Prediction of residual stress depth profiles in case-carburized gears

Abstract

Residual stresses have a significant influence on the load carrying capacity of gears. High near-surface compressive residual stresses can increase pitting strength and tooth root bending strength up to 50%. Tooth flank fracture (TFF) is a gear fatigue failure mode, which is in contrast to pitting or tooth root breakage not initiated at the surface or close to the surface but in larger material depths beneath the active gear flank. Therefore, the residual stresses in larger material depths are decisive for TFF. In these larger material depths, the residual stress conditions are almost unknown up to now. It is assumed that tensile residual stresses are present in these areas, which can negatively influence the TFF load carrying capacity. So far, no validated methods are known to estimate or predict these tensile residual stresses. As a result, the tensile residual stresses could not be considered in the calculation methods for the risk of TFF at all until recently.

This paper presents a method to predict the residual stresses in case-carburized gears based on comprehensive numerical simulations of case-carburizing processes of different-sized gears. The calculated residual stress profiles also consider the tensile residual stresses in larger material depths.

1 Introduction

Residual stresses (RS) are a major influencing factor on the load carrying capacity of case-carburized gears. For the gear failure modes tooth root breakage and pitting, a significant improvement of the load carrying capacity is possible with higher compressive residual stresses at or close to the surface. The gear fatigue failure modes tooth flank fracture (TFF) and tooth interior fatigue failure (TIFF) are usually initiated in larger material depths below the surface of the active gear flank. The residual stresses probably have a similar effect on the load carrying capacity regarding TFF and TIFF, like for pitting and bending strength. However in these cases, the residual stresses at larger material depths are decisive. In these regions, the higher compressive residual stresses are not present anymore and the residual stresses are possibly tensile. So far, the residual stresses in larger material depths are hardly known and only a few measurements exist.

In the research project FVA 835 I [1] the residual stresses in larger material depths of case-carburized gears were investigated. The objective in the research project was to research the residual stress condition at larger material depths and to enhance the calculation of the TFF risk with consideration of tensile residual stresses. In Müller [2], the result, an enhanced calculation method based on ISO TS 6336‑4 [3] was presented. With this method tensile residual stresses can now also be taken into account in the calculation of TFF risk.

In the research project FVA 835 I, also a simulation model was built and validated with residual stress measurements of different sized gears. Based on the simulated residual stress depth profiles, a set of equations for the prediction of residual stress depth profiles was derived. However, the results from the calculation approach showed deviations from the simulations for some larger gears. The reason for this was the small number of simulated depth profiles for larger gears.

In this work, the accuracy of the calculation approach developed in FVA 835 I [1] is improved, especially for larger gears. The equations of this work can be used to predict the residual stress depth profiles including also tensile residual stresses.

2 Residual stresses in case-carburized gears

2.1 Residual stress states and residual stresses measurements

Residual stresses in case-carburized gears occur during the heat treatment process. In this process, typically compressive residual stresses are formed in the near-surface region and tensile residual stresses in the core region of the tooth. The residual stresses at the surface or up to a certain depth in case-carburized gears are additionally influenced by further processing or finishing steps, such as mechanical cleaning (blasting), shot peening, or gear grinding.

The residual stress depth profile up to a depth of approx. 0.5 mm can be measured reliably by standardized methods, e.g. with X‑ray diffraction. For the region close to the component surface, countless measurements already exist, e.g. for different finishing conditions [4,5,6] or different grinding conditions [7, 8]. Typical near-surface residual stress depth profiles for case-carburized gears are shown in Fig. 1. The near-surface residual stresses state has a significant influence on the load carrying capacity of gears. For example the tooth bending strength can be increased up to 50% with shot peening [4]. In addition, the crack initiation of tooth root breakage is shifted slightly into larger material depth when high compressive residual stresses are present at the surface of the tooth root [9, 10].

Fig. 1

Comparison of the residual stress depth profiles in the unpeened, mechanical cleaned, and shot peened condition (measurement results, exemplary presentation) according to [4, 11] for the near-surface up to 0.32 mm

For sub-surface initiated gear failures, such as tooth flank fracture (TFF) or tooth interior fatigue failure (TIFF), where the crack is usually initiated in considerably larger material depths ($\gg 0.5$ mm), the residual stresses in these larger depths are relevant. In these regions, the compressive residual stresses of the near-surface region are decreased and there are possibly tensile residual stresses.

