Int. J. Fatigue Vol. 16, No. 9, pp.

661–667, 1998
 1998 Published by Elsevier Science Ltd. All rights reserved
Printed in Great Britain
0142–1123/98/$—see front matter

PII: S0142-1123(98)00028-0

Super-long life tension–compression fatigue

properties of quenched and tempered 0.46%
carbon steel
Yukitaka Murakami*, Masayuki Takada† and Toshiyuki Toriyama‡
*Department of Mechanical Science and Engineering, Kyushu University, 6-10-1,
Hakozaki, Higashi-ku, Fukuoka 812-8581, Japan
†Mitsubishi Chemistry Co. Ltd., Bannosu-cho, Sakaide, Kagawa 762, Japan
‡Department of Mechanical and Engineering, Ritsumeikan University, Noji-higashi,
Kusatu, Shiga 525, Japan
(Received 3 October 1997; revised 23 February 1998; accepted 26 March 1998)

To investigate the effect of non-metallic inclusions on fatigue properties of quenched and tempered
0.46% carbon steel (HV ⬵ 650) in super-long life fatigue range (107 ⱕ N ⱕ 5.0 × 108), tension–
compression fatigue tests were carried out. The fatigue strength was discussed based on the √area
parameter model. The results obtained are:
1. Fatigue fracture origins were mostly at a non-metallic inclusion.
2. However, the location of the inclusion at fracture origin were not uniformly distributed over the
specimen section due to the non-uniform distribution of residual stress induced by heat treatment.
3. The fatigue limit by the cycle N = 5.0 × 108 can be predicted by the √area parameter model, i.e.
with three parameters, the Vickers hardness, HV, of the matrix, the square root of the projected area
of inclusions, √area, and residual stress.
4. The expected value of inclusion size √areamax of 0.46% carbon steel in a definite number of
specimens can be estimated using the statistics of extreme values. The lower bound of the scatter
of fatigue strength was predicted with the combination of the √area parameter model and the value
of √areamax.  1998 Published by Elsevier Science Ltd. All rights reserved.

(Keywords: √area parameter model; residual stress; super-long life fatigue; tension–compression fatigue)

INTRODUCTION size of non-metallic inclusion defects, and distribution

of residual stress and the fatigue test machine charac-
Recently, attention has been given to the fact that the
teristics all influence the fatigue properties. An accurate
S–N curves of high strength steels display a second
grasp of these factors and detailed consideration of
drop, after about 108 cycles. The first plateau occurs
experimental results are very important to understand
in the long life range (N = 106 苲 107), the second
fatigue properties correctly.
appears in the super-long life range (N ⱖ 107)1–5. A
This paper discusses the effect of matrix hardness,
number of factors influence the fatigue properties of
non-metallic inclusion size, and residual stress on the
high strength steels; evidence suggests that this
fatigue properties of quenched and tempered 0.46%
phenomenon is due to the difference in position of the
carbon steel in the super-long life fatigue range (N ⱕ
fracture origin. For specimens subjected to high stress
5.0 × 108). The appropriateness of √area parameter
it is near the surface; whereas at a lower stress, it is
model in super-long life fatigue range is examined.
in the interior. Researchers6 have established the
√area parameter model by which we can predict the
fatigue strength of components whose fracture origin THE √area PARAMETER MODEL
is a small defect or non-metallic inclusion. It has,
however, been pointed out5 that when the second drop Murakami6 demonstrated that if the fatigue fracture
appears in the S–N curve, the super-long life fatigue origin is a small defect or non-metallic inclusion, the
limit predicted by the √area parameter model is higher fatigue limit is determined by the matrix Vickers hard-
than experimental data. ness, HV, and the square root of the projected area of
The scatter of Vickers hardness of the matrix, the the defects, √area6,7. Next, they proposed the fatigue
limit prediction equations (the √area parameter model)
taking the residual stress into consideration. They
Corresponding author. Tel: + 81 92 642 3380; Fax: + 81 92 641 9744 showed that the fatigue limit of high strength steels

661
662 Y. Murakami et al.

