Following Spur Gear Crack Propagation in the Tooth Foot by Finite Element Method

The objective of this study was to follow the crack propagation in the tooth foot of a spur gear by using Linear Elastic Fracture Mechanics (LEFM) and the Finite Element Method (FEM). The tooth foot crack propagation is a function of Stress Intensity Factors (SIF) that play a very crucial role in the life span of the gear. A two-dimensional quasi-static analysis is carried out using a program that determines the gear geometry, coupled with the Finite Element Code (ANSYS). The study estimates the stress intensity factors and monitors their variations on the tooth foot according to crack depth, crack propagation angle, and the crack position. An appropriate methodology for predicting the crack propagation path is applied by considering gear tooth behavior in bending fatigue. The results are used to predict/prevent catastrophic rim fracture failure modes from occurring in critical components.     

Influence of different load models on gear crack path shapes and fatigue lives

A computational model for determination of the service life of gears with regard to bending fatigue at gear tooth root is presented. In conventional fatigue models of the gear tooth root, it is usual to approximate actual gear load with a pulsating force acting at the highest point of the single tooth contact. However, in actual gear operation, the magnitude as well as the position of the force changes as the gear rotates. A study to determine the effect of moving gear tooth load on the gear service life is performed. The fatigue process leading to tooth breakage is divided into crack‐initiation and crack‐propagation period. The critical plane damage model has been used to determine the number of stress cycles required for the fatigue crack initiation. The finite‐element method and linear elastic fracture mechanics theories are then used for the further simulation of the fatigue crack growth.     

Simulation of crack propagation in spur gear tooth for different gear parameter and its influence on mesh stiffness

Most of the gear dynamic model relies on the analytical measurement of time varying gear mesh stiffness in the presence of a tooth fault. The variation in gear mesh stiffness reflects the severity of tooth damage. This paper proposes a cumulative reduction index (CRI) which uses a variable crack intersection angle to study the effect of different gear parameters on total time varying mesh stiffness. A linear elastic fracture mechanics based two dimensional FRANC (FRacture ANalysis Code) finite element computer program is used to simulate the crack propagation in a single tooth of spur gear at root level. A total potential energy model and variable crack intersection angle approach is adopted to calculate the percentage change in total mesh stiffness using simulated straight line and predicted crack trajectory information. A low contact ratio spur gear pair has been simulated and the effect of crack path on mesh stiffness has been studied under different gear parameters like pressure angle, fillet radius and backup ratio. The percentage reduction of total mesh stiffness for the simulated straight line and predicted crack path is quantified by CRI. The CRI helps in comparing the percentage variation in mesh stiffness for consecutive crack. From the result obtained, it is observed that the proposed method is able to reflect the effect of different gear parameters with increased deterioration level on total gear mesh stiffness values.     

Study on Random Fracture and Crack Growth of Gear Tooth Waist

The gear tooth fracture usually occurs at the root, but sometimes also occurs at the waist, or even at the top. The random fracture is defined as the rupture at the waist or top of the tooth. The random fracture at the waist is studied in this paper. In order to simulate a manufacturing defect on the tooth surface, a mic-notch (a minute notch) was cut at the waist of the twelve teeth in the two test gears. A gear-running test was carried out under ladder loading till a gear tooth fractured. The fracture appearance illuminates that the failure is fatigue fracture. The initial crack of the notch grew in the five teeth, and no crack propagation was not found in the other seven teeth. The stress intensity factor and the crack propagation length are comparatively studied by three methods such as linear elastic fracture mechanics theory (LEFMT), FRANC3D simulation and the test. In the early stage of crack propagation, the theory values of LEFMT are close to the simulation, but the difference gets larger and larger with the increase in crack length till the gear tooth is broken. However, the difference of crack propagation length between simulation and the test is less, and the error is in the range of 2.4–13.3%. Therefore, the simulation could truly predict the crack growth length.     

