Gears are case hardened to produce residual stresses at the surface — improving wear resistance, bending fatigue, and contact fatigue strength. These beneficial compressive stresses are balanced by tensile stresses within the core. This poses an increased risk of fatigue crack growth below the surface. Both tooth interior fatigue fracture (TIFF) and tooth flank fracture (TFF) — also known as tooth flank breakage (TFB) — describe a failure mode where a subsurface fatigue crack initiates close to case core boundary, at approximately mid-height of the tooth.
Previous research [1–8] has established that the direction in which the crack propagates and the appearance of the associated fracture is dependent on the flank loading (i.e. single-stage loading vs. idler usage). Although there does not appear to be total agreement in the literature, TIFF (failure with reverse loading) and TFF (failure with single flank loading) appear to have similar characteristics and crack initiation mechanisms. However, as shown in Figure 1, the final fracture shape is different, due to TIFF having near-symmetric total stresses along the tooth center line (with two possible initiation points per tooth). The location of the crack initiation distinguishes this failure mode from other fatigue failure modes.
TIFF and TFF failures can appear at loads below the allowable loading conditions for pitting and bending fatigue failure modes based on internationally accepted calculation procedures (such as ISO 6336  and AGMA 2101-D04 ). Therefore, understanding of TIFF and TFF failure modes is required at the design stage to avoid durability issues in the field. Previous research has shown the TIFF and TFF risk is dependent on the gear macro geometry, material, loading, and hardening properties.
As of the time of writing, there is no standardized method to assess the probability of this type of failure and the relative importance of the influencing factors. It is, however, worth noting TFF is an active topic within the ISO gearing committee, which is working on a draft standard, ISO/DTR 19042, for the calculation of tooth flank fracture performance.
In this manuscript, we give a brief summary of the calculation methods found in the literature for both TIFF and TFF, as recently discussed by Al and Langlois  and Al, et al. . The proposed approaches for TIFF and TFF all have similar fundamental approaches consisting of four stages:
- Calculation of stress history.
- Calculation or specification of residual stresses.
- Calculation of equivalent stresses using some fatigue criterion.
- Comparison with some initiation thresholds based on field experience or experiments.
Each of the calculation methods described below differs in some of the details of the above steps. Further, the applicability of the methods depends on the assumptions made and the implementation details of each stage. Therefore, the implementation of each of these steps could be changed (different models or assumptions may be used), creating a number of permutations for engineers to choose from in order to achieve to a design requirement.
Tooth Interior Fatigue Fracture Calculation Methods
MackAldener [1–3] has shown that an analysis method based on 2D FEA can be used to analyze the risk of TIFF and determine optimum macro-geometry, material, and case-hardening properties. In this analysis, MackAldener used the gear load distribution analysis program LDP (Ohio State University Load Distribution Program) to calculate the total force on one tooth at different phases within the mesh cycle. The calculated force was then applied to a 2D FE model of a single pair of teeth in contact as a torque after normalizing with the face width. A contact analysis was then run on the 2D FE model in order to calculate the stress history. MackAldener's papers show the evolution of the methodology used to estimate residual stresses and material properties. While MackAldener's early papers  described a methodology where transformation strain and material fatigue properties were assumed constant throughout the case, in his later papers these modeling assumptions were replaced with non-homogeneous profiles (see Sections 3.2 and 3.3).
Due to complexity of setting up and running MackAldener's FE-based method within a general FE package, MackAldener  also proposed a simpler semi-analytical method. This method was proposed for rapid calculation, design parameter studies, and optimization but with some compromise in the accuracy of the results. In the analysis of results presented in MackAldener , the crack-initiation risk factor result was seen to be over-predicted, as compared to MackAldener's FE-based method, by a maximum of 20 percent.
MackAldener used a factorial design to evaluate the effect of gear design parameters on TIFF risk and concluded that TIFF failure can be avoided if the slenderness ratio is reduced, tensile residual stresses are reduced, the gear is not used as an idler gear, and optimum case and core properties are used.
