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\begin{document}

\title{An enhanced fatigue residual life prediction model based on fatigue
driving stress by considering loading interaction effects}



\author[1]{Xu Zhao}%
\author[1]{Kaiwen Wang}%
\author[1]{Xuming Niu}%
\author[1]{Zhigang Sun}%
\author[1]{Yingdong Song}%
\affil[1]{Nanjing University of Aeronautics and Astronautics Department of Strength and Vibration Engineering Department of Power Engineering}%


\vspace{-1em}



  
  \date{May 18, 2024}


\begingroup
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\selectlanguage{english}
\begin{abstract}
It has been demonstrated that the loading history takes into account the
loading sequence and loading interaction effects. Nonlinear damage
models based on fatigue driving stress theory consider the loading
sequence but exclude the loading interaction effects. In this study, a
novel evolution curve for fatigue driving stress was created by
including the loading interaction factor in the equation of driving
stress evolution. Through using fatigue driving stress equivalence, the
remaining fatigue life under varying amplitude loading was then
predicted by an enhanced fatigue driving stress model. Compared with
Miner's rule, the K-R model, Zhu's model, and Li's model, this new model
gave more accurate and reliable predictions.%
\end{abstract}%



\sloppy


\textbf{An enhanced fatigue residual life prediction model based on
fatigue driving stress by considering loading interaction effects}

Kaiwen Wang\textsuperscript{a,b}, Xu Zhao\textsuperscript{a,b},Xuming
Niu\textsuperscript{a,b}, Zhigang Sun\textsuperscript{a,b,*} , Yingdong
Song \textsuperscript{a,b,c}

\textsuperscript{a} College of Energy \& Power Engineering, Nanjing
University of Aeronautics and Astronautics, Nanjing, 210016, China

\textsuperscript{b} Key Laboratory of Aero Engine Thermal Environment
and Thermal Structure, Ministry of Industry and Information Technology,
Nanjing, 210016, China

\textsuperscript{c} State Key Laboratory of Mechanics and Control of
Mechanical Structures, Nanjing University of Aeronautics and
Astronautics, Nanjing, 210016, China

\textsuperscript{*} Corresponding author. E-mail: szg\_mail@nuaa.edu.cn;
ydsong@nuaa.edu.cn

\textbf{Abstract}

It has been demonstrated that the loading history takes into account the
loading sequence and loading interaction effects. Nonlinear damage
models based on fatigue driving stress theory consider the loading
sequence but exclude the loading interaction effects. In this study, a
novel evolution curve for fatigue driving stress was created by
including the loading interaction factor in the equation of driving
stress evolution. Through using fatigue driving stress equivalence, the
remaining fatigue life under varying amplitude loading was then
predicted by an enhanced fatigue driving stress model. Compared with
Miner's rule, the K-R model, Zhu's model, and Li's model, this new model
gave more accurate and reliable predictions.

\textbf{Keywords}

Fatigue, life prediction, fatigue driving stress, loading interactions,
loading sequence