The measurement of the residual stresses in larger material depths is less validated. Reliable residual stress measurements at larger material depths are currently only possible with more complex or expensive measuring methods, such as for example the contour method or neutron diffraction. Nevertheless, some residual stresses measurements up to larger material depths have been documented in some works. These works show the transition from compressive to tensile residual stresses or the tensile residual stresses inside the tooth. The following is a summary of some important work on the findings regarding residual stresses in larger material depths of case-carburized gears.

The influence of case-hardening depth (CHD) on the depth distribution of residual stresses has already been demonstrated by Tobie [12], showing that the near-surface residual stresses decrease with increasing case-hardening depth. Furthermore, it is assumed that increasing CHD shifts the transition from compressive to tensile residual stresses to larger depths within the component, resulting in higher tensile residual stresses in the smaller core cross-section. The shift of the transition from compressive to tensile residual stresses to larger depths for larger CHDs and the consequent increase in tensile residual stresses due to a smaller volume for the tensile residual stresses for the mechanical equilibrium is state of the art and observed in several other works [13,14,15].

Fig. 2 shows residual stress depth profiles of a case-carburized gear with module 5 mm and a CHD of 1.0 mm measured by Weber [15, 16]. For the X‑ray measurements of the residual stresses two correction methods (M&E [17] and finite element correction) are applied to consider the influence of the layer by layer removal on the residual stress state. In addition to X‑ray diffraction measurements, measurements were also carried out using the contour method (CM). With the contour method, a component is split into two halves. The deformations caused by residual stresses after the separation are measured and then applied in the opposite direction to an FEM model. The elastic deformation of the model allows conclusions to be drawn about the residual stress state of the component. In the near-surface regions of the tooth, the residual stresses could not be determined with sufficient accuracy by contour method. In the center of the tooth, the axial residual stresses are approximately 100…250 MPa depending on the tooth thickness. Other works [18,19,20,21,22] also show tensile residual stresses of approx. 100…200 MPa in larger material depths of case-carburized gears.

Fig. 2

Residual stress depth profile of a case-carburized spur gear with module 5 mm and a CHD of 1.0 mm. For the contour method average and deviation of two measurements for each side is presented [15]

The residual stresses in case-carburized gears are present in different orientations or coordinate directions. For the compressive residual stresses in the near-surface region, the tangential (tooth height direction) and axial (tooth width direction) residual stresses are usually comparable and the residual stresses normal to the tooth flank are zero. Weber [16] assumes this also for the residual stresses in larger material depth. According to Witzig [21], the residual stresses normal to the component surface can be considered negligibly small. The magnitude of the residual stresses in the tangential and axial direction can be also assumed to be equal. However, in some cases, measurements or simulations show lower tensile residual stresses in the axial direction than in the tangential direction [1, 18].

2.2 Prediction of residual stresses in case-carburized gears

To date, only a few equations exist to predict residual stress depth profiles of case-hardened parts. The residual stress depth profile is often derived from the hardness depth profile, as for example, according to Lang [23]. Lang proposed equations to calculate the hardness depth profile and from this, the residual stress depth profile for case-carburized, induction hardened and nitrided parts. This calculation approach shows good agreement with measured values for the compressive residual stresses near the surface and is used in the standard calculation methods of ISO 6336. The calculation of the residual stress depth profile from the hardness depth profile is comparatively simple and can also be performed with only the parameters which are specified respective are the result of the heat treatment: surface hardness, core hardness and CHD. In this case, the hardness depth profile must first be measured or calculated, e.g., according to Lang, from the parameters mentioned above. The tensile residual stresses inside the tooth, in larger material depth are not considered by the approach of Lang [23]. According to Lang, the tensile residual stresses are negligible for sufficiently large core cross sections. However, in the case of small core cross sections (with a large case hardening depth, slender teeth or in the region of the tooth tip), significant tensile residual stresses may be present in the core area based on the presumed mechanical stress equilibrium. In addition, in some cases, even lower residual stresses in larger depth can be relevant, especially for the fatigue limit prediction for sub-surface initiated failures, such as tooth flank fracture (TFF). The hardness and residual stresses are decisive influencing variables in the TFF risk calculation. With the modifications proposed in [2] tensile residual stresses can be considered within the ISO 6336‑4 calculation of the TFF risk.

