A NEW MODEL FOR FATIGUE FAILURE DUE
TO CARBIDE CLUSTERS

A Melander and S Haglund

Swedish Institute for Metals Research

Drottning Kristinas väg 48

S-114 28 Stockholm

Sweden

Abstract

A new model was developed for fatigue failure due to clusters of carbide

particles in hardened steels. The model calculates the failure probability for
fatigue specimens due to the clusters. The model considers a random distribu-
tion of spherical carbide particles. A cluster is defined as a spherical volume
of material where the local carbide fraction is above a certain minimum level.
The probability to find such clusters of critical sizes from fatigue crack prop-
agation point of view is calculated. The critical cluster size is defined by the
threshold value for the stress intensity of a crack through the cluster.

The model is used to predict the effect of carbide size and carbide volume

fraction on the failure probability for fatigue specimens. The model is applied
to a number of tool and high speed steels. In most cases the model predicts
fatigue strengths due to carbide clusters above 800MPa ie on the level of or
above the fatigue strength levels observed experimentally. This is believed
to indicate that clustering can be significant from the fatigue point of view
for grades of tool and high speed steels with carbide fractions from 20% and
upwards.

INTRODUCTION

In three previous reports [1, 2, 3] a model was developed for calculation

of the failure probability of fatigue specimens containing distributions of
particles like inclusions and carbides. The model was based on fracture
mechanics and it was assumed that a specimen would fail if it contains at least

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6TH INTERNATIONAL TOOLING CONFERENCE

one carbide or inclusion which has a critical size. It was assumed that cracks
exists at all particles already before fatigue testing and that they have the same
size as the particles. The critical particle/crack size is reached if the stress
intensity of the crack is above the threshold value for crack propagation.
The model only contains parameter values which can be determined with
independent experiments like fracture mechanics tests and image analysis
combined with optical and SEM microscopy for inclusion and carbide size
distributions. The model was applied to data for tool and high speed steels.
In this model only cracks around single particles are considered. This is fully
relevant for inclusions in clean steels due to the low density of inclusions.
The carbide fractions in tool and high speed steels can however become
so large that more than one particle is involved in the fracture process. In
such steels crack formation and propagation in clusters of carbides should
be considered.

We first make a simple illustration of the likelihood for interactions be-

tween cracks at neighbouring carbides by considering the distances between
carbides in different close packed configurations. For spherical carbides
of one single size packed in cubes with one carbide per cube the carbides
will touch when the carbide volume fraction is close to 50%. At the car-
bide fraction of 25% the shortest distances between carbide surfaces is only
28% of the carbide diameter. In both these cases strong interaction between
cracks at neighbouring carbides can be expected. In the most densely packed
configuration, the so called fcc packing, the carbides will touch when the
carbide fraction approaches 75%. The implication of these considerations
is that interaction between cracks at neighbouring carbides can be expected
in clusters with local carbide fractions from some 25% and upwards.

In the present report the same basic ideas from fracture mechanics and

statistics are used as in [1, 2, 3] but instead of single carbides particles we
consider clusters of several carbides which together generate a crack which
can lead to failure of a fatigue specimen. Such models could be expected to
be applicable for steels with high average carbide contents.

THE MODEL

A model was formulated for fatigue failure caused by a cluster of carbide

particles. The model is based on the assumption that a cluster of carbides
can cause a fatigue crack of the same size as the cluster size early in fatigue
life. That crack will be permanently arrested if the range of stress intensity

A New Model for Fatigue Failure due to Carbide Clusters

831

of the crack at the cluster is below the threshold value for fatigue crack
propagation,

∆K

th

. If however the range of stress intensity is above the

threshold value then the crack will grow and the specimen will eventually
fail.

∆K = 2.0 · ∆σ ·

s

DCR

2π

(1)

∆K = ∆K

th

(2)

The expression for the stress intensity

∆K in eq. (1) is based on a penny

shaped crack in an infinite medium.

