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$: Parameter study on the calculated risk of tooth flank fracture of …€¦ · Besides the properties of the heat influenced near-surface zone, different residual stress profiles and$
Bulletin of the JSME

Journal of Advanced Mechanical Design, Systems, and ManufacturingVol.11, No.6, 2017

Paper No.17-00194© 2017 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2017jamdsm0074]

Parameter study on the calculated risk of tooth flank fracture

of case hardened gears

Michael HEIN*, Thomas TOBIE* and Karsten STAHL* * FZG – Gear Research Centre, Technical University of Munich

Boltzmannstraße 15, 85748 Garching, GERMANY

E-mail: [email protected]

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, Fig. 1 (a)) or tooth interior fatigue fracture

(TIFF, Fig. 1 (b)).

Tooth flank fracture is usually observed on the driven gear of a single gear stage with one-sided flank loading

whereas tooth interior fatigue fracture often occurs on idler gears with reverse loading.

1

Received: 27 March 2017; Revised: 20 April 2017; Accepted: 7 May 2017 Abstract Due to better material qualities, new surface finishing methods and better heat-treatment process reliability, flank surface damages, such as pitting or micropitting, can be prevented in a reliable manner. This results in an increase of unexpected flank damages with crack initiation below the surface of the loaded gear flank, 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 direction of both the active flank surface and the core area. Damages caused by tooth flank fracture usually result in a total breakdown of the gear unit. The main mechanisms leading to tooth flank fracture have been investigated in different research projects. By now, an ISO technical specification for calculation of tooth flank fracture load capacity for case hardened spur and helical gears is in preparation. Based on the available draft technical specification, a parameter study on the load capacity calculation of the damage mechanism tooth flank fracture has been performed in order to identify characteristic influence factors. Furthermore, the tooth flank fracture load capacity was compared to the pitting load carrying capacity for different example gearings. Based on the parameter study, the influence of surface hardness, hardness gradient and core hardness on the damage mechanism tooth flank fracture is characterized. With these findings, different heat treatment processes and material characteristics can be quantified regarding their susceptibility for tooth flank fracture damages. Besides the properties of the heat influenced near-surface zone, different residual stress profiles and geometrical parameters (radii of curvature, module …) have been analyzed, too. Based on the performed parameter study, design limits for practical application have been derived and are presented in the present paper. These derived design limits allow a fast estimation of the exposure concerning tooth flank fracture for a given gear unit design.

Keywords : Tooth flank fracture, Load capacity, Basic calculation principles, Parameter study, ISO technical specification, Tooth flank breakage, Tooth interior fatigue fracture

2© 2017 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2017jamdsm0074]

Hein, Tobie and Stahl, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.11, No.6 (2017)

Fig. 1 Exemplary pictures of tooth flank fracture (a) (Witzig, 2012) on the left and tooth interior fatigue fracture (b)

(MackAldener and Olsson, 2002) on the right

Tooth flank fracture is characterized by a crack initiation below the active flank surface due to shear stresses

caused by the Hertzian contact pressure and a crack propagation in the direction of both the active flank surface and the

core area (see Fig. 2 (a)). Typically, the primary crack is orientated in an angle of 40-50° to the flank surface. Tooth

interior fatigue fracture is also characterized by a crack initiation below the active flank surface but has a different

fracture shape caused by two potential crack initiation points due to the reverse loading (Al et al., 2016) (see Fig. 2 (b)).

Damages caused by tooth flank fracture or tooth interior fatigue fracture usually result in a total breakdown of the gear

unit.

Fig. 2 Crack propagation of Tooth flank fracture (a) and Tooth interior fatigue fracture (b) (Witzig, 2012). The crack

initiation for tooth flank fracture is below the active flank surface due to shear stresses caused by the Hertzian contact

pressure whereas tooth interior fatigue fracture shows a different fracture shape caused by two potential crack initiation

points due to reverse loading.

In this paper, the damage mechanism tooth flank fracture is treated in detail as it is gaining more and more

industrial importance. Since tooth flank fracture can also occur if the load carrying capacity regarding pitting

(according to ISO 6336-2 (ISO 6336-2, 2006)) and tooth root breakage (according to ISO 6336-3 (ISO 6336-3, 2006))

is sufficient, the demand for a standardized calculation method for assessing the risk of tooth flank fracture is growing.

By now, a standardized method for the calculation of the tooth flank fracture load capacity of cylindrical, case hardened

spur and helical gears is in preparation (ISO/DTS 19042 (ISO/DTS 19042, in preparation)). This simplified method,

established by FZG/Witzig (Witzig, 2012), has been derived from and the results are in good correlation with already

existing high sophisticated calculation methods (Hertter, 2003; Höhn et al., 2010; Oster, 1982).

