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**Research article**
19 Jun 2019

**Research article** | 19 Jun 2019

Effect of Quenching Media and Tempering Temperature on Fatigue Property and Fatigue Life Estimation Based on RBF Neural Network of 0.44 % Carbon Steel

^{1}Information & Control Engineering Faculty, Shenyang Jianzhu University, Shenyang, China^{2}School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110004, China

^{1}Information & Control Engineering Faculty, Shenyang Jianzhu University, Shenyang, China^{2}School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110004, China

**Correspondence**: Changyou Li (chyli@mail.neu.edu.cn)

**Correspondence**: Changyou Li (chyli@mail.neu.edu.cn)

Abstract

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In this work, the effect of the quenching media (brine,
water, and two types of naphthenic mineral oils) and the tempering
temperature (200, 400, 600 ^{∘}C) on the static mechanical properties
and the fatigue life has been investigated using 300 fatigue and 36 static
tension tests. S–N curves and standard deviations of fatigue life under each
heat treatment condition were calculated and shown. The fracture surfaces of
the selected 11 specimens were observed by the scanning electron microscope
and the reasons of affecting the fatigue life were discussed. To estimate
the mean fatigue life under the conditions of any given tempering
temperature and cycle stress amplitude based on 300 fatigue tests, the mean
fatigue life estimation method based on RBF neural network was presented and
verified by 12 other fatigue tests. The test results have shown that (1) the
mean fatigue life decreases with the increase of tempering temperature for
the same quenching media, (2) the mean fatigue life using brine is more than
water which is more than naphthenic mineral oils for the same tempering
temperature, and (3) the proposed method based on RBF neural network could
accurately estimate the mean fatigue life when the tempering temperature and
cyclic stress amplitude are given for each quenching medium.

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Guo, S., Li, C., Shi, J., Luan, F., and Song, X.: Effect of Quenching Media and Tempering Temperature on Fatigue Property and Fatigue Life Estimation Based on RBF Neural Network of 0.44 % Carbon Steel, Mech. Sci., 10, 273–286, https://doi.org/10.5194/ms-10-273-2019, 2019.

1 Introduction

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In general, the fatigue life of the carbon steel could be improved by the
quenched and tempered process. For example, it has been reported by Mei and
Morris (1990) that the fatigue crack growth rate of the austenitic stainless
steels 304L was 10 times slower than that of 304LN due to martensite coating
the growing crack of 304L. Moreover, Fatigue life of structures is mainly
determined by the surface roughness, residual stress and microstructure
(Suraratchai et al., 2008). It has been observed that the microstructure and
the residual stress relief of the quenched and tempered dual phase steels
were affected by the tempering temperature (Anazadeh Sayed and Kheirandish,
2012; Kuang et al., 2014). For the tool steel, retained austenite
percentage, hardness, and microstructure are affected by the different
quenching media (Amini et al., 2013). It has been experimentally verified by
Kumar et al. (2014) that during the tempering temperature range of
200–400 ^{∘}C, there is sudden increase in impact
strength, ductility and toughness of the ductile cast iron, as the
temperature and time increase. The hardness and ultimate tensile strength
gradually decrease, and the percent elongation of the spring steel increases
by increasing tempering time and temperature (Htun et al., 2008). Therefore,
it is important to analyze the fatigue property of the quenched and tempered
0.44 % carbon steel with the different quenching media and tempering
temperature to ensure the high safety and reliability of the part of the
type of material in service.

The effect of the quenching media on the fatigue property of AISI 4340 has
been reported by Hamza et al. (2016) where the material was austenized at
the different temperature and subsequently tempered at 350 ^{∘}C for
1 h. The test results showed that fatigue resistance with the quenching
media of air was superior to that with the quenching media of water which
was superior to the oil when the austenized temperature was 850 or
800 ^{∘}C. This was different from the case with the austenized
temperature of 900 or 950 ^{∘}C. It has been observed by Harichandra
et al. (2016) that the specimens of EN31 steel quenched in gingili oil have
10 times more fatigue life than the specimens quenched in the water and 100
times more than unquenched specimens in high-cycle rotating bending fatigue
tests.