DCR is the diameter of the crack of the

cluster and

∆σ is the stress range. The critical crack diameter at the cluster

corresponding to the threshold value for crack propagation is called

DCR

c

.

A cluster was defined as a volume,

V , of material within which the volume

fraction of carbide phase was larger than a critical value,

f

c

, see Fig. 1. The

critical carbide fraction should be so large that a crack could form on the
equator plane of the cluster early in fatigue life, Fig. 2. It is estimated that
this would mean that the volume fraction

f

c

is between 0.25 and 0.75. If

cracks are easily generated around the individual carbides in the cluster these
cracks would then be expected to grow rapidly the short distances between
the carbides so an equatorial crack in the cluster is formed, see Fig. 3.

We make the simplifications that all carbides have the same size and that

the carbides and clusters are spherical. The carbide diameter is,

s, and the

cluster diameter is,

D. All carbides which have centres inside the cluster

sphere belong to the cluster. Some of these carbides will extend partly
outside the cluster sphere. This means that the crack which is generated by
the carbides of the cluster has a diameter of

DCR = D + s.

We can then formulate the probability that the cluster volume is filled

with

N carbides by using Poisson statistics for the space distribution of

carbides. The Poisson distribution is characterised by the expected number
of particles,

λ, in a specific volume. If the average fraction of carbides in

the material is

f

p

then the expected number of carbides in the cluster volume

can be written

λ = (D/s)

3

· f

p

(3)

If we require that the carbide fraction is

f

c

inside the volume

V in order

to define a cluster then a certain minimum number of carbides is needed to
fill the cluster

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6TH INTERNATIONAL TOOLING CONFERENCE

s

Figure 1.

Cluster of carbides.

s

Figure 2.

Cracks at carbides in cluster.

D

DCR

Figure 3.

Crack through cluster.

N

c

= λ

f

c

f

p

(4)

A New Model for Fatigue Failure due to Carbide Clusters

833

This number is often several times larger than the average expected num-

ber. The most likely number of carbides in the cluster volume is the expected
number. This means that the probability is small to find a significantly larger
number of carbides in the cluster volume.

The probability to find

N

c

or more carbides in the cluster volume

V can

be written

P (D) =

∞

X

n

=N

c

(D)

λ

n

(D)

n!

· exp

−

λ

(D)

(5)

this sum is converted to a finite sum in the following way

P (D) = 1 −

N

c

(D)−1

X

n

=0

λ

n

(D)

n!

· exp

−

λ

(D)

(6)

The expected number of critical clusters in the specimen volume, Vspec,

which have cracks which exceed the threshold criterion , eqs 1 and 2, is then

L

c

=

Z

∞

D

c

dD

Z

V

spec

dV

spec

· P (D)

1

V (D)

=

Z

∞

D

C

dD · l(D)

(7)

where

∆K

th

= 2.0 · ∆σ ·

s

(D

c

+ s)

2π

(8)

D

c

= 2π



∆K

th

2 · ∆σ



2

− s

(9)

where the expected number of clusters is first calculated per each cluster

volume

V (D) and this is subsequently integrated to obtain the expected

number of clusters in the whole specimen volume

V

spec

, see Table 1 for

model parameters.

The failure probability for the specimen is equal to the probability that

at least one critical cluster exists in the volume. If the clusters are Poisson
distributed the probability can be written

P = 1 − exp

−

L

c

(10)

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6TH INTERNATIONAL TOOLING CONFERENCE

Table 1.

Reference values of parameters in the model

Parameter

Reference value

∆K

th

4 Mpa

√m

s

2.5 µm

f

p

0.20

f

c

0.50

V

spec

1.0 · 10

−

6

m

3

PREDICTIONS OF THE MODEL

In this chapter a number of calculations will be presented to illustrate how

the predictions of the model depend on different model parameters. In the
calculations the reference case was set as presented in Table 1.

