Engineering Fracture Mechanics 73 (2006) 1947–1958

www.elsevier.com/locate/engfracmech

Determination of stress intensity factors

for three-dimensional subsurface cracks in hypoid gears
M. Guagliano, L. Vergani, M. Vimercati *

Dipartimento di Meccanica, Politecnico di Milano, Via La Masa 34, 20156 Milan, Italy

Received 25 January 2006; accepted 1 April 2006

Available online 9 June 2006

Abstract

Aim of this paper is to propose a numerical tool able to compute the stress intensity factor for the Mode I, II and III
along the front of any subsurface cracks found in spiral and hypoid gear teeth, allowing to assess the criticality of the crack
itself. The described approach is based on two steps: firstly the contact pressures in the un-cracked teeth are efficiently com-
puted by an advanced contact solver; in this way the complex tooth geometry and the intricate meshing condition can be
accurately considered; then, the displacements field due to those pressures is applied as boundary conditions to a finite
element model of the cracked zone and the SIFs are efficiently computed. This information, together with the knowledge
of the DK threshold value of the material, makes possible to decide whether the defect is acceptable and to evaluate the
conditions of propagation.
Ó 2006 Elsevier Ltd. All rights reserved.

Keywords: Hypoid gear; Rolling contact fatigue; Spalling; Pitting; Subsurface crack

1. Introduction

The reliability of a mechanical system is strictly related to the survival of the several components which
made up the device itself. The easiest way to guarantee the mentioned reliability is to oversize these compo-
nents, even though nowadays, due to the ever more severe requirements which the machines have to accom-
plish, this policy is not satisfactory. For example, in the automotive and aerospace power transmission, the
engineers are required to design gears which are able to carry heavy load at high speed and, at the same time,
have the minimum size and weight. These goals can be achieved only by applying a ‘‘design by analysis’’ pro-
cess, i.e., a more refined design approach including very accurate simulation able to reproduce the actual
working condition of the mating elements has to be adopted. With this aim, it is fundamental to investigate
and to properly understand the mechanisms of the damage and, starting from this knowledge, to be able to
predict the component failure.

*
Corresponding author. Tel.: +39 0223998249; fax: +39 0223998202.
E-mail address: martino.vimercati@polimi.it (M. Vimercati).

0013-7944/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.engfracmech.2006.04.002
1948 M. Guagliano et al. / Engineering Fracture Mechanics 73 (2006) 1947–1958

