# Rapid Determination of theS-N Curve for Steel by means of the Thermographic Method

Authors: Adam Lipski
Journal: Advances in Materials Science and Engineering
 (2016)

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## Abstract

The paper presents a thermographic method of accelerated determination of theS-N curve. In the presented method, theS-N curve was developed based on energy-related parameter with the assumption of its dependency on the stress amplitude. The tests made on C45 steel and X5CrNi18-10 steel under reversed bending revealed that theS-N curve obtained by accelerated thermographic method fits inside the 95% confidence interval for theS-N curve obtained from the full test.


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## Body

## 1. Introduction

TheS-N curve is the basic parameter defining fatigue strength properties of the material under cyclic load, particularly in the case of the so-called stress approach. The fatigue tests are long-lasting and expensive. For example, the minimum number of specimens required to obtain theS-N curve depends on the type of test program conducted and ranges from 12 to 24 for data on design allowables or reliability data (ASTM E 739 91 (reapproved 2004)). This requires testing of specimens at tens or hundreds of thousands of cycles, which significantly influences duration of tests. A test of an individual specimen depends on load frequency and can last even several days, thus making it expensive as well. Hence researchers struggle to develop new methods or to improve existing ones for rapid determining ofS-N curve. Thanks to those methods it is possible to significantly cut cost of fatigue tests by reduction of the quantity of specimens and shortening test duration.Thanks to modern, fast, and high sensitivity thermographic cameras it is possible to record surface temperature distribution of the tested object (specimen) and thus obtain among other things new information about physical phenomena connected with loading and the change of strain state in structural members. This research focuses on phenomena related to temperature changes induced by monotone tension and compression load (e.g., [1–3]) and fatigue load under both uniaxial and multiaxial loading (e.g., [4–7]). Based on the above and other works it is possible to introduce new strength testing methods, including those related to the fatigue of materials and structures.The effect of temperature changes induced by fatigue loading was initially used for development of accelerated methods for determining fatigue limit. The researcher who first used thermography to determine a surface temperature distribution of a tested specimen and thereby determine the fatigue limit was Risitano [8]. The method proposed by him was used or modified, among others, in the studies [9–12]. According to a similar concept, the fatigue limit is determined using the lock-in method [13, 14]. In these studies, the fatigue limit is usually determined for the axial load based on the assumption that there is a temperature stabilization period in Phase 2 of the fatigue life (Figure 1, line A). Luong [9, 10] has also applied Risitano’s methodology to determine the fatigue limit of XC55 steel under four-point rotating bending at 100 Hz loading frequency.Figure 1
The change in the temperatureT during the fatigue test for material featuring Phase 2 temperature stabilization (curve A) and a constant rate of temperature rise (curve B) (t: time).The temperature evolution can be used for fatigue life prediction. Huang et al. [15] showed that for materials with high ductility under rotating bending the slope of the temperature rise at the beginning of Phase 3 (Figure 1) can be correlated to the fatigue life. The presented methodology may be useful for terminating the operation of machinery in order to prevent catastrophic failure. Jiang et al. [16] showed that the life of specimen undergoing axial fatigue load is related to the temperature rise from the initial temperature. Jiang et al. suggested that the temperature rise during Phase 2 can be used as an index of fatigue life. However, it is inapt for some materials that do not exhibit steady-state temperature in Phase 2 [17]. Meneghetti [18] presented a technique for assessment of fatigue life, which involves measuring the cooling rate correlated to the specific heat energy after a sudden interruption of the fatigue test. He showed that fatigue life is correlated to the specific heat energy.Effects of temperature changes induced by fatigue loading can be used for development of accelerated methods for determining the fatigueS-N curve. Fargione et al. [19] proposed for uniaxial loading that the integration of the area under the temperature rise curve over the entire number of cycles of a specimen (area under curve A, Figure 1, especially under curve A in Phase 2) remains constant and is independent of the stress amplitude, thus representing a characteristic of the material. Amiri and Khonsari [20, 21] used the rate of temperature rise in Phase 1 for prediction of fatigue failure.However not all materials show temperature stabilization period during fatigue life in Phase 2. That was observed, for example, by Krewerth et al. [22], Plekhov et al. [23], and Wang et al. [24]. This fact significantly limited the use of previously developed thermographic methods of rapid determination of the fatigue curve.A thermographic method for determination of theS-N curve was presented in this paper. This method is based on the parameter proposed by Fargione et al. [19] and used in [25–27] but with assumption of its dependency on the stress amplitude as indicated by Crupi et al. [28]. Because this parameter is proportional to energy dissipated under cyclic load which is dependent on the stress level [29] then this assumption seems to be more appropriate.Application of the proposed method was presented in [30], where it was used to determine the inclination of theS-N curve which is important from the point of view of the accuracy of the fatigue limit determination by the Locati method [31, 32]. This paper presents the improved version of the analysis of the accelerated test results, allowing outside determination of the inclination of theS-N curve determination of its position on theS-N plane.


