Influence of inclusion content on rolling contact fatigue in a gear steel: Experimental analysis and predictive modelling

Rolling contact fatigue tests were carried out on ring specimens made of quenched and tempered SAE 5135 gear steel with three different steel-production processes, through a bi-disc machine under pure rolling condition and water lubrication. Early formation of micro-pits then coalescing into macro-pits was observed on the rolling surface, while the final failure was caused by subsurface originated spalling phenomena. Microscope analysis of specimens section highlighted the complex surface and subsurface crack layout, and permitted to recognise sulphides as preferential sites for cracks initiation. The inclusion content was analysed throughout the extreme value statistics and the maximum expected inclusion in the Hertzian contact zone was introduced in a failure assessment diagram recently proposed, which resulted effective in predicting the specimen failures.     

Static finite element stress intensity factors for annular cracks

Results of finite element static stress intensity factor calculations for an annular crack around a spherical inclusion (void) are presented and compared with those from approximate analytical methods.     

Analysis of stress intensity factors for three-dimensional cruciform cracks

The stress intensity factors for three-dimensional cruciform surface cracks in a semi-infinite body are numerically calculated by the body force method. Mindlin's point force solution is used for the derivation of basic equations to express the influence coefficient of triangular elements, into which the crack is divided. The interactions between crossed crack planes as well as contact between crack surfaces are considered in the iterative manner. Stress intensity factors for a cruciform median crack and a cruciform semicircular crack under a point force on the surface of a semi-infinite solid are analyzed. The possibility of growth of a median crack toward the free surface of the semi-infinite solid is discussed. A cruciform semicircular surface crack under remote uniaxial tension, or under combined tension and compression is also analyzed. The effect of contact of crack surfaces on stress intensity factors is discussed.     

Standing contact fatigue testing of a ductile material: surface and subsurface cracks

During standing contact fatigue testing of case hardened steel plates, four different fatigue crack types are found: ring/cone; lateral; radial; and median cracks. Fatigue results are presented as load versus cycle number, with endurance limits and initiation laws for the ring/cone and lateral cracks. The behaviour of the radial surface strain outside the contact is altered by the presence of cracks. In particular this makes in situ crack detection possible for the lateral crack. The ductility of the tested material is found to be important for fatigue crack initiation. Numerical elastoplastic computations are used to derive the stress cycles responsible for each crack type. Stress cycles at different locations and in different directions are compared in order to explain why a particular crack type initiates. It is noted that cracks are produced normal to principal stresses of sufficient range, which are tensile sometime during the load cycle. Implications for spalling are discussed.     

Analysis of subsurface crack propagation under rolling contact loading in railroad wheels using FEM

A general subsurface crack propagation analysis methodology for the wheel/rail rolling contact fatigue problem is developed in this paper. A three-dimensional elasto-plastic finite element model is used to calculate stress intensity factors in wheels, in which a sub-modeling technique is used to achieve both computational efficiency and accuracy. Then the fatigue damage in the wheel is calculated using a previously developed mixed-mode fatigue crack propagation model. The advantages of the proposed methodology are that it can accurately represent the contact stress of complex mechanical components and can consider the effect of loading non-proportionality. The effects of wheel diameter, vertical loading amplitude, initial crack size, location and orientation on stress intensity factor range are investigated using the proposed model. The prediction results of the proposed methodology are compared with in-field observations.     

Calculation of stress intensity factors for curved cracks in anisotropic FGMs

Abstract

A combined analytical and numerical method is proposed for computation of mixed-mode stress intensity factors (SIFs) for arbitrary curved cracks in anisotropic functionally graded materials (FGMs). By developing a pair of closed-form expressions that relate the SIFs and the J_k-integrals, it is anticipated that the SIFs can be properly extracted should the J_k-integrals be accurately evaluated. To this end, a novel method for calculating the J_k-integrals is presented and has proved reasonably accurate in numerical computations. Since neither a priori information nor extra auxiliary solutions corresponding to the singular behavior is required, this proposed scheme appears to be applicable to problems containing arbitrary shapes of curvature in generally anisotropic FGMs.     

Mode I stress intensity factors for curved cracks in gears by a weight functions method

The most recent trend in power transmission design considers the damage-tolerant approach as one of the methods to obtain safe, reliable and light systems. This means that components containing cracks must be considered and analysed to understand the conditions that cause critical cracks and defects and their dimensions.
In this paper a cracked tooth of an automotive gearwheel is considered. A numerical procedure (based on the slice synthesis weight function method) to calculate the stress intensity factors of curved cracks due to bending loads is illustrated. The results are compared with those obtained by expensive finite element calculations. The agreement is satisfactory and the proposed technique proves to be very attractive from the point of view of time saving.
One example of an application to fatigue design practice is provided, namely the analysis of fatigue crack propagation in surface-treated gears. The results show the role played by residual stresses induced by carburizing and shot peening.     