Figure 4 illustrates how the critical crack size,

DCR, at the threshold value

depends on the stress range,

∆σ, eqs.(1) and (2). A cluster size around 20

µm is needed in order to produce a critical stress range of 1000 MPa.

The critical number of carbides needed to fill a cluster of diameter

D is

plotted in Fig. 5. The calculation is performed with eqs. (3) and (4) for
a carbide size of s = 2.5 µm and a filling degree of

f

c

= 0.5. When the

cluster size is 100 µm the number of carbide particles approaches 100. If the
carbide fraction in the material is 20 % then the average number of particles
in the same volume as the 100 µm cluster is 40 i.e. 60 carbides less than in
the cluster.

Figure 6 illustrates how the integrand

l(D) of eq.(7) varies with the cluster

size for the reference set of parameters. It can be seen that the integrand
falls rapidly with increasing cluster diameter. This means that the expected
number of critical clusters decreases rapidly with increasing cluster size. At
a diameter of approximately 15 µm the number of carbides in the cluster
becomes so large, i.e. over 100, so that numerical overflow occurs in the
calculation of several factors in the equation. However for smaller cluster
diameters it can be seen that the curve shows a stepwise variation. This is
caused by the summation in integral numbers of carbides.

The failure probability for carbide volume fractions from 5 % up 40 % are

illustrated in Fig. 7. For the carbide fraction of 40 % the failure probability
increases rapidly around the stress range 1200 MPa. For the carbide fraction
of 5 % this increase comes around 1800 MPa. It should be mentioned that

A New Model for Fatigue Failure due to Carbide Clusters

835

the fatigue strength of tool or high speed steels of hardness around 60 HRC
often is slightly below 1000 MPa. In these cases the failures are often caused
by inclusions. The results of Fig. 7 thus indicate that clusters of carbides
will only influence the fatigue strength in cases where the average carbide
fraction is well above 40 %. This statement of course applies only to the
specific selection of model parameters of Table 1.

The degree of filling of carbides in the cluster has the effect on the failure

probability illustrated in Fig. 8. The failure probability is quite sensitive
to the exact value of fc. The argument presented in the introduction of the
report illustrates that the distance between carbide surfaces of neighbouring
particles is quite small even for a

f

c

value around 25 %. This means that all

three values of

f

c

plotted in Fig. 8 are fully realistic but

f

c

= 0.75 is probably

too conservative. It is interesting to note that the failure probability goes up
rapidly around the stress range 1000 MPa for the critical filling degree of
f

c

= 0.25. The diagram was calculated with the average particle fraction of

f

p

= 0.2 see Table 1. This means that provided that f

c

= 0.25 is a realistic

value then clusters will be a significant cause of fatigue failure for steels with
f

p

= 0.2 carbides and higher.

The effect of the carbide size on the failure probability is illustrated in Fig.

9. It can be seen that a change from a carbide size of 1.5 µm to 5 µm gives
a shift in the stress range where the failure probability goes up of almost
1000 MPa.

CALCULATIONS FOR MEASURED CARBIDE SIZE
DISTRIBUTIONS

The present model is formulated for carbide distributions of one single

size. In those cases where information is available on the carbide size dis-
tribution the parameters in the model can be calculated in the following
way.

A carbide size distribution has been determined from sections through a

material. The accumulated size distribution has been given an exponential
representation

N

A

(x) = f · exp

−

kx

(11)

where

f and k are constants and x is the equivalent size of a carbide

section in the planar section through the material. Equivalent size means

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6TH INTERNATIONAL TOOLING CONFERENCE

that the real observed carbide section has the same section area as a circle
with diameter

x.

This planar distribution can be converted to a three dimensional size dis-

tribution by assuming that the particles are spherical [2].

g

V

(x) =

s

2k

πx

· f · k · exp

−

kx

(12)

where

s is the diameter of the spherical particle.