## 2. Test Description

### 2.1. Test Station

The tests were performed at the Department Laboratory for Research on Materials and Structures (certified by the Polish Centre for Accreditation, PCA AB 372) of the Faculty of Mechanical Engineering at the UTP University of Science and Technology using the reversed bending fatigue testing machine (with rotational frequency of 77 Hz).The main item of the test station was the thermographic camera CEDIP Silver 420M (FLIR SC5200) equipped with high sensitivity InSb matrix cooled using Stirling pump.The main parameters of the camera are as follows: resolution of sensor 320 × 256 pixels, spectral range 3.6 ÷ 5.0μm, sensitivity below 20 mK (available: 8 mK), and maximum frequency 140 Hz for the entire matrix (up to 25 kHz at lower resolution).The thermographic camera recorded (at the constant frequency from 100 to 450 Hz depending on load level) the surface temperature distribution of the tested object (specimen) fixed in the testing machine. Camera images were transmitted via USB interface to PC with appropriate software, where they were digitally stored directly on HDD in form of PTW format files.The following software was installed on the computer: VirtualCAM allowing two-way communication between PC and the thermographic camera via USB interface, CIRRUS Front End which was the user interface used to control the CEDIP camera, and ALTAIR allowing downloading, storage, and advanced processing of thermographic images.


### 2.2. Test Specimens

Test specimens were made of cold-drawn bars from the following:(i)
Medium carbon C45 (1.0503) steel.
(ii)
Corrosion resistant austenitic X5CrNi18-10 (1.4301) steel.Chemical composition of the specimen material is specified in Tables1 and 2. The geometry of applied specimens is shown in Figure 2.Table 1
Chemical composition of the specimens made of C45 steel used for tests (wt%).
C
Si
Mn
P
S
Cr
Mo
Ni
Cr + Mo + Ni
Acc. to PN-EN 10083-2:2008
0.42–0.50
<0.40
0.50–0.80
<0.045
<0.045
<0.40
<0.10
<0.40
<0.63
For tested specimens
0.43
0.15
0.60
0.010
0.012
0.04
0.01
0.01
0.06Table 2
Chemical composition of the specimens made of X5CrNi18-10 steel used for tests (wt%).
C
Mn
Si
P
S
Cr
Ni
N
Acc. to PN-EN 10088-1:2014-12
≤0.07
≤2
≤1
≤0.045
≤0.015
17 ÷ 19.5
8 ÷ 10.5
≤0.11
For tested specimens
0.02
1.53
0.61
0.07
0.016
18.75
8.22
—Figure 2
Geometry of specimens used for tests.


