Crack initiation from micro surface holes in bearings under rolling contact fatigue

Investigations concerning surface crack growth are necessary for understanding the mechanism of rolling contact fatigue (RCF) of bearings because the surface defects cause flaking failures. In the present work, micro holes were artificially made prior to the RCF tests and the initiation of the surface cracks from the micro holes was observed in order to find the key factors for understanding their features. Crack initiation directions were compared to the stress intensity factors calculated by a simple method based on the theory. The extent to which ‘contact pressure (wedge effect)’ and ‘contact stresses’ are applicable for understanding the correlations between the crack initiation directions and stress intensity factors is discussed. The crack initiation directions are strongly correlated to the stress intensity factors caused by the contact stresses alone. We concluded that the crack growth and initiation are dominated by stress intensity factors caused by contact stresses rather than the wedge effect.     

A numerical approach to subsurface crack propagation assessment in rolling contact

Subsurface crack mode II propagation parallel to the contact surface is a damage mechanism leading to dramatic failure in many components subjected to cyclic loading. A weight function (WF) was elaborated for calculating the applied mode II stress intensity factor (SIF) of a crack in a two‐dimensional half‐space in plane strain condition, for crack completely closed and frictionless contact between the crack faces. With respect to other methods, the WF allows faster SIF calculation, thus being suitable for simulation of many repeated load cycles and fatigue crack propagation. The WF was applied for simulating a case of rolling contact experiments found in the literature, and good agreement between experimental and numerical results was obtained, showing the effectiveness of the WF method in damage tolerant design.     

A mixed mode crack growth model taking account of fracture surface contact and friction

Fracture surface interactions, of whatever origin, can significantly affect the stress intensity factor, and consequently, can also be relevant to fatigue crack propagation. In the occurrence of interaction between fracture surfaces, the effective loading cycle experienced by material near the crack tip may be very different from that evaluated on the basis of the external loadings only. The purpose of the work described in this paper is to obtain the effective mode II stress intensity factor, k _IIeff, in a surface cracked elasto-plastic plate with a factory roof fracture surface subjected to an in-plane shear (mode II) loading. A new model estimating the magnitude of the frictional mode II stress intensity factor, k _f, arising from the mismatch of the fracture surface roughness during in-plane shear, is developed. Furthermore, the results of this study are employed in modeling the fatigue response of the surface cracked plates subjected to mixed mode loading.     

Characterization of the computational accuracy in surface crack problems

The paper focuses on the comparison of different methods for calculating stress intensity factors (K_I) in surface crack problems based on results of numerical analyses of elastic crack-tip fields. The computational accuracy is quantified by means of the so-called ‘averaged error estimation technique’ which is extended to the evaluation of local errors in the determination of stress intensity factors at characteristic points of the crack front. Numerical data involved in the present study are obtained from boundary-element calculations. Three values of the stress intensity factor, i.e. those defined from nodal tractions, displacements and energy-release rate, are provided. The highest error level is found for the displacement-based data, while the energy-release calculations yield the best accuracy. A considerable increase in the error value is noticed near the intersection of the crack front with a body surface where the conventional assumption on the square-root stress singularity is, in general, not applied. It is shown that the accuracy of stress intensity factor analysis can be improved by eliminating uncertainties associated with the local stress state along the crack front. © 1998 John Wiley Sons, Ltd.     

On propagation of short rolling contact fatigue cracks

Recent accidents involving railway rails have aroused demand for improved and more efficient rail maintenance strategies to reduce the risk of unexpected rail fracture. Numerical tools can aid in generating maintenance strategies: this investigation deals with the numerical modelling and analysis of short crack growth in rails. Factors that influence the fatigue propagation of short surface‐breaking cracks (head checks) in rails are assessed. A proposed numerical procedure incorporates finite element (FE) calculations to predict short crack growth conditions for rolling contact fatigue (RCF) loading. A parameterised FE model for the rolling‐sliding contact of a cylinder on a semi‐infinite half space, with a short surface breaking crack, presented here, is used in linear‐elastic and elastic–plastic FE calculations of short crack propagation, together with fracture mechanics theory. The crack length and orientation, crack face friction, and coefficient of surface friction near the contact load are varied. The FE model is verified for five examples in the literature. Comparison of results from linear‐elastic and elastic–plastic FE calculations, shows that the former cannot describe short RCF crack behaviour properly, in particular 0.1–0.2 mm long (head check) cracks with a shallow angle; elastic–plastic analysis is required instead.     

