Calculation of Tooth Flank Fracture Load Capacity

1: Introduction

Due to improved material qualities, new surface finishing methods and increased heat-treatment process reliability, flank surface damages, such as pitting or micropitting, can be prevented more and more in a reliable manner. This results in an increase of unexpected flank damages with crack initiation below the surface of the loaded gear flank(s), for example tooth flank fracture (TFF, also known as tooth flank breakage, Figure 1 (a)) or tooth interior fatigue fracture (TIFF [28, 29], Figure 1 (b)).

Due to their unexpected occurrence, damages caused by tooth flank fracture often lead to high costs for gearbox manufacturers. Since tooth flank fracture can also occur if the load carrying capacity regarding pitting (according to ISO 6336-2 [2]) and tooth root breakage (according to ISO 6336-3 [3]) is sufficient [8], the demand for a reliable standardized calculation method for assessing the risk of tooth flank fracture is growing. As the formation mechanism of tooth flank fracture damages has been analyzed and understood in detail over the last two decades, the first standardized methods for assessing the risk of tooth flank fracture are available. In this paper, the draft technical specification ISO/DTS 6336-4 [6] (formerly registered as ISO/DTS 19042 [5]), which is now in preparation, is treated in detail. This simplified method, established by FZG/Witzig [38], has been derived from, and the results are in good agreement with already existing, highly sophisticated calculation methods [21, 22, 31].

As the calculation of tooth flank fracture load capacity is highly dependent on sufficient input data, this paper aims to provide guidelines on how to use the developed ISO draft technical specification correctly. Therefore, the typical characteristics of tooth flank fracture failures are described initially, followed by a comprehensive description of the calculation method. Afterwards, a parameter study on main influence parameters on tooth flank fracture is presented in order to increase the user’s awareness of these parameters, which can mainly be influenced during the design process of the gearbox. Finally, the practical applicability of the presented calculation method and its limitations are shown.

2: Failure Description – Tooth Flank Fracture

Tooth flank fracture is a severe fatigue failure on gears with crack initiation below the flank surface. Tooth flank fracture failures are reported from different industrial gear applications, such as spur or helical gears for car, truck, and bus transmissions, wind turbines or turbo transmissions, as well as from bevel gears for water turbines (e.g., [10]). Failures are also known from specially designed test gears for gear running tests [14, 36, 38] and typically occur on the driven partner of case-carburized gears but have also been observed on nitrided and induction hardened gears.

Tooth flank fracture is characterized by a primary fatigue crack in the region of the active contact area, initiated below the surface due to shear stresses caused by the flank contact. This primary crack is often located at approximately half the height of the tooth. The crack starter is in the material depth, typically in the area of the case-core interface, and is often — but not always — associated with a small non-metallic inclusion. The primary crack grows in both directions, toward the surface, as well as in the material depth in an angle of approximately 40–50° to the tooth flank surface. Subsequent cracks growing from the surface may occur (see Figure 2). The final breakage is due to forced rupture. The fractured surface shows typical fatigue characteristics. [37]

Observed tooth flank fractures usually occurred after more than 107 load cycles, which points out the fatigue character of this failure mechanism and is a typical differentiating factor to tooth root breakage, which usually occurs after <106 load cycles (but can also occur at higher running times in some cases [13]).

3: Calculation Methods for the Assessment
of the Risk of Tooth Flank Fracture

Tooth flank fracture damages are initiated below the surface due to shear stresses caused by the Hertzian flank contact. Therefore, the locally occurring shear stresses are a main influence parameter for the assessment of the risk of tooth flank fracture. For an ideal Hertzian contact without relative movement of the contacting surfaces, the maximum shear stress is located at a depth of 0.78·bH, where bH is half of the Hertzian contact width. Regarding a typical rolling and sliding contact of mating gears with rough surfaces and residual stresses induced by manufacturing or load, the calculation of the maximum stress is much more complex.

For assessing the risk of tooth flank fracture damages, basically two different types of calculation models exist:

The first approach is a material-physically based one, which takes into account different effects influencing the whole stress state from the surface to the material depth. Such calculation methods are presented by FZG/Oster [31], FZG/Hertter [21], or Ghribi et. al [16, 17]. Usually, these methods deliver the best results regarding the assessment of the risk of tooth flank fracture damages and may be also applicable to flank damages with crack initiation on the surface, such as pitting. The material-physically based calculation models need very comprehensive and detailed input data and require some integrations and iteration steps. Therefore, these models are not appropriate to assess the risk of tooth flank fracture in the design process or in a standardized calculation method.                                              

As a second type, practical calculation models exist which have been derived from material-physically models. These calculation methods are often calibrated with experimental results and therefore have a limited applicability. Inside these boundaries, the practical calculation models deliver results comparable to those of the material-physically ones. The calculation process is less complex and consequently suitable for a standardized method. Regarding tooth flank fracture, such calculation models are presented by DNV [1], FZG/Witzig [38], or in ISO/DTS 6336-4 [6].

For calculating the tooth flank fracture load capacity, both types of calculation methods use the comparison of the local occurring shear stresses and the local permissible shear stress.

In the following, both types of calculation models are briefly presented by an example calculation method (material-physically based FZG-model and practical calculation approach according FZG/Witzig – ISO/DTS 6336-4). More detailed information can be found in [6, 12, 17, 19, 20, 21, 22, 31, 37, 38].

