE). The finite element method is used for discretization of the near field and the boundary element method is employed to model the semi‐infinite far field. Because of the non‐linear nature of the discretizied equations of motion modified Newton–Raphson approach has been used at each time step to linearize them. Damage evolution based on tensile principal strain using mesh‐dependent hardening modulus technique is adopted to ensure the mesh objectivity and to calculate the accumulated damage. The methodology employed is shown to be computationally efficient and consistent in its treatment of both damage growth and damage propagation in gravity dams. Other important features considered in the analysis are: (1) realistic damage modelling for concrete that allows isotropic as well as anisotropic damage state and exhibits stiffness recovery upon load reversals. (2) softening initiation and strain softening criteria for concrete, and (3) proper modelling of semi‐infinite foundation using FE–BE method that allows to consider dam–foundation interaction analysis. As an application of the proposed formulation a gravity dam has been analysed and the results are compared with different foundation stiffnesses. The results of the analysis indicate the importance of including rock foundation in the seismic analysis of dams. Copyright © 1999 John Wiley & Sons, Ltd.     

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.     

The distributed dislocation technique for calculating plasticity-induced crack closure in plates of finite thickness

An analytical method for calculating plasticity-induced fatigue crack closure in plates of finite thickness is presented. The developed method utilizes the distributed dislocation technique (DDT) and Gauss-Chebyshev quadrature. Crack tip plasticity is incorporated by adopting a Dugdale type strip yield model. The finite plate thickness effects are taken into account by using a recently obtained three-dimensional solution for an edge dislocation in an infinite plate. Numerical results for the ratio of the size of the crack tip plasticity zones are presented for the cases of uniform thickness wake and linearly increasing wake for a range of plate thickness to crack length ratios and applied load ratios. The results show a very good agreement with previous analytical solutions in the limiting cases of very thick and very thin plates. Further results for the opening stress to maximum stress ratio are also provided and are compared with known three-dimensional finite element (FE) solutions. A good agreement is observed. The developed method is shown to be an effective and very powerful tool in modeling the crack closure phenomenon.     

Internal–external circumferential crack behaviour in the cement layer of total hip replacement

This study aimed to investigate crack behaviour at the internal and external surfaces of the cement layer in total hip replacement. A three‐dimensional model of the femur with the cemented prosthesis was developed and analysed. Cracks were placed on the internal, external and both internal and external surfaces of the cement layer. Stress intensity factors were measured during gait. Results revealed that the stress intensity factors modes I and III were the most dominant in the crack propagation in the cement layer. The domain of mode I was the medial and lateral sides of the cement layer. Meanwhile, the domain of mode III was the anterior and posterior sides of the cement layer. The stress intensity factor and distance from the distal end indicated an inverse relationship. The internal and external cracks had no significant interaction. Moreover, stress intensity factors at the external surface of the cement layer were higher than those on the internal surface.     

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.     

The partition of unity finite element method for elastic wave propagation in Reissner–Mindlin plates

This paper reports a numerical method for modelling the elastic wave propagation in plates. The method is based on the partition of unity approach, in which the approximate spectral properties of the infinite dimensional system are embedded within the space of a conventional finite element method through a consistent technique of waveform enrichment. The technique is general, such that it can be applied to the Lagrangian family of finite elements with specific waveform enrichment schemes, depending on the dominant modes of wave propagation in the physical system. A four‐noded element for the Reissner–Mindlin plate is derived in this paper, which is free of shear locking. Such a locking‐free property is achieved by removing the transverse displacement degrees of freedom from the element nodal variables and by recovering the same through a line integral and a weak constraint in the frequency domain. As a result, the frequency‐dependent stiffness matrix and the mass matrix are obtained, which capture the higher frequency response with even coarse meshes, accurately. The steps involved in the numerical implementation of such element are discussed in details. Numerical studies on the performance of the proposed element are reported by considering a number of cases, which show very good accuracy and low computational cost. Copyright © 2006 John Wiley & Sons, Ltd.     