Tooth flank fracture (TFF) (a) Tooth interior fatigue fracture (TIFF) (b)

Loaded flanks

Po

ints

on

pat

ho

fco

nta

ct

Loaded flank

Po

ints

on

pat

ho

fco

nta

ct

Direction of crack

propagation

Crack initiation

and primary crack

Secondary crack

Tooth flank fracture (TFF) (a) Tooth interior fatigue fracture (TIFF) (b)

2



2. Failure description

Tooth flank fracture is a typical 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

transmissions, wind turbines or turbo transmissions and bevel gears for water turbines (e.g. (Bauer and Böhl, 2011)).

Failures are also known from specially designed test gears for gear running tests (Bruckmeier, 2006; Tobie, 2001;

Witzig, 2012) 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 approx. half

the height of the tooth and the crack starter is in the material depth, typically in the area of the case-core interface, and

often – but not always – associated with a small non-metallic inclusion. The primary crack grows in both directions,

towards the surface as well as in the material depth in an angle of approx. 40-50° to the tooth flank surface. Subsequent

cracks growing from the surface may occur (see Fig. 2 (a)). The final breakage is due to forced rupture. The fractured

surface shows typical fatigue characteristics. (Tobie et al., 2013)

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.

3. Material-physically based FZG-model for the assessment of the risk of tooth flank fracture

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 basically developed by Oster (Oster, 1982) and Hertter (Hertter, 2003). This model is based on the comparison

of a local occurring 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 (Tobie et al., 2013) the calculation model is basically able to take into

account the following influences also shown in Fig. 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 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 (Ghribi et al., 2015).

A detailed description of the material-physically based FZG-model for the assessment of the risk of tooth flank

fracture can be found in (Boiadjiev et al., 2014; Hertter, 2003; Höhn et al., 2010; Oster, 1982; Tobie et al., 2013;

Witzig, 2012). The calculation procedure needs comprehensive and detailed input values and some integrations and

iteration steps are necessary. Therefore, this method is quite complex. For this reason, a more practical-oriented

calculation approach was derived by FZG/Witzig (Witzig, 2012) and is presented in the following.

3



Fig. 3 Main parameters for the load & stress condition of a loaded tooth flank according to Tobie (Tobie et al., 2013)

are the stresses induced by the Hertzian contact and the EHD-contact as well as the influence of the friction load,

residual stresses, surface roughness and thermal load.

4. Practical calculation approach for tooth flank fracture load capacity

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 (Witzig, 2012). 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 (Witzig, 2012)

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, EHD contact, surface asperities or thermal load

Valid for case-carburized gears

The basic formulae for assessing the risk of tooth flank fracture according to the practical approach of

Witzig (Witzig, 2012) are similar to the ones used in the FZG-model. The decisive parameter for evaluating the risk of

tooth flank fracture according to Witzig (Witzig, 2012) is the local material exposure AFF which is the quotient of local

equivalent stress state in the material depth y, τeff(y), and the local material strength τper(y):

4



04,0)(

)(

y

yA

per

eff

FF

(1)

The local equivalent stress state τeff(y) according to Witzig (Witzig, 2012) considers the local equivalent stress

state without consideration of residual stresses τeff,L(y), the influence of residual stresses on the local equivalent stress

state Δτeff,L,RS(y) and the quasi-stationary residual stress state τeff,RS(y). The local material strength τper(y) is a function of

the local hardness HV(y) and the material, which is defined by the hardness conversion factor Kτ,per and the material

factor Kmaterial.

)()()()( ,,., yyyy RSeffRSLeffLeffeff (2)

)()( , yHVKKy materialperper

(3)

The local equivalent stress state without consideration of residual stresses τeff,L(y) according to

FZG/Witzig (Witzig, 2012) is based on the calculation of the stress state of an Hertzian line-contact between

semicircle and half plane according to Föppl (Föppl, 1947). FZG/Witzig (Witzig, 2012) 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.

54,14

4,0

416

41488,0

)(

22

2

2

,

Hred

r

redr

H

red

r

red

r

H

Leff

p

Ey

y

E

p

EyEyp

y

(4)

Therefore the approximation given in equation (4) was derived without consideration of residual stresses and shear

stresses induced by friction, EHD contact, surface asperities or thermal load. It is dependent on the Hertzian stress pH,

the reduced modulus of elasticity Er, the local radius of relative curvature ρred and the material depth y.