Tempering temperature is another important factor affecting the fatigue
characteristics of the quenched and tempered carbon steel. It has been
reported by London et al. (1989) that growth rates of the small surface
crack was gradually slowing down at the same cyclic stress intensity when
the tempering temperature was increased from 200 to 700 ^{∘}C for
cantilevered bending fatigue samples. Moreover, the threshold of stress
intensity factor of small cracks growth was greater than that of their
corresponding long crack when the tempering temperatures was 200 ^{∘}C or 400 ^{∘}C, and this was exactly the opposite of the case when
the quenching temperature is 500 and 700 ^{∘}C. It was observed by
Amirat and Chaoui (2003) that the fatigue crack growth rates of the mining
chain steel tempered at 500 ^{∘}C was slightly lower than the one tempered at
200 ^{∘}C in the near the threshold of the stress intensity factor. Similarly,
the work of Tsay et al. (1997) also showed that the fatigue crack growth
rates would increase when the tempering temperature was decreased from 600
to 400 ^{∘}C for both D6AC steel plates and laser welds. However, the
conclusions were obtained by Sultan (2013) that the fatigue resistance of
the carburized steel was decreased with the decrease of the tempering
temperature. Here, the carburized specimens were austenized at
760 ^{∘}C, and quenched in water, and then tempered at temperatures
200, 300 and 400 ^{∘}C for 1 h. The
fatigue limit was also affected by the tempering temperature. For example,
it was reported by Yu et al. (1988) that the fatigue limit decreased as the
tempering temperature increased from 315, 482 to 649 ^{∘}C for the
centre-notched specimens, and the notch sensitivity and absolute notch
fatigue strength were not significantly affected by the tempering
temperature. It was found by Oguma and Nakamura (2009) that in the 10^{5}–10^{8} cycle range, the fatigue strength of the high strength steel
tempered at 433 K was lower than that of the material tempered at 573 K where
the uniaxial tension-compression fatigue tests were carried out. It was
observed by Williams et al. (2006) that the tempering temperature increase
of the 4300 sintered steel from 205 to 315 ^{∘}C resulted in an
increase of the fatigue strength of 2 %. The tempering temperatures of the
optimal fatigue resistance have been obtained for some types of steel. For
example, the fatigue limit, ductility and toughness of the high strength
spring steel were highest when the tempering temperature was 450 ^{∘}C. Here, the specimens were austenized at 900 ^{∘}C for 40 min
and quenched into oil, and then tempered for 30 min at the temperatures
of 300, 350, 400, 450 and 500 ^{∘}C (Lee et al., 1997). The high
strength steel tempered at 250 ^{∘}C exhibited superior fatigue
properties in the short life regions. The steel was heated up to
950 ^{∘}C for 5 min and quenched using water. Tempering
treatments at the temperatures of 100, 250 and 340 ^{∘}C were
performed for 30 min and followed by air cooling (Kwon et al., 2014).
For 0.45 % carbon steel, it was observed by Siddiqui et al. (2006) that
the fatigue life increased with the increase of the tempering temperature
from 100 to 200 ^{∘}C, and the resistance to fatigue failure reduced
when the tempering temperature was further increased. It was also seen by
Anctil and Kula (1970) that the crack-growth rates of 4340 steel decreased
as the tempering temperature increased to 600 F, and then increased again
with higher tempering temperatures.

In this work, the effect of quenching media and tempering temperature on the
fatigue life of the quenched and tempered 0.44 % carbon steel were
analyzed and discussed, and the mean fatigue life estimation method based on
RBF neural network was presented. The specimens were austenized at
850 ^{∘}C for 20 min and quenched into water, brine, and two
types of naphthenic mineral oils respectively. And they were subsequently
tempered at different temperature. The remainder of this paper is organized
as follows: In Sect. 2, the original material, the geometry of specimens,
static tensile and fatigue tests were described, and the static tensile
experiment results were shown and analyzed by the statistical method. In
Sect. 3, the scatter plots of the fatigue life were given for all fatigue
tests, S–N curves were formulated and their curve-fitting parameters were
listed, and the standard deviations of fatigue life under each heat
treatment condition were shown in one figure. The fracture surfaces of the
selected specimens were observed by the scanning electron microscope and the
laws of fatigue life were discussed. In Sect. 4, the mean fatigue life
estimation method based on RBF neural network was presented and verified by
12 other fatigue tests. Finally, the conclusions were drawn in Sect. 5.