The present model contains the input parameters

f

p

and

s which depend

on the size distribution of the carbides. We define

f

p

=

Z

∞

0

g

V

(x) ·



4π

3



·



x

2



3

dx

(13)

which can be solved with the aid of the gamma function

f

p

=

√

2π

6k

2

· f · Γ(3.5)

(14)

We chose the particle size which corresponds to the maximum in the

particle volume distribution ie. in

h(x) = g

V

(x) ·



4π

3



·



x

2



3

(15)

This maximum occurs for

s = x

max

= 2.5/k

(16)

In the previous reports [1, 2, 3] carbide size distributions were determined

for a number of tool and high speed steels. The experimental data were fitted
to the carbide distribution expression of eq. 11. The fitted parameters are
reproduced in Table 9 for the four steels ASP2014, PM23, VANADIS10 and
M2.

Failure probabilities were calculated for the carbide distributions of Table

9 with the parameters (

f

c

= 0.5 respectively 0.25, ∆K

th

= 4 MPa

√m and

V

spec

= 1.0·10

−

6

m

3

) and the results are presented in Figs. 10 and 11. Firstly

the results for PM23 62

HRC are disregarded since the average carbide frac-

tion is above the fc value in both cases. This means that the present cluster
definition loses its meaning. For the other steels the predicted failures due

A New Model for Fatigue Failure due to Carbide Clusters

837

Table 2.

Carbide distribution parameters [2]

Material

f [mm

2

]

K [µm

−

1

]

s [µm]

f

p

PM23 62HRC

5064567

3.665

0.68

0.52

PM23 66HRC

1417143

3.075

0.81

0.21

ASP2014

456313

3.922

0.64

0.067

VANADIS10

174349

1.012

2.47

0.23

M2

7720

0.653

3.83

0.025

to clusters occur between 1500 MPa and 3500 MPa in the case of

f

c

= 0.5

and between 800 MPa and 3200 MPa in the case of

f

c

= 0.25. This can be

compared to the experimentally determined fatigue strengths at 2 million cy-
cles of 1053 MPa for M2, 1294 MPa for VANADIS10, 1438 MPa for PM23
66 HRC and 1605 MPa for ASP2014. The steels thus fail at lower stress
ranges than predicted from carbide clustering with one exception namely
VANADIS10 calculated with

f

c

= 0.25 where the predicted fatigue strength

falls below the experimentally observed one. In the case of VANADIS10
studies of fracture surfaces showed that 1 out of 12 failures were caused
by carbide clusters. The carbide cluster in VANADIS10 had a size around
30 µm. In the case of the other three steels no carbide clusters were observed
to initiate fatigue failure. The failures were instead caused by inclusions or
individual large carbides. It can be concluded that carbide clusters play a
role in the fatigue failure of VANADIS10 where the average carbide fraction
is around 25 % and the carbide size around 2.5 µm but that clusters play a
minor role in the other steels. The results of Figs. 10 and 11 indicate that a
reasonable value of

f

c

lies somewhere mid between

f

c

= 0.25 and 0.50.

CONCLUSIONS

A new model for the failure probability of fatigue specimens containing

random distributions of carbides was presented. Crack generation at clusters
of carbides was considered.

The fatigue strength due to clusters decreases with increasing carbide
volume fraction (carbide size and critical filling degree in cluster are
constant).

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6TH INTERNATIONAL TOOLING CONFERENCE

The fatigue strength due to clusters decreases with decreasing critical
filling degree in a cluster (carbide size and volume fraction carbides
are constant).

The fatigue strength due to clusters decreases with increasing carbide
size (critical filling degree in cluster and volume fraction carbides is
constant).

The model predicts fatigue strengths due to clusters down to 800 MPa
for carbide fractions of around 20 % and carbide sizes around 2 µm.
This is on the same level as experimentally determined fatigue strengths
for tool and high speed steels. This indicates that carbide clusters are
of significance for fatigue failure in such materials.