3.1: Material-Physically Based FZG-Model

The assessment of the risk of tooth flank fracture and other surface or subsurface initiated fatigue failures on gear flanks is possible according to the material-physically based FZG-model for the assessment of the risk of tooth flank fracture developed by Oster [31] and Hertter [21]. This model is based on the comparison of a local occurring equivalent shear stress in a volume element at or below the flank surface and the local material strength at the considered material depth. According to Tobie [37], the calculation model is basically able to take into account the following influences (also shown in Figure 3) for determining the resulting stress condition for the contact of two mating gear flanks (rolling/sliding contact):

• Normal contact force due to the applied torque, resulting in a pressure distribution and stresses according to the Hertzian theory.
• Modified pressure distribution due to lubricated contact, described by EHL-theory.
• Shear and bending load.
• Tangential load caused by friction force, resulting in additional shear stresses.
• Thermal load due to friction force.
• Stress peaks due to rough surface.
• Residual stresses due to mechanical processes and heat treatment.

The resulting local occurring equivalent shear stress for the FZG-model is determined by help of the shear stress intensity hypothesis (SIH). A similar method for assessing the risk of tooth flank fracture based on a finite element analysis and Crossland’s or Dang Van criterion, respectively, is presented by Ghribi [17].

A detailed description of the material-physically based FZG-model for the assessment of the risk of tooth flank fracture can be found in [12, 21, 22, 31, 37, 38]. The calculation procedure needs comprehensive and detailed input values, and some integrations and iteration steps are required. Therefore, this method is quite complex. For this reason, a more practical-oriented calculation approach was derived by FZG/Witzig [38] and is presented in Figure 3.

3.2: Practical Calculation Approach
According to FZG/Witzig

Based on the previously shown material-physically based FZG-model for the assessment of the risk of tooth flank fracture, a practical calculation approach was derived by Witzig [38]. In contrast to the previously shown FZG-model, this approach enables the user to assess the risk of tooth flank fracture already in the design phase of a new type of transmission because it is based on only a few specific input values which are typically available at design stage and does not need complex integrations or calculation steps.

Whereas the complex material-physically based FZG-model is capable of assessing fatigue damages on the tooth flank surface as well as in the material depth, the practical calculation approach according to Witzig [38] only aims at assessing the risk of tooth flank fracture in the material depth. The following approximations compared to the material-physically based FZG-model have been made:

• Calculation method in closed form solution.
• No consideration of tensile residual stresses.
• No consideration of shear stresses induced by friction, EHL contact, surface asperities or thermal load.
• Valid only for case-carburized gears.

The basic formulae for assessing the risk of tooth flank fracture according to the practical approach of FZG/Witzig [38] are similar to the ones used in the FZG-model. The decisive parameter for evaluating the risk of tooth flank fracture according to FZG/Witzig [38] is the local material exposure A_FF which is the quotient of local equivalent stress state in the material depth y, τ_eff(y), and the local material strength τ_per(y):

The local equivalent stress state τ_eff(y) according to Witzig [38] considers the local equivalent stress state without consideration of residual stresses τ_eff,L(y), the influence of residual stresses on the local equivalent stress state ∆_τeff,L,RS(y) and the quasi-stationary residual stress state τ_eff,RS(y). The local material
strength τ_per(y) is a function of the local hardness HV(y) and the material, which is defined by the hardness conversion factor K_τ,per and the material factor K_material.

The local equivalent stress state without consideration of residual stresses τ_eff,L(y) according to FZG/Witzig [38] is based on the calculation of the stress state of an Hertzian line-contact between semicircle and half plane according to Föppl [15]. FZG/Witzig [38] showed that the calculated main shear stress is converging to the shear stress intensity according to the material-physically based FZG-model particularly in greater material depths where the crack initiation of tooth flank fracture damages usually is found.

Residual stresses in the carburized layer may influence the total stress state. Therefore, they have to be considered for the calculation of the local equivalent stress state τ_eff(y). The practical approach according to FZG/Witzig [38] only considers compressive residual stresses as tensile residual stresses in the core for typical tooth profiles are assumed to be small and are therefore neglected. Higher tensile stresses in the core region may increase the risk of tooth flank fracture but are hardly determinable by existing measuring methods and are therefore not included in the calculation approach so far. Furthermore, it is assumed that tangential and axial to the tooth flank orientated residual stress components show similar values and normal orientated residual stresses can be neglected.

Witzig [38] performed a detailed parameter study to validate the practical approach for the assessment of the risk of tooth flank fracture with the material-physically FZG-model. The following parameters were varied:

• Hertzian stress p_H.
• Radius of relative curvature ρ_red.
• Case hardening depth CHD.
• Residual stress depth profile.

For all parameter combinations, the material exposure was evaluated in the material depth range of 0·b_H ≤ y ≤ 9·b_H, where b_H is half of the Hertzian contact width, regarding the calculated maximum material exposure A_FF,max, the material exposure depth profile and the depth coordinate of the maximum material exposure y_max. In summary, the results provided by the practical approach derived by FZG/Witzig [38] are in very good accordance to the results of the material-physically based FZG-model for material depths
y ≥ 1·b_H. According to FZG/Witzig [38], the practical approach for assessing the risk of tooth flank fracture as described herein is only applicable for case-carburized gears in material depths y ≥ 1·b_H and is validated only for parameters within the following ranges:

• 500 N/mm² ≤ pH ≤ 3000 N/mm².
• 5mm ≤ ρ_red ≤ 150mm.
• 0.3 mm ≤ CHD ≤ 4.5 mm.

Based on experimental investigations on case-carburized gears and industrial examples [12, 37, 38], it is known, that a maximum material exposure A_FF,max ≥ 0.8 can lead to tooth flank fractures for gear materials of typical quality and cleanness (quality MQ according to ISO 6336-5 [4]). Detailed information on the practical calculation approach for assessing the risk of tooth flank fracture can be found in [12, 19, 20, 38].