As known [1], the service life of gears is mainly influenced by the teeth damage which can occur due to fati-
gue loading conditions; in particular the researcher’s attention has been focused on the bending fatigue failure
at the tooth root and on the rolling contact fatigue (RCF) which induced damage on the tooth mating sur-
faces. In this paper this latter phenomenon will be considered. Experimental investigation shows that RCF
can produce two types of surface damage: pitting, which starts from cracks just on the surface of the teeth,
and spalling, which finds its origin in cracks placed at some distance from the surface, usually in correspon-
dence of an internal material defect sited near the position of maximum shear stress. Both these mechanisms
cause the removal of the surface layer of material as a consequence of the stable propagation of the cracks; the
size of the damage due to spalling is larger and causes more rapid surface deterioration than the pitting does
[2].
Many researches have dealt with this issue trying to understand the actual mechanisms of the RCF dam-
ages; this task is quite complex because many aspects, such as material properties, size and distribution of the
internal defects, surface geometry, loading condition, lubrication, etc, have to be considered. The investigation
started with the experiment of Way [3] who assumed that the lubricant pressure into a crack was a possible
mechanism of Mode I crack growth; Fleming et al. [4] and Hills et al. [5] made the first attempt to use fracture
mechanics to characterize subsurface crack growth and Keer et al. [6] computed the SIFs for Mode I and II for
a subsurface crack in an elastic half-space loaded by Hertzian contact pressure; then, Sin et al. [7] and Kaneta
et al. [8] have shown that Mode II propagation is dominant for the growth of horizontal subsurface crack
parallel to the surface.
More recently, many papers about RCF damage expressly in the gear field, which is also the object of the
present research, have been proposed; Blake et al. developed a pitting life model based on fracture mechanics
in order to estimate service lives and failure probabilities in spur gear [9]; Glodez et al. presented several mod-
els for simulation of the surface fatigue process in the contact area, allowing a proper determination of the
spur gear pitting/spalling resistance [10,11]; Flodin et al. proposed models for wear prediction in spur and heli-
cal gears [12,13]; Ding et al. found in the ligament collapse the mechanism for spalling formation in spur gear
[2]; Guagliano et al. described a Weight Function based approach to predict spur gear spalling [14] and carried
out a contact fatigue failure analysis of shot-peened spur gears [15]. As evident, the gears considered in all of
these works are cylindrical, that is, they are characterized by simple tooth geometry and it is possible to handle
them using 2D schematization. On the contrary, RCF in gear pairs having crossing or intersecting axes - such
as spiral or hypoid gear which are widely employed in aerospace or automotive field - have not yet significantly
investigated; nowadays the only way to handle this subject is to refer to the International Standards [16] which
are very conservative and do not allow to accomplish the previously mentioned ‘‘design by analysis’’. The
cause of this lack of knowledge is reasonably related to the fact that it is a really tough task to reproduce
the complicated tooth geometry and to simulate the intricate meshing condition which typically occur for
these categories of gear; in particular, when the aim is to compute the contact pressure distribution over
the tooth flank, referring to the Hertz theory is not enough.
This paper would like to make a first attempt to deal with this challenging topic proposing a versatile
approach to calculate the stress intensity factors of internal cracks in hypoid gears. It derives from a previously
developed study which is originally aimed to analyze cracked railway wheels under Hertzian loading condition
[17]. By properly modifying that approach, non-Hertzian contact pressure can be also taken into account and
it become possible to handle cracked hypoid gear. The computation procedure it is based on the following
steps (see Fig. 1): firstly, starting from an accurate 3D geometry description of the gear tooth [18], a con-
tact/stress analysis, which is carried out by means of an advanced numerical solver [19,20], allows to compute
very precisely the contact pressure distribution over the un-cracked teeth during the whole meshing cycle [21];
then, this non-Hertzian pressure distribution is provided as loading condition for the calculation, according to
the Boussinesq theory, of the displacement field in the un-cracked tooth which is properly reduced to a half-
space; finally, the calculated displacement components are applied as boundary conditions to a finite element
model of the zone surrounding the crack allowing the computation of the stress intensity factors along the
crack front.
The main advantage of this method consists in the capability of obtaining easily and with limited calcula-
tion time results concerning different values of the parameters (mainly position, dimensions and shape of the
crack) that influence the stress intensity factors in a complete loading cycle.
 M. Guagliano et al. / Engineering Fracture Mechanics 73 (2006) 1947–1958 1949

Fig. 1. Computational procedure for determining the stress intensity factors of internal cracks in hypoid gears.

In the present paper, the results obtained for the hypoid gearset belonging to a real differential truck trans-
mission will be described: by computing the trend of stress intensity factor for the Mode I, II and III over the
whole meshing cycle, the mentioned approach has allowed to asses the criticality of internal defects of the
material, which have been simulated as circular cracks. These data, together with the knowledge of the DK
threshold value of the material, will also give the chance to evaluate the conditions of internal defects
propagation.

2. The method of analysis

This section is devoted to a detailed description of the computational procedure for calculating the stress
intensity factors of subsurface cracks in hypoid gears. As mentioned, the approach consists in three main steps
(see Fig. 1): computation of the contact pressure distribution in the un-cracked tooth, evaluation of the dis-
placement field under the tooth surface and calculation of the SIFs along the crack front.
The reliability of the method has been previously verified [17] by comparing it with models from literature;
in particular, the results showed a difference less than 10% with respect of the data collected by Kaneta et al.
[8].