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 τeff(y). The practical approach according to FZG/Witzig (Witzig,

2012) 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. 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. Following, the empirically

determined formulae for calculating the influence of residual stresses on the local equivalent stress state Δτeff,L,RS(y)

according to FZG/Witzig (Witzig, 2012) are shown. Influence factors are the residual stress depth profile σRS(y), the

case hardening depth CHD at 550 HV and the maximum value of the residual stresses σRS,max. With help of the

adjustment factors K1 and K2 it is possible to describe the influence of the residual stresses on the local equivalent stress

state Δτeff,L,RS(y) in a closed form. The following described adjustment factors K are derived from calculations with

sophisticated calculation methods.

2

1,1

1,. 9tanh32100

)()( Ky

yKy

RS

RSLeff

(5)

max,max, ,

58,4

,1 tanh)1(RSHRSH pCHDp KyKKK

(6)

5



The quasi-stationary residual stress state τeff,RS(y) can be calculated according to the following formula.

Concerning the residual stresses, the same assumptions as previously stated have to be made.

)(15

2)(, yy RSRSeff

(9)

Fig. 4 shows the components of the local equivalent stress state τeff(y) according to FZG/Witzig (Witzig, 2012) for

an example test gearing used for tooth flank fracture tests specified in Table 1. The calculation was performed for the

lower point of single tooth contact B. It clearly can be seen that the influence of the residual stresses is significant only

in the hardened case layer where a significant hardness gradient is present.

Table 1 Parameters of an example test gearing according to FZG/Witzig (Witzig, 2012) used for tooth flank fracture

tests – example 1

Pinion Gear

Normal module 3,0

Normal pressure angle

(deg)

20.0

Helix angle (deg) 0

Centre distance (mm) 200

Material 18CrNiMo7-6

Number of teeth 67 69

Surface hardness (HV) 700 700

Core hardness (HV) 440 440

Case hardening depth

(mm)

0,50 0,50

Fig. 5 shows the local equivalent stress state τeff(y), the local material strength τper(y) and the resulting local

material exposure AFF(y) for the above mentioned example. As the material strength τper 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∙bH. This is also

the depth where the maximum local equivalent stress state τeff,max and therefore the maximum material exposure AFF,max

is calculated. As AFF,max ≈ 1,1, this test gearing has a high calculated risk of failing by tooth flank fracture for the given

load. Actually, this example failed in the test run by flank fracture.

6

CHDCHDCHD eeK 834,1151,5 115,025,11 (7)

10100

200100

2tanh

10161101,0tanh

max,max,2

2

RS

H

RS

red

p

yCHD

K

(8)



Fig. 4 Components (influence of residual stresses on the local equivalent stress state Δτeff,L,RS(y), local equivalent stress

state without consideration of residual stresses τeff,L(y), quasi-stationary residual stress state τeff,RS(y)) of the local

equivalent stress state τeff(y) for the example test gearing; half of the Hertzian contact width bH = 0,37 mm; Hertzian

stress pH = 1475 N/mm²

Fig. 5 Local equivalent stress state τeff(y), local material strength τper(y) and local material exposure AFF(y) for the

example test gearing

τeff

τeff,L

τeff,RS

Δτeff,L,RS

τeff

AFF

τper

7



Witzig (Witzig, 2012) 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 pH

Radius of relative curvature ρred

Case hardening depth CHD

Residual stress depth profile

For all parameter combinations, the material exposure was evaluated in the material depth range of 0∙bH ≤ y ≤ 9∙bH,

where bH is half of the Hertzian contact width, regarding the calculated maximum material exposure AFF,max, the

material exposure depth profile and the depth coordinate of the maximum material exposure ymax. In summary, the

results provided by the practical approach derived by FZG/Witzig (Witzig, 2012) are in very good accordance to the

results of the material-physically based FZG-model for material depths y ≥ 1∙bH. Fig. 6 exemplary shows the

comparison of the maximum material exposure calculated according to the practical approach by FZG/Witzig (Witzig,

2012) (AFF,max,Witzig) and the respective values according to the FZG-model (AFF,max,FZG).

According to FZG/Witzig (Witzig, 2012), 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∙bH and is validated only for

parameters within the following ranges:

500 N/mm² ≤ pH ≤ 3000 N/mm²

5 mm ≤ ρred ≤ 150 mm

0,3 mm ≤ CHD ≤ 4,5 mm

Based on experimental investigations on case-carburized gears and industrial examples (Boiadjiev et al., 2014;

Tobie et al., 2013; Witzig, 2012), it is known, that a maximum material exposure AFF,max ≥ 0,8 can lead to tooth flank

fractures for gear materials of typical quality and cleanness (quality MQ according to ISO 6336-5 (ISO 6336-5, 2016)).