Experimental and numerical investigation of fatigue crack growth in the cracked gear tooth

The numerical simulation was conducted to analyse the fatigue crack growth in gear with the finite element codes ansys (ANSYS, Inc. Canonsburg, Pennsylvania, USA.) and franc 3d (Fracture Analysis Consultants, Inc. Ithaca, New York, USA.), and the corresponding fatigue test was also carried out. During the simulation, the location of maximal stress induced by the external force was first determined by the code ansys , and then the obtained results were imported into the franc 3d to analyse the crack growth. The analysed results were input into the codes ansys and franc 3d again to compute the stress and the stress intensity factor in the following steps. After several rounds of analysis, the results of the fatigue crack propagation were obtained. The investigations show that the crack mode I is dominant during the crack growth and the stress intensity factor K_I raises with increase of crack growth length and a series of quarter‐elliptical cross sections of the ruptured gear tooth are obtained. The simulation results are in good accordance with experimental findings.     

Influence of high speed on crack propagation path in thin rim gears

Thin rim gears need accurate design as cracks nucleated at the tooth root fillet may propagate in a safe way (through the tooth) or in catastrophic way (through the rim). Crack propagation direction is mainly influenced by both wheel geometry parameters and crack initiation point. For specific geometry configurations, crack propagation path may be influenced also by other parameters such as the centrifugal load. For this reason, in this work the effect of the centrifugal load (proportional to wheel speed), related to the bending one, has been investigated. The stress field at the tooth root fillet and near the crack has been considered to evaluate the crack initiation point and to explain the propagation direction. This research activity has been carried out by means of numerical models (traditional 2D and 3D finite elements and extended finite elements (XFEM)). Results show that both crack initiation point and crack propagation path are strongly influenced by the centrifugal load entity; this effect is mainly evident in the uncertainty zone of the backup ratio.     

The effects of spur gear tooth spatial crack propagation on gear mesh stiffness

Due to the presence of non-uniform load distribution, local non-homogeneity of material quality and potential misalignment of gear shafts and bearings, etc., spatial cracks may occur in the fillet region of spur gear teeth. These cracks will eventually propagate in three distinct directions either individually or simultaneously. These directions are the crack depth direction, the tooth width direction and the tooth profile direction. In this paper, an analytical investigation of the influence of spatial crack propagation on the time-varying Gear Mesh Stiffness (GMS) and also the Load Sharing Ratio (LSR) is presented. In order to quantitatively define the spatial crack propagation scenario, the involute spur gear tooth geometry cut with a typical double rounded rack is first determined using two parametric equations. The effects of some gear design parameters and initial crack locations on GMS and LSR are determined and compared with the results from previous papers that used Finite Element Analysis (FEA) in order to verify the proposed analytical model. Finally, a quasi-parabolic crack propagation scenario is assumed, in which 7 propagation cases and 3 typical crack growth paths on the tooth surface are investigated to determine their effect on the maximum reduction of GMS when compared to normal conditions. The results are important for the dynamic simulation of gear transmission behavior, and consequently helpful for the monitoring of gearbox working condition and detection of early crack damage that may exist in gear sets.     

A modification of UniGrow 2‐parameter driving force model for short fatigue crack growth

In this paper, a modification of the UniGrow model is proposed to predict total fatigue life with the presence of a short fatigue crack by incorporating short crack propagation into the UniGrow crack growth model. The UniGrow model is modified by 2 different methods, namely the “short crack stress intensity correction method” and the “short crack data‐fitting method” to estimate the total fatigue life including both short and long fatigue crack propagations. Predicted fatigue lives obtained from these 2 methods were compared with experimental data sets of 2024‐T3, 7075‐T56 aluminium alloys, and Ti‐6Al‐4V titanium alloy. Two proposed methods have shown good fatigue life predictions at relatively high maximum stresses; however, they provide conservative fatigue life predictions at lower stresses corresponding high cycle fatigue lives where short crack behaviour dominates total fatigue life at lower stress levels.     

