A standardizable approach for Tooth Flank Fracture

The herein presented work is part of a wider research program, aimed at the prediction and prevention of subsurface-initiated fatigue failures in large maritime bevel gears. It continues the efforts of the ‘Improved reliability of thrusters’ joint industry project—a cooperation between thruster suppliers, gear manufacturers, steel suppliers, forging companies and the maritime classification society Det Norske Veritas (DNV). The article combines the previously published findings on the material properties of carburized CrNiMo steels [1], the stress and fatigue predictions in the mean cross-section of spiral bevel gear teeth [3], the test results from back-to-back gear testing [4] and the observations from early-stage tooth flank fracture on large maritime bevel gears [2, 5] to a standardizable approach for the prediction and ultimately prevention of tooth flank fracture (TFF).

The failure mode TFF is characterized by crack initiation in the interface between the hard case and soft core of case hardened, but also nitrided or induction hardened gear teeth [20]. Cracks initiate typically in mode II (in-plane shear) underneath the load-carrying tooth flank and propagate through the gear tooth in mode I (opening) due to the acting bending stresses in a 40 to 50° angle to the loaded flank. The primary crack front can be accompanied by secondary and tertiary cracks. When final rupture occurs, a large part of the weakened gear tooth breaks off, leaving traces of macroscopic fatigue features on the fracture surfaces (see Fig. 3). Those features are partially removed due to the relative motion of the two fracture surfaces and the resulting plastic deformation.

TFF is caused by a non-proportional, multiaxial stress state in a material depth characterized by locally changing constitutive parameters. The acting stress tensor is complex, consisting of dynamic Hertzian, out-of-phase bending, transverse shear, transverse normal and frictional shear stresses, and imposed by static residual stresses [5]. With failures occurring in the high to very high cycle fatigue regime, stress-based multiaxial fatigue criteria can suitably be applied. The basis for the herein proposed TFF approach forms the Dang Van (DV) criterion due to its widespread use for the analysis of rolling contact fatigue [10, 13, 23] and its ability to yield a standardizable approach. The DV criterion assumes local plasticity in the mesoscale, even if the material is only loaded elastically in the macroscale. Governing the conversion between the meso- and macroscale is a time-invariant residual stress according to Melan’s shakedown theorem [12]. The DV’s statistically most accurate definition is outlined in Eq. 1 [25], relying on the shear stress amplitude \(\tau_{a}\) as a function of the Euler angles \(\theta\), \(\phi\) and the maximum hydrostatic stress \(\sigma_{hyd,max}\). It weighs \(\tau_{a}\) and \(\sigma_{hyd,max}\) through the model constants \(a_{DV}\) and \(b_{DV}\) and compares the maximum equivalent stress against the uniaxial fatigue strength under alternating load \(f_{-1}\). With TFF occurring in carburized gear teeth in a depth of 1 to 2 times the case hardening depth (\(CHD\)), the stresses, fatigue properties and model constants across this specific hardness range need to be quantified. In the case of the quoted DV criterion, \(a_{DV}\) and \(b_{DV}\) are derived from the material constant \(f_{-1}\) and \(t_{-1}\) (i.e. the uniaxial and shear fatigue strengths under alternating load). The ratio \(f_{-1}/t_{-1}\) is commonly referred to as the fatigue ratio \(\kappa\). For case hardened gear teeth, \(\kappa\) ensures a brittle material behavior in the hard case and a ductile behavior in the tooth interior or core. As \(\kappa\) increases from case to core, so do \(a_{DV}\) and the effect of the shear stress amplitude \(\tau_{a}\) on \(A_{DV}\). Contrary, \(b_{DV}\) and the effect of \(\sigma_{hyd,max}\) on \(A_{DV}\) decrease.