For case-hardened gears, extensions to Lang’s calculation approach have therefore already been proposed, in which the in reality occurring tensile residual stresses in the tooth core are considered. Residual stress depth profiles with tensile residual stresses inside the tooth can be calculated, for example, according to Weber [16], Konowalczyk [24, 25] or Böhme [22]. The principles of these calculations are shown schematically in Fig. 3. Most calculation approaches are based on the compressive residual stresses calculated with the approach of Lang up to a certain depth (mostly 0.5 $\cdot$ CHD) but show in part significant differences in their iterative calculated tensile residual stresses based on mechanical equilibrium.

Fig. 3

Schematic overview of residual stress depth calculation according to Lang [23] (left) and based on the calculation approach of Lang for the near-surface up to 0.5 $\cdot$ CHD according to Weber [16] (middle) and Konowalczyk [24, 25] (right)

Further methods for the prediction of residual stress depth profiles have been proposed in the past. Hereby, the focus was mostly on the near-surface residual stresses. For example, Abersek and Flasker [26] propose two different models to calculate a residual stress depth profile based on hardness or/and retained austenite depth profile.

3 Simulation model for residual stress depth profiles

In the research project FVA 835 I [1] a simulation model for the residual stresses formation during the case carburizing process was built by the Chair and Institute for Materials Applications in Mechanical Engineering—IWM at RWTH Aachen University. The model is based on extensive simulative investigations as well as experimental research carried out for the most part by the Leibniz-IWT in Bremen. To characterize the material behavior of the material 18CrNiMo7‑6 (1.6587), investigations with dilatometer specimens and gears were carried out. These include the determination of flow behavior, transformation kinetics, transformation strains and thermal strains, quenching and tempering behavior. These properties were determined for samples with carbon contents between 0.18 and 0.82 mass percent carbon as a function of the relevant micro-structural compositions and thermal conditions. The simulation model was validated with residual stresses measurements of module 3 mm and module 18 mm gears.

With this simulation model, the tangential residual stress depth profiles at the pitch point C (approx. mid tooth height) were simulated for different gear geometries and several CHDs. In addition, the influence of the carbon content depth profile and the quenching medium was investigated. Four different types of calculated carbon content profiles were investigated. For larger material depths the carbon content is 0.2 mass percent for all profiles. The carbon content depth profiles differ only in the near-surface region and the slope of the decline. The investigated liquid quenching media are oil-, saltwater- and polymer bath. In addition, gas quenching was investigated.

The simulated residual stress depth profiles were validated with measured residual stress depth profiles of module 3 mm and 18 mm gears and different carbon content depth profiles or quenching media. The residual stresses were measured by X‑ray diffraction and different correction methods were applied to consider the influence of the layer removal on the residual stress state.

After the research project the simulation model was further used to simulate the residual stresses distributions especially for larger sized gears. For the additional gear geometries with a module in the range of 7.5…27 mm the residual stresses were simulated with CHDs of 0.15, 0.2 and 0.3 $\cdot$ module. These CHDs are relevant for industrial gears because a maximum tooth bending and pitting strength is usually expected in this range of the CHD [27]. All the performed simulations that are the bases of this work are listed in Appendix A.

Further information regarding the simulation model is published in [1, 28].

4 Derivation of equations for the prediction of residual stress depth profiles

4.1 Calculation of residual stress depth profiles based on given characteristic points

First, characteristic points of the simulated residual stress depth profile were defined. These characteristic points can be described by the material depth and the residual stress value. Fig. 4 shows an example of the characteristic points for a simulated residual stress depth profile. The compressive residual stress maximum is described by $\sigma_{D}$ at a material depth $y_{D}$. This point near the surface is not directly relevant for the TFF evaluation, since the crack origin is located at a larger material depth. However, using these characteristic points showed a higher agreement to the simulated residual stress depth profile in the region between maximal compressive residual stresses and the transition from compressive to tensile residual stresses. The transition from compressive to tensile residual stresses, where the residual stresses are zero, is described by $y_{DZ}$. The maximal slope of the residual stress depth profile is considered with the characteristic point or value $s$. The tensile residual stresses inside the tooth are described by $\sigma_{Z}$. The position of the maximal tensile residual stresses is usually in the center of the tooth. For some cases, the maximal tensile residual stresses are reached closer at $y_{DZ}$. In all cases, however, there is a plateau with comparable values in larger material depth.