2 Material and test

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The material investigated in this work is the 0.44 % carbon steel and its
composition is presented in Table 1. The raw material is a hot rolled steel
bar of which the diameter is 14 mm. It is difficult that the hardening depth
obtained by quenching is up to 7 mm for the steel bar. So, the hot rolled
steel bars were firstly rough machined to the shape being similar to the
final specimen. The diameters of the clamping part and the dangerous section
are 8.5 and 6.0 mm respectively. Then, they were austenized at 850 ^{∘}C for 20 min and quenched into water, brine, oil 1 or oil 2. In the
brine, the mass ratio of the food grade salt to the tap water is 20 %. Oil 1 and Oil 2 are the naphthenic mineral oil with several types of additives
for accelerating cooling, preventing oxidation, preventing rust and so on.
The flash points and the kinematic viscosities of Oil 1 and Oil 2 are 205
and 208 ^{∘}C, 29.8 and 31.06 mm^{2} s^{−1} respectively. Subsequently,
the specimens were tempered at 200, 400, and 600 ^{∘}C for 60 min respectively. Next, they were machined into the funnel specimen. The
geometry of the specimen is shown in Fig. 1. The central part of the
specimen was polished to the average surface roughness (*R*_{a}) 0.4 µm.

The static tensile tests for the specimens with the different quenching
media, tempering temperature were firstly carried out. They were completed
by the hydraulic servo tension and compression testing machine. In order not
to destroy the clamping blocks of the test machine, due to the high hardness
of the specimens after the quenching and tempering treatment, the additional
clamp was designed and shown in Fig. 2. The material of additional clamp
has the high tensile strength and the low hardness. The specimen was
connected to the additional clamp by the thread which could be seen in
Fig. 2. The means of the tensile strength *σ*_{b}, yield strength *σ*_{s}, and elongation at break *δ*_{5} were listed in Table 2. There were 3 samples for
each static tensile test. According to Table 2, it was observed that (1) almost all of means of tensile strength decreased and the elongation at
break increased with the increase of tempering temperature from 200 to 600 ^{∘}C for each kind of quenching media, (2) the order of the
quenching medium are brine, water, oil 1 and oil 2 in accordance with the
rule of the tensile strength from large to small, (3) the brine has the
maximal decline rate of tensile strength and oil 2 has the smallest one with
the increase of tempering temperature from 200 to 600 ^{∘}C, (4) oil 2 has the lowest sensitivity of *σ*_{b},
*σ*_{s}, and *δ*_{5} with
respect to the tempering temperature, and then oil 1, water, and finally
brine. This is the reason that martensite has high strength and low
toughness, the order of the quenching media are brine, water, oil 1 and oil 2 by the rule of the cooling capacity from high to low, and the hardening
depth obtained by quenching increased with the increase of the cooling
capacity of a quenching media (Thelning, 1984; Ye et al., 2011).

The effect of the quenching media and tempering temperature on the fatigue
life of 0.44 % carbon steel were discussed by high and low cycle rotating
bending fatigue tests. And they could be compared as much as possible under
the same cyclic stress amplitudes. Therefore, at the tempering temperature
200 ^{∘}C, the cycle stress amplitudes included 750, 650, 550, 500,
and 450 MPa for Oil 1 and Oil 2, included 750, 700, 680, 660, and 650 MPa for
water, and 1000, 950, 900, 850, and 800 MPa for brine. At the tempering
temperature 400 ^{∘}C, the cycle stress amplitudes for water, oil 1
and oil 2 were composed of 700, 650, 550, 500, and 450 MPa, and for brine,
850, 750, 650, 550, and 500 MPa. At the tempering temperature 600 ^{∘}C, the cycle stress amplitudes for brine, oil 1 and oil 2 were 600, 550,
500, 450, and 400 MPa, and for water, 550, 500, 450, 430, and 400 MPa. The
cycle frequency was 50 Hz. Five tests were carried out at each cycle stress
amplitude. When the crack of the specimen grows to the length which resulted
in the test machine to not rotate properly due to excessive bending
deformation or the specimen failed due to fracture, the corresponding cycle
number is defined as the fatigue life of the specimen in this work. If the
cycle number exceeds than 10^{6} and does not reach the corresponding
fatigue life, the fatigue test will be stopped.