3.3: Standardized Method for Calculating
the Tooth Flank Fracture Load Capacity

To date, no general, standardized method for the calculation of the tooth flank fracture load capacity is available. For marine transmissions, the simplified approach according to DNVGL-CG-0036 [1] may be used to calculate a “subsurface safety against fatigue” for surface hardened pinions and wheels. To satisfy the growing demand of the industry regarding a standardized method for assessing the risk of tooth flank fracture, an ISO technical specification is now in preparation [5]. ISO/DTS 6336-4 “Calculation of load capacity of spur and helical gears – Calculation of tooth flank fracture load capacity” [6] (formerly work title: ISO/DTS 19042 “Calculation of tooth flank fracture load capacity of cylindrical spur and helical gears” [5]) is mainly based on the previously presented practical approach for assessing the risk of tooth flank fracture for case-carburized gears according to FZG/Witzig [38]. It is supplemented by some advice for practical use regarding the determination of hardness and residual stress depth profiles and calculating the Hertzian stress. The calculation of a maximum material exposure A_FF,max and a resulting safety factor SFF can be performed by method A or method B where method A is the more accurate method which needs more input values and more complex calculations or measurements, respectively.

The calculation has to be performed at not less than seven calculation points along the path of contact in the profile direction. In the direction of the face width, the most critical section has to be chosen. For every specified contact point CP (see Figure 4), the local material exposure A_FF is calculated for a reasonably chosen number of material depth coordinates y. The risk of tooth flank fracture and subsequent the resulting safety factor is determined with the maximum calculated local material exposure A_FF,max for all analyzed contact points CP over the material depth y where y is equal to or greater than half of the Hertzian contact width b_H. According to ISO/DTS 6336-4 [6], method B, seven contact points are specified along the path of contact: A (lower end point on the path of contact), AB (midway point between A and B), B (lower point of single pair tooth contact), C (pitch point), D (upper point of single pair tooth contact), DE (midway point between D and E) and E (upper end point on the path of contact).

The calculation of the Hertzian stress p_H according to ISO/DTS 6336-4 [6] is performed by means of a detailed contact analysis, for example based on a full 3D elastic contact model (method A) or according to a simplified method similar to the one used in ISO 6336-2 [2] (method B). If there is no reliable measured hardness or residual stress depth profile (method A) available, the hardness depth profile can be approximated (method B) by the approach of Lang [25] or Thomas [34], and the residual stress depth profile can be approximated by the approach of Lang [26].

4: Parameter Study on Main Influence Parameters on the Calculation
of Tooth Flank Fracture Load Capacity According to ISO/DTS 6336-4

This section focuses on the main influence parameters on the assessment of the risk of tooth flank fracture on case-carburized gears according to ISO/DTS 6336-4 [6] and the practical applicability of the presented DTS. Figure 5 provides an overview on these main influence parameters on the calculation of the local material exposure A_FF, taking into account the load parameters and the gear geometry, as well as the material data.

Beermann [11] has already performed a parameter study on the influence of the tooth flank macro geometry on the risk of tooth flank fracture according to an early draft of ISO/DTS 19042 [5]. For his parametric study, the calculation approach delivers reasonable results. Pinnekamp [32] made some example calculations with a high-speed gear, and an industrial gear failed by tooth flank fracture in field. His calculations state that ISO/DTS is capable of assessing the risk of tooth flank fracture correctly for these examples. This is in good agreement with several FZG internal calculations for transmissions from different application areas with or without tooth flank fracture damages. The results of the calculations with the simplified calculation approach always match the detected damages.

In the following described parameter study, the influence of changes of the main influence parameters on the calculated risk of tooth flank fracture according to ISO/DTS 6336-4 [6] is investigated in detail. The following parameters are varied:

• Load parameters and Hertzian stress.
• Surface hardness.
• Core hardness.
• Hardness depth profile.
• Residual stress depth profile.

Every parameter is varied on its own in order to eliminate crossover influences with other parameters. For this study, a reference test gearing has been chosen. The geometry data of the used reference gearing can be found in Table 1. This test gearing was used for tooth flank fracture tests carried out by FZG/Witzig [38].

In a first step, a basic calculation of the tooth flank fracture load capacity was performed as it would be done at the design stage of a new gearbox. In this case, only the main gearing data as well as the materials and approximated heat treatment parameters are defined. Therefore, only Method B as specified in ISO/DTS 6336-4 can be used for calculation of Hertzian contact stress p_dyn, hardness depth profile HV(y) (method B2 according to FZG/Thomas [34] as the sum of OH and KH is different from 1,100 HV), and residual stress depth profile (method B according to Lang [26] and Witzig [38]).

Figure 6 shows the calculated hardness and residual stress depth profile according to method B of ISO/DTS 6336-4 [6] for the reference gearing on the left. The Hertzian contact stress p_dyn was calculated according to method B assuming an optimum profile modification. The values of p_dyn along the path of contact are shown on the right in Figure 6. Additionally, the half of the Hertzian contact width b_H is shown. All calculations were performed for a pinion torque of T₁ = 1,200 Nm.

Figure 7 shows the components of the local equivalent stress state τ_eff(y) according to ISO/DTS 6336-4 [6] for the reference gearing on the left. As the lower point of single tooth contact B is the most critical one, the values are shown for this calculation point. It clearly can be seen that the influence of the residual stress is significant only in the hardened case layer where a significant hardness gradient is present. Figure 7 on the right shows the local equivalent stress state τ_eff(y), the local material strength τ_per(y), and the resulting local material exposure A_FF(y) for the aforementioned reference gearing. As the material strength τ_per is a function of the hardness depth profile, it can be noticed that the case-core interface for this example is in a material depth y ≈ 2·b_H. This is also the depth where the maximum local equivalent stress state τ_eff,max, and therefore the maximum material exposure A_FF,max, is calculated. As A_FF,max ≈ 0.9, the reference gearing has a high calculated risk of failing by tooth flank fracture for the given load of T₁ = 1,200 Nm.