Fig. 4

Selected characteristic points for the description of a residual stress depth profile

With the characteristic points, the depth profile of the residual stresses $\sigma_{RS}(y)$ can be calculated over the material depth y. A logistic function was adjusted in such a way that the characteristic points can be inserted directly or via the parameters $k$, $r$ and $\delta$. The adjusted logistic equation is given by Eq. 1. With this equation, the calculation or description of a representative residual stress depth profile is possible.

$$\sigma_{RS}(y,\,\sigma_{D},\,k,\,\delta,\,\sigma_{Z})=\sigma_{D}+\frac{\sigma_{Z}-\sigma_{D}}{1+e^{-k(y+\delta)}}$$

(1)

For the calculation of a residual stress depth profile based on given characteristic points with Eq. 1 the parameters $k$, $r$ and $\delta$ are calculated according to Eq. 2, Eq. 3 and Eq. 4 based on the selected characteristic points: $\sigma_{D}$, $y_{D}$, $y_{DZ}$, $s$, $\sigma_{Z}$.

$$k=\frac{{1}}{y_{D}-y_{DZ}}\cdot\log\left(\frac{{-(\sigma_{Z}-\sigma_{D})/\sigma_{D}-1}}{{(\sigma_{Z}-\sigma_{D})/((r-1)\cdot\sigma_{D})-1}}\right)$$

(2)

$$r=\frac{{\sigma_{Z}-\sigma_{D}}}{{\sigma_{D}-\sigma_{Z}\cdot e^{\frac{{4\cdot s\cdot(y_{D}-y_{DZ})}}{{\sigma_{D}-\sigma_{Z}}}}}}+1$$

(3)

$$\delta=-\frac{{1}}{{k}}\cdot\log\left(-\frac{{\sigma_{Z}}}{{\sigma_{D}}}\right)-y_{DZ}$$

(4)

4.2 Calculation of the characteristic points

The characteristic points can be calculated based on equations derived from simulated residual stress depth profiles of different case-carburized gears (see Appendix A). For the calculation of the characteristic points, the points are considered individually. For each characteristic point, all simulated values are taken into account. Correlation analyses was performed to identify influences of the gear properties on the values of the relevant characteristic point. In this way, the main influences on a specific characteristic point could be identified.

In Fig. 5 an exemplary part of a correlation matrix is shown. The depth of the transition from compressive to tensile residual stresses $y_{DZ}$ is correlated with the gear properties: CHD, module and number of teeth. It is visible that CHD and module have an influence on this characteristic point, whereas the number of teeth has a lower or negligible influence. For most characteristic points CHD and module showed a clear influence on the value. However, for the maximal slope $s$ only the CHD showed a clear direct correlation. Further gear properties such as the tooth thickness, the flank curvature, the pitch deviation or the normal pressure angle are investigated for each characteristic point to identify the main influencing gear properties.

Fig. 5

Exemplary correlation of the characteristic point $y_{DZ}$ with CHD, module and number of teeth

The results show that the residual stress depth profile depends mainly on the CHD and the gear size. The gear size is best described by the gear module. However, in some cases, a better correlation was found when considering additional geometrical data such as the tooth height or tooth thickness. Hereby, the accuracy of the agreement between simulations and calculated residual stress depth profile was compared. In the following, in some equations the correlation of the value of the characteristic point and the gear size could be described with a more simple equation when the tooth thickness was used. The tooth thickness $s_{t}$ at the reference diameter (see Eq. 8) was chosen because here the tooth thickness is calcuated most easily and with the fewest geometrical data. The calculated residual stress depth profile is nevertheless only valid for the pitch point, since the simulations were carried out here.

In Fig. 6 the maximal tensile residual stresses ($\sigma_{Z}$) of all simulations are shown. The maximal tensile residual stresses are within the range of 50…300 N/mm². Based on the investigated correlations the simulation results can be brought into a linear correlation as a function of CHD and module.

Fig. 6

Maximal tensile residual stress $\sigma_{Z}$ of all simulated residual stress depth profiles as a function of CHD and module

Two functions are also drawn in Fig. 6. One for maximal tensile residual stresses (90%) and one for average (50%) tensile residual stresses. Higher residual tensile stresses increase the risk of flank failure. With the 90% function, it is possible to carry out a design with regard to tooth flank fracture with increased safety. The functions in the diagram correspond to Eq. 5 and 6.