## 2.1. Test Station

The tests were performed at the Department Laboratory for Research on Materials and Structures (certified by the Polish Centre for Accreditation, PCA AB 372) of the Faculty of Mechanical Engineering at the UTP University of Science and Technology using the reversed bending fatigue testing machine (with rotational frequency of 77 Hz).The main item of the test station was the thermographic camera CEDIP Silver 420M (FLIR SC5200) equipped with high sensitivity InSb matrix cooled using Stirling pump.The main parameters of the camera are as follows: resolution of sensor 320 × 256 pixels, spectral range 3.6 ÷ 5.0μm, sensitivity below 20 mK (available: 8 mK), and maximum frequency 140 Hz for the entire matrix (up to 25 kHz at lower resolution).The thermographic camera recorded (at the constant frequency from 100 to 450 Hz depending on load level) the surface temperature distribution of the tested object (specimen) fixed in the testing machine. Camera images were transmitted via USB interface to PC with appropriate software, where they were digitally stored directly on HDD in form of PTW format files.The following software was installed on the computer: VirtualCAM allowing two-way communication between PC and the thermographic camera via USB interface, CIRRUS Front End which was the user interface used to control the CEDIP camera, and ALTAIR allowing downloading, storage, and advanced processing of thermographic images.


## 2.2. Test Specimens

Test specimens were made of cold-drawn bars from the following:(i)
Medium carbon C45 (1.0503) steel.
(ii)
Corrosion resistant austenitic X5CrNi18-10 (1.4301) steel.Chemical composition of the specimen material is specified in Tables1 and 2. The geometry of applied specimens is shown in Figure 2.Table 1
Chemical composition of the specimens made of C45 steel used for tests (wt%).
C
Si
Mn
P
S
Cr
Mo
Ni
Cr + Mo + Ni
Acc. to PN-EN 10083-2:2008
0.42–0.50
<0.40
0.50–0.80
<0.045
<0.045
<0.40
<0.10
<0.40
<0.63
For tested specimens
0.43
0.15
0.60
0.010
0.012
0.04
0.01
0.01
0.06Table 2
Chemical composition of the specimens made of X5CrNi18-10 steel used for tests (wt%).
C
Mn
Si
P
S
Cr
Ni
N
Acc. to PN-EN 10088-1:2014-12
≤0.07
≤2
≤1
≤0.045
≤0.015
17 ÷ 19.5
8 ÷ 10.5
≤0.11
For tested specimens
0.02
1.53
0.61
0.07
0.016
18.75
8.22
—Figure 2
Geometry of specimens used for tests.


## 3. Full Test Results

### 3.1. C45 Steel

21 specimens at 8 load levels were tested for the fatigueS-N curve determination. The lowest alternating stress level S
a was 300 MPa and the highest one was 495 MPa. The test results and theS-N curve with confidence interval for the significance level α
=
0.05 were presented in Figure 3.Figure 3
TheS-N curve for C45 steel under reversed bending.The Basquin relationship obtained from linear regression (R
2
=
0.98) of test results is(1)
S
a
m
·
N
=
C
,where m
=
7.57 and C
=
2.97
·
10
24.The fatigue limit for this steel obtained from Dixon and Mood relations [33] by Staircase method S
e is 261 MPa with standard deviation s
s
=
9.1 MPa.


### 3.2. X5CrNi18-10 Steel

18 specimens at 6 load levels were tested for the fatigueS-N curve determination. The lowest alternating stress level S
a was 320 MPa and the highest one was 495 MPa. The test results and theS-N curve with confidence interval for the significance level α
=
0.05 were presented in Figure 4.Figure 4
TheS-N curve for X5CrNi18-10 steel under reversed bending.Parameters of Basquin relationship (1) obtained from linear regression (R
2
=
0.98) of test results are m
=
10.38 and C
=
5.12
·
10
31.The fatigue limit for this steel obtained from Dixon and Mood relations [33] by Staircase method S
e is 278.2 MPa with standard deviation s
s
=
7.4 MPa.