2.1. Computation of the contact pressure distribution

In order to compute the pressure distribution over the tooth flanks, it is necessary to develop the gear con-
tact analysis. As prescribed by the theory of gearing [22], this aim can be accomplished in a reliable way only if
the geometry of the mating surfaces is very accurately described; this is true especially when complex tooth
shape, as the one object of this study, are handled. In this paper, an articulate algorithm based on the numer-
ical simulation of the gear cutting process [18] allowed to compute very precisely the mathematical represen-
tation of hypoid gear tooth surfaces. Fig. 2 shows the representation of the pinion meshing with the driven
gear member which has been obtained for the studied hypoid gear drive by means of the mentioned mathe-
matical model.
Then, these gear tooth surfaces have been provided as input for an advanced contact solver which combines
a semi-analytical surface integral theory (for solving the contact problem) and the traditional finite element
method (for computation of gross deflections associated with tooth bending) [19,20]. This approach makes
possible to carry out very accurate contact analysis and stress calculation employing a relative coarse mesh;
in particular, unlike the usual solvers based only on the Finite Element Method, a locally refined mesh around
the contact region is not required. This latter characteristic is very advantageous when, as it happens in the
present study, the researcher is interested in studying the whole meshing cycle trying to capture the contact
zone which travels fast over the two mating bodies. Fig. 3 reports the contact pressure 3D plots computed
for the pinion and for the driven gear in one meshing instant [21]; it is evident that the load is shared between
more than one tooth pair and that the pressure distribution shows a typical sharp and oblong shape. These
pictures make also clear that such complex loading conditions could not be reproduced by the Hertz theory
which, on the contrary, is widely employed to solve successfully contact problem in spur gear.
1950 M. Guagliano et al. / Engineering Fracture Mechanics 73 (2006) 1947–1958

Fig. 2. Representation of the geometry for the studied hypoid gear: (a) pinion mating with the gear member; (b) zoom of the mating teeth.

Fig. 3. 3D contact pressure plotted in one meshing instant: (a) pinion; (b) gear.

2.2. Computation of the subsurface displacement field

The second step of the procedure requires the calculation of the displacement field which is induced under
the tooth surface by the previously obtained contact pressure distribution; with this aim, it is convenient to
schematize the tooth as a half-space. This assumption can be considered realistic for the tooth of the driven
member. In fact, unlike the pinion, the cutting process usually adopted for manufacturing this member [23]
produces a simpler tooth geometry allowing to neglect the curvature along the tooth profile (this evidence
 M. Guagliano et al. / Engineering Fracture Mechanics 73 (2006) 1947–1958 1951

Fig. 4. Some viewing points of the contact pressure distribution on the gear tooth convex surface which is reduced to a plane: (a) 3D plot;
(b) 2D plot—above view; (c) 2D plot—front view.

is easily noted—see Fig. 3—by comparing the transverse tooth profile of two members). For these reasons, the
following discussion is referred to the gear member; future developments of this work will be aimed to assess
the extension of that assumption also to the driving member.
According to these considerations, Fig. 4 shows from different points of view the contact pressure distribu-
tion computed in one meshing instant applied to the two dimensions geometric development of the gear tooth
convex surface; x-axis and y-axis are the measures of the curvilinear coordinate respectively along the face
width and along the profile of the tooth. Doing this way, the loading condition is being reduced to a pressure
distribution applied to the free plane of a half-space.
This representation allows to straightforwardly obtain the displacement field under the tooth surface. With
this aim, the contact pressure distribution is firstly schematized as a set of finite number of point forces normal
to the free surface of the half-space. Then, according to the Boussinesq theory [24,25] and employing
the scheme reported in Fig. 5, the displacement components induced by each of those point loadings are

Fig. 5. Scheme for computing the displacement field.

1952 M. Guagliano et al. / Engineering Fracture Mechanics 73 (2006) 1947–1958

analytically computed; the reference frame (r, h, z) is cylindrical and has the origin in the point of loading
application. Omitting the analytical calculations, the following relations allow to obtain the state of stress
under the tooth surface:
   
P 1 z 2 2 1=2 2 2 2 5=2
rr ¼ ð1  2mÞ 2  2 ðr þ z Þ  3r zðr þ z Þ ð1Þ
2p r r
3P
rz ¼  z3 ðr2 þ z2 Þ5=2 ð2Þ
2p  
P 1 z
rh ¼ ð1  2mÞ  2 þ 2 ðr2 þ z2 Þ1=2 þ zðr2 þ z2 Þ3=2 ð3Þ
2p r r
P 5=2
srz ¼ 3 rz2 ðr2 þ z2 Þ ð4Þ
2p
Then, the terms u and w, which are the components of the displacement respectively along r and z axes, can
be computed:
 