In the following section, the results of the aforementioned parameter variations are shown and discussed in detail. Unless otherwise specified, the following assumptions were used for the performed calculations and the – shown figures as reference:

Calculation of tooth flank fracture load capacity according to ISO/DTS 6336-4 [6].

Torque at pinion T₁ = 1200 Nm.

Calculation of p_dyn according to method B with load sharing factors X_CP for optimum profile corrections and load factors K_A = K_Hβ = K_Hα = K_Hγ = K_v = 1.0.

Calculation of the hardness depth profile HV(y) according to method B2 (Thomas [34]) with hardness as specified in Table 1.

Calculation of the residual stress depth profile σ_RS(y) according to method B (Lang [26] and Witzig [38]) from hardness depth profile HV(y).

A_FF,max is calculated in the lower point of single tooth contact B, which is the most critical contact point. Therefore, all figures are shown for contact point B unless otherwise stated.

4.1: Load Parameters and Hertzian Stress

As a first step, the load parameters have been varied, starting with the nominal torque at the pinion T₁. Based on the reference value of T₁ = 1,200 Nm, two other variants — T_1,low = 800 Nm and T_{1, high} =1,600 Nm — have been chosen. The results of this variation are shown in Figure 8. The resulting distribution of the Hertzian contact stress p_dyn along the path of contact is shown on the left. The local occurring equivalent shear stress τ_eff, the local material strength τ_per, and the resulting local material exposure A_FF are shown on the right. 

As expected, the maximum material exposure decreases with decreasing external load (e.g. torque). By reducing the torque by 50 percent, the maximum material exposure is also nearly reduced by 50 percent (decreases from ∼1.2 to ∼0.65) for the given reference gearing. The local material strength is only dependent on the hardness depth profile and therefore remains unchanged, while the local occurring equivalent shear stress τ_eff is decreasing with decreasing load. Referenced to half of the Hertzian contact width, the material depth of maximum material exposure y_max/b_H remains unchanged. Taking into account the changing value of b_H, the absolute material depth of maximum material exposure y_max is reduced with reduced load. 

In a next step, the method for obtaining the Hertzian contact stresses was changed from method B (calculation based on load sharing factors) to method A. Therefore, a detailed calculation model of the given reference gearing with shafts and bearings was built up in the program system RIKOR [33]. With the help of this tool, a sophisticated analysis of the loaded contact can be performed. The input torque was set to T₁ = 1,200 Nm, and the flanks were modeled with a short tip relief of Ca = 30 µm, both pinion and gear. The maximum local Hertzian contact stresses p_H calculated by RIKOR along the facewidth have been chosen for calculation at the seven specified points of ISO/DTS 6336-4 [6]. Figure 9 shows the calculated load distribution along the path of contact compared to the one according to method B on the left. As it can be seen, the values for contact points A, C, and E are higher for method A, whereas the Hertzian contact stress for contact points B and D is higher for method B. Overall, the estimation according to method B is quite good compared to the sophisticated calculation based on a full 3D elastic contact model for the herein considered reference gearing.

Due to the slight difference in load distribution in the area of single tooth contact, the most critical contact point changes from B to C by using the calculation of p_H according to method A. Nevertheless, the calculated maximum material exposures calculated according to method B for the relevant contact points with regard to TFF (B…D) along the path of contact are in good correlation with those calculated according to method A. Considering contact points A and E (begin and end of contact), the differences between method A and B are more significant, yet these points are usually not relevant for tooth flank fracture load capacity calculations.

Though this example shows a very good correlation of the calculated distributions of the Hertzian contact stress p_dyn according to method A and method B, respectively, method A should be the preferred method wherever possible. For calculation of the Hertzian contact stress according to method A, every program system using a full 3D elastic contact model for the loaded tooth contact analysis, such as RIKOR [33], can be used. Similar investigations have also been carried out by Al, regarding tooth interior fatigue fracture [7] and tooth flank fracture [9].

4.2: Surface Hardness

Furthermore, the surface hardness was also varied with three values, OH = 660 HV, OH = 700 HV, and OH = 740 HV. To calculate the hardness depth profile, method B2 according to Thomas [34] was chosen. The case hardening depth CHD550 = 0.5 mm, as well as the core hardness KH = 440 HV, remained unchanged.

Figure 10 shows the calculated hardness and residual stress depth profiles on the left. The local occurring equivalent shear stress τ_eff, the local material strength τ_per and the resulting local material exposure A_FF are shown on the right. As the hardness depth profile is changing, also the local material strength is changing and due to the change of the residual stress depth profile, also the local occurring equivalent shear stress τ_eff is changing. The calculated maximum material exposure A_FF,max is unexpectedly decreasing with decreasing surface hardness in this example.

The reason for this can be found in the different shape of the hardness depth profile in the case-core interface. Due to the estimation according to method B2 and a fixed value of CHD₅₅₀, the shape of the case-core interface is changing with changing surface hardness. Taking into account the complex cross influences regarding residual stresses and therefore the occurring stresses, this unexpected change of A_FF,max can be explained. If CHD₅₅₀ would be adapted to keep the shape of the case-core interface the same, a change of the surface hardness would not have any influence on the calculated local material exposure.

4.3: Core Hardness

In the next step, the core hardness was varied with three values—KH = 400 HV, KH = 440 HV, and KH = 480 HV. For calculation of the hardness depth profile, method B2 according to Thomas [34] was chosen. The case hardening depth CHD₅₅₀ = 0.5 mm, as well as the surface hardness OH = 700 HV, remained unchanged.