3 Fatigue Test Results and Discussion

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The scatter plots of the fatigue test results with the tempering temperature
200, 400 and 600 ^{∘}C are shown in Fig. 3. Here, the data point
with the arrow is not the fatigue life, but is the cycle number at which the
corresponding fatigue test was stopped. The S–N curves are shown in Fig. 4. The S–N curve was processed as fellows: (1) cycle stress amplitudes and
means of the fatigue life were taken the logarithm with respect to base 10,
(2) the linear regression model was employed to obtain the formulation
between the base 10 logarithm of means of the fatigue life and cycle stress
amplitudes,

$$\begin{array}{}\text{(1)}& \mathrm{log}N=\mathrm{log}c-n\mathrm{log}S,\end{array}$$

*N*, *c*, *n* and *S* are fatigue life, two material constants and cycle stress amplitude respectively, and *N* and *S* can be described by

$$\begin{array}{}\text{(2)}& N{S}^{n}=c\end{array}$$

and then S–N curve can be plotted. The material constants *c* and *n* are listed
in Table 3. Standard deviation of fatigue life is shown in Fig. 5. In
processing S–N curve and calculating the standard deviation of fatigue life,
the data points with the arrow in Fig. 3 were uniformly set to 10^{6}.

Set *N*_{BM}, *N*_{WM}, *N*_{O1M} and
*N*_{O2M} for the mean fatigue life of the specimens with the
quenching media being brine, water, oil 1 and oil 2 repectively. According
to Figs. 3, 4 and 5, it can be observed that (1) at the tempering
temperature 200 ^{∘}C, *N*_{BM} is far more than
*N*_{WM}, *N*_{WM} is more than
*N*_{O1M}, and *N*_{O1M} is more than *N*_{O2M},
(2) at 400 ^{∘}C, *N*_{BM} is also more than
*N*_{WM}, *N*_{O1M} and *N*_{O2M} and there is no
significant difference among *N*_{WM}, *N*_{O1M} and
*N*_{O2M}, (3) at 600 ^{∘}C, *N*_{BM} is still more
than *N*_{WM} which is almost equal to *N*_{O1M} and
*N*_{WM}, and *N*_{O1M} are slightly larger than
*N*_{O2M}, (4) with the tempering temperature increase from 200 to
600 ^{∘}C, the difference of the mean fatigue life between one
quenching media and other is less and less, (5) the mean fatigue life with
oil 2 is the least affected by the tempering temperature, followed by that
with oil 1, then that with water, and finally that with brine, (6) the
dispersion of the fatigue life almost increase with a decrease of the
applied cycle stress amplitude for the same tempering temperature and
quenching media (Li et al., 2017a, b), but it cannot be followed by the
fatigue life data of the specimens with tempering temperature 200 ^{∘}C and the quenching media of brine, and (7) there are some data points with
the arrow only when the cycle stress amplitude is low for oil 1, oil 2 and
water, while no matter when the cyclic stress is high or low under the
condition of the quenching media of brine and the tempering temperature
200 ^{∘}C.

The fractures of some samples in the fatigue tests were observed by scanning electron microscope. The test parameters and fatigue life of those samples were listed and the corresponding mean, standard deviation and coefficient of variation were given in Table 4. Figure 6 showed the scanning electron micrograph (SEM) of the whole fracture surface (WFS) and local fracture surface (LFS) of those specimens listed in Table 4. The SEMs of LFS of the specimen 9, 11 and 12 were not shown in Fig. 6 because they were similar with that of these specimens except specimen 1, 2 and 5. The black spots were oxides, because the fractures were not protected well. It was observed that the fracture surfaces of the specimens 1, 2 and 5 were very flat and did not have the obvious fatigue source area and fatigue striation. According to the local high power micrographs of their fracture surfaces, many facets and steps could be observed. The features implied that the brittle fracture and transgranular fracture have occurred and there was no obvious crack growth process in the sample 1, 2 and 5. However, for the other specimens, the fracture surfaces had more than one obvious fatigue source areas and a lot of the fatigue striations and the fracture dimples. The nucleation and propagation process of cracks were very obvious, and the final unstable extension fracture surface has the obvious fracture dimple.