ð1  2mÞð1 þ mÞ 1
u¼ zðr2 þ z2 Þ1=2  1 þ r2 zðr2 þ z2 Þ3=2 ð5Þ
2pEr 1  2t
P h 3=2 1=2
i
w¼ ð1 þ tÞz2 ðr2 þ z2 Þ þ 2ð1  t2 Þðr2 þ z2 Þ ð6Þ
2pE
These terms can be also expressed in a rectangular reference frame (x, y, z) by using the following relation:

ux ¼ uðr; zÞ cos h ð7Þ

uy ¼ uðr; zÞ sin h ð8Þ
uz ¼ wðr; zÞ ð9Þ
Repeating these calculations for each point loading Pi and by adding each contribute, the displacement field
everywhere in the half-space is known:
X
n
ux;tot ¼ ux;i
i¼1
Xn
uy;tot ¼ uy;i ð10Þ
i¼1
Xn
uz;tot ¼ uz;i
i¼1

In this paper, friction between the mating surfaces has been neglected. Aim of future work will be to take
into account also this aspect by implementing in the model the equations of the displacement field related to
the tangential loads which appear due to the friction.

2.3. Computation of the stress intensity factors

The results obtained by the previous step are used as boundary conditions of a three-dimensional finite ele-
ment model of the zone surrounding the crack: in particular, the displacements at the points corresponding to
the nodes of the boundary surface of the FE model are applied to these latter and are used as boundary con-
ditions of the analysis. In this way it is simple to accurately evaluate the SIF for Mode I, II and III along the
crack front.
The FE model consists in a cylinder of radius R and height H containing the three-dimensional crack whose
front can be circular or elliptical (Fig. 6); a is the half-length of the crack. The dimensions of the FE model are
critical parameters. In fact, it is necessary to verify that the dimensions are large enough to guarantee that the
boundaries of the model are not influenced by the presence of the crack; doing so, it is possible to apply the
 M. Guagliano et al. / Engineering Fracture Mechanics 73 (2006) 1947–1958 1953

Fig. 6. Finite element model of the cracked zone: (a) the whole model; (b) particular of the model showing the crack.

displacements analytically calculated in the uncracked half-space to the FE model. After a trial and error pro-
cedure, the chosen dimensions were the following ones: R = 4.5a and H = 3a.
Brick elements with 20 nodes and second order shape functions were used. Special elements were employed
to simulate the contact between the crack faces, avoiding their overlapping. The friction between the crack
faces can be included in the model. This model allows a very refined mesh near the crack front, where the
1/4 point technique was used to better simulate the stress singularity according to LEFM. The stress intensity
factors along the crack front are calculated by using direct methods (through the nodal displacements) or the
virtual crack extension technique.

3. A case study

The described procedure has been applied in order to asses the criticality of internal cracks in a real hypoid
gear drive which belongs to a truck differential system. Table 1 summarizes the basic geometric data of the
studied gearset: the transmission has a 15 teeth pinion and a 44 teeth gear-member; pinion offset is equal
to 25 mm. The material is 21NiCrMo4 steel (UTS = 1650 MPa, YS = 1100 MPa); the tooth surfaces are sub-
jected to case hardening.
With the aim to perform contact analysis, a value of input torque equal to 250 N m has been chosen; this
loading level is placed in the middle of the application range of the truck.
Focusing the attention on one tooth, the whole meshing cycle has been divided and analyzed in 50 steps
(details about the setting of this contact analysis are fully reported in [21]).
Fig. 7a shows the complete loading history computed for the convex side of one gear tooth. In detail, for
sake of clarity, the contact pattern of only 13 instants extracted from the previously mentioned 50 analysis

Table 1
Main geometric characteristics of the studied hypoid gear drive
Parameter Pinion Gear
Module [1/mm] 5.11
Shaft angle [°] 90
Offset [mm] 25
Number of teeth 15 44
Mean spiral angle [°] 43.00 28.90
Hand of spiral Left Right
Face width [mm] 41.43 38.00
Outer cone distance [mm] 106.40 126.10
Pitch angle [°] 26.88 62.41
Addendum [mm] 5.09 2.96
Dedendum [mm] 3.91 6.04
1954 M. Guagliano et al. / Engineering Fracture Mechanics 73 (2006) 1947–1958

Fig. 7. Contact pressure distribution computed during the whole meshing cycle for the studied hypoid gear: (a) whole contact pattern
history; (b) maximum pressure vs meshing instant.