The legend on the right of Fig. 6 is also valid for the following figures and shows the simulated quenching media and carbon content profiles for the different gears with different CHDs in the range of 0.1…0.3 $\cdot$ module. In addition, two simulations with a CHD outside of this range were performed. One with a very small CHD of 0.05 $\cdot$ module and one with a very large CHD of 0.67 $\cdot$ module, to verify the behavior of the simulation model, for example, for the residual stresses for almost trough hardened gears.

$$\sigma_{Z,\mathrm{max}}=60.2+\frac{{395.25\cdot\mathrm{CHD}}}{{\sqrt{m_{n}}}}-51.38\cdot\mathrm{CHD}$$

(5)

$$\sigma_{Z}=22.88+395.25\left(\frac{{\mathrm{CHD}}}{{\sqrt{m_{n}}}}-0.13\cdot\mathrm{CHD}\right)$$

(6)

In Fig. 7 the simulation results for the depth of the transition from compressive to tensile residual stresses ($y_{DZ}$) are shown. A clear exponential correlation with CHD times module was observed. However, this correlation is only accurate for smaller sized gears with CHD $\cdot$ module $<$ 10 mm². For larger gears the calculated depth of $y_{DZ}$ differs increasingly from the values of the simulations.

Fig. 7

Depth of the transition from compressive to tensile residual stress $y_{DZ}$ of all simulated residual stress depth profiles as a function of CHD and module

For the smaller sized gears with CHD $\cdot$ module up to 10 mm², shown in Fig. 8 (left) only one simulation shows a larger deviation from Eq. 7.

$$\begin{aligned}[b]&y_{DZ}=0.662\cdot(\mathrm{CHD}\cdot m_{n})^{0.504}\\&\text{for}\;\mathrm{CHD}\,\cdot\,\mathrm{module}\leq 10\,\mathrm{mm}^{2}\end{aligned}$$

(7)

For the larger sized gears with CHD $\cdot$ module larger than 10 mm² a better correlation was found with consideration of the tooth thickness $s_{t}$. Module and tooth thickness are nearly proportional. For a larger module the tooth thickness increases. However, in some cases, the tooth thickness showed a better or simpler correlation with a characteristic point. In some cases, also a combination of both led to the smallest deviations of function and simulated values. The tooth thickness is calculated at the reference diameter according to Eq. 8 and not at the pitch point since the equation is much simpler. Also, only a few geometric data are necessary, such as module $m_{n}$, helix angle $\beta$, profile shift $x_{i}$ and profile angle $\alpha_{n}$ according to ISO 21771 [29].

$$s_{t}=m_{n}/\cos(\beta)\cdot(\pi/2+2\cdot x_{i}\cdot\tan(\alpha_{n}))$$

(8)

In Fig. 8 (right) the values of $y_{DZ}$ of the simulations with CHD $\cdot$ module $>$ 10 mm² are shown. The depth of the transition from compressive to tensile residual stresses of the simulations of gears with the liquid quenching media can be described by Eq. 9.

$$\begin{aligned}[b]&\displaystyle y_{DZ}=2.358+1.205\cdot(\mathrm{CHD}+0.0006\cdot\mathrm{CHD}\cdot s_{t})\quad\\& \displaystyle\text{for}\;\mathrm{CHD}\,\cdot\,\mathrm{module}> 10\,\mathrm{mm}^{2}\\& \textnormal{and liquid quenching media}\end{aligned}$$

(9)

It is visible that the larger sized gears with gas quenching show the lowest values in Fig. 8. For these variants the transition from compressive to tensile residual stresses is closer to the surface. For these gears Eq. 10 gives an adequate description.

$$\begin{aligned}[b]&\displaystyle y_{DZ,\mathrm{gas}}=0.647+1.205\cdot\mathrm{CHD}+0.0007\cdot\mathrm{CHD}\cdot s_{t}\quad\\ &\displaystyle\text{for}\;\mathrm{CHD}\,\cdot\,\mathrm{module}> 10\,\mathrm{mm}^{2}\; \textnormal{and gas quenching}\end{aligned}$$

(10)

Fig. 8

Depth of the transition from compressive to tensile residual stresses $y_{DZ}$ of the simulations for gears with CHD $\cdot$ module $<$ 10 (left) and CHD $\cdot$ module $\geq$ 10 (right)

All maximal slopes ($s$) of the simulated residual stress depth profiles are shown in Fig. 9 (left). Here, a clear exponential correlation of CHD and the maximal slope $s$ is visible. For all CHDs in the range of 0.5 mm up to 8 mm the Eq. 11 gives a good approximation of the maximal slope $s$.