## 3.1. C45 Steel

21 specimens at 8 load levels were tested for the fatigueS-N curve determination. The lowest alternating stress level S
a was 300 MPa and the highest one was 495 MPa. The test results and theS-N curve with confidence interval for the significance level α
=
0.05 were presented in Figure 3.Figure 3
TheS-N curve for C45 steel under reversed bending.The Basquin relationship obtained from linear regression (R
2
=
0.98) of test results is(1)
S
a
m
·
N
=
C
,where m
=
7.57 and C
=
2.97
·
10
24.The fatigue limit for this steel obtained from Dixon and Mood relations [33] by Staircase method S
e is 261 MPa with standard deviation s
s
=
9.1 MPa.


## 3.2. X5CrNi18-10 Steel

18 specimens at 6 load levels were tested for the fatigueS-N curve determination. The lowest alternating stress level S
a was 320 MPa and the highest one was 495 MPa. The test results and theS-N curve with confidence interval for the significance level α
=
0.05 were presented in Figure 4.Figure 4
TheS-N curve for X5CrNi18-10 steel under reversed bending.Parameters of Basquin relationship (1) obtained from linear regression (R
2
=
0.98) of test results are m
=
10.38 and C
=
5.12
·
10
31.The fatigue limit for this steel obtained from Dixon and Mood relations [33] by Staircase method S
e is 278.2 MPa with standard deviation s
s
=
7.4 MPa.


## 4. Full Thermographic Test Results

### 4.1. C45 Steel

21 specimens at 8 load levels were tested using IR camera under constant-amplitude load conditions. The lowest stress amplitude levelS
a
=
300 MPa and the highest one was 495 MPa. The change of the surface temperature of selected specimens during tests was presented in Figure 5, where T
=
0°C for N
=
0 cycles corresponds to the initial specimen temperature of 23–25°C.Figure 5
Samples of theT-N curves for C45 steel under reversed bending.The specimen mean temperature changes presented in Figure5 result from the internal energy increase which can be estimated based on the following relation:(2)
Δ
I
=
c
v
·
ρ
∫
0
N
f
a
i
l
u
r
e
T
N
·
d
N
,where c
v is specific heat at constant volume, ρ is density, d
N is cycle number increment, and N
f
a
i
l
u
r
e is number of loading cycles until fatigue failure.Sincec
v and ρ can be assumed as constant, then for specimens from the same material the internal energy change can be estimated based on the area under T
=
f
(
N
) curve (Figure 11):(3)
Φ
=
∫
0
N
f
a
i
l
u
r
e
T
N
·
d
N
.The dependence of energy-related parameterΦ on load level S
a (Figure 7) was described using the relationship similar to the Basquin formula (R
2
=
0.90):(4)
S
a
k
·
Φ
=
B
,where k
=
5.18 and B
=
6.94
·
10
17.


### 4.2. X5CrNi18-10 Steel

12 specimens at 5 load levels were tested using thermographic camera in constant-amplitude loading conditions. The lowest alternating stress levelS
a was 320 MPa and the highest one was 480 MPa.The change of the surface temperature of selected specimens during tests was presented in Figure6, where T
=
0°C for N
=
0 cycles corresponds to initial specimen temperature of 22–25°C.Figure 6
Samples of theT-N curves for X5CrNi18-10 steel under reversed bending.Figure 7
The dependence ofΦ parameter on load level S
a for C45 steel under reversed bending.The dependence of energy-related parameterΦ on load level S
a was shown in Figure 8 and described using relationship (4) (R
2
=
0.93) with parameters k
=
5.18 and B
=
6.94
·
10
17.Figure 8
The dependence ofΦ parameter on load level S
a for X5CrNi18-10 steel under reversed bending.