\textbf{Nomenclature}\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
fatigue strength constant & cumulative damage\tabularnewline
\midrule
\endhead
fatigue strength index & fatigue damage state of 1\textsuperscript{st}
stage loading\tabularnewline
fatigue life & fatigue damage state of 2\textsuperscript{rd} stage
loading\tabularnewline
the number of cycle & new equivalence equation\tabularnewline
equivalence equation & applied stress\tabularnewline
fatigue driving stress function & applied stress of
1\textsuperscript{st} stage loading\tabularnewline
fatigue life of 1\textsuperscript{st} stage loading & applied stress of
2\textsuperscript{nd} stage loading\tabularnewline
the number of cycle of 1\textsuperscript{st} stage loading & equivalent
driving stress of 1\textsuperscript{st} stage loading\tabularnewline
fatigue driving stress function of 1\textsuperscript{st} stage loading &
critical fatigue driving stress\tabularnewline
fatigue life of 2\textsuperscript{nd} stage loading & equivalent driving
stress of 2\textsuperscript{nd} stage loading\tabularnewline
the equivalent number of cycle of 2\textsuperscript{nd} stage loading &
applied stress of 3\textsuperscript{rd} stage loading\tabularnewline
the number of cycle of 2\textsuperscript{nd} stage loading & equivalent
driving stress of 3\textsuperscript{rd} stage loading\tabularnewline
fatigue life of 3\textsuperscript{rd} stage loading & applied stress of
i\textsuperscript{th} stage loading\tabularnewline
the equivalent number of cycle of 3\textsuperscript{rd} stage loading &
equivalent driving stress of i\textsuperscript{th} stage
loading\tabularnewline
the number of cycle of 3\textsuperscript{rd} stage loading &
temperature-dependent fatigue strength coefficient\tabularnewline
fatigue life of i\textsuperscript{th} stage loading & the new equivalent
number of cycle of 2\textsuperscript{nd} stage loading\tabularnewline
the number of cycle of i\textsuperscript{th} stage loading & ratio of
1\textsuperscript{st} and 2\textsuperscript{nd} stage loading
stresses\tabularnewline
damage of i\textsuperscript{th} stage loading & ratio of
i-1\textsuperscript{th} and i\textsuperscript{th} stage loading
stresses\tabularnewline
\bottomrule
\end{longtable}

\textbf{Introduction}

Mechanical components generally operate under cyclic stresses of varying
amplitude during their lifetime. Fatigue phenomena are generated by
these stresses and it is the main cause of failure of mechanical
components during operation. Assessing fatigue damage is a critical
issue, and it is one of the most common structural engineering problems.
In general, fatigue damage accumulation theories can be divided into two
categories: (1) linear damage accumulation theories and (2) nonlinear
damage accumulation theories.

To predict the remaining life of these components, it is important to
establish a method for assessing the accumulation of fatigue damage. For
many years, design engineers have used Miner's rule and its
modifications to predict the fatigue life of components under variable
loading\textsuperscript{11}. In the linear approach, the work absorbed
for each fatigue cycle is assumed to be constant and independent of one
another. Nor does it take into account stress damage below the fatigue
limit and the interaction between applied loads. This can lead to an
order of magnitude difference between the predicted life and the test
life, and such calculations may be unconservative\textsuperscript{2-4}.
Several researchers have tried to modify Miner's rule, but predictions
of life expectancy based on it are often unsatisfactory because of its
inherent flaws\textsuperscript{5}. Marco and
Starkey\textsuperscript{6}first proposed a nonlinear load-related damage
rule, denoting the damage accumulation as , whereis a factor depending
on theth loading.

More recently, Kwofie and Rahbar\textsuperscript{8-9} developed a novel
approach to nonlinear damage accumulation based on S-N curves, referred
to as the K-R model. Kwofie and Rahbar introduced the concept of fatigue
driving forces that lead to fatigue damage in this model. The fatigue
driving stress is a function of the cyclic stress, the number of loading
cycles, and the fatigue life. This value increases with increasing
loading cycles until a critical maximum driving stress is reached at the
time of fracture. Residual fatigue life can be predicted by equating the
fatigue driving stresses to produce cycles equivalent to the previous
loading. It does not require much characterization of the material and
the parameters of the model are only related to the S-N curve of the
material. Later, Zuo\textsuperscript{10} et al. made a further study
based on the K-R model and proposed a new nonlinear damage accumulation
rule to improve the inherent flaws in the linear damage accumulation
rule and to keep it simple in its application. The main advantages of
the model are its ease of use, it only requires S-N curves, and it does
not require any additional material properties to account for the
effects of the loading history of the material.

It has been found that in addition to the loading sequence effects to be
taken into account, the load interaction effects can also impact fatigue
damage at multiple stages. Abrupt changes in load amplitude under
different loading sequences can cause changes in the damage evolution of
subsequent load cycles, which in turn affects the remaining life of the
structure. Schijve, Yarema\textsuperscript{11} and
Batsoulas\textsuperscript{12} noted that the number of damage nuclei
will lead to a load interaction effect that is greater at higher loading
stresses. In addition, the number of damage nuclei produced by different
loading stress levels will lead to variations in damage accumulation and
this effect will increase with larger differences between stress levels.