Figure 11 shows the calculated hardness and residual stress depth profiles on the left. The local occurring equivalent shear stress τ_eff, the local material strength τ_per, and the resulting local material exposure A_FF are shown on the right. With increasing core hardness, the material strength τ_per is increasing — at least from a certain material depth on — but also the local equivalent stress state τ_eff is influenced because the approximated residual stresses according to method B are decreasing due to the modified hardness profile. The resulting values of A_FF,max are not significantly changing, but the depth coordinates of the maximum material exposure ymax change. Also, this example shows that the calculation of tooth flank fracture load capacity is characterized by complex interactions between different load, geometry, and material parameters which are included in the presented model but do not allow a further simplification.

4.4: Hardness Depth Profile

As already shown in the last two sections, the hardness depth profile plays a crucial role for assessing the risk of tooth flank fracture damages (but also regarding the flank load carrying capacity [23, 24]). Whereas only some parameters of the hardness depth profile were varied on their own beforehand, the next step aims at varying the hardness depth profile as a whole. At the design stage of a gearbox, only basic information about the heat treatment is available, e.g., surface hardness, core hardness, and case hardening depth for case-carburized gears. Regarding ISO/DTS 6336-4 [6], the hardness depth profile of case-carburized gears can be approximated by method B1 (Lang [25]) or method B2 (Thomas [34]). Method A specifies a measured hardness depth profile as calculation input value.

Figure 12 shows the comparison of a measured hardness depth profile [38] of the reference gearing specified in Table 1 with the according approximation by Lang and Thomas. The input parameters were chosen in accordance with Table 1. The calculated hardness and residual stress depth profiles are shown on the left. The residual stress depth profile was derived from the hardness depth profile with method B. The local occurring equivalent shear stress τ_eff, the local material strength τ_per, and the resulting local material exposure A_FF are shown on the right.

For this example, the approximated hardness depth profiles are in quite good correlation to the measured one. Near the surface, the approximation according Thomas shows a slightly better correlation, whereas the approximation according to Lang [25] is describing the case-core interface in a better way. The approximation according to Lang [25] does not hit CHD₅₅₀ correctly, as this only works if the sum of surface hardness and core hardness is 1,100 HV. Based on the small differences in the hardness depth profile, the calculated maximum material exposure varies from A_FF,max ∼0.7 to ∼0.9, depending on the used hardness depth profile. For this example, the approximation according to Thomas is on the safe side, as it is calculating higher maximum material exposures compared to the one calculated with the measured data.

As it can be seen, tooth flank fracture phenomena are very sensitive to changes in the hardness depth profile; especially in the area of the case-core interface. As a consequence, it is very important to get reliable data on the hardness depth profile of the gearing being investigated. Wherever possible, method A (measured data) is preferred. If the calculation of tooth flank fracture load capacity proves a susceptibility of the considered gearing for tooth flank fracture, the hardness values used have to be considered carefully and tightly tolerated for manufacturing. If a determination of the hardness depth profile with a sufficient precision is not possible (for example during the preliminary design process), the minimum safety factor should be increased in order to avoid TFF reliably.

A general recommendation on the method used for approximating the hardness depth profile cannot be given. Which method is the best depends on the individual application, the specifications for heat treatment, and the heat treatment process. Figure 13 shows an additional example for the approximation of the hardness depth profile for a helical gearing (module m_n = 4.25 mm). The measured hardness depth profile [18] is compared to the ones estimated according to Lang [25], Thomas [34], and MackAldener [30]. For this example, all methods match the point of CHD₅₅₀. Lang and MackAldener show a good correlation in the area of the case-core interface, whereas Thomas shows a quite good correlation near the surface.

Overall, the hardness depth profile used for calculating the risk of tooth flank fracture is a crucial point for obtaining reliable results. Wherever possible, measurement should be performed to at least verify the approximations used. According to ISO/DTS 6336-4 [6], other analytical methods, such as the ones according to MackAldener [30] or Tobe [35] for calculating the hardness depth profile may also be applied upon agreement of supplier and customer.

4.5: Residual Stress Depth Profile

In a last step, the residual stress depth profile is investigated in detail. Up to now, only the hardness depth profile has been considered, and the residual stress depth profile was approximated by method B according to Lang [26] and Witzig [38]. Method A according to ISO/DTS 6336-4 [6] specifies a measured residual stress depth profile. For case-carburized gears, it is assumed that the residual stresses are balanced between case and core. In the surface near area, compressive residual stresses can be found which switch to tensile residual stresses in the core region of the tooth.

Residual stress depth profiles can be determined by X-ray diffraction with electrochemical erosion or hole-drilling method, for example. Both methods allow only reliable measurements up to a material depth of 0.5 … maximum 1.0 mm. A further method for obtaining residual stress depth profiles is neutron diffraction, which allows a fully non-destructive testing but generates high costs.

As previously shown, the case-core interface of case-carburized gears often is the most critical area regarding the risk of tooth flank fracture. As residual stress measurements at such material depths are costly and time consuming, the use of approximation equations is necessary. Nevertheless, existing residual stress depth profile measurements should be used at least for verifying the approximation equations if they are available. In addition to the existing method according to Lang [26] and Witzig [38], further research work is ongoing inside the German research association FVA.

Figure 14 shows an exemplary comparison of a measured residual stress depth profile [38] for the reference gearing with an estimated profile according to method B. Additionally, the measured hardness depth profile as well as from derived residual stress depth profile is shown. The residual stress depth profile was measured up to a material depth of y ∼ 0.3 mm. The measured residual stress depth profile and the estimated one show quite a good correlation.