The fatigue crack growth behaviour of 2524‐T3 aluminium alloy in an Al2O3 particle environment

The effect of the Al₂O₃ dust environment on the crack propagation behaviour of 2524‐T3 Al alloy was investigated. The results show that the Al₂O₃ dust environment reduces the fatigue crack growth rate (FCGR) of alloy especially at low ΔK. Many Al₂O₃ particles are deposited and stuck in the crack during fatigue loading which promotes crack closure, while this effect is gradually weakened with the increase of ΔK. The deposited Al₂O₃ particles induce the disorderly arranged slip bands (SBs) ahead of the crack tip which deflects the crack path making it more tortuous in the Al₂O₃ dust.     

Long crack growth behavior of implant material Ti–5Al–2.5Fe in air and simulated body environment related to microstructure

The fatigue crack propagation behavior of Ti–5Al–2.5Fe with various microstructures for biomedical applications was investigated in air and in a simulated body environment, Ringer's solution, in comparison with that of Ti–6Al–4V ELI and that of SUS 316L stainless steel. The crack propagation rate, da/dN, of Ti–5Al–2.5Fe in the case of each microstructure is greater than that of the Widmanstätten structure in Ti–6Al–4V ELI in air whereas da/dN of Ti–5Al–2.5Fe is nearly equal to that of the equiaxed structure in Ti–6Al–4V ELI in air when da/dN is plotted versus the nominal cyclic stress intensity factor range, ΔK. da/dN of the equiaxed structure and that of the Widmanstätten structure in Ti–5Al–2.5Fe are nearly the same in air when da/dN is plotted versus ΔK. da/dN of Ti–5Al–2.5Fe is nearly equal to that of SUS 316L stainless steel in the Paris Law region, whereas da/dN of Ti–5Al–2.5Fe is greater than that of SUS 316L stainless steel in the threshold region in air, when da/dN is plotted versus ΔK. da/dN of Ti–5Al–2.5Fe or Ti–6Al–4V ELI is nearly the same in air and in Ringer's solution when da/dN is plotted versus the effective cyclic stress intensity factor range, ΔK_eff, whereas da/dN of Ti–5Al–2.5Fe or Ti–6Al–4V ELI is greater in Ringer's solution than in air when da/dN is plotted versus ΔK.     

Non‐proportional size scaling of strength of concrete in uniaxial and biaxial loading conditions

A procedure for non‐proportional size scaling of the strength of concrete based on the weakest‐link statistics is proposed to synchronize strength data from specimens of different geometries and different loading modes. The procedure relies on proportional size scaling of strength to determine the parameters of the statistical model and often on finite element analysis to calculate the coefficient of the equivalent strength. The approach for non‐proportional size scaling is capable to synchronize the uniaxial strength data of concrete from uniaxial tensile specimens and 3‐point bending specimens, or the biaxial tensile strength data of circular plates in different loading mode. The non‐transference of the uniaxial strength data to the biaxial strength data is unclear in its mechanism but possibly due to the variation of statistical distribution of microcracks with stress states in different specimens.     

Effects of various environments on fatigue crack growth in Laser formed and IM Ti–6Al–4V alloys

The fatigue crack growth behaviors of Laser formed and ingot metallurgy (IM) Ti–6Al–4V alloys were studied in three environments: vacuum, air and 3.5% NaCl solution. Taking the Unified Fatigue Damage Approach, the fatigue crack growth data were analyzed with two intrinsic parameters, stress intensity amplitude ΔK and maximum stress intensity K_max, and their limiting values ΔK^* and . Fatigue crack growth rates da/dN were found increase with stress ratio R, highest in 3.5% NaCl solution, somewhat less in air and lowest in vacuum, and higher in IM alloy than in Laser formed one. In 3.5% NaCl solution, stress corrosion cracking (SCC) was superimposed on fatigue at R=0.9 for where K_max>K_ISCC, the threshold stress intensity for SCC. This and environment-assisted fatigue crack growth were evidenced by the deviation in fatigue crack growth trajectory (ΔK^* vs. curve) from the pure fatigue line where . Furthermore, the fractographic features, identified along the trajectory path, reflected the fatigue crack growth behaviors of both alloys in a given environment.     