Considering the loading interaction effects, with reference to the
literature\textsuperscript{13,14}, Zhu\textsuperscript{15} introduced a
load ratio between each loading stage to describe the loading
interaction effects, making an improvement to the K-R model. In order to
study the loading interaction effects on damage accumulation,
Peng\textsuperscript{16} analyzed that after undergoing high-loading
cycling, the damage evolution curve under low-loading cycling will be
shifted based on its damage evolution curve of constant amplitude
loading and this shift has a promoting effect on damage accumulation,
while the opposite is true for low-high loading, providing an intuitive
theoretical explanation for loading interaction effects.

The objective of this study is to present a new modified fatigue driving
stress model that accounts for the loading interaction effects and is
used to predict the remaining life under variable amplitude fatigue
loading. The validity of this model is also verified by comparing it
with Miner's rule and other models based on fatigue driving stress
theory through fatigue tests of variable amplitude on several materials.

\textbf{Models based onfatigue driving stress theory}

\textbf{2.1. Fatigue driving stress theory}

It is well known that the S-N curve of a material can be expressed using
a power function as follows:

\par\null

whereis the fatigue strength constant, is the fatigue strength index, is
the applied cyclic stress, andis the fatigue life.

In the new approach to nonlinear damage accumulation developed by Kwofie
and Rahbar\textsuperscript{8,9}, it is argued that although the applied
loading is constant, the transient driving loading, which varies
continuously with fatigue loading, is the primary causative factor for
damage accumulation and is defined as the fatigue driving stress.

For a given stress loading, the fatigue driving stress function can be
expressed as:

Where is the life-fraction of load and is an increasing function with
respect to , as the number of loading cycles increase from zero to ,
increases from to Thus, at we have

Thus it is expected that while the applied cyclic load may be constant,
the fatigue driving stress will increase with cycling until the fatigue
strength constant is reached, at which point fracture of the specimen is
expected to occur\textsuperscript{9}.

\textbf{2.2.} \textbf{Existing models based on fatigue driving stress
theory}

\textbf{2.2.1.K-R model}

By equivalence of fatigue driving stress, models based on fatigue
driving stress theory have been developed. Taking a two-stage cyclic
loading as an example, the fatigue lives corresponding to different
stresses loading and are and respectively, and the growth law of fatigue
driving stress is presented in Figure 1.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image61/image61}

\end{center}
\end{figure}

FIGURE 1 Growth law of fatigue driving stress under two-stage cyclic
loading

According to Figure 1, under a high-low loading sequence, the member is
first loaded under stress for cycles, and the fatigue driving stress
reaches point E. Then we have

Assuming that the member is loaded under stress for cycles to point F
such that points E and F have the same fatigue driving stress and is the
equivalent number of cycles under stress loading, the fatigue driving
stress at point F can be expressed as

Because the E and F points have the same fatigue driving stress,
equations (4) and (5) are required to be equivalent. Then equating the
fatigue driving stress reached by cycles of loading at stress to the
fatigue driving stress reached by cycles of loading at stress, according
to equation (2), the equivalence equation can be expressed as

\par\null

The equivalent driving stress at points E and F can be expressed as:

Equation (7) shows that the fatigue driving stress achieved by loading
stressat life-fraction of is equal to the fatigue driving stress
achieved by loading stress at life-fraction of .