Evaluation of stress intensity factors for multiple surface cracks in bi-material tubes

This paper presents stress intensity factors (SIFs) of multiple semi-elliptical surface cracks in bi-material tubes subjected to internal pressure by boundary element method. In this case the water-tube boiler with oxide scale formed on the inner surface due to prolonged exposure at elevated temperature is considered as the bi-material tubes. Variations of modulus of elasticity and thickness for the oxide scale are used to evaluate their effects on the stress intensity factors. The increasing of thickness of the oxide scale causes decreasing values of the normalized stress intensity factor as the modulus of elasticity for the oxide scale is greater than that of the tube metal. Conversely, if the modulus of elasticity for the oxide scale is smaller, the increasing of thickness of the scale would also give increasing values of the normalized stress intensity factor.     

Stress intensity factors for three‐dimensional weld toe cracks using weld toe magnification factors

In welded components, particularly those with complex geometrical shapes, evaluating stress intensity factors is a difficult task. To effectively calculate the stress intensity factors, a weld toe magnification factor is introduced that can be derived from data obtained in a parametric study performed by finite element method (FEM). Although solutions for the weld toe magnification factor have been presented, these are applicable only to non‐load‐carrying cruciform or T‐butt joints, due possibly to the requirement of very complicated calculations. In the majority of cases for various welded joints, the currently used weld toe magnification factors do not adequately describe the behaviour of weld toe cracks. In this study, the weld toe magnification factor solutions for the three types of welded joints such as cruciform, cover plate and longitudinal stiffener joints were provided through a parametric study using three‐dimensional finite elements. The solutions were formed with exponents and fractions that have polynomial functions in terms of a/c and a/t – that is, crack depths normalised by corresponding half crack lengths and specimen thickness. The proposed weld toe magnification factors were applied to evaluate the fatigue crack propagation life considering the propagation mechanisms of multiple‐surface cracks for all welded joints. It showed good agreement within a deviation factor of two between the experimental and calculated results for the fatigue crack propagation life.     

Determination of dynamic stress intensity factors of multiple cracks

The dynamic stress field around a crack or cracks embedded in an infinite isotropic elastic medium subjected to a SH wave are determined. Based on the qualitatively similar features of crack and dislocation, the stress wave emitted from a vibrating screw dislocation can be regarded as a Green's function. By superimposing an array of dislocation and adjusting the distribution density to fulfill the boundary condition, a singular integral equation with kernels containing Bessel functions is derived, which can be solved by the Galerkin method. Dynamic stress intensity factors, which can be as much as 28 percent higher than the static value, are found to be the same as those results obtained by other investigators. The stress intensity factors of a set of infinite cracks of equal length are also calculated as an application of this method.

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Résumé On détermine le champ des contraintes dynamiques entourant une fissure ou des fissures noyées dans un milieu élastique infini et isotrope soumis à une onde SH. En se basant sur les caractéristiques qualitativement similaires que présentent une fissure et une dislocation, on peut considérer que l'onde de tension émise par une dislocation-vis en vibration constitue une fonction de Green. En superposant une famille de dislocation et en ajustnt la densité de la distribution de manière à satisfaire une condition aux limites, on dérive une équation intégrale singulière comportnt comme kernels des fonctions de Bessel, qui peut être résolue par la méthode de Galerkin. Les facteurs d'intensité dynamique, qui peuvent atteindre une valeur 28 pour cent plus élevée qu'une valeur statique, sont trouvés être les mêmes que ceux identifiés par d'autres chercheurs. Le facteur d'intensité de contraintes correspondant une série de fissures infinies de longueur égale est également calculé en application de la méthode.

     

On the calculation of derivatives of stress intensity factors for multiple cracks

In this paper, the work of Lin and Abel [Lin SC, Abel JF. Variational approach for a new direct-integration form of the virtual crack extension method. Int J Fract 1988;38:217-35] is further extended to the general case of multiple crack systems under mixed-mode loading. Analytical expressions are presented for stress intensity factors and their derivatives for a multiply cracked body using the mode decomposition technique. The salient feature of this method is that the stress intensity factors and their derivatives for the multiple crack system are computed in a single analysis. It is shown through two-dimensional numerical examples that the proposed method gives very accurate results for the stress intensity factors and their derivatives. It is also shown that the variation of mode I and II displacements at one crack-tip influence the mode I and II stress intensity factors at any other crack. The computed errors were about 0.4-3% for stress intensity factors, and 2-4% for their first order derivatives for the mesh density used in the examples.     