Witzig [38] showed a comparison between residual stress depth profiles across a whole tooth measured by neutron diffraction and the estimated profiles according to method B. The measured residual stresses are near zero in the area of the case-core interface and are turning to tensile residual stresses in the core area. The approximation according to method B is in good correlation with the measured values up to the case-core interface.

5: Practical Applicability of the Standardized Method for Calculating
the Tooth Flank Fracture Load Capacity According to ISO/DTS 6336-4

The standardized method for assessing the risk of tooth flank fracture according to ISO/DTS 6336-4 [6] was presented briefly in this paper. A more detailed description can be found by Hein et al. [19, 20]. Following, a practical guideline for the future users of ISO/DTS 6336-4 is given based on the previously described parameter study.

5.1: Guideline for Assessing the Risk of Tooth Flank Fracture

ISO/DTS 6336-4 can be used for designing gears in order to avoid tooth flank fracture damages and for recalculating existing gearboxes to assess their risk of tooth flank fracture. Depending on the purpose, different calculation methods may be useful. Hence, the following guideline differentiates between designing and recalculating a (probably failed) gear stage. Regardless of the purpose, the following data have to be known:

• Gear geometry
• Number of teeth z₁ / z₂
• Helix angle β
• Pressure angle α_n
• Normal module m_n
• Tip circle diameter d_a1 / d_a2
• Profile shift factors x₁ / x₂
• Center distance a
• Material data
• Modulus of elasticity E₁ / E₂
• Poisson’s ratio ν₁ / ν₂
• Tensile strength of material R_m
• Load parameters
• Torque
• Load factors K
• Basic information about heat treatment
• Surface hardness OH
• Core hardness KH
• Case hardening depth CHD₅₅₀

Contact Points Along the Path of Contact

ISO/DTS 6336-4 demands the calculation at not less than 7 points (A, AB, B, C, D, DE, E) along the path of contact. In general, this is sufficient. Nevertheless, more calculation points can be chosen, if considered to be useful. They should be distributed evenly along the path of contact.

Calculation of Local Hertzian Contact Stress p_dyn

The local Hertzian contact stress can be calculated according to method A (full 3D elastic contact model) or method B (considering load factors and load sharing factor). For the given spur gear example in section 4.1, a detailed contact analysis did not add much benefit to the calculation results. Nevertheless, a detailed contact analysis is recommended when helical gears are used or the gearbox design allows significant deflections. Data for such a detailed contact analysis are often already available in the design process of a gearbox. Therefore, it is recommended to calculate the local Hertzian contact stress with help of a complex loaded tooth contact analysis (LTCA) whenever possible with little effort.

Hardness Depth Profile HV(y)

The hardness depth profile plays a crucial role in assessing the risk of tooth flank fracture damages correctly. Therefore, it is strongly recommended to use measured values according to method A, if they are available. In the design phase of a gear stage, measured data are usually not available. Consequently, a reliable method for estimating the hardness depth profile according to method B1 or B2 has to be chosen in accordance with previously gained experience on the material used. For recalculating a gear stage, measured data should be used, if possible. If no measurements or practical knowledge are available at all, it is recommended to compare the available methods for estimating the hardness depth profile (for example [25, 30, 34, 35]) regarding the calculated risk of tooth flank fracture. After careful consideration of the calculated results, the most conservative method should be chosen for further calculations to be on the safe side.

Residual Stress Depth Profile σ_RS(y)

As the hardness depth profile is of importance, the residual stress depth profile is also of high relevance for assessing the risk of tooth flank fracture. Since reliable residual stress depth profile measurements are very expensive, estimation methods usually have to be used for designing and recalculating gear stages. For normal-sized case-carburized gears, the approximation method according to Lang [26] and Witzig [38] (method B) delivers suitable results. If failed gear stages are recalculated and the results do not explain the damage behavior, it might be useful to use measured residual stress depth profiles in order to exclude an insufficient heat treatment.

Minimum Safety Factor

The minimum safety factor should be chosen to S_FF,min ≥ 1.2 according to ISO/DTS 6336-4, if a detailed calculation is possible (use of method A). If the calculation contains any uncertainties (such as brief approximations of hardness or residual stress depth profiles according to method B), in consequence, a higher minimum safety factor should be chosen.

5.2: Limitations Regarding the Applicability of the Standardized Method According to ISO 6336-4

As ISO/DTS 6336-4 is based on the practical calculation approach to assess the risk of tooth flank fracture according to FZG/Witzig [38], the same limitations are also applicable to ISO 6336-4.

In summary, the results provided by the practical approach derived by FZG/Witzig [38] are in very good accordance to the results of the material-physically based FZG-model for material depths y ≥ 1·b_H. According to FZG/Witzig [38], the practical approach for assessing the risk of tooth flank fracture is only applicable for case-carburized gears in material depths y ≥ 1·b_H and is validated only for parameters within the following ranges:

• 500 N/mm² ≤ p_H ≤ 3000 N/mm².
• 5 mm ≤ ρ_red ≤ 150 mm.
• 0.3 mm ≤ CHD ≤ 4.5 mm.

Based on experimental investigations of case-carburized gears and industrial examples [12, 37, 38], it is known that a maximum material exposure A_FF,max ≥ 0.8 can lead to tooth flank fractures for gear materials
of typical quality and cleanness (quality MQ according to ISO 6336-5 [4]).