For an existing defect with the length of 2*a* in the brittle solid, the
theoretical tensile strength is (Cheng and Ohr, 1981)

$$\begin{array}{}\text{(3)}& {\mathit{\sigma}}_{\mathrm{Tb}}=\sqrt{{\displaystyle \frac{\mathit{\gamma}E}{\mathrm{4}a}}}\end{array}$$

*γ* and *E* are the surface energy of unit defect area and Young's modulus
respectively. The local cleavage fracture stress is (Curry and Knott, 1978;
Chen and Cao, 2015)

$$\begin{array}{}\text{(4)}& {\mathit{\sigma}}_{\mathrm{f}}=\sqrt{{\displaystyle \frac{\mathrm{2}{\mathit{\gamma}}_{\mathrm{p}}E}{\mathit{\pi}(\mathrm{1}-{\mathit{\nu}}^{\mathrm{2}})f}}}\end{array}$$

*γ*_{p}, *ν* and *f* are the effective surface energy of the ferrite matrix, Poisson
ratio and the half-length of the just initiated crack nucleus in a
second-phase particle or in a grain with critical size respectively. When
the applied normal stress is more than the local cleavage fracture stress
*σ*_{f}, the grain-sized crack propagates into the contiguous
ferrite grains and it results in the cleavage fracture of the specimen
finally (Chen and Cao, 2015). It could be seen that the local cleavage
fracture stress is proportion to the theoretical tensile strength according
to Eqs. (3) and (4). Therefore, it could be implied that the maximal normal
stress which is applied to the specimen with the larger tensile strength
*σ*_{b} is larger than that of the specimen with the
lower tensile strength *σ*_{b}, when the specimen
was fractured by cleavage.

According to the dislocation model for fatigue crack initiation (Tanaka and
Mura, 1981), when the applied cycle shear stress amplitude is Δ*τ*
and the cycle number is *n*, the tensile stress is

$$\begin{array}{}\text{(5)}& {\mathit{\sigma}}_{xx}={\displaystyle \frac{\mathrm{3}\sqrt{\mathrm{ln}}(\mathrm{\Delta}\mathit{\tau}-\mathrm{2}k)}{\sqrt{\mathrm{2}h}}}\end{array}$$

where *l*, *k* and *h* are the half length of the grain size, the frictional stress and the distance between two adjacent slip plane. When the tension and
pressure cycle stress is applied to the specimen, the slip plane is inclined
at 45^{∘} to the applied stress. If *σ*_{xx} is increased to the
local cleavage fracture stress *σ*_{f}, the crack will be
nucleated and the corresponding cycle number is defined as the crack
initiation life of the specimen. Therefore, the crack initiation life of the
specimen with the larger tensile strength *σ*_{b} is
larger than that of the specimen with the lower tensile strength
*σ*_{b}.

According to Fig. 6, especially panels (a), (c) and (e), the fracture
will soon occur, once a crack was nucleated. This implies that the crack
initiation life accounts for most of the total fatigue life (includes crack
initiation life and crack propagation life). So, due to the larger tensile
strength *σ*_{b} of the specimens with the quenching
media being brine and water and the tempering temperature 200^{∘}, the
fatigue life is far more than that of other specimens sometimes. However,
Δ*τ* is not only determined by the applied
tension and pressure cycle stress, but also is be affected by the stress
concentration due to the internal defects and the surface roughness of the
specimens. Generally, the geometric parameters of the internal defects and
the surface roughness have the strong randomness. It might leads to the very
large Δ*τ* in some specimen, and *σ*_{xx} is quickly
increased to the local cleavage fracture stress *σ*_{f} when the applied cycle stress amplitude is identical. In
this case, due to almost no crack growth process, there is very low fatigue
life for the specimens with the quenching media being brine and water and
the tempering temperature 200^{∘}. While, the fatigue life of the
specimens with the obvious nucleation and propagation process of cracks,
such as the specimen 3, 4, 6, 7, 8, 9, 11 and 12 listed in Table 4, was less
affected because the crack propagation life is not affected. This could also
explain that there was larger coefficient of variation of the fatigue life
under the heat treatment condition of the specimen 1, 2, and 5 listed in
Table 5.