Effect of crack face contact and friction on Brazilian disk specimens—A finite difference solution

A full curvilinear transformation is employed to study the effect of contact and friction on Brazilian disk specimens containing a crack and subjected to concentrated loads at angles 0° <  < 90°. Homogeneous and bimaterial disks made of glass and epoxy are considered. The effect of loading angle and friction coefficient on the stress intensity factors, as well as the contact length is studied. The results are compared to available semi-analytical and finite elements results. It is found that when the crack faces are in contact without stick zones, an increase in friction causes a decrease of the normal gap, tangential shift and stress intensity factors. When stick conditions appear in the contact zone, an increase in the coefficient of friction also results in increasing the stick zone within the contact zone.     

A solution method for frictional contact problems of a crack in FGMs under mechanical and thermal loads

This article provides a numerical treatment of a finite crack in an interfacial layer with spatially varying elastic properties under in-plane mechanical and thermal loading conditions. The variation of stress intensity factors and energy release rates with the functions which are governing the material properties of the interfacial layer is studied. Transient and steady-state response of a central crack in FGMs subjected to the mechanical and thermal loads are investigated. Unlike earlier studies which consider the cracks encountered as open, the current investigation studies cracks in an essentially compressive environment in which the crack faces are in contact and frictional effects play an important role. To solve this contact problem, a simple and efficient, iterative finite element method developed by authors is used. Numerical examples are provided to verify the technique and the results are compared with those of the published papers. This revised version was published online in July 2006 with corrections to the Cover Date.     

A mixed augmented Lagrangian-extended finite element method for modelling elastic–plastic fatigue crack growth with unilateral contact

The complete modelling of fatigue crack growth is still an industrial challenging issue for numerical methods. A new technique for the finite element modelling of elastic–plastic fatigue crack growth with unilateral contact on the crack faces is presented. The extended finite element method (X-FEM) is used to discretize the equations, allowing for the modelling of arbitrary cracks whose geometries are independent of the finite element mesh. This paper presents an augmented Lagrangian formulation in the X-FEM framework that is able to deal with elastic–plastic crack growth with treatment of contact. An original formulation, which takes advantages of two powerful numerical methods, is presented. Next the numerical issues such as contact treatment and numerical integration are addressed, and finally numerical examples are shown to validate the method. Copyright © 2007 John Wiley & Sons, Ltd.     

Critical study of existing solutions for a penny-shaped interface crack, comparing with a new boundary element solution allowing for frictionless contact

A numerical study of the fundamental problem of a pressurized penny-shaped crack at the interface of two dissimilar half spaces is carried out allowing for the possibility of frictionless contact between crack faces. A new, highly accurate axi-symmetric formulation of the boundary element method (BEM) for the solution of elastic contact problems is employed. The correctness and accuracy of available predictions of different kinds for several key characteristics of the solution of this problem are checked. First, comparison of the BEM results for the near-tip contact length shows a very good agreement with some existing predictions. Second, the global solution obtained by BEM is compared with existing asymptotic solutions, obtained with both the open and the frictionless contact models. BEM results show that at the closest neighborhood to the crack tip the global solution of the problem is governed by the first term of the asymptotic solution of the frictionless contact model (up to a distance of the order of a fraction of the near-tip contact length). After a small transition region, in an adjacent surrounding zone whose extent is almost independent of the near-tip contact length, the global solution of the problem is governed by the first term of the asymptotic solution of the open model. As a result of the comparison presented, the regions in which the classical fracture parameters, stress intensity factor (SIF) and energy release rate, can be accurately obtained from the global numerical solution of a crack of this kind have been determined. Third, BEM results and previous estimations show certain discrepancies with a recently published closed form solution of the near-tip contact length and the mode II SIF of the frictionless contact model. A new closed form expression of this mode II SIF, derived from the asymptotic solution of the open model, is proposed in this paper.     

Surface crack growth of silicon nitride bearings under rolling contact fatigue

Surface crack growth of silicone nitride ceramic bearings under rolling contact fatigue has been investigated from the viewpoints of contact stresses (ring crack model) and fluid pressure (wedge effect model). The mechanisms of these two models have been investigated independently; however, it was impossible to separate the effects of contact stresses and fluid pressure on surface crack growth. In this paper the effects of contact stresses (ring crack model) on surface crack growth are investigated. In the ring crack model the crack growth is caused by contact stresses around the circumference of the contact circle. The growth of surface cracks located inside and outside the contact track was observed in order to obtain data from which we could reexamine the ring crack model. The outside cracks under rolling contact fatigue were propagated by contact stresses alone and also the inside cracks grew as slowly as the outside cracks. We concluded that the cracks are propagated by the single effect of contact stresses. Preliminary observations of surface crack growth showed that the cracks were unaffected by wear and residual stresses.     