$$\begin{aligned}[b]\displaystyle s=494.5\cdot\mathrm{CHD}^{-0.898}\end{aligned}$$

(11)

Fig. 9

Maximal slope $s$ of the simulated residual stress depth profiles as a function of CHD

In Fig. 9b only the simulated values for larger CHDs are shown. In Fig. 9b it is visible that especially for larger CHDs (CHD $>$ 1.6 mm) the Eq. 12 is a better approximation for the not gas-quenched gears. For the gas-quenched gears, Eq. 13 is a better approximation. Simulation no. 15 shows an extraordinarily high slope. Simulation no. 15 is with a relatively small a CHD of 0.1 $\cdot$ module for a module 18 mm gear. Overall, the non-gas quenched variants with lower CHD tend to show higher slopes.

$$\begin{aligned}[b]&\displaystyle s=291.7\cdot\mathrm{CHD}^{-0.696}\\ &\displaystyle\text{for}\;\mathrm{CHD}\,\geq 1.6\,\mathrm{mm}\; \textnormal{and liquid quenching media}\end{aligned}$$

(12)

$$\begin{aligned}[b]&\displaystyle s=926.4\cdot\mathrm{CHD}^{-0.967}\\ &\displaystyle\text{for}\;\mathrm{CHD}\,\geq 1.6\,\mathrm{mm}\; \textnormal{and gas quenching}\end{aligned}$$

(13)

For smaller CHDs in the range of 0.5 mm up to 1.6 mm also a better approximation is possible. The description of the slope $s$ for these smaller CHDs in smaller sized gears can be significantly increased by additional consideration of the gear size in form of the tooth thickness $s_{t}$. In Fig. 10 (left) the slope $s$ for gears with 0.5 mm $<$ CHD $<$ 1.6 mm is shown. In Fig. 10 (left) also Eq. 14 is shown. The corresponding deviations of equation and simulated values can not be read from the diagram and are therefore shown in Fig. 10 (right).

$$\begin{aligned}[b]&\displaystyle s=e^{7.466-0.835\cdot\mathrm{CHD}-1.798\cdot\frac{\mathrm{CHD}}{s_{t}}}\\ &\displaystyle\text{for }\mathrm{CHD}\,<1.6\,\mathrm{mm}\end{aligned}$$

(14)

Fig. 10

Maximal slope $s$ of the simulated residual stress depth profiles as a function of CHD and $s_{t}$ for CHD $<$ 1.6 mm (left) and the corresponding deviations (right)

In Fig. 11 the depth values of the maximal compressive residual stresses of the simulated residual stress depth profiles are shown. The values show a linear correlation depending on CHD and the tooth thickness $s_{t}$. Only for simulation no. 35 (quenched with polymer) is the shown function (Eq. 15) less applicable. This was simulated with a measured carbon profile and a polymer quenching and is the only depth profile with this specific combination of properties.

$$\begin{aligned}[b]&y_{D}=0.35\\&{}+0.6\cdot\left(\mathrm{CHD}-\frac{12.35\cdot\mathrm{CHD}}{s_{t}}+0.00022\cdot\mathrm{CHD}\cdot s_{t}^{2}\right)\end{aligned}$$

(15)

Fig. 11

Depth of maximal compressive residual stresses $y_{D}$ of the simulated residual stress depth profile as a function of CHD and the tooth thickness

In Fig. 12 the maximal compressive residual stresses of the simulated residual stress depth profiles are shown. The values are predominantly in the range of $-200\ldots{}-450\,\mathrm{N/mm^{2}}$. Overall, however, a comparatively poorer description of the correlation is visible. Other influences besides CHD and module or tooth thickness are also suspected here, which could not be identified due to the small number of simulations. However, these points are comparatively less relevant for the calculation of the tooth flank fracture risk. In addition, there are already validated calculation approaches for the residual stresses in the surface or near-surface region of the component, such as the approach according to Lang [23]. In addition, the near-surface residual stress state in gears also or mostly depends on the process steps after the heat treatment.