On estimating stress intensity factors and modulus of cohesion for fractal cracks

Existing solutions for the singular stress field in the vicinity of a fractal crack tip have been adapted for a somewhat modified problem. Since the integration along the fractal curve is prohibitive and does not lend itself to the presently available mathematical treatments, a simplified one has replaced the original problem. The latter involves a smooth crack embedded in a singular stress field, for which the order of singularity is adjusted to match exactly the one obtained from the analyses pertaining to the fractal crack. Of course, this is only an approximation, and we may only hope that it leads toward correct results, at least in a cursory sense. The advantage of such an approach becomes obvious when one inspects the final closed-form solutions for (a) the stress intensity factor in mode I fractal fracture, and (b) cohesion modulus, which results from the cohesive zone model and serves as a measure of the material resistance to crack propagation. As expected for the fractal geometry employed here, our results are strongly dependent on the fractal dimension D (or roughness exponent H).     

An efficient stress intensity factor solution scheme for corner cracks at holes under bivariant stressing

This paper summarizes the development of an efficient stress intensity factor (SIF) solution scheme applicable to a corner crack (CC) in a rectangular section subjected to arbitrary stressing on the crack plane. A general bivariant weight function (WF) formulation developed previously for a CC in a plate was extended to address a CC at a hole. Two supplemental algorithms were developed to achieve a substantial reduction in the computational time necessary for practical application. The new SIF solution scheme was validated by comparison with more than 180 three‐dimensional (3D) boundary element (BE) solutions.     

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.     

Determination of effective stress intensity factors under mixed-mode from digital image correlation fields in presence of contact stresses and plasticity

Digital image correlation (DIC) is more and more popular to monitor fatigue crack growth and to determine the stress intensity factors. However, the posttreatment of the recorded displacement fields becomes tricky when the crack faces are not stress-free and when crack tip plasticity becomes significant. Several posttreatment methods to locate the crack tip and measure the effective stress intensity factors in such cases are compared, using finite element method-computed displacement fields, and then used on real DIC fields. An approach coupling DIC and finite element method is proposed to estimate the contact stresses along the crack.     

Mode II stress intensity factors for central slant cracks with frictional surfaces in uniaxially compressed plates

The effect of crack surface friction on mode II stress intensity factor (SIF) of a central slant crack in a plate uniformly loaded in uniaxial compression is quantified. A previously developed two-dimensional finite element analysis was utilised after its modification to accommodate the friction between the crack surfaces. The plane strain state was assumed. A new numerical technique was devised to avoid the iteration procedures, which had to be employed due to the existence of frictional forces.

The crack inclination angle varied between zero and 75° measured from the horizontal direction. The coefficient of friction of the crack surfaces changed from zero to 1. In case of relatively sliding crack surfaces, mode II SIF existed. As is well known, the resulting mode II SIF decreased with increasing the coefficient of friction of the crack surfaces. Further, mode II SIF increased with increasing crack line inclination angle and then decreased after reaching a maximum value. The angle corresponding to that maximum SIF increased as the coefficient of friction of the crack surfaces increased.     

Computing stress intensity factors for curvilinear cracks

The use of the interaction integral to compute stress intensity factors around a crack tip requires selecting an auxiliary field and a material variation field. We formulate a family of these fields accounting for the curvilinear nature of cracks that, in conjunction with a discrete formulation of the interaction integral, yield optimally convergent stress intensity factors. In particular, we formulate three pairs of auxiliary and material variation fields chosen to yield a simple expression of the interaction integral for different classes of problems. The formulation accounts for crack face tractions and body forces. Distinct features of the fields are their ease of construction and implementation. The resulting stress intensity factors are observed converging at a rate that doubles that of the stress field. We provide a sketch of the theoretical justification for the observed convergence rates and discuss issues such as quadratures and domain approximations needed to attain such convergent behavior. Through two representative examples, a circular arc crack and a loaded power function crack, we illustrate the convergence rates of the computed stress intensity factors. The numerical results also show the independence of the method from the size of the domain of integration. Copyright © 2015 John Wiley & Sons, Ltd.