4 Fatigue Life estimation based on RBF neural network

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For the quenched and tempered 0.44 % carbon steel, their mean fatigue life
under the condition of any given cycle stress amplitude can be estimated by
S–N curves which have been obtained in Sect. 3, when the tempering
temperature is 200, 400 or 600 ^{∘}C and the quenching media is oil 1, oil 2, water or brine. However, the mean fatigue life can not been
estimated by S–N curves in Sect. 3 if the tempering temperature is not
200, 400 or 600 ^{∘}C, for example 300 ^{∘}C. If the mean
fatigue life with the tempering temperature 300 ^{∘}C must be
estimated using S–N curves, a lot of additional fatigue tests must be
carried out again. Therefore, the fatigue life estimation method will be
presented and is based on RBF neural network. The RBF neural networks have
shown to perform well in many problems of practical interests (Khan et al.,
2018). It can approximate any continuous function with an arbitrary
precision. Moreover, RBF neural network has only one hidden layer and the
number of neurons in the hidden layer is automatically increased until the
mean square error reaches the given accuracy or the number of neurons is up
to the given value. Therefore, RBF neural network can avoid the
arbitrariness in the number of hidden layers and the number of neurons per
hidden layer (Mahanty and Gupta, 2004).

For the given quenching media, the structure of the presented fatigue life
estimation method based on RBF neural network is shown in Fig. 7. It
includes three layer, input layer, hidden layer and output layer. The input
vector is composed of two elements, tempering temperature and cycle stress
amplitude. In the hidden layer, there are *n* neurons. *n* is determined by the
given accuracy or the set number of neurons and is calculated automatically
by the algorithm of RBF neural network. The RBF of each neuron is the
Gaussian function and can be expressed as

$$\begin{array}{ll}\text{(6)}& {\displaystyle}& {\displaystyle}{\mathrm{\varnothing}}_{i}=\mathrm{exp}\left(-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}{\mathit{\sigma}}_{i}^{\mathrm{2}}}}\mathrm{|}\mathrm{|}{x}_{j}-{c}_{i}\mathrm{|}{\mathrm{|}}^{\mathrm{2}}\right),{\displaystyle}& {\displaystyle}i=\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{\dots},n,\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}j=\mathrm{1},\mathrm{2}\end{array}$$

*σ*_{i} is the spread of GRBF (Gaussian RBF) ∅_{i}, *x*_{j} is the input
parameter including the tempering temperature and cycle stress amplitude,
*c*_{i} is the centre for the *i*th hidden neuron, and $\left|\right|\cdot \left|\right|$ denotes the Euclidian norm. *ω*_{i}
($i=\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{\dots},n$) is the weight connecting the hidden layer to the
output (mean fatigue life) *y*. Then, the output *y* can be expressed as (Mahanty and Gupta, 2004; Yang et al., 2016)

$$\begin{array}{}\text{(7)}& y=\prod _{j=\mathrm{1}}^{\mathrm{2}}\prod _{i=\mathrm{1}}^{n}{\mathit{\omega}}_{i}\mathrm{exp}\left(-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}{\mathit{\sigma}}_{i}^{\mathrm{2}}}}\mathrm{|}\mathrm{|}{x}_{j}-{c}_{i}\mathrm{|}{\mathrm{|}}^{\mathrm{2}}\right)\end{array}$$