## 4.1. C45 Steel

21 specimens at 8 load levels were tested using IR camera under constant-amplitude load conditions. The lowest stress amplitude levelS
a
=
300 MPa and the highest one was 495 MPa. The change of the surface temperature of selected specimens during tests was presented in Figure 5, where T
=
0°C for N
=
0 cycles corresponds to the initial specimen temperature of 23–25°C.Figure 5
Samples of theT-N curves for C45 steel under reversed bending.The specimen mean temperature changes presented in Figure5 result from the internal energy increase which can be estimated based on the following relation:(2)
Δ
I
=
c
v
·
ρ
∫
0
N
f
a
i
l
u
r
e
T
N
·
d
N
,where c
v is specific heat at constant volume, ρ is density, d
N is cycle number increment, and N
f
a
i
l
u
r
e is number of loading cycles until fatigue failure.Sincec
v and ρ can be assumed as constant, then for specimens from the same material the internal energy change can be estimated based on the area under T
=
f
(
N
) curve (Figure 11):(3)
Φ
=
∫
0
N
f
a
i
l
u
r
e
T
N
·
d
N
.The dependence of energy-related parameterΦ on load level S
a (Figure 7) was described using the relationship similar to the Basquin formula (R
2
=
0.90):(4)
S
a
k
·
Φ
=
B
,where k
=
5.18 and B
=
6.94
·
10
17.


## 4.2. X5CrNi18-10 Steel

12 specimens at 5 load levels were tested using thermographic camera in constant-amplitude loading conditions. The lowest alternating stress levelS
a was 320 MPa and the highest one was 480 MPa.The change of the surface temperature of selected specimens during tests was presented in Figure6, where T
=
0°C for N
=
0 cycles corresponds to initial specimen temperature of 22–25°C.Figure 6
Samples of theT-N curves for X5CrNi18-10 steel under reversed bending.Figure 7
The dependence ofΦ parameter on load level S
a for C45 steel under reversed bending.The dependence of energy-related parameterΦ on load level S
a was shown in Figure 8 and described using relationship (4) (R
2
=
0.93) with parameters k
=
5.18 and B
=
6.94
·
10
17.Figure 8
The dependence ofΦ parameter on load level S
a for X5CrNi18-10 steel under reversed bending.


## 5. Accelerated Test Results

TheS-N curve for both steels was determined in gradually increasing loading test. The lowest alternating stress level S
a
1 was 230 MPa and the stress increment Δ
S amounted to 10 MPa. Δ
N for each load level was 30 000 cycles. After each load level the test was stopped for about 30 seconds for the purpose of load increase. The test was made at the same load frequency of 77 Hz as the full test. The change of the surface temperature of selected specimen during test was presented in Figure 9, where T
=
0°C for N
=
0 cycles corresponds to initial specimen temperature of 19°C.Figure 9
Sample of theT-N curve for X5CrNi18-10 steel under accelerated test.Theoretically, accelerated test can be made using only one specimen, but, in practice, tests on three samples can increase the accuracy of estimates. Tables3 and 4 show total fatigue life achieved by each sample for both tested materials.Table 3
Summary of fatigue life for the individual specimens made of C45 steel in the accelerated tests.
Specimen
Number 1
Number 2
Number 3
Level of fatigue failure
i
16
16
15
Failure stress, MPa
S
k
380
380
370
Total fatigue life, in cycles
N
c
474 500
475 300
429 800Table 4
Summary of fatigue life for the individual specimens made of X5CrNi18-10 steel in the accelerated tests.
Specimen
Number 1
Number 2
Number 3
Level of fatigue failure
i
18
17
16
Failure stress, MPa
S
k
400
390
380
Total fatigue life, in cycles
N
c
518 400
487 300
457 600