Fig. 12

Maximal compressive residual stresses $\sigma_{D}$ of the simulated residual stress depth profile as a function of module, CHD and the tooth thickness

The average maximal compressive residual stresses are here described by Eq. 16. For the gas-quenched variants the average maximal compressive residual stresses are higher and are calculated by Eq. 17. With these equations a rough estimate of the compressive residual stresses after case carburizing is possible.

$$\begin{aligned}[b]&\displaystyle\sigma_{D}=-409.82+644.55\cdot\left(\frac{{\mathrm{CHD}}}{{m_{n}}}-0.0011\cdot s_{t}\right)\\ &\displaystyle \textnormal{for liquid quenching media}\end{aligned}$$

(16)

$$\begin{aligned}[b]&\displaystyle\sigma_{D,\text{gas}}=-467.62+644.55\cdot\left(\frac{{\mathrm{CHD}}}{{m_{n}}}\right)-0.73\cdot s_{t}\\ &\displaystyle \textnormal{for gas quenching}\end{aligned}$$

(17)

5 Verification

The equations for the prediction of the characteristic points, show overall high agreement with the simulated values. For the respective points the individual deviations were shown in the previous chapter. The verification and determination of the accuracy of the derived calculation approach to predict residual stress depth profiles in comparison to the simulated residual stress depth profiles is described in the following. For this purpose, the deviation of the profiles based on the two methods is calculated. However, since the development focuses on the stresses in greater depth, the evaluation was carried out in a weighted manner. Up to $0.5\cdot y_{DZ}$, the near-surface residual stresses are not considered (w = 0). From $0.5\cdot y_{DZ}$ up to a depth of $y_{DZ}$ the weighting factor is w = 0.5 and in larger material depth the weighting factor is w = 1.0.

With this evaluation criterion, the improved approximation formula achieves an average RMSE for the deviation of 39.6 N/mm². Compared to the equations in FVA 835 I [1] the accuracy is improved by approxematly 50%. For 80% of the residual stress depth profiles the deviation is between 0 and 50 N/mm² and only for one depth profile above 100 N/mm² (no. 35). This was simulated with a measured carbon profile and a polymer quenching and is the only depth profile with this specific combination of properties.

An exemplary comparison of the calculated and simulated residual stress depth profile is shown in Fig. 13. The properties of the gear and the case-carburizing process of the simulation are shown above the plot. The data used to calculate the residual stress depth profile are shown at the top of the plot. Simulation no 14. has a greater deviation than average with a weighted RMSE of 96 N/mm². At the near-surface region the calculated residual stress depth profile differs significantly. The selected logistic-function can not describe the increase in compressive residual stresses in the near-surface region. However, this research focuses on the residual stress depth profiles in larger material depth, especially the tensile residual stresses. For the residual stresses in larger depths, a fairly good agreement is visible.

Fig. 13

Comparison of the calculated residual stress depth profile with the simulated depth profile for simulation no. 14

6 Validation

For the validation of the calculation approach, additional measurements are compared to calculated residual stress depth profiles. In Fig. 14, a residual stresses measurement for a bevel gear is shown. The quenching medium is not documented in [22]. In Fig. 14 the calculated residual stress depth profiles for gas-quenched gears and for not gas-quenched gears are also shown. The maximal tensile residual stresses are comparable and there is also good agreement of the compressive residual stresses in the region near the surface. However, the slope and depth of the transition from compressive to tensile residual stresses differ significantly of the two calculated residual stress depth profiles and the measured values. With the application of the equations for gas-quenched gears a better agreement with the measured values can be achieved. By using the equations for not gas-quenched gears the calculated residual stress depth profile does not show a good agreement with the measured values.

This could be due to the fact that the measurement was made on a bevel gear and the equations were developed for spur and helical gears. In addition, the quenching medium for the bevel gear is not known and therefore, this example could be a gas-quenched bevel gear. In principle, however, there are also further uncertainties in the developed equations. Different case carburizing processes and case carburizing parameters at different manufacturers lead to different residual stress depth profiles. It is therefore recommended to adapt the calculation of the characteristic points to measurements, if available.

In this case, an adaption of the equations for the slope $s$ and the depth of the transition from compressive to tensile $y_{DZ}$ is recommended. Hereby, the adaptions should be made by shifting functions by addition or subtraction of the deviations.