cases are plotted; for each pattern the point of maximum pressure is highlighted by a black point; the hori-
zontal segment is the pitch line. Fig. 7b reports the trend of maximum pressure value versus the meshing step
and makes clear that in the 25th step the highest pressure value (1016 MPa) is reached. The shape of each con-
tact pattern is similar to an ellipse; the value of the parameters describing each pattern (length of the ellipse
axes and the maximum pressure value) differs from the other patterns; that is, the pressure pattern changes
while is travelling over the tooth flank.
As shown, once the loading history is known, the proposed approach allows to calculate the stress intensity
factors trends for internal cracks having any shape, dimension and position during the whole meshing cycle. In
this paper, the authors are interested to evaluate the criticality of internal defects of the material such as voids
or inclusions; with this aim a circular crack with radius a equal to 0.015 mm, which is a typical inclusion size,
has been considered. Making a conservative assumption, in the analysis friction between crack faces has been
neglected.
The attention has been focused on the most critical zone of the tooth that is the region just under the con-
tact pressure computed in the 25th meshing step; this area is also in proximity of the pitch line which, as
known, is a preferential place for pitting/spalling formation. Considering this loading case, it is useful to place
a reference frame having the origin in the point of maximum pressure and the x and y axes parallel to the axes
of the pseudo-contact ellipse (Fig. 8a).
Table 2 summarizes the crack positions which have been analyzed: referring to the crack center, the crack
has been moved along the x-axis and the y-axis; moreover, being the aim of the authors to study the spalling
formation, the crack is placed always on a plane parallel to the free surface to a depth where the tangential
stresses due to the loading step 25th are maximum (z = 0.25 mm). In Fig. 8a, the crack in the four extreme
positions (corresponding to an eccentricity on x-axis equal to ±1.0 mm and on y-axis equal to ±10 mm) is
 M. Guagliano et al. / Engineering Fracture Mechanics 73 (2006) 1947–1958 1955

Fig. 8. Definition of the main parameters of the analysis: (a) crack position on the tooth surface; (b) calculation points along the crack
front.

Table 2
Positions of the crack considered in the analysis
Eccentricity Crack center coordinate Eccentricity Crack center coordinate
x [mm] y [mm] z [mm] x [mm] y [mm] z [mm]
x  1.0 1.00 0.00 0.25 y  10 0.00 10.00 0.25
x  0.8 0.80 0.00 0.25 y8 0.00 8.00 0.25
x  0.6 0.60 0.00 0.25 y6 0.00 6.00 0.25
x  0.4 0.40 0.00 0.25 y4 0.00 4.00 0.25
x  0.2 0.20 0.00 0.25 y2 0.00 2.00 0.25
x 0.0 0.00 0.00 0.25 y0 0.00 0.00 0.25
x + 0.2 0.20 0.00 0.25 y+2 0.00 2.00 0.25
x + 0.4 0.40 0.00 0.25 y+4 0.00 4.00 0.25
x + 0.6 0.60 0.00 0.25 y+6 0.00 6.00 0.25
x + 0.8 0.80 0.00 0.25 y+8 0.00 8.00 0.25
x + 1.0 1.00 0.00 0.25 y + 10 0.00 10.00 0.25

reported; it also evident that all the crack positions are analyzed considering the meshing steps from the 19th
through 31st; in fact, it has been verified that, for such crack positions, all the remaining loading steps have no
significant influence on the SIFs value.
Finally, it is convenient to define four points on the crack front (Fig. 8b): points A and B are aligned with
the x-axis; points C and D are aligned with the y-axis. These points are the ones considered for the stress inten-
sity factors calculation.

3.1. Results and discussion

Once the calculation grid over the gear tooth surface is set up, it is possible to compute the stress intensity
factors allowing to map the criticality of the defects.
Observing the results obtained for all the analyzed cases, it is easy to note that everywhere in the tooth the
stress intensity factor for Mode I is null; this is due to the compressive nature of the load. According to this
evidence, in the following, results in terms of Mode II and III are discussed.

3.1.1. Defect critically versus crack x-positions

The first crack position which has been considered is the one having eccentricity along x-axis equal to
1 mm. Referring to the point A of this crack, Fig. 9a reports the trend of KII during the meshing cycle; this
graph reveals that the KII value is not null when the loading step is located in proximity of the crack. Fig. 9b
1956 M. Guagliano et al. / Engineering Fracture Mechanics 73 (2006) 1947–1958

Fig. 9. Trends of stress intensity factor for Mode II in point A for cracks along x-axis.