A NEW MODEL FOR THE NUMERICAL DETERMINATION OF PITTING RESISTANCE OF GEAR TEETH FLANKS

Abstract— A numerical model for determining the pitting resistance of gear teeth flanks is presented in this paper. The model considers the material fatigue process leading to pitting, i.e. the conditions required for crack initiation and then simulation of fatigue crack propagation. The theory of dislocation motion on persistent slip bands is used to describe the process of crack initiation, where the microstructure of a material plays a crucial role. The simulation of crack growth takes into account both short crack growth, where the modified Bilby, Cottrell and Swinden model is used for simulation of dislocation motion, and long crack growth, where the theory of linear elastic fracture mechanics is applied. The stress field in the contact area of meshing spur gear teeth and the functional relationship between the stress intensity factor and crack length are determined by the finite element method. For numerical simulations of crack initiation and crack propagation in the contact area of spur gear teeth, an equivalent model of two cylinders is used. On the basis of numerical results, and with consideration of some particular material parameters, the service life of gear teeth flanks is estimated. The developed model is applied to a real spur gear pair, which is also experimentally tested. The comparison of numerical and experimental results shows good agreement and it can be concluded that the developed model is appropriate for determining the pitting resistance of gear teeth flanks.     

Effects of Shot‐Peening on Fretting Fatigue Crack Growth Behavior in Ti‐6Al‐4V

Abstract: The effects of shot‐peening on fretting fatigue crack growth behaviour in titanium alloy, Ti‐6A1‐4V were investigated. Three shot‐peening intensities: 4A, 7A and 10A were considered. The analysis involved the fracture mechanics and finite element sub‐modelling technique to estimate crack propagation lives. These computations were supplemented with the experimentally measured total fretting fatigue lives of laboratory specimens to assess the crack initiation lives. Shot‐peening has significant effect on the initiation/propagation phases of fretting fatigue cracks; however this effect depends upon the shot‐peening intensity. The ratio of crack initiation and total life increased while the ratio of the crack propagation and total life decreased with an increase of shot‐peening intensity. Effects of residual compressive stress from shot‐peening on the crack growth behaviour were also investigated. The fretting fatigue crack propagation component of the total life with relaxation increased in comparison to its counterpart without relaxation in each shot‐peened intensity case while the initiation component decreased. Improvement in the fretting fatigue life from the shot‐peening and also with an increase in the shot‐peening intensity appears to be not always due to increase in the crack initiation resistance from shot‐peened induced residual compressive stress.     

Analysis of crack propagation during tooth interior fatigue fracture

Tooth interior fatigue fracture is a failure mode that is initiated as a fatigue crack in the interior of the tooth of a gear. TIFF cracks have been observed in case hardened idler gears. A fracture mechanical analysis of a TIFF crack is performed utilising FEA. A 3D TIFF crack is modelled at a position in the tooth that corresponds with an observed crack surface. The different material properties in the case and the core, determined by mechanical testing, are considered, as well as the residual state of stress due to case hardening. Various crack lengths are analysed to estimate crack propagation both into the core and into the case. The following results have been found:

• A TIFF crack initiated slightly under the case layer will propagate into the case layer where it stops.

• The main crack propagation will take place in the core.

• The crack propagation is only a small portion of the total life (order of 10⁵ cycles).

• After reaching the case layer the TIFF crack eventually deflects toward the tooth root and the upper part of the tooth falls off. The crack deflection is due to redistribution of contact loading. Several gear teeth pairs are simultaneously in contact and the cracked tooth is loaded less than the uncracked during this stage of life.

Prediction of short fatigue crack growth of Ti‐6Al‐4V

It is observed that the short fatigue cracks grow faster than long fatigue cracks at the same nominal driving force and even grow at stress intensity factor range below the threshold value for long cracks in titanium alloy materials. The anomalous behaviours of short cracks have a great influence on the accurate fatigue life prediction of submersible pressure hulls. Based on the unified fatigue life prediction method developed in the authors' group, a modified model for short crack propagation is proposed in this paper. The elastic–plastic behaviour of short cracks in the vicinity of crack tips is considered in the modified model. The model shows that the rate of crack propagation for very short cracks is determined by the range of cyclic stress rather than the range of the stress intensity factor controlling the long crack propagation and the threshold stress intensity factor range of short fatigue cracks is a function of crack length. The proposed model is used to calculate short crack propagation rate of different titanium alloys. The short crack propagation rates of Ti‐6Al‐4V and its corresponding fatigue lives are predicted under different stress ratios and different stress levels. The model is validated by comparing model prediction results with the experimental data.     