Shear loaded interface crack under the influence of friction: a finite difference solution

Formulation of the elastic two‐dimensional problem of contact with friction is presented. Two‐dimensional equilibrium equations and boundary conditions in an orthogonal curvilinear co‐ordinate system are written explicitly. The above formulation is solved with the aid of the finite difference technique. An iterative algorithm which does not require load increments is employed for solving interface fracture problems with contact and friction subjected to a monotonically increasing load. The J‐integral is extended for problems in which there is friction along the crack faces. Stress intensity factors are calculated by means of the J‐integral, as well as an asymptotic expansion of the tangential shift. Two problems are analysed: (1) a crack in homogeneous material in the presence of friction involving stationary contact; and (2) an interface crack in the presence of friction involving receding contact. Results are compared to those found by analytical and semi‐analytical methods which are presented in the literature, as well as to those obtained by means of the finite element method. The accuracy of the results establishes the reliability of the finite difference analysis, as well as the post‐processors. In addition, a problem involving stick conditions is considered. It is observed that with increasing friction, the normal gaps and tangential shifts decrease. The size of the contact zone increases and values of the stress intensity factor decrease. Copyright © 2004 John Wiley & Sons, Ltd.     

Investigation of rolling contact crack initiation in bainitic and pearlitic rail steels

The stress–strain history and the crack initiation lives of bainitic and head‐hardened pearlitic rail steels were determined under rolling contact loading by implementing the semi‐analytical Jiang–Sehitoglu rolling contact model that incorporates both ratchetting and multiaxial fatigue damage. The calculations revealed that the bainitic steel withstands higher loads than the pearlitic steel at low shear tractions, however; both materials behave in an increasingly similar manner as the shear tractions increase. Furthermore, maximum damage occurs in both steels when ratchetting and fatigue damage coincide on the surface. In addition to shedding light on the rolling contact fatigue (RCF) performance of bainitic and pearlitic rail steels, the current work also establishes a methodology for the realistic prediction of crack initiation under RCF.     

Variation of mixed modes stress intensity factors of an inclined semi-elliptical surface crack

In this paper, a singular integral equation method is applied to calculate the stress intensity factor along crack front of a 3D inclined semi-elliptical surface crack in a semi-infinite body under tension. The stress field induced by displacement discontinuities in a semi-infinite body is used as the fundamental solution. Then, the problem is formulated as a system of integral equations with singularities of the form r ^–3. In the numerical calculation, the unknown body force doublets are approximated by the product of fundamental density functions and polynomials. The results show that the present method yields smooth variations of mixed modes stress intensity factors along the crack front accurately for various geometrical conditions. The effects of inclination angle, elliptical shape, and Poisson's ratio are considered in the analysis. Crack mouth opening displacements are shown in figures to predict the crack depth and inclination angle. When the inclination angle is 60 degree, the mode I stress intensity factor F _I has negative value in the limited region near free surface. Therefore, the actual crack surface seems to contact each other near the surface.     

Stress concentration around a strongly oblate cavity in a cubic crystal and its associated crack problems

The stress distribution around a strongly oblate spheroidal cavity in a cubic crystal is determined by the equivalent inclusion method. The stress concentration factor is shown to be a product of two factors: one factor is purely geometric; the other factor depends on the material properties. By allowing the aspect ratio of the cavity to approach zero, the stress intensity factor of the associated penny-shaped crack is deduced. The energy release rates of cracks on {1 0 0} planes are computed for different growth directions. Theses results are found to be correlated well with Zener’s anisotropy factor.     

Stress intensity factors of a central slant crack with frictional surfaces in plates with biaxial loading

The stress intensity factor is a traditional topic in mechanics and there have been many solutions for many different cases. The closed frictional crack problem has been modeled in the rock mechanics field where fractures are mostly under compression. Further, the effect of finite plate dimensions under biaxial loading has not been considered in the literature. The key contribution of the present paper is to evaluate the effect of the crack length to plate width ratio on the mode I and II stress intensity factors (SIF) of a central slant crack with frictional surfaces in plates with biaxial loading of different patterns, i.e. tension-tension, tension-compression, compression-tension or compression-compression. A plane strain elastic two-dimensional finite element analysis was adopted. Crack length to plate width ratios equal to 0.1, 0.3 and 0.5 with biaxial ratios from –1 to 1, crack angles from 0° to 90° and friction coefficients from 0 to 1 were considered. Contact regimes and the effect of the crack length to plate width ratio were found dependent on biaxial ratio and pattern, friction coefficient and crack angle.     