Table 1 Chemical composition (wt %)

C Si Mn P S Cu Ni Cr

0.46 0.22 0.74 0.024 0.026 0.01 0.02 0.15

can be predicted accurately by this √area parameter the end of the process. However this releases the
model6,8. residual stress, and grips, made in advance, will warp
The √area parameter model is given as follows: and bend the specimen. If such a bent specimen is
(For surface defects and inclusions) used in a fatigue test, the experimental data will always
understate the correct value, sometimes consider-
␴w = 1.43(HV + 120)/(√area)1/6·[(1 − R)/2]␣ (1)
ably9,10. Therefore care must be taken when prepar-
ing specimens.
(For internal defects and inclusions) The surface of the tested part of the specimens was
electropolished to a depth of 50 ␮m, after polishing
␴w = 1.56(HV + 120)/(√area)1/6·[(1 − R)/2]␣ (2)
with 600 emery paper. The fatigue tests were carried
out under the following conditions: 80 Hz of tension–
Where compression fatigue test, force control system, and a
␴w: Prediction of fatigue limit, MPa stress ratio R = − 1.0. The stress ratio, R, means
HV: Vickers hardness of the matrix, kgf/mm2 nominal or apparent value. If there are residual stresses
√area: Square root of the projected area of small in the specimen, the stress ratio, R, changes locally.
defects or inclusions, ␮m An accuracy value of R, taking the residual stress into
R: Stress ratio, ␴min/␴max consideration, has to be used in Equations (1)–(3).
Four strain gauges were stuck on each specimen to
check the bending of the specimen under load.
␣ = 0.226 + HV × 10−4
After testing, fracture surfaces were observed by
microscope to check the presence of fish-eye marks.
The lower bound, ␴wᐉ, fatigue strength is obtained The hardness was taken to be the average of four
when the maximum size defect is just in contact with values measured near the fracture origin. Vickers hard-
the surface of a specimen. The prediction equation for ness, HV, was measured using a 200 g load. The
␴wᐉ is given as: scatter of the four values of Vickers hardness was
typically 5%. The difference of Vickers hardness before
␴wᐉ = 1.41(HV + 120)/(√area)1/6·[(1 − R)/2]␣ (3)
and after fatigue testing was approximately 5%.
The expected value6–8,12 of inclusion size √areamax
of 0.46% carbon steel was estimated using the statics
of extreme values.11
EXPERIMENTAL
The material used was 0.46% carbon steel rolled cylin- RESULTS AND DISCUSSION
drical bars of 26 mm diameter. Table 1 shows the
chemical composition. The specimens were machined Distribution of non-metallic inclusions at the fracture
to 14 mm diameter after annealing for 1 h at a origin
temperature of 845°C. Next, after quenching (845°C, Figure 2 shows the S–N curve, which shows as the
1 h, 20°C water cooling) and tempering (200°C, 1 h, second drop as reported in recent research. Table 2
20°C water cooling), the specimens were machined to shows the results of fatigue tests. Almost all specimens
the final dimensions. broke at a non-metallic inclusion, and the fracture
Figure 1 shows the dimensions of the specimens. surface showed fish-eye marks.
Taking deformation of the specimens by machining
into consideration, the grips of the specimens were
manufactured after specimen machinery. In many
investigations the gauge length has been machined at

Figure 1 Tension–compression fatigue specimen, mm Figure 2 S–N curve (tension–compression, f = 80 Hz, R = − 1)

 Fatigue properties of quenched and tempered 0.46% carbon steel 663

Table 2 Size and location of inclusions at fatigue fracture origin and the fatigue limit predicted by Equations (1) and (2)

Specimen no. HV (kgf/mm2) ␴ (MPa) Nf √area (␮m) h (␮m) Shape of ␴⬘w (MPa) ␴/␴⬘w
inclusions or
pits

1 652 686 5.30 × 104 33.8 0 (pit) 614 1.12

2 644 646.8 1.25 × 107 61.3 460 600.2 1.08

3 665 627.2 9.27 × 106 95.7 1455 572.6 1.10

4 660 607.6 2.05 × 107 79.3 1070 587 1.04

5 637 588 3.11 × 107 73.9 509 576.5 1.02

6 650 568.4 2.71 × 107 69.4 281 592.5 0.959

7 659 548.8 1.71 × 107 64.9 237 606.2 0.905

8 642 529.2 8.96 × 107 60.2 140 600.5 0.881

9 647 509.6 9.70 × 107 71.8 430 586.9 0.868

10 666 490 5.0 × 108→ – – – –

11 667 499.8 5.0 × 108→ – – – –

12 666 666.4 3.83 × 106 59.3 102 620.9 1.07

13 659 676.2 5.94 × 104 36.1 0 (pit) 612.8 1.10

The value of R is assumed to be − 1.0.

HV, Vickers hardness; ␴, stress; Nf, cycles to failure; √area, inclusion size; h, distance from surface; ␴⬘w, fatigue limit predicted by Equations
(1) and (2); R = − 1.0; →, not broken.