Al and Langlois  demonstrated a modification to the analysis of TIFF based on MackAldener's FE-based method, in which the loaded tooth contact analysis (LTCA) results from a specialized 3D elastic contact model, has been used to determine the load boundary conditions for analysis of TIFF. This replaces a computationally expensive, explicitly modeled FE contact analysis with simple load boundary conditions obtained by a separate specialized gear LTCA. This method has been validated against MackAldener's FE results.
Tooth Flank Fracture Calculation Methods
The first model has been developed by FZG. This method has been published in Witzig , Tobie et al. , and Boiadjev et al. . It relies on the calculation of the local stress history based on a shear stress intensity hypothesis of Hertter . The method has significant empirical contributions and is limited in applicability due to the empirical nature of the equation used in calculating local material exposure. In the literature, this method has been presented for single flank loading only. It could, in theory be extended to consider double-flank loading (i.e. idler usage), but this is not trivial. As described by Witzig, this method requires Hertzian contact stresses as inputs, and these stresses can be calculated using a gear-load distribution program or via simplified analytical calculations such as those available in the standards. The method as published is also restricted to case-hardened gears, due to the assumptions related to the residual stress calculation. It should be noted that this method in its current form can underestimate the critical-fatigue stresses if resulting residual stresses within the core are not negligible, since these tensile stresses within the core are not taken into account. This assumption for the residual stresses is only valid when the core section is much larger when compared to the thickness of the case. This introduces limitations on applicability for slender teeth and extensive case hardening depths.
Ghribi and Octrue  proposed an alternative calculation method for TFF load capacity. This method is more generic than that of Witzig  and can be applied to both TFF and TIFF. The method proposes use of a multi-axial fatigue criterion and considers the importance of including tensile stresses in the core. The stress history is calculated using the Hertzian contact stress calculations of ISO/TR 15144-1  micropitting load capacity calculation standard, together with a proposal of Johnson , to calculate stress at a depth inside the tooth. Method A of ISO/TR 15144-1  is based on using the results of a 3D gear loaded tooth contact analysis; however, only Hertzian contact stresses calculated by the standard have been considered in the analysis. Addition of stresses due to bending has been mentioned as planned future work, as these stresses could have an impact on the calculated stress states.
Neither of these methodologies is based on finite element analysis (FEA), although they clearly could be adapted to do so. However, using general FE packages requires considerable time and computational power to set up and run analyses.
It is the author's opinion that the critical effect of material quality and inclusions is the key factor missing in the types of analyses presented. We would expect this could be addressed as a factor applied to, e.g., the material thresholds; however, significant field experience and further experimental studies are required to address this point.
Recently, Al, et al.  applied MackAldener's modified TIFF methodology  to TFF and showed a good correlation against the calculation method proposed by Witzig . In contrast to methodologies presented by Witzig  and Ghribi and Octrue , this calculation method considered the whole stress tensor, including bending stresses. It should also be noted that failure thresholds obtained from Witzig's calculation differ from those obtained using the approach presented by Al, et al. , where a threshold close to 1 was found. Al, et al.  confirmed that ignoring tensile residual stresses within the core would cause underestimation of the critical fatigue stresses and showed the effect of tensile residual stresses increases with torque.
TIFF and TFF Risk Evaluation Method
This section reintroduces the methodology that was previously described and validated in our preliminary work (Al and Langlois , and Al, et al. ).
This methodology has been derived from MackAldener's finite element method, but the need for full FE tooth contact analysis has been removed by using loading conditions calculated with a specialized loaded tooth contact analysis. MackAldener also simplified his FE analysis in a later stage, not for calculation of crack initiation risk factor, but when investigating the crack propagation mechanism . These methods remove the complexity of the contact analysis and therefore speed up the calculation while reducing the computational requirements.