Fig. 6 Comparison of the maximum material exposure calculated according to simplified calculation approach and

according to material-physically based FZG-model (Witzig, 2012); The results provided by the practical approach

derived by FZG/Witzig (Witzig, 2012) are in very good accordance to the results of the material-physically based

FZG-model for material depths y ≥ 1∙bH.

5. 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 (DNVGL -CG-0036, 2015) 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 in preparation by now (ISO/DTS 19042, in preparation). ISO/DTS 19042 "Calculation of tooth flank

fracture load capacity of cylindrical spur and helical gears" 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 (Witzig,

Maxim

um

mate

rial exposure

AF

F,m

ax,W

itzig

acc.

tosim

plif

ied

calc

ula

tion

appro

ach

Maximum material exposure AFF,max,FZG acc. to FZG-model

8



2012) and is supplemented by some advises 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 AFF,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 7 calculation points along the path of contact in profile

direction. In direction of the facewidth, the most critical section has to be chosen. For every specified contact point CP

(see Fig. 7), the local material exposure AFF 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 AFF,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 bH.

Fig. 7 Definition of local contact point on the tooth flank (ISO/DTS 19042, in preparation)

The calculation of the Hertzian stress pH according to ISO/DTS 19042 (ISO/DTS 19042, in preparation) 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 (ISO 6336-2, 2006) (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 (Lang, 1979) or Thomas (Thomas, 1998) and the residual stress

depth profile can be approximated by the approach of Lang (Lang, 1979).

6. Main influence parameters on tooth flank fracture and practical applicability

The chapter focuses on the main influence parameters on the assessment of the risk of tooth flank fracture on

case-carburized gears according to ISO/DTS 19042 (ISO/DTS 19042, in preparation) and the practical applicability of

the presented DTS. Fig. 8 gives an overview on these main influence parameters on the calculation of the local material

exposure AFF, taking into account the load parameters, the gear geometry as well as the material data.

Beermann (Beermann, 2015) 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 (ISO/DTS 19042, in

preparation). For his parameter study, the calculation approach delivers reasonable results. Pinnekamp (Pinnekamp and

Heider, 2015) 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 correct for

these examples. This is in good accordance 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 are always matching the detected damages.

9



Fig. 8 Influence parameters on the local material exposure

Fig. 9 shows a variation of the core hardness for given values of surface hardness and CHD for the test gearing

presented in Table 1. With increasing core hardness, the material strength τper is increasing –at least from a certain

material depth on- but also the local equivalent stress state τeff is influenced because the approximated residual stresses

according to Lang (Lang, 1979) are decreasing due to the modified hardness profile. The resulting values of AFF,max are

not significantly changing, but the depth coordinates of the maximum material exposure ymax change. 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.

Fig. 9 Variation of core hardness for example 1 (test gearing presented in Table 1); With increasing core hardness, the

material strength τper is increasing –at least from a certain material depth on- but also the local equivalent stress state τeff

is influenced because the approximated residual stresses according to Lang (Lang, 1979) are decreasing due to the

modified hardness profile.

External load (Torque, K-

Factors)

Tooth flank geometry: Micro

geometry Macro

geometry

Local material exposure

FFA

Local equivalent stress state

eff

Local Hertzian

contact stress

dynp

Normal radius of relative curvature

red

Local material shear strength

per

Material: Material

constants Hardness

Depth Profile

Residual Stress Profile

Core Hardness (CH): 450 HV

Surface Hardness: 720 HV

CHD: 0,5 mm

430 HV

410 HV

390 HV

370 HV

350 HV

Core Hardness (CH): 450 HV

430 HV

410 HV

390 HV

370 HV350 HV

10



7. 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 be more and more prevented 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. 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 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 19042 "Calculation of tooth flank fracture load capacity of

cylindrical spur and helical gears" is currently prepared. It is mainly based on the simplified approach of FZG/Witzig

(Witzig, 2012) and is supplemented by some advises for practical use regarding the determination of hardness and

residual stress depth profiles and calculating the Hertzian stress. Main influence parameters on the risk of tooth flank

fracture are the Hertzian stresses, the radii of curvature as well as the hardness and residual stress depth profile. As up

to now some influence parameters are covered empirically only, further research work is performed to include more

specific material parameters, e.g cleanness and tensile residual stresses, into the calculation method.

Acknowledgement

This work was supported by the German Research Foundation (DFG) and the Technical University of Munich

(TUM) in the framework of the Open Access Publishing Program.

This paper is based on the presentation held at the "Motion and Power Transmissions – MPT 2017" Conference in

Kyoto (Hein et al., 2017).

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hardened gears. ASME 2013 DETC, Portland, Oregon. ASME (2013).

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- Eine Grenze der Zahnradtragfähigkeit in der Werkstofftiefe). Doctoral Thesis, Technical University of Munich

(2012).

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