ACKNOWLEDGMENTS

The present research was financed jointly by the Uddeholm Research

Foundation, Jernkontoret (TO80) and Erasteel Kloster which is gratefully
acknowledged.The authors are indebted to the members of the industrial
steering group for their great interest and support. The steering group con-
sists of Leif Westin, Erasteel Kloster, Odd Sandberg, Uddeholm Tooling,
and Kerstin Fernheden, Jernkontoret.

REFERENCES

[1] F. MEURLING, Influence of carbide distribution, inclusion contents and surface ma-

chining processes on the fatigue properties of powder metallurgically produced high
speed steels and tool steels for cold work applications, Swedish Institute for Metals
Research, Report IM-1999-017(1999).

[2] F. MEURLING, FATSIMR – Fatigue limit prediction software for high speed steel and

cold work steel specimens, Swedish Institute for Metals Research, Report IM-1999-
547(1999).

[3] F. MEURLING, A. MELANDER, M. TIDESTEN and L. WESTIN, Influence of carbide

and inclusion contents on the fatigue properties of high speed steels and tool steels,
International Journal of Fatigue 23(2001)215-224.

A New Model for Fatigue Failure due to Carbide Clusters

839

0

5

10

15

20

25

1000

1500

2000

2500

∆σ

(MPa)

Cr

itical cluster diameter

, D

c

(

µ

m)

Figure 4.

Critical crack size as a function of stress range.

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6TH INTERNATIONAL TOOLING CONFERENCE

0

20

40

60

80

100

120

140

4

6

8

10

12

14

16

Cluster diameter, D (

µ

m)

Cr

itical n

umber of par

ticles

, N

c

Figure 5.

Number of carbides in a cluster of diameter D when the carbide size is 2.5 µm

and f

c

= 0.5.

A New Model for Fatigue Failure due to Carbide Clusters

841

0

2

4

6

8

10

12

14

16

18

20

10

10

10

10

10

10

10

10

10

10

30

25

20

15

10

5

0

5

10

15

Cluster diameter, D (

µ

m)

l(D)

Figure 6.

The integrand l

(D) of eq 7.

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6TH INTERNATIONAL TOOLING CONFERENCE

1000

1500

2000

2500

fp=0.40

fp=0.20

fp=
0.10

fp=
0.05

∆σ

(MPa)

F

ailure probability

, P

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 7.

Failure probability of specimen for different carbide volume fractions. Param-

eters are given in Table 1.

A New Model for Fatigue Failure due to Carbide Clusters

843

800

1000

1200

1400

1600

1800

2000

∆σ

(MPa)

F

ailure probability

, P

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

fc=0.50

fc=0.75

fc=0.25

Figure 8.

Failure probability of specimen for different degrees of filling of carbides in

each cluster. Please note that the horizontal axis extends to lower stresses than in the other
diagrams. Parameters are given in Table 1.

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6TH INTERNATIONAL TOOLING CONFERENCE

1000

1500

2000

2500

∆σ

(MPa)

F

ailure probability

, P

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

s=1.5

µ

m

s=2.5

µ

m

s=5

µ

m

Figure 9.

Failure probability of specimen for different carbide sizes. Parameters are given

in Table 1.

1000

1500

2000

2500

3000

3500

4000

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

∆σ

(MPa)

F

ailure probability

, P

PM23 66HRC

V

ANADIS10

M2

ASP 2014

Fc=0.50

Figure 10.

Failure probabilities for the steels VANADIS10, M2, PM23 and ASP2014

calculated with f

c

= 0.5.

A New Model for Fatigue Failure due to Carbide Clusters

845

500

1000

1500

2000

2500

3000

3500

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

∆σ

(MPa)

F

ailure probability

, P

PM23 66HRC

V

ANADIS10

M2

ASP 2014

Fc=0.25

Figure 11.

Failure probabilities for the steels VANADIS10, M2, PM23 and ASP2014

calculated with f

c

= 0.25.