Analysis of Stress History
The specialized loaded tooth contact analysis (LTCA) model combines an FE representation of bending and base rotation stiffness of the gear teeth and blank with a Hertzian contact formalism for the local contact stiffness . This calculation includes the effect of extended tip contact where the effective contact ratio is increased under load due to tooth bending. This effect can be particularly important for slender tooth gears, which are also more at risk of TIFF.
This model is used to determine load boundary conditions at a selected number of time steps through the mesh cycle. At each time step, the load distribution between and across the teeth is calculated, and at each of the contact lines, load positions, load magnitudes, and Hertzian half widths are obtained.
A separate fine 2D mesh of the gear tooth is then built automatically using plane-strain elements. At each time step within the mesh cycle, the position and distribution of the load is determined from the results of the 3D tooth contact analysis and applied to the 2D FE mesh, using the average load position and Hertzian half width. In the results presented in Section 4, the finite element mesh was sized according to the Hertzian half width, and a refinement study was performed to check the convergence of the results.
Hardness Profile and Material Properties
The variation of the material properties within the case and core play an important role in TIFF. However, many assumptions have been made in previous analyses in this area. Since the analyzed gear is case hardened, the material properties are not constant throughout the tooth. The critical shear stress, and the fatigue sensitivity to normal stress, in the critical plane criterion, are also expected to vary with location. As with MackAldener, for our analysis we have assumed these properties vary in the same way as an assumed hardness profile.
Hardness profile definition used by MackAldener 
where, Hsurface and Hcore are the hardness at the surface and core, respectively, g is a function which determines the variation between the case and the core defined by MackAldener, z is the normal depth at the point considered and z is the total case depth.
MackAldener's method relies on measurement of the total case depth, which is often neither measured nor known. Therefore, as an alternative, a hardness measurement at a defined effective case depth is used. In such cases, a different hardness profile may be used, i.e. Lang , which is, in fact, the hardness profile used by Witzig .
Hardness profile definition used by Lang  and adopted by Witzig 
in which CHD is the effective case depth, where hardness drops below 550 HV.
Comparison of hardness profile definitions
Figure 2 shows a comparison of the hardness profile measurement and curve fit proposed by MackAldener with other hardness profile models found in the literature. Unless otherwise stated, for this article, MackAldener's curve fit has been used. It is interesting to note the hardness profile model proposed by Thomas  has been found to give the best comparison against MackAldener's measurement, and that of Tobe et al.  is also close.
The method of Lang could lead to a difference in the predicted risk of TIFF and TFF, since fatigue properties and residual stresses are expected to differ near the case-core boundary. This is demonstrated and further discussed in the results section.
3.2.4 Determination of material properties required by multiaxial fatigue analysis
For the cases discussed in this manuscript, the material properties are assumed to vary continuously between case and core in the same manner as the hardness profile. This assumption is not required if variations of the material properties are known.
Critical shear stress
Fatigue sensitivity to normal stress
3.3 Residual Stress Analysis
Residual stresses influence the stress states within the gear tooth. These stresses are not load dependent and are assumed to be constant over time. Residual stresses due to case hardening and shot peening are superimposed.
Residual stress calculation according to MackAldener 
Using the 2D mesh used to calculate the stress history due to flank loading, residual stresses are predicted by performing a separate FE analysis. The volume expansion in the surface layer due to the case-hardening process is modeled by applying a temperature profile to the FE model. The temperature profile applied is the same as the transformation strain profile when the coefficient of thermal expansion is set to 1. All side nodes are allowed to move only in the radial direction.
The transformation strain profile is isotropic and measured relative to the core. This profile has been presented as a piecewise polynomial with smooth connections by MackAldener .
ε1 is the transformation strain at the surface,
ε2 is the maximum transformation strain.
Residual stress analysis according to Lang  and modified by Witzig 
The calculation method proposed by Lang simply requires the heat-treatment type and depth from the surface to be known in order to calculate tangential residual stresses. As can be seen from the equations, only compressive residual stresses are calculated via this method. Note that HV(z) in the equation refers to Lang's hardness profile as opposed to MackAldener's.