Fig. 14

Comparison of measurements of Böhme [22] and the calculated residual stress depth profiles with the herein proposed equation

In Fig. 15 X‑ray measurements and measurements with the contour method are shown for a further example and compared to the calculated residual stress depth profile. The residual stress depth profiles were measured at the lower path (compare Fig. 2). The calculation of the residual stress depth profile was done for the module 5 mm spur gear with a CHD of 1 mm. The calculated maximal compressive residual stresses correspond well with the X‑ray measurements. For the transition from compressive to tensile residual stresses all methods show a comparable depth, where here the calculated transition is somewhat closer to the surface than in the measurements. The maximal tensile residual stresses are higher for the calculated values. However, the residual stress depth profile is calculated for the pitch point at approx. mid tooth height. Therefore the higher tensile residual stresses meet the expectations.

Fig. 15

Comparison of measurements of Weber [15] and the calculated residual stress depth profiles with the herein proposed equation for the module 5 mm spur gear with a CHD of 1 mm

7 Summary and conclusion

For the design of gears regarding sub-surface initiated fatigue failures, such as tooth flank fracture (TFF) or tooth interior fatigue failure (TIFF), it must be possible to estimate the residual stress depth profiles up to larger material depths.

In this work, equations for the prediction of residual stress depth profiles in case-carburized gears are presented. The main equations (Eqs. 1–4) describe residual stress depth profiles based on specified characteristic points such as the maximal tensile residual stresses $y_{Z}$. In addition, equations (Eqs. 5–17) were defined to calculate the values of these characteristic points based on gear module, CHD, tooth thickness and quenching medium. These equations are derived from 51 residual stress depth profiles in gears that were simulated with a complex simulation model developed in the research project FVA 835 I [1]. The investigated gears have different sizes ranging from module 3 mm to module 27 mm. For the gears the CHD was varied within 0.1…0.3 $\cdot$ module. In addition, the quenching medium and the carbon content profile was varied.

The simulated residual stress depth profiles were further investigated regarding correlations with the gear properties:

• The main influences on the residual stress depth profile are the gear size (module) and the CHD. Further influences of the gear geometry or tooth form (considered here with the combination of module and tooth thickness) are observed as well.

• For larger CHDs, the surface residual stresses decrease, the depth of transition from compressive to tensile residual stresses increases and the tensile residual stresses increase. This behavior depends on the gear size and geometry.

• The simulated residual stress depth profiles for gas-quenched gears differ from the residual stresses of the other investigated liquid quenching media (oil, saltwater and polymer). The difference can be explained by the lower heat transfer for gas quenching.

• The carbon content profile shows an influence on the residual stress depth profile. However, no direct correlation could be observed due to few simulated residual stress depth profiles with different carbon content profiles.

The residual stress depth profiles in larger material depth can be described with a logistic function. In this work, a sigmoid function is adapted, so that the defined characteristic points of a residual stress depth profile are input variables. Based on the simulated residual stress depth profiles, equations for predicting these characteristic points are derived.

• For the calculation of residual stress depth profiles with the herein proposed equations (Eqs. 1–4), the characteristic points $y_{D}$, $\sigma_{D}$, $y_{DZ}$, $s$ and $\sigma_{Z}$ are input variables.

• For the calculation of the characteristic points (Eqs. 5–17), the input variables are the CHD, module of the gear $m_{n}$ and the tooth thickness $s_{t}$ at the reference diameter.

With the proposed equations (also see Appendix B) it is possible to calculate residual stress depth profiles at the pitch point C in case-carburized gears with consideration of the tensile residual stresses in larger material depths. Therefore, only the main gear geometry and the CHD is necessary as input for the non-iterative calculation. It is recommended to back up calculation results with measurements if measurements are available. The calculation approach can be adapted and modified by specifying single characteristic points or modification of the equations for single characteristic points to better fit measured residual stress depth profiles.

• The calculated compressive stresses in the near-surface region show good agreement with measurements as well as calculated residual stresses with the approach of Lang. However, the residual stresses for different grinding or finishing conditions can differ significantly and can be considered with a specified $\sigma_{D}$ in the proposed equations.

• In some cases the calculated transition from compressive to tensile residual stresses is closer to the surface and in some cases, it is in larger material depth compared to measurements.

• The calculated tensile residual stresses in larger material depth show good agreement with measured values.

Overall a simple calculation approach for comparatively accurate prediction of residual stress depth profiles based on extensive research is presented. In combination with the proposed enhancement of ISO/TS 6336‑4 in [2] it is now possible to predict and consider tensile residual stresses in the assessment of tooth flank fracture risk.

Appendices

Appendix A—Simulated gears

Table 1

Appendix B—Matlab code for the calculation of residual stress depth profiles