Non‐destructive evaluation of fatigue damage and fatigue crack initiation in type 316 stainless steel by positron annihilation line‐shape and lifetime analyses

Rotating bending fatigue tests were conducted using type 316 stainless steel. The fatigue tests were periodically terminated, and fatigue damage and fatigue crack initiation were non‐destructively and sequentially evaluated by positron annihilation line‐shape and lifetime analyses. The counter‐jig and anticoincidence methods were used for positron annihilation line‐shape and lifetime analyses, respectively, to enhance the analytical precision. The fatigue crack lengths were monitored by a plastic replication technique, and related to the parameters in both analyses. S‐parameter obtained in the line‐shape analysis increased with increasing fatigue damage, while it was difficult to detect fatigue crack initiation and subsequent small fatigue crack growth. That was because the precision of line‐shape analysis was limited. On the other hand, both fatigue damage and fatigue crack initiation were successfully detected by lifetime analysis. Positron annihilation lifetime also increased with increasing fatigue damage, and lifetime was longer at the notch root with fatigue crack than at the smooth section without crack. It was considered that the precision of lifetime analysis was high enough to detect high dislocation density areas at the fatigue crack tips.     

Crystal plasticity finite‐element modelling of cyclic deformation and crack initiation in a nickel‐based single‐crystal superalloy under low‐cycle fatigue

Nickel‐based single‐crystal superalloys are predominantly used for turbine blades in aircraft engines and land‐based gas turbines. Understanding and predicting the fatigue failure of Ni‐based single‐crystal superalloys are critical to ensure the safety of these components during operation. In this paper, low‐cycle fatigue experiments were carried out to investigate cyclic deformation of a nickel‐based single‐crystal superalloy MD2, recently developed by GE Power, with different crystallographic orientations. Specialty in situ scanning electron microscope (SEM) tests were also conducted to study the slip‐controlled initiation of short cracks under low‐cycle fatigue. In particular, the stress–strain response for both [001] and [111] orientations was used to calibrate a crystal plasticity model, which allowed us to simulate the activation of crystallographic slip systems and predict the initiation of short fatigue crack. Using the accumulated shear strain as a criterion, the simulations confirmed that the slip system with the maximum accumulated shear strain appeared to control the crack initiation. The location and direction of slip traces and short cracks, captured by the crystal plasticity finite‐element simulations, agreed with the in situ SEM observations. The modelling tool will be valuable for assessing the structural integrity of critical gas turbine blades.     

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.     

Numerical simulation of the coupling between thermal dissipation and fish‐eye crack growth in very high cycle fatigue regime

In this paper, we study the temperature field associated with the propagation of a fatigue crack in a very high cycle fatigue regime during ultrasonic fatigue testing. We use a Paris–Hertzberg crack growth law to compute the evolution of the crack and a perfectly elastic–plastic constitutive law to compute the plastic dissipation per cycle at the tip of the crack. A thermomechanical finite element model is proposed to estimate the evolution of the temperature field during the crack propagation. Numerical results obtained agree fairly well with experimental results.     

The effect of ferrite phases on the micromechanical response and crack initiation in the intercritical heat‐affected zone of a welded 9Cr martensitic steel

This paper presents a crystal plasticity model to predict the tensile response and crack initiation in a mixed ferrite‐martensite material with a low volume fraction of pro‐eutectoid ferrite, representative of a welding‐induced intercritical heat‐affected zone. It is shown that small volume fractions of ferrite can have a significant effect on material strength and ductility depending on the ferrite grain orientation. For relatively “soft” ferrite grains, microcracks can grow across interferrite ligaments with damage accumulating in the ferrite, leading to a reduction in strength and strain hardening, but with little influence on ductility; in contrast, relatively “hard” ferrite grains act to accelerate microcrack initiation, leading to reduced ductility, with negligible influence on strain hardening up to the maximum load.