This model has been used by both the TFF calculation methods proposed by Witzig  and Ghribi and Octrue . The implementation described by Ghribi and Octrue can, however, also calculate tensile residual stresses by considering a force balance across the teeth. Note, this improvement on Lang's model has not been considered for this article.
Comparison of residual stress calculation methods
Figure 3 shows the results presented by MackAldener for the variation of residual stresses with depth beneath the surface, both using the analysis method above and from measurements carried out by MackAldener. Figure 3 further compares this residual stress profile with that proposed by Lang  and used by Witzig  in the investigation of TFF. Interestingly, the profiles differ quite notably. This may be due to a significant material dependency not considered, but the exact reason is unknown, and further understanding is required. It should be noted the resulting calculated residual stresses can change from one mesh position to another due to the variation in tooth thickness.
Final Stress State and Fatigue Crack Initiation Criterion
The effective stress state within the gear teeth during its load cycle is calculated — without calculating residual stresses at each step — by superimposing the calculated stress history states and the initially estimated residual stresses.
The Findley multi-axial fatigue criterion  is then used to analyze the stress history and assess the possibility of failure. Within our analysis, the Findley critical plane stress has been calculated for every 5 degrees of inclination at each node. The value of 5 degrees was chosen, instead of every 1 degree used by MackAldener , as results did not show a significant dependency on this value. This is confirmed by the cases presented in the results section of this manuscript where differences between using an inclination increment of 2.5 degrees over 5 degrees is less than 0.05 percent.
The Findley stress is calculated as: σF = τa + acp × σn,max
where τa is the shear stress amplitude, and σn,max is the maximum normal stress. Variation of the material properties within the tooth is related to the hardness profile as described above.
The ratio between the maximum Findley critical plane stress and critical shear stress is a measure of the risk of crack initiation. This metric is called the crack initiation risk factor (CIRF).
Summary of author's method
Table 1 provides a brief summary of the methodology used throughout the rest of this manuscript.
Methodology described in this section has been derived from MackAldener's finite element method and has been previously validated in our preliminary work (Al and Langlois , and Al, et al. ).
Discussion and future work
This section covers three parts: The first investigates the effect of residual stress calculation methods on the crack initiation risk factor to establish if tensile residual stresses can be neglected; the second part aims to quantify the effect of possible design options for avoiding TIFF on the pitting and bending fatigue safety factors, and the last part investigates the torque ranges for each type of failure.
Effect of neglecting tensile residual stresses
MackAldener conducted a factorial design with five factors. In total, 32 designs have been considered by varying critical plane stress within the core (A), fatigue sensitivity to normal stress within the core (B), gear tooth geometry (C), total case depth (D), and torque on the pinion (E). For each of the factors, two levels, low and high, have been considered. Their values are presented in Table 2. Details of the gear tooth geometries are given in Table 3. For each of the designs, the CIRF throughout the tooth was calculated. Al and Langlois  previously used this study to validate the proposed methodology. The same factorial design experiment has been carried out using Lang  to specify residual stresses, where residual tensile stresses within the core are not considered.
Figure 4 shows a comparison of the calculated maximum CIRF for all 32 designs. From Figure 4, it is clear there is a good overall correlation between CIRF calculated by the author's method and that calculated by MackAldener .
Figure 5 displays the average CIRF results for each factor at its low and high level together with the average for some interactions. It can be seen that good agreement exists for factors A, B, D, and E, and reasonable agreement for factor C.
Details regarding these comparisons are discussed in Al and Langlois . The more interesting observation here comes from examining the cases where the author's method is used with Lang  for residual stresses. As can clearly be seen from Figure 4, this approach underestimates the maximum CIRF in all designs investigated. Furthermore, using Lang  for residual stresses changes the relationships and some interactions expected from the factors, seen in Figure 5. This change can be attributed to differences in the hardness profiles and/or neglected tensile residual stresses within the core.