and c which report the same results for two other values of eccentricity along x-axis (x = 0 mm and
x = +1 mm) make clear that the trend, without changing the shape, follows the crack position and shifts
on the left.
For each crack position, it is helpful to introduce the parameter DKII as the difference between the maxi-
mum and the minimum value of KII which have been recorded over the meshing cycle. As well shown in
Fig. 9d, where the trend of DKII in A versus the crack positions along p the x-axis is summarized, it is simple
to note that the value of DKII slightly varies between 2 and 2.5 MPa m. The trends of KII in point B has been
verified to be equal to the ones reported for point A. DK for p Mode III in points A and B can be neglected
because it has been found to be always lower than 0.06 MPa m. p
As expected KII values for point C (or D) are negligible (<0.06 MPa m). On the contrary, regarding KIII in
points C (or D), the results showed in Fig. 10 have been collected. The trends are very similar to thepones
obtained for KII in point A; the main difference is that, now, the DKIII value is slightly lower (1.5–2 MPa m).
All these results allow to affirm that moving the crack along x-axis does not affect significantly the criticality
of the crack itself. This evidence is reasonable due to the fact that, although the cracks have a different position
with respect to the contact pressure history, all the cracks are subjected to a similar loading history.
Comparing the SIF values in points A (or B) and C (or D) with threshold values of the material, it is easy to
figure out whether the crack propagation is possible.

Fig. 10. Trends of stress intensity factor for Mode III in point C for cracks along x-axis.
 M. Guagliano et al. / Engineering Fracture Mechanics 73 (2006) 1947–1958 1957

Fig. 11. Trends of stress intensity factor for Mode II in point A for cracks along y-axis.

3.1.2. Defect critically versus crack y-positions

Analyzing the cracks placed along the y-axis, the following results have been obtained; Fig. 11a, b and c
show the trend of KII in the point A for the three characteristic crack positions along y-axis (eccentricity
y = 10 mm, y = 0 mm, y = +10 mm).pFor the case with eccentricity y = 10 mm (Fig. 11a), p the absolute
value of the maximum KII (0.5 MPa m) is quite different from the minimum one (0 MPa m); in fact
that, for this crack position, contact pressure distribution on the left of point A is heavier than the one on
the right. As expected, for the case with eccentricity
p y = 0 mm (Fig. 11b), the same results of the case with
eccentricity x = 0 mm is computed (2.5 MPa m). KII value for the case with eccentricity y = +10 mm is
null because the contact pressure distribution is quite far from the crack and does not affect the crack itself
(Fig. 11c). The trend of DKII versus the crack position along y-axis well summarizes that situation
(Fig. 11d): as the crack is coming closer to the origin of the reference frame, DKII increases, achieving the maxi-
mum value just in the point of maximum pressure. Point B can be p handled in analogous way. Value of DK for
Mode III in points A and B is always lower than 0.125 MPa m and consequently in this case it can be
neglected.
Referring to KIII in points
p C and D, it is possible to make a similar discussion (Fig. 12); just the maximum
DKIII is lower (2 MPa m).
The analysis along y-axis confirms that the most critical region is located in proximity to the most loaded
zone of the tooth.

Fig. 12. Trends of stress intensity factor for Mode III in point C for cracks along y-axis.
1958 M. Guagliano et al. / Engineering Fracture Mechanics 73 (2006) 1947–1958

4. Conclusions

In this paper, a numerical tool able to assess the criticality of any subsurface cracks found in spiral and
hypoid gear teeth has been proposed. The main characteristics of the computational approach have been
described: firstly the contact pressure in the un-cracked teeth are efficiently computed by an advanced contact
solver, then the displacements field due to that pressure is applied as boundary conditions to a finite element
model of the cracked zone, being the aim the stress intensity factors calculation for the Mode I, II and III
along the crack front.
As application of the procedure, the criticality of internal defect of material in a real hypoid gear drive
which belongs to a truck differential system has been mapped. Firstly, the complex loading history has been
analyzed and schematized as several pseudo-elliptical contact pressure distributions which are characterized
by a shape which changes while is travelling over the tooth flank; then, the material defects have been simu-
lated as a circular crack and it has been tested, over the whole meshing cycle, in several positions located in the
region where the contact pressure distribution is the heaviest. As expected, in all the analyzed cases the stress
intensity factor for Mode I was null, while remarkable values of KII and KIII have been obtained. In detail, it
has been noted that the maximum DKII values are reached in correspondence of the short axis of the pseudo-
ellipse corresponding to the heaviest loading step; this area is also located in proximity of the pitch line which
is a typical region for the pitting/spalling formation.

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