Lebensdauerberechnung festgewalzter Kerbproben unter Axialbelastung auf der Basis eines Rißfortschrittskonzepts

Axial fatigue life calculation of fillet rolled specimens by means of a crack growth model Fillet rolling is a method which significantly improves the fatigue strength of members. Residual compressive stresses induced in the surface layer during the fillet rolling process are able to retard or prevent crack propagation. An elastic‐plastic on the J‐integral based crack growth model considering the crack opening and closure phenomenon in nonhomogeneous plastic stress fields is described. Experimentally determined crack growth curves and fracture fatigue life curves at constant amplitude loading were used to verify the developed model.     

Predicting corner crack fatigue propagation from cold worked holes

We predict the fatigue propagation of corner cracks from cold worked holes using three dimensional finite element models. The models account for the through thickness variation in residual stress left after cold working. The predictions are compared to experimental results in aluminum 2024-T351 and 7075-T651. The models show the evolution of P-shaped crack fronts similar to those observed in experiments. Predictions based on the initial residual stress field left after cold working were nonconservative, predicting either slower than experimental crack growth or crack growth that arrests. Predictions based on an estimate of the stable relaxed residual stress field near the hole were conservative, and predicted 5-10 times greater life than the current Department of Defense reduced initial flaw size approach.     

Numerical simulation of fatigue crack propagation in compressive residual stress fields of notched round bars

In this paper, the influence of the residual compressive stresses induced by roller burnishing on fatigue crack propagation in the fillet of notched round bar is investigated. A 3D finite element simulation model of rolling has allowed to introduce a residual stress profile as an initial condition. After the rolling process, fatigue loading has been applied to three‐point bending specimens in which an initial crack has been introduced. A numerical predictive method of crack propagation in roller burnished specimens has also been implemented. It is based on a step‐by‐step process of stress intensity factor calculations by elastic finite element analyses. These stress intensity factor results are combined with the Paris law to estimate the fatigue crack growth rate. In the case of roller burnished specimens, a numerical modification concerning experimental crack closure has to be considered. This method is applied to three specimens: without roller burnishing, and with two levels of roller burnishing (type A and type B). In all these cases, the computational finite element predictions of fatigue crack growth rate agree well with the experimental measurements. The developed model can be easily extended to crankshafts in real operating conditions.     

SMALL CRACK GROWTH AND FATIGUE LIFE PREDICTIONS FOR HIGH-STRENGTH ALUMINIUM ALLOYS: PART I—EXPERIMENTAL AND FRACTURE MECHANICS ANALYSIS

The small crack effect was investigated in two high-strength aluminium alloys: 7075-T6 bare and LC9cs clad alloy. Both experimental and analytical investigations were conducted to study crack initiation and growth of small cracks. In the experimental program, fatigue tests, small crack and large crack tests were conducted under constant amplitude and Mini-TWIST spectrum loading conditions. A pronounced small crack effect was observed in both materials, especially for the negative stress ratios. For all loading conditions, most of the fatigue life of the SENT specimens was shown to be crack propagation from initial material defects or from the cladding layer. In the analysis program, three-dimensional finite element and weight function methods were used to determine stress intensity factors and to develop SIF equations for surface and corner cracks at the notch in the SENT specimens. A plasticity-induced crack-closure model was used to correlate small and large crack data, and to make fatigue life predictions. Predicted crack-growth rates and fatigue lives agreed well with experiments. A total fatigue life prediction method for the aluminium alloys was developed and demonstrated using the crack-closure model.     