Numerical assessment of the loading of rolling contact fatigue cracks close to rail surface irregularities

Rolling contact fatigue damage of railway rails in the form of squats, characterised by local depressions and cracks located at the rail surface, has been linked to the occurrence of local rail surface irregularities. This study concerns rolling contact fatigue cracks in the vicinity of fairly smooth surface irregularities, here denoted dimples. The influence of factors such as dimple geometry, cluster effects, and crack size is evaluated. To this end, dynamic vehicle–track simulations featuring realistic wheel and rail profiles are employed to characterise the dynamic impact during a wheel passage. The contact load in the vicinity of the dimples is then mapped onto a 3D finite element model of a rail section containing a crack in the rail head. The crack loading is finally quantified by multimodal stress intensity factors. The analyses establish that also shallow dimples might have a significant impact on the crack loading. This effect is increased for larger or multiple irregularities but decreases as the crack grows.     

Simulation of surface pitting due to contact loading

A computational model for simulation of surface pitting of mechanical elements subjected to rolling and sliding contact conditions is presented. The two-dimensional computational model is restricted to modelling of high-precision mechanical components with fine surface finishing and good lubrication, where the cracks leading to pitting are initiated in the area of largest contact stresses at certain depth under the contacting surface. Hertz contact conditions with addition of friction forces are assumed and the position and magnitude of the maximum equivalent stress is determined by the finite element method. When the maximum equivalent stress exceeds the local material strength, it is assumed that the initial crack develops along the slip line in a single-crystal grain. The Virtual Crack Extension method in the framework of finite element analysis is then used for two-dimensional simulation of the fatigue crack propagation under contact loading from the initial crack up to the formation of the surface pit. The pit shapes and relationships between the stress intensity factor and crack length are determined for various combinations of contacting surface curvatures and loadings. The model is applied to simulation of surface pitting of two meshing gear teeth. Numerically predicted pit shapes in the face of gear teeth show a good agreement with the experimental observations. © 1998 John Wiley & Sons, Ltd.     

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.     

Similarities of stress concentrations in contact at round punches and fatigue at notches: implications to fretting fatigue crack initiation

A linear elastic model of the stress concentration due to contact between a rounded flat punch and a homogeneous substrate is presented, with the aim of investigating fretting fatigue crack initiation in contacting parts of vibrating structures including turbine engines. The asymptotic forms for the stress fields in the vicinity of a rounded punch-on-flat substrate are derived for both normal and tangential loading, using both analytical and finite element methods. Under the action of the normal load, P , the ensuing contact is of width 2 b which includes an initial flat part of width 2 a . The asymptotic stress fields for the sharply rounded flat punch contact have certain similarities with the asymptotic stress fields around the tip of a blunt crack. The analysis showed that the maximum tensile stress, which occurs at the contact boundary due to tangential load Q , is proportional to a mode II stress intensity factor of a sharp punch divided by the square root of the additional contact length due to the roundness of the punch, Q /(√( b − a )√ π b ). The fretting fatigue crack initiation can then be investigated by relating the maximum tensile stress with the fatigue endurance stress. The result is analogous to that of Barsom and McNicol where the notched fatigue endurance stress was correlated with the stress intensity factor and the square root of the notch-tip radius. The proposed methodology establishes a 'notch analogue' by making a connection between fretting fatigue at a rounded punch/flat contact and crack initiation at a notch tip and uses fracture mechanics concepts. Conditions of validity of the present model are established both to avoid yielding and to account for the finite thickness of the substrate. The predictions of the model are compared with fretting fatigue experiments on Ti–6Al–4V and shown to be in good agreement.     

Fractal finite element method based shape sensitivity analysis of multiple crack system

This paper presents fractal finite element based continuum shape sensitivity analysis for a multiple crack system in a homogeneous, isotropic, and two dimensional linear-elastic body subjected to mixed-mode (modes I and II) loading conditions. The salient feature of this method is that the stress intensity factors and their derivatives for the multiple crack system can be obtained efficiently since it only requires an evaluation of the same set of fractal finite element matrix equations with a different fictitious load. Three numerical examples are presented to calculate the first-order derivative of the stress intensity factors or energy release rates.