Effect of Factors on Pitting and Bending Safety Factors
Factor (C) gear design and Factor (E) torque on the pinion are two parameters investigated in the factorial design, which would have a direct effect on pitting and bending fatigue calculations, according to ISO 6336 . It has been assumed all gears have a flank tolerance class of 5 according to ISO 1328-1  and material quality grade of ME to get a representative theoretical comparison. It should be noted that other parameters could also potentially have an effect on the pitting and bending safety factors; however, internationally accepted calculation standards have assumptions based on the ISO material type selected.
Figure 6 shows how crack initiation risk factor, bending safety factor, and pitting safety factor vary with the change in the common factors that affect all three calculations. It should be noted that resistance to all three failure types can be improved by reducing the torque. For the cases investigated, a non-slender gear shows an improved safety against bending, however reduced safety for TIFF and pitting. As can be seen from Figure 6, both 1/CIRF and gear-bending fatigue are more sensitive to both geometry and loading compared to pitting.
Design domain for gears investigated by Witzig
Details of gear tooth geometries and cutters specified by Witzig  are provided in Table 4. Gear set 67/69 with pressure angle 15° (details not provided here) could not be created from supplied tooth thickness and center distance information within Witzig's  thesis. For each design, the CIRF throughout the tooth was calculated, and trends have been compared to those obtained by Witzig . Both of the gear-tooth geometries used within this article do not include any profile modification other than a generous tip relief.
Based on the data provided by Witzig , transformation strains, surface and core critical stresses, and total case depth (see Table 2) for gears defined in Table 4 are estimated. Details of this analysis are provided in Al, et al. .
Figure 7 summarizes the results from Witzig  for spur gear set 67/69 (Figure 7a) and 40/41 (Figure 7b). It should be noted that the Y-axis on the right, for the maximum material exposure, is shifted to give comparable results (i.e. the critical value for the Findley criterion is expected to be 1 while for Witzig this critical value is 0.8).
The maximum crack initiation risk factors calculated using the author's method and author's method with Lang for residual stresses are extracted from Figure 7 and plotted together with pitting and bending fatigue safety factors in Figure 8. As can be seen in Figure 8, ISO Material Quality ML does not provide adequate resistance to either pitting or bending. As the material quality increases, the torque range over which TIFF failure could potentially occur, in comparison to pitting-and-bending failure, is seen to broaden. However, it should be noted that TIFF and TFF calculations, at present, do not take material quality into account. Using MackAldener's approach, this parameter could be included within the critical fatigue strength.
Figure 9 shows the effect of the flank tolerance class on pitting and bending fatigue safety factor. As can be seen from Figure 9, the increasing flank tolerance class reduces the safety factor at each load level and potentially reduces the torque interval over which TFF type of failure could occur. Also, it should be noted that TIFF and TFF calculations are not affected by any change to flank tolerance class (since friction between contacting surfaces has been assumed negligible, see Langlois et al.  for further details).
This study aimed to improve the existing understanding of Tooth Interior Fatigue Fracture load capacity and compare calculated load capacity to the allowable loading conditions for bending and pitting fatigue failure based on ISO 6336-1  standard calculation procedures. Possible methods that could be used to mitigate TIFF risk have been presented, and the effect of these methods on the performance with respect to the other failure modes were quantified.
The key methodologies and conclusions from this manuscript are:
A parametric study initially conducted by MackAldener  to investigate which parameters influence the risk of TIFF and used for validation in Al and Langlois , has been revisited to investigate neglecting tensile stresses within the core using Lang  to specify residual stresses. As a result of this assumption, it has been found that the expected relationships between the factors are modified.
The common variables (i.e. gear geometry and torque) used in the calculation of the crack initiation risk factor, pitting resistance, and bending resistance have been investigated to extract the overall effects on all three failure types.
It has been shown that the torque range across which TFF failure can be seen could be relatively small compared to the operating range.
Further investigations on the critical effect of material quality and inclusions are required to improve numerical calculation methods and standards. Further investigations into the design space to investigate a large set of gears comparing TIFF, pitting fatigue, and bending fatigue also is planned.
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