Figure 3 shows the fish-eye fracture patterns in over the specimen section13. However, in this work
0.46% carbon steel observed under the microscope. locations of the fracture origin were not uniformly
The chemical composition of non-metallic inclusions distributed over the specimen section. It is unlikely
at fracture origin were Al, S, and Ca. Judging from that the reason for this is specimen bending load.
the shape of the inclusions, they were assumed to be Rather it is likely that:
Al2O3 or Al2O3·CaO. Almost all inclusions at the
1. The reason may be the scatter of hardness distri-
fracture origin were eccentric and not at the centre of
bution in specimen section. There may be lower
fish-eye marks. Some of the fish-eye marks were not
hardness regions in the interior of the specimen by
perfect circles, for example the fish-eye mark of speci-
heat treatment, which become the fracture origin.
men no. 12 shown in Figure 3. The reason for this is
2. The residual stress may concentrate the distribution
discussed in Section 4.3.
of fracture origins. There may be residual stress in
Figure 4 shows the distribution of the distance from
the near surface region which may promote fracture.
the surface of inclusions at fracture origin. The location
of the inclusions of fracture origin were not uniformly In almost all reports of super-long life fatigue these
distributed over the specimen section. Inclusions at the matters were not examined sufficiently. These funda-
fracture origin tended to be shallow (distance from mental problems must be considered to understand
surface = 102 苲 1455 ␮m) compared with the radius fully the properties of super-long life fatigue fracture.
( = 3.5 mm) of the specimen. If these data are used This paper addresses this issue.
to predict the fatigue limit by √area parameter model
with nominal R ratio, R = − 1.0, in some cases the Vickers hardness distribution at specimen section
values of ␴/␴w (applied stress/predicted fatigue limit) The Vickers hardness distribution was measured
become less than 1.0. This means the specimens broke across a section from specimens no. 1 and no. 2
at the stress lower than predicted fatigue limit (see previously broken in fatigue tests (see Figure 5).
Table 2). If pure tension–compression fatigue tests of There are some lower hardness readings near the
high strength steels are carried out ensuring no bending, centre of specimen no. 1, but hardness near the fracture
the fracture origins ought to be uniformly distributed origin was approximately 600 苲 660 in both specimens
664 Y. Murakami et al.

Figure 5 Vickers hardness distribution in heat-treated specimens

no. 1 and no. 2. The scatter in hardness data is

approximately 10%. Therefore, it is unlikely that only
the scatter in hardness distribution is responsible for
Figure 3 Fish-eye fracture patterns in 0.46% carbon steel concentration of the fracture origin.
Residual stress distribution and predicted fatigue
limit taking residual stress into consideration, ␴⬘w
Almost all inclusions at the fracture origin were off
the exact centre of the fish-eye mark as shown in
Figure 3. The fish-eye shape of specimen no. 12 is an
ellipse rather than a circle. The fish-eye of specimen
no. 12 is almost the same as that observed in specimens
with residual compression at the surface due to shot
penning14. It is suggested that when the residual stress
near the fracture origin is higher than that of the
surface, the fish-eye will take an elliptic shape. Before
fatigue tests residual stress was neglected. However,
the residual stress obviously affects fatigue strength
judging from the fish-eye shape and concentrated distri-
bution of the fracture origin. Hence, it is necessary to
measure surface residual stress. The residual stress on
the surface of a specimen surface (for which a fatigue
test was not carried out) was measured by X-ray. The
method of preparation of the specimen and the surface
Figure 4 Distribution of the inclusions at fracture origin
was the same as described in Section 3. This specimen
survived in fatigue test (applied stress ␴ = 499.8 MPa,
number of cycles N = 5.0 × 108), after the residual
stress had been measured.
 Fatigue properties of quenched and tempered 0.46% carbon steel 665