Throughout this manuscript, a rationalized version of the DV criterion will be compared against other TFF criteria, specifically the Witzig criterion [34]. Witzig proposed his TFF-criterion in the FVA556 I project, which has since been adapted in the ISO/TS 6336‑4 for the prediction of TFF in cylindrical gears [20]. The subsequent FVA556 II and FVA556 III projects verified its applicability to large cylindrical and spiral bevel gears [9, 32]. As the bevel gear-focused FVA556 III project was led by Boiadjiev, the criterion will herein be referred to as Witzig-Boiadjiev (WI-BO). It calculates the local material utilization in a specific material depth \(A_{FF}(x)\) from the local equivalent stress without consideration of residual stresses \(\tau_{eff,L}\), the influence of the residual stresses on the local equivalent stress \(\Delta\tau_{eff,L,res}\) and the quasi-stationary residual stress \(\tau_{eff,res}\) and compares them against the local material shear strength \(\tau_{per}\) along a surface-perpendicular path. The WI-BO criterion does not rely on model constants to weigh its stresses and differs thereby from conventional multiaxial fatigue criteria like DV, Findley, Liu & Zenner or Crossland [11, 12, 15, 35]. While these criteria combine the planar shear and normal stresses \(\tau_{a}\), \(\tau_{m}\), \(\sigma_{na}\), \(\sigma_{nm}\), the second invariant of the deviatoric stress tensor \(J_{2,a}\) or the hydrostatic stress \(\sigma_{hyd}\) to an equivalent stress, the WI-BO criterion utilizes gear-specific stresses. Both, the lack of model constants and the use of gear-specific stresses make it impossible to assess the WI-BO criterion’s accuracy against other stress-based multiaxial fatigue criteria. The accuracy of the herein quoted DV criterion was quantified in a study by Papuga et. al [26] against 284 multiaxial fatigue tests under proportional and non-proportional stresses. It achieved an average fatigue error index (\(A_{DV}-1\)) of -2.2% and a standard deviation of the fatigue error index of 15.9%.

The WI-BO criterion has been defined for a specific range of input parameters. It will be shown that the 5 mm lower limit for the equivalent radius of curvature \(\rho_{eq}\) is non-conservative and leads to inaccurate results for one of the studied gears. The herein outlined, rationalized version of the DV criterion allows for a simple stress prediction and works for all studied input parameters.

Similar to the WI-BO criterion, the herein outlined TFF criterion can either be applied to the gear’sThe analytical equations outlined in this manuscript study the material strengths and stresses underneath the gear’s design point \(MB\), but can similarly be applied to the entire tooth flank. Assuming constant hardness and residual stress profiles across the tooth, only knowledge of the local Hertzian contact stress \(\sigma_{H,i}\) and the local, equivalent radius of curvature \(\rho_{eq,i}\) is required. Figure 1 outlines the model in its entirety.

Of the calculations that were run on maritime gears [2, 3, 5], FVA gears [33] and medium-sized test gears [4], three cases are highlighted here. With them, the rationalized DV criterion is compared against the WI-BO [9, 34], the Hertter-Wirth (HE-WI) [16, 33] and the DNV subsurface fatigue criterion [14]. Table 1 specifies the macrogeometries of the two studied maritime bevel gears (G1, G2) with a history of TFFs and of the medium-sized test gear (B2-1) with 6 documented TFFs on 10 performed tests [4]. Listed are the gears’ pressure angle \(\alpha\), the number of teeth z, the outer pitch diameter \(d_{e}\), the tooth width b, the mean spiral angle \(\beta_{m}\), the profile shift \(x_{hm}\) and the addendum/dedendum factors \(k_{hap}/k_{hfp}\). The produced hardness profile is described through the case hardening depth \(CHD\), the surface/core hardness \(HV_{s}/HV_{c}\) and the depth of the hardness peak \(x_{HV,max}\). Its use to describe not the depth of the hardness peak but the hardness gradient in the \(CHD\) will be outlined later in the manuscript. The listed load parameters are the power P, the RPM n and the cycles to failure \(N_{f}\). All gears were manufactured from case hardened 18CrNiMo7‑6 steel.

Gear set G1 will be used to visualize and verify the proposed analytical equations against an accurate, 2D plane strain, numerical stress prediction in the gear’s mean cross-section [3]. Figure 2 plots the LTCA-derived contact stresses \(\sigma_{H,i}\) and radii of curvature \(\rho_{eq,i}\) across a G1 pinion tooth and its mean cross-section.

A standardizable approach for the prediction of TFF in spiral bevel gears has been presented. It relies on the well-established DV criterion and assumes TFF crack initiation parallel to the load-carrying flank. The DV criterion’s iteration over all material planes is thereby forgone to enable a standardizable TFF prediction. The fractographic analysis of early-stage TFFs on large maritime bevel gears and the shear stress comparison between \(\tau_{MRH,max}\) and \(\tau_{xy,a}\) substantiate the rationalization. In the relevant material depth, the stress difference between the orthogonal shear stress amplitude and the true maximum is less than 3%. TFF cracks initiate approximately parallel to the load-carrying flank under the acting orthogonal shear stresses and alter their trajectory once the crack has grown to a specific size, adopting the typically observed 45° angle. As shown in the manuscript, the DV criterion’s maximum hydrostatic stress can sufficiently be approximated as a function of Lang’s residual stress [22].