For the quenching media is oil 1, oil 2 or water, the sample size of the
training set is 2403. The tempering temperatures are 200, 400 and
600 ^{∘}C and there are 801 cycle stress amplitudes from 400 to
800 MPa with the increment of 0.5 for each tempering temperature. For brine,
the sample size is 4203 and cycle stress amplitudes are from 400 to 1100 MPa.
Each element of the output vector (mean fatigue life) in the training sets
is calculated by the estimated S–N curves in Sect. 3. The testing set is
similar with the corresponding training set except that cycle stress
amplitudes are from 400.25 to 799.75 MPa for oil 1, oil 2 and water, and from
400.25 to 1099.75 MPa for brine. The mean squared error goal is 10^{−6}.
For oil 2, oil 1, water and brine, the test and estimated mean fatigue life
by the trained RBF neural network are shown in panels (a), (c), (e) and (g) in
Fig. 8 respectively. The estimation error surfaces of the mean fatigue
life for the testing set are shown in panels (b), (d), (f) and (h) in Fig. 8
respectively. The mean fatigue life estimation error *E*_{i} given the
quenching medium, tempering temperature and cyclic stress amplitude is
calculated by

$$\begin{array}{}\text{(8)}& {E}_{i}={\displaystyle \frac{\left|{y}_{i}-{N}_{i}\right|}{{N}_{i}}}\times \mathrm{100},\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}i=\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{\dots}\end{array}$$

*y*_{i} and *N*_{i} are estimated mean fatigue life by the presented method and the
calculated mean fatigue life by S–N curve in Sect. 3. According to Fig. 8, it can be observed that (1) the estimated mean fatigue life by the
presented method is in good agreement with the mean fatigue life of the
test, and (2) there is the large estimation error of the mean fatigue life
for the test set when the cyclic stress amplitude is close to the upper or
lower bound of the training set, but the maximum estimation error is less
than 0.4 % for four kinds of quenching media.

To validate the presented method, 12 fatigue tests were carried out again. The cycle frequency was still 50 Hz. Three fatigue tests were carried out at each cycle stress amplitude. The cycle stress amplitudes and the tempering temperature were different from those of the previous fatigue tests. The fatigue test conditions and results are shown in Table 5. It can be seen that the maximum estimation error of the presented method is 18.19 %. It implies that the proposed method can estimate the mean fatigue life in the case of the given tempering temperature and cyclic stress amplitude with high precision, and can avoid the disadvantages of using S–N curve to estimate the mean fatigue life.

5 Conclusions

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The fatigue life and the static mechanical parameters data of 0.44 %
carbon steel specimens were analyzed and the mean fatigue life estimation
method based on RBF neural network was presented in this work. The
experiment results showed that (1) for these four kinds of quenching medium,
there is a downward trend in the mean of *σ*_{b} and *N* and the
standard deviation of *N*, and there is a upward trend in the mean of
*δ*_{5} with the increase of tempering temperature
from 200 to 600 ^{∘}C, (2) the mean of *σ*_{b} and
*N* is least influenced by the tempering temperature for the specimens with oil 2, then oil 1, next water, finally brine, (3) for the same tempering
temperature, according to the rule that the mean and the standard deviation
of *N* is from large to small, the order of quenching medium is brine, water,
oil 1 and oil 2. The brittle, transgranular and cleavage fracture occurred
in fatigue test specimens. For the designed testing set, the maximum
estimation error of the mean fatigue life estimation method based on RBF
neural network is less than 0.4 % for four kinds of quenching media.
Verified fatigue test results show the maximum estimation error of the
presented method is 18.19 %. Therefore, the proposed method can accurately
estimate the mean fatigue life when the tempering temperature and cyclic
stress amplitude are given for each quenching medium.

Data availability

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Data availability.

All the data used in this paper can be obtained upon request from the corresponding author.

Author contributions

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Author contributions.

SG, CL and JS conducted experiments, analyzed data and wrote the manuscript. FL and XS were contributed to ideas for this research work and fatigue life estimation model.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Financial support

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Financial support.

This research has been supported by the National Natural Science Foundation of China (grant no. 51575095, 51675089), the science and technology plan of ministry of housing and urban-rural development of China (grant no. zjbwt2016002), and the science and technology foundation of Liaoning province of China (grant no. 20170540767).

Review statement

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Review statement.

This paper was edited by Zbigniew Gronostajski and reviewed by two anonymous referees.

References

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