The equations stated in ISO/DTS 6336-4 have been experimentally and theoretically verified by “standard-sized” case-carburized gears made of gear materials of typical quality and cleanness with parameters within the above-mentioned limitations. For certain boundary conditions, such as materials of low quality and cleanness, or uncertainties in the input parameters (e.g. operating conditions), TFF might also occur for calculated maximum material exposures A_FF,max < 0.8. As a result, the minimum safety factor S_FF,min should be increased in such cases. For “optimized” materials of high quality or cleanness, even lower minimum safety factors might be allowable, if an adequate proof of the increased load carrying capacity regarding tooth flank fracture can be given. For further developments on the calculation method according to ISO/DTS 6336-4, it is conceivable that different material qualities comparable to ML, MQ, or ME according to ISO 6336-5 [4] for pitting or tooth root breakage are introduced, if new results from ongoing research works are available.

6: Conclusion

Due to improved material qualities, new surface finishing methods and increased heat-treatment process reliability, flank surface damages, such as pitting or micropitting, can more and more be prevented in a reliable manner. This may result in an increase of unexpected flank damages with crack initiation below the surface of the loaded gear flank(s), for example tooth flank fracture. Tooth flank fracture is characterized by a crack initiation below the active surface due to shear stresses caused by the Hertzian flank contact and crack propagation in the direction of both the active flank surface and the core area, and usually results in a total breakdown of the gear unit.

Sophisticated methods for assessing the risk of tooth flank fracture have already been available for a long time. These methods usually need comprehensive and detailed input values, which are not always available at an early stage in the design process of a new transmission, and the methods also need complex integrations or calculation steps.

As the industrial demand for simplified and standardized calculation methods for assessing the risk of tooth flank fracture is growing, a draft technical specification ISO/DTS 6336-4 “Calculation of load capacity of spur and helical gears – Calculation of tooth flank fracture load capacity” is currently being prepared. It is mainly based on the simplified approach of FZG/Witzig [38].

In this paper, a detailed parameter study has been performed on the main influence parameters on the calculated risk of tooth flank fracture according to ISO/DTS 6336-4. The hardness and residual stress depth profile has been identified as a main influence parameter on the flank fracture load capacity. Therefore, a reasonable way for gaining reliable hardness and residual stress depth profiles for the given gear material and application has to be found. Furthermore, a practical guideline for assessing the risk of tooth flank fracture has been given.

Depending on the reliability of the input parameters and the practical knowledge of the designer, appropriate minimum safety factors S_FF,min diverging from the proposed one may be chosen. For this process, available calculation results regarding TFF from gearboxes already in service (with or without failures) can also be used.

Because, up until now, some influence parameters on the risk of tooth flank fracture damages are covered only empirically, further research work is needed to include more specific material parameters, i.e. cleanness and tensile residual stresses, into the calculation method. 