## 6. Analysis of the Accelerated Test Results

TheT-N curve (Figure 9) obtained from gradually increasing loading test was cut into individualT-N curves for the different load levels (Figure 10). The reference level of the temperature was the initial temperature of the specimen.Figure 10
Sample of cutT-N curves for different load levels for specimen from X5CrNi18-10 steel.Figure 11
Schematic representation of energy-based parametersΦ and φ.It was assumed that under the gradually increasing loading test at each load level only part of the total internal energy represented by parameterφ was measured (Figure 11):(5)
φ
=
∫
0
N
s
t
e
p
T
N
·
d
N
,where N
s
t
e
p
=
Δ
N
=
30
000 cycles.Palmgren-Miner’s hypothesis in its energy-related version can assume the following form:(6)
∑
i
=
1
q
φ
i
Φ
i
=
1
f
o
r
S
a
i
≥
S
e
.Assuming that the relationship between energy-related parameterΦ
i for full test and load S
a
i at the level i is described by formula (4)(7)
S
a
i
k
·
Φ
i
=
B
⟹
Φ
i
=
B
S
a
i
kand putting (7) into (6),(8)
∑
i
=
1
q
φ
i
B
/
S
a
i
k
=
1
⟹
∑
i
=
1
q
S
a
i
k
·
φ
i
=
Bis obtained:(9)
∑
i
=
1
q
S
a
i
k
·
φ
i
-
B
=
0
,where there are two unknown parameters B and k of the S
a
=
f
(
Φ
) curve which can be determined numerically. Then it is possible to determine the values of Φ
i at the level i by using (7).Fatigue life at the leveli can be estimated based on the following proportions:(10)
φ
i
Φ
i
=
n
i
N
i
⟹
N
i
=
Φ
i
φ
i
·
n
i
.Tables5 and 6 show parameters (see (1) and (4)) obtained by the presented method for both tested materials.Table 5
Parameters obtained for the individual specimens of C45 steel under reversed bending in the accelerated thermographic tests.
k
B
m
C
Full test
5.18
6.94
⋅
10
17
7.57
2.97
⋅
10
24
Specimen number 1
5.10
7.17
⋅
10
21
7.35
1.23
⋅
10
24
Specimen number 2
5.15
8.41
⋅
10
21
7.18
4.48
⋅
10
23
Specimen number 3
5.27
8.40
⋅
10
21
7.66
5.15
⋅
10
24
Specimens numbers 1–3
—
—
7.40
1.42
⋅
10
24Table 6
Parameters obtained for the individual specimens of X5CrNi18-10 steel under reversed bending in the accelerated thermographic tests.
k
B
m
C
Full test
6.914
4.03
⋅
10
22
10.377
5.12
⋅
10
31
Specimen number 1
6.408
4.00
⋅
10
21
10.157
2.63
⋅
10
31
Specimen number 2
6.500
4.10
⋅
10
21
9.196
6.75
⋅
10
28
Specimen number 3
6.500
4.00
⋅
10
21
9.456
2.30
⋅
10
29
Specimens numbers 1–3
—
—
9.603
5.87
⋅
10
29TheS-N curves estimated for the individual specimens were presented in Figures 12 and 13. The individual estimated fatigue life data obtained from (10) for all three specimens has been used as data to estimate oneS-N curve (Figures 14 and 15).Figure 12
TheS-N curves estimated for individual specimens made of C45 steel on the background of the full test results.Figure 13
TheS-N curves estimated for individual specimens made of X5CrNi18-10 steel on the background of the full test results.Figure 14
Comparison of estimatedS-N curves obtained by the accelerated thermographic method and full test for C45 steel.Figure 15
Comparison of estimatedS-N curves obtained by the accelerated thermographic method and full test for X5CrNi18-10 steel.


## 7. Conclusions

The presented new thermographic method enables rapid determination of theS-N curve. This method is based on the energy-related parameter with the assumption of its dependency on the stress amplitude.Major findings and conclusions of the presented tests and analysis are summarized as follows:(1)
TheS-N curve obtained by accelerated thermographic method fits inside the 95% confidence interval for theS-N curve obtained from the full test for both tested steels.
(2)
The estimation of the parameters of theS
a-Φ curve and theS-N curve for each specimen for both materials is characterized by repeatability.
(3)
The accelerated thermographic test for one specimen lasted ca. 1.5 hours only. For comparison, the full test lasts several days.
(4)
The accelerated thermographic test can be made on one specimen. Three specimens for increasing the accuracy of estimates were assumed in this work. The full test requires a minimum of a dozen or so specimens.



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*Source: 4134021-2016-04-20.xml*