Figure 6 Residual stresses on the surface of a specimen, MPa

Figure 6 shows the residual stresses measured. The Considering Figures 2 and 8, the first horizontal part
residual stress in the middle of the surface of the of the S–N curve is 10% higher than the predicted
specimen was equal to + 150 MPa. Measured residual fatigue limit, ␴⬙w, and though N will exceed 5.0 × 108,
stress was not affected by machining or polishing by data for two of the unbroken specimens in Figure 2
emery paper, because the surface of the specimen was is predicted to be ␴/␴⬙w = 0.9 苲 1.0. Therefore, it may
electropolished approximately 50 ␮m par diameter. be said that the prediction of fatigue limit by the
Therefore, it is quite natural that a residual stress of √area parameter model is effective in the case of
approximately + 150 MPa should exist in the interior. internal inclusions.
Only one specimen was measured. It is likely that
the distribution of residual stress will be different in Prediction of the range of fatigue strength scatter
other specimens. The main reason why fatigue strength and fatigue
Figure 7 shows a modified S–N curve that compares life varies in each specimen is that the size of the
the ratio of applied stress, ␴, to fatigue limit, ␴⬘w, inclusion, which becomes the fracture origin, depends
predicted by √area parameter model (Equations (1) and on the specimen (as shown in Table 2)6–12. To predict
(2)) with the stress ratio R = − 1.0, ␴/␴⬘w, and the the fatigue strength scatter, we predict the maximum
number of cycles to failure, Nf, without taking the size of inclusions, √areamax, contained in a definite
residual stress into consideration. The minimum value numbers of specimens and adapt to the √area parameter
of ␴/␴⬘w is 0.868. model6–8,14. The prediction of the range of the scatter
Figure 8 shows the modified S–N curve taking of fatigue strength is useful to estimate the strength of
residual stress into consideration, assuming the tensile a large number of parts like mass-produced car compo-
residual stress (+150 MPa) existed at the fracture ori- nents15.
gin. In Figure 8, the value of R is equal not to − 1.0 Figure 9 shows plots of the cumulative probability
and depends on applied stress. The minimum value of of √areamax in the probability graph of extreme value
␴/␴⬘w is 0.932, but it is possible that the residual stress for 0.46% carbon steel produced in 1985. The standard
at the fracture origin is higher than + 150 MPa, and inspection area is S0 = 0.384 mm2 and number of
hardness at the fracture origin is lower than the value inspections is n = 40. Estimating the return period
used to predict fatigue limit. Therefore, the actual T, for N specimens, the expected maximum sizes of
value of ␴/␴⬙w may be regarded as higher than 0.932. inclusions √areamax are 34.9, 40.4, 45.9 ␮m for N =

Figure 7 ␴/␴⬘w vs. Nf without taking the residual stress into con- Figure 8 ␴/␴⬙w vs. Nf with taking the residual stress into consider-
sideration. ␴, Applied stress; ␴⬘w, predicted fatigue limit ation. ␴, Applied stress; ␴⬙w, predicted fatigue limit
666 Y. Murakami et al.

Figure 9 Statistical distribution of the maximum size of inclusions Figure 10 Statistical distribution of the inclusions at fracture origin.
of quenched and tempered 0.46% C steel produced in 1985 and 0.46% C steel (1985)
1996. S0 = 0.384 mm2, n = 40

1, 10 and 100 specimens, respectively. For determining problem. Therefore the statistical distribution of 0.46%
√areamax for N specimens from Figure 9, for example, carbon steel produced in 1985 would be measured with
the return period T for one specimen was calculated6,16 S0 = 66.37 mm2, n = 20. The results are shown in
by dividing the risk volume V of a specimen by the Figure 11. It can be assumed that these data are
standard inspection volume V0 which is the product of sufficiently linear and follow statistical distribution of
S0 and the mean value (√areamax,m) of 40 (n = 40)
√areamax in Figure 9. The risk volume of a specimen
is the volume of the specimen with a diameter of 7
mm in Figure 1, namely, V = ␲/4 × 72 × 20 = 769.69
mm3. The value of √areamax,m in Figure 9 is 5.8116
␮m. Thus, we have V0 = S0 × √areamax,m = 2.23 × 10−3
mm3, T = V × N/V0 = 3.45 × 105 and it follows that
√areamax = 34.92 ␮m for N = 1 from Figure 9 or
from the distribution equation obtained by the least
squares method (y = − ln[ − ln{(T − 1)/T}] and areamax
= 2.3849y + 4.51511).
Figure 9 also shows plots of 0.47% carbon steel
produced in 1996. The √areamax of 0.46% carbon steel
produced in 1985 is larger than 0.47% carbon steel
produced in 1996.
Figure 10 shows the cumulative probability of the
extreme value of non-metallic inclusions at the fracture
origin in a broken specimen. It may not be fair to use
data of eccentric distribution of inclusion at the fracture
origin on a specimen section, but data may be suf-
ficiently linear and follow some statistical distribution.
These √area are much larger than 45.9 ␮m predicted
by statistical distribution of the extreme values in
Figure 9 for 100 specimens. The largest size of
√areamax measured in plots of Figure 9 (17.5 ␮m) is
a lot larger than the second largest size (11.5 ␮m) and
is located far from the least squares line. This problem
may be caused by the fact that there were few inspec-
tions or the standard inspection area was not suf- Figure 11 Statistical distribution of the maximum size of inclusions
ficiently large. It might be that using a larger size of of quenched and tempered 0.46% C steel produced in 1985. S0 =
S0 compared with S0 = 0.384 mm2 would improve this 66.37 mm2, n = 20
 Fatigue properties of quenched and tempered 0.46% carbon steel 667

to extend the standard inspection area, S0. If suf-

ficient linear data are obtained, the results can be
used as a relative quality comparison between
materials produced at different times or localities.
Moreover, fatigue strength scatter can be predicted
easily by this method. The prediction based on this
method in this paper was in good agreement with
experimental results.

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