References

1. DNVGL, 2015, “Calculation of gear rating for marine transmissions”, DNVGL-CG-0036.
2. ISO, 2006, “Calculation of load capacity of spur and helical gears – Part 2: Calculation of surface durability (pitting)”, ISO 6336-2.
3. ISO, 2006, “Calculation of load capacity of spur and helical gears – Part 3: Calculation of tooth bending strength”, ISO 6336-3.
4. ISO, 2016, “Calculation of load capacity of spur and helical gears – Part 5: Strength and quality of materials”, ISO 6336-5.
5. ISO/DTS, in preparation, “Calculation of tooth flank fracture load capacity of cylindrical spur and helical gears”, ISO/DTS 19042.
6. ISO/DTS, in preparation, “Calculation of load capacity of spur and helical gears – Calculation of tooth flank fracture load capacity”, ISO/DTS 6336-4.
7. Al, B. C. and Langlois, P., 2016, “Analysis of Tooth Interior Fatigue Fracture Using Boundary Conditions from an Efficient and Accurate LTCA”, Gear Solutions, 2016(02).
8. Al, B. C., Patel, R. and Langlois, P., 2016, “Comparison of Tooth Interior Fatigue Fracture Load Capacity to Standardized Gear Failure Modes”, FTM (Fall Technical Meeting) 2016, American Gear Manufacturers Association, Pittsburgh, USA.
9. Al, B. C., Patel, R. and Langlois, P., 2016, “Finite Element Analysis of Tooth Flank Fracture Using Boundary Conditions from LTCA”, Gear Technology, 2016(09/10), pp. 62–68.
10. Bauer, E. and Böhl, A., 2011, “Flank Breakage on Gears for Energy Systems”, Gear Technology, 28(8), pp. 36–42.
11. Beermann, S., 2015, “Tooth Flank Fracture – Influence of Macro and Micro Geometry”, International Conference on Gears 2015, VDI.
12. Boiadjiev, I., Witzig, J., Tobie, T. et al., 2014, “Tooth flank fracture – basic principles and calculation model for a sub surface initiated fatigue failure mode of case hardened gears”, International Gear Conference, Woodhead Publishing, Lyon Villeurbanne, France, pp. 670–80.
13. Bretl, N., Schurer, S., Tobie, T. et al., 2013, “Investigations on Tooth Root Bending Strength of Case Hardened Gears in the Range of High Cycle Fatigue”, AGMA 2013 Fall Technical Meeting, AGMA, Indianapolis, USA.
14. Bruckmeier, S., 2006, “Tooth flank fracture at spur gears” (In German: “Flankenbruch bei Stirnradgetrieben”). Ph.D. thesis, Technical University of Munich.
15. Föppl, L., 1947. Desire and force, Volume III – Higher strength of materials for engineers. (In German: Drang und Zwang, Band III – Eine höhere Festigkeitslehre für Ingenieure). Leibnitz Verlag (publisher), Munich.
16. Ghribi, D. and Octrue, M., 2014, “Some theoretical and simulation results on the study of the tooth flank”, International Gear Conference, Woodhead Publishing, Lyon Villeurbanne, France, pp. 659– 69.
17. Ghribi, D., Octrue, M. and Sainsot, P., 2015, “Comparative study of the tooth flank fracture in cylindrical gears – Efficient method to asses the risk of the tooth flank fracture on the cylincrical gears”, International Conference on Gears 2015, VDI.
18. Hein, M., Tobie, T. and Stahl, K., 2016. IGF Nr. 17347, FVA-Nr. 554 II – Issue 1178 – System service life testing II, final report, shortened test method for gearboxes and drive components for verification of the service life (In German: Systemlebensdauerprüfung II Abschlussbericht – Verkürztes Testverfahren für Getriebe und Antriebselemente zur Bestätigung der Betriebsfestigkeit der Bauteile). Research Association for Drive Technology (FVA), Frankfurt/Main.
19. Hein, M., Tobie, T. and Stahl, K., 2017, “Parameter study on the calculated risk of tooth flank fracture of case hardened gears”, Journal of Advanced Mechanical Design, Systems, and Manufacturing.
20. Hein, M., Tobie, T. and Stahl, K., 2017, “Parameter study on the calculated risk of tooth flank fracture of case hardened gears”, Motion and Power Transmissions – MPT 2017, JSME, Kyoto, pp. 324–29.
21. Hertter, T., 2003, “Theoretical strength load capacity calculation of through-hardened and case- hardened spur gears (In German: “Rechnerischer Festigkeitsnachweis der Ermüdungstragfähigkeit vergüteter und einsatzgehärteter Stirnräder”). Ph.D. thesis, Technical University of Munich.
22. Höhn, B.-R., Oster, P. and Hertter, T., 2010, “A Calculation Model for Rating the Gear Load Capacity Based on Local Stresses and Local Properties of the Gear Material”, International Conference on Gears, VDI, Garching, Germany, pp. 87–98.
23. Höhn, B.-R., Stahl, K., Oster, P. et al., 2011, “Flank-load-carrying capacity of case hardened gears with grinding burn”, International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, ASME, Washington, DC.
24. König, J., Koller, P., Tobie, T. et al., 2015, “Correlation of relevant case properties and the flank load carrying capacity of case-hardened gears”, ASME 2015 International Design engineering Technical Conferences & Computers and, ASME, Boston, Massachusetts, USA.
25. Lang, O. R., 1988, “Calculation and Dimensioning of inductive hardened elements” (In German: “Berechnung und Auslegung induktiv randschichtgehärteter Bauteile”), AWT-Tagung, AWT – Arbeitsgemeinschaft Wärmebehandlung und Werkstofftechnik, Darmstadt, pp. 332–49.
26. Lang, O. R., 1979, “Dimensioning of complex steel elements for finite and infinite life” (In German: “Dimensionierung komplizierter Bauteile aus Stahl im Bereich der Zeit- und Dauerfestigkeit”), Zeitschrift für Werstofftechnik, 10, pp. 10–29.
27. MackAldener, M. and Olsson, M., 2002, “Analysis of crack propagation during tooth interior fatigue fracture”, Engineering Fracture Mechanics, 69(18), pp. 2147–62.
28. MackAldener, M. and Olsson, M., 2000, “Design Against Tooth Interior Fatigue Fracture”, Gear Technology, 2000(11/12), pp. 18–24.
29. MackAldener, M. and Olsson, M., 2000, “Interior fatigue fracture of gear teeth”, Fatigue & Fracture of Engineering Materials & Structures, 2000(23), pp. 283–92.
30. MackAldener, M. and Olsson, M., 2001, “Tooth Interior Fatigue Fracture – computational and material aspects”, International Journal of Fatigue, 23(4), pp. 329–40.
31. Oster, P., 1982, “Gear flank stresses under EHL-conditions” (In German: “Beanspruchung der Zahnflanken unter Bedingungen der Elastohydrodynamik”). Ph.D. thesis, Technical University of Munich.
32. Pinnekamp, B. and Heider, M., 2015, “Tooth Flank Fracture – Research, Standardization and Practical Experience”, International Conference on Gears 2015, VDI.
33. Stiller, S., Otto, M. and Stahl, K. “Extension of RIKOR for LTCA of gear drives” (In German: “Erweiterung Ritzelkorrekturprogramm (RIKOR) zur Bestimmung der Lastverteilung von Stirnradgetrieben”) – Issue 1077. Research Association for Drive Technology (FVA), Frankfurt/Main, 2013.
34. Thomas, J., 1998, “Flank load carrying capacity and running behavior of hard-finished bevel bears” (In German: “Flankentragfähigkeit und Laufverhalten von hart-feinbearbeiteten Kegelrädern”). Ph.D. thesis, Technical University of Munich.
35. Tobe, T., Kato, M., Inoue, K. et al., 1986, “Bending Strength of Carburized SCM420H Spur Gear Teeth”, Bulletin of JSME, 29(247), pp. 273–80.
36. Tobie, T., 2001, “Pitting and tooth root load carrying capacity of case-hardened gears – Influences from CHD, heat treatment and manufacturing for different gear sizes” (In German: “Zur Grübchen- und Zahnfußtragfähigkeit einsatzgehärteter Zahnräder – Einflüsse aus Einsatzhärtungstiefe, Wärmebehandlung und Fertigung bei unterschiedlicher Baugröße”). Ph.D. thesis, Technical University of Munich.
37. Tobie, T., Höhn, B.-R. and Stahl, K., 2013, “Tooth flank Breakage – Influences on subsurface initiated fatigue failures of case hardened gears”, ASME 2013 DETC, ASME, Portland, Oregon.
38. Witzig, J., 2012, “Flankenbruch – Eine Grenze der Zahnradtragfähigkeit in der Werkstofftiefe.” Ph.D. thesis, Technical University of Munich.