A study of fatigue crack growth from artificial corrosion pits at welded joints under complex stress fields

The paper studies the effects of artificial corrosion pits and complex stress fields on the fatigue crack growth of full penetration load‐carrying fillet cruciform welded joints with 45° inclined angle. Parameters of fatigue crack growth rate of welded joints are obtained from S‐N curves under different levels of corrosion. A numerical method is used to simulate fatigue crack growth using different mixed mode fatigue crack growth criteria. Using polynomial regression, the crack shape correction factor of welded joints is fitted as a function of crack depth ratios. Because the maximum circumferential stress criterion is simple and easy to use in practice, fatigue crack growth rate is modified using this criterion. The relationship of effective stress intensity factor, crack growth angle and crack depth is studied under different corrosion levels. The simulated crack growth path obtained from the numerical method is compared with the actual crack growth path observed by fatigue tests. The results show that fatigue cracks do not initiate at the edge or bottom of pits but at the weld toes where the maximum stress occurs. The artificial corrosion pits have little effect on the effective stress intensity factor ranges and crack growth angle. The fatigue crack growth rates of welded joints with pits 1 and 2 are 1.15 times and 1.40 times larger than that of the welded joint with no pit, respectively. The simulated crack growth path agrees well with the actual one. The fatigue life prediction accuracy using the modified formulation is improved by about 18%. The crack shape correction factor obtained using the maximum circumferential stress criterion is recommended being used to calculate fatigue life.     

Fatigue crack growth in railway axles: Assessment concept and validation tests

Railway axles are safety relevant components which are usually designed for up to 30 years of service. Besides the experience based definition of inspection intervals, the application of fracture mechanics tools is currently being introduced as an appropriate method. Basic fatigue crack growth data both in the range of stable crack propagation and near the threshold have been experimentally determined for the heat-treated railway axle steels 25CrMo4 (EA4T) and 34CrNiMo6+QT under constant and variable amplitude loading at relevant stress ratios (predominantly fully reversed load cycles, R = −1). For the computational modelling of fatigue crack propagation, a generally applicable stress intensity factor solution has been derived by finite-element analyses. The results are employed for predicting fatigue crack growth in a reference railway axle within the shaft and in the fillet zone near a press fit. Additionally, the influence of press fitting on the crack propagation behaviour in a fillet is discussed. Finally, fatigue crack growth curves experimentally determined on 1:3 and 1:1 scaled axles at constant and variable amplitude loading are compared to the test results for standard M(T) specimens, as well as to respective analytical predictions.     

Stress intensity factor solutions for estimation of fatigue lives of laser welds in lap-shear specimens

Fatigue behavior of laser welds in lap-shear specimens of high strength low alloy (HSLA) steel is investigated based on experimental observations and two fatigue life estimation models. Fatigue experiments of laser welded lap-shear specimens are first reviewed. Analytical stress intensity factor solutions for laser welded lap-shear specimens based on the beam bending theory are derived and compared with the analytical solutions for two semi-infinite solids with connection. Finite element analyses of laser welded lap-shear specimens with different weld widths were also conducted to obtain the stress intensity factor solutions. Approximate closed-form stress intensity factor solutions based on the results of the finite element analyses in combination with the analytical solutions based on the beam bending theory and Westergaard stress function for a full range of the normalized weld widths are developed for future engineering applications. Next, finite element analyses for laser welded lap-shear specimens with three weld widths were conducted to obtain the local stress intensity factor solutions for kinked cracks as functions of the kink length. The computational results indicate that the kinked cracks are under dominant mode I loading conditions and the normalized local stress intensity factor solutions can be used in combination with the global stress intensity factor solutions to estimate fatigue lives of laser welds with the weld width as small as the sheet thickness. The global stress intensity factor solutions and the local stress intensity factor solutions for vanishing and finite kinked cracks are then adopted in a fatigue crack growth model to estimate the fatigue lives of the laser welds. Also, a structural stress model based on the beam bending theory is adopted to estimate the fatigue lives of the welds. The fatigue life estimations based on the kinked fatigue crack growth model agree well with the experimental results whereas the fatigue life estimations based on the structural stress model agree with the experimental results under larger load ranges but are higher than the experimental results under smaller load ranges.