Two of the three presented gear sets suggest a close match in material utilization but not failure probability between the WI-BO [9, 34] and the simplified DV criterion. Gear set B2‑1 outlines an inaccuracy in the WI-BO criterion, in particular \(K_{2}\) and the lower limit of the \(\rho_{eq}\) range. While the plotted results for the DV criterion study a path at approximately half tooth height, the WI-BO criterion predicts the maximum utilization along a path towards the tooth tip on the wheel and the tooth root on the pinion, where \(\rho_{eq}\) is small and \(K_{2}\) exceedingly large. The following section is used to further highlight the differences between the WI-BO and DV criteria.

The presented results document comparable stress levels for the shear stress intensity derived, analytical equations of the WI-BO criterion and the herein promoted orthogonal shear stress amplitude. The integration of the shear stress across all material planes and the calculation of a single shear stress amplitude in surface-parallel material planes yield similar results. But without the need for a stress integration, the standardized DV criterion relies on two comprehensible stress equations and avoids the need for an input parameter range.

The herein outlined material model is based on numerous fatigue tests [1] and details the correlation between a hardness decrease and a ductility increase from case to core (see Eq. 12 for \(\kappa\)). The DV criterion considers that transition by increasing the impact of the acting shear stresses on A in the tooth interior and the effect of the compressive hydrostatic stresses in the hard case. As a result, it suggests the application of high hardenability steels that yield a higher core hardness after quenching and tempering to prevent TFF. When recalculating the maximum material utilization A for gear set G1, a core hardness increase from 393 to 450 HV decreases A from 1.06 to 1.05. For the same case, the WI-BO criterion predicts an increase in material utilization from 1.08 to 1.11, despite the criterion’s linear correlation between hardness and shear strength. The increase in \(A_{FF}\) can be attributed to the definition of Lang’s residual stress model [22], which sets the compressive residual stresses to 0 MPa for \(x> x_{c}\) and estimates the local residual stress from the hardness difference (\(HV(x)-HV_{c}\)). Despite relying on the same residual stress model, the DV criterion correctly predicts a reduction in material utilization for an increase in core hardness.

According to the FVA556 I and III projects [9, 34], the WI-BO criterion predicts a 50% failure probability for a material utilization of \(A_{FF}=0.8\) under constant load. What utilization is suitable for an ISO-typical 1% failure probability is unclear. The conversion factor \(f_{xK}\) promoted in this manuscript allows for the TFF-free design of industrial gears and the necessary load adjustment to reliably produce TFF on test gears. Also the technological size factor, represented through the hardness depth ratio \(T450/T550\), the statistical size factor \(K_{X}\) and lifetime factor \(K_{NT}\) help quantify the differences between small, highly-loaded test gears and large industrial gears subjected to moderate loads. The steep hardness gradient observed on large gears, the large, highly stressed volume, the lower core hardness and the load cycle accumulation over multiple years of operation elevate the TFF likelihood on maritime gears. Gear set G1 is reanalyzed at a more operationally-typical load of P = 1760 kW or 80% of the nominal load. The modified DV criterion considers now the conversion factor \(f_{xK}\) and plots the material utilization for a 1% failure probability, being much closer to the practically observed failure frequencies than the initially plotted 50% probability.

The DV criterion yields its 1.06 utilization peak in the TFF-typical depth of 4.4 mm, indicating a substantial risk for TFF even under the considered moderate loads. The high utilization corresponds well with the large number of observed crack-like indications from ultrasonic scanning on G1 [3]. To demonstrate the criterion’s applicability to the study of all LTCA contact points between meshing pinion and wheel teeth, the material utilization of a G1 pinion tooth is plotted in Fig. 12.

The article draws on the results of the ‘Improved reliability of thrusters’ joint industry project and summarizes the conducted research on the material properties of carburized CrNiMo steels, the performed gear tests and the multiaxial stress and fatigue predictions to a, for spiral bevel gears applicable, standardizable TFF approach. The highlights and main findings of this article are:The presented TFF criterion can readily be applied to a single contact point, typically the gear’s design point or the entire gear tooth, if an LTCA is carried out. The model has been set up to yield a failure probability of 1% under constant load at A = 1 when considering \(f_{xK}\) and 50% at A = 1 without \(f_{xK}\). Due to the quadratic relationship between \(\sigma_{H}\) and A, \(A_{max}\) should be set conservatively depending on the reliability of the considered load data. While extensive gear calculations were carried out in the context of this research, further studies are necessary to verify the proposed influence factors.