Subsequently, if the loading is continued for cycles under stress to
achieve the equivalent driving stress, we have

If fatigue failure occurs under the second stage of loading, then the
fatigue driving force will follow the path FG from until the critical
fatigue driving stress is reached. From equations (7) and (8) we can get

\par\null

Taking equation into the equation, the remaining life fraction of the
second stage of loading is obtained as

Similarly, if there is a third stage of loading, then the fatigue
driving stress reached by cycles of loading stress is equal to the
fatigue driving stress reached by ()cycles of loading stress , then
the equivalent equation can be expressed as

The equivalent driving stress is expressed as

Then if the loading is continued for cycles under stress to reach the
equivalent driving stress, we have

Combining equation(7), equation (8), and equation (12), equation (13)
can be transformed as

Thus, for a multi-stage cyclic loading with stresses of , loading cycles
of , and fatigue failure lives of , the equivalent fatigue driving
stress takes the form of

At fracture, the critical fatigue driving stress will be equal to the
fatigue strength constant

\par\null

Taking the equation into the equation, a new damage model is defined in
the form of equation (17)

\par\null

Therefore, each item on the left-hand side of the equation defines the
damagedue to applied loading stressas

where is the cycle life fraction under loading stress, is the fatigue
failure life of loading , and is the life of the initially applied
loading . Thus, the cumulative damage can be expressed as

When fatigue failure occurs at =1, the remaining life fraction of the
ith load is

\textbf{2.2.2.} \textbf{Modified K-R model}

\textbf{Zhu model}

In order to consider the load interaction effects,
Zhu\textsuperscript{15} et al. proposed a modified K-R model by imposing
a load ratio between two stages to obtain a multi-stage residual life
prediction model as

\textbf{Li model}

Since the K-R model only considers fatigue damage at a constant
temperature, Li\textsuperscript{17} et al proposed a modified K-R model
at variable temperatures. Fatigue driving stress was initially
normalized to

where is the temperature-dependent fatigue strength coefficient.

The remaining life fraction for the second-stage loading at a
temperature different from the first stage can be obtained based on the
equivalence of fatigue driving stress.

where and are the temperature-dependent fatigue strength coefficients
for the first stage and the second stage respectively.

In order to consider the effect of the first-stage loading on the
remaining life fraction of the second stage, Li et al obtained the
following expression by proposing to use as a pre-cycling factor.

Extending the model to multi-stage loading as

\par\null

Li and Zhu give a modified form of the K-R model by adding a power
exponential term to the remaining life model respectively, but do not
give a specific explanation of the reason for the correction.

\textbf{3.} \textbf{A new residual life prediction model}

\textbf{3.1.Loading interaction theory}

According to damage equivalence, variable amplitude loadings can be
equated to a constant amplitude loading, as shown in Figure 2.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image117/image117}

\end{center}
\end{figure}

FIGURE 2 Damage equivalence

Figure 2 shows that cycles at high loading have exactly the same damage
state as cycles at low loading . However, in view of the rationality of
fatigue damage equivalence under different stress levels,
studies\textsuperscript{18-19} point out that the traditional damage
equivalence state does not exist under different stress levels.
According to this point of view, in the fatigue driving stress model,
equivalence according to equation (6) is not sufficient as well. In
particular, for the high-low loading sequence, it tends to speed up the
damage evolution process under low loading, whereas for the low-high
loading sequence it is the reverse. At the same time, the greater the
difference between the stress levels applied, the greater the effect.
This phenomenon (loading interactions) is usually described in terms of
load ratio and has been reported by Corten and
Dolan\textsuperscript{20}, Freudenthal and Heller\textsuperscript{21},
Morrow\textsuperscript{22}, and Huang\textsuperscript{23} et al.

More recently, targeting the flaws of traditional damage equivalence
methods, Peng\textsuperscript{16} proposed a fatigue damage equivalence
rule by considering the loading interaction effects. Generally,
according to the principle of damage equivalence, points A and B in
Figure 3 have the same damage state, which can be represented by the
concept of fatigue damage state as:

The damage evolution path is then OABE, but the loading of the second
stage retains its damage evolution trend under constant amplitude
loading, ignoring the interaction effects between the loadings. After
undergoing high-loading cycling, the damage evolution curve under
low-loading cycling will be shifted based on its damage evolution curve
of constant amplitude loading and this shift has a promoting effect on
damage accumulation, while the opposite is true for low-high loading.
The damage curve after the offset can be assumed to be AN, and the
damage curve AN is shifted and extended to give an equivalent damage
curve OFE with low loading applied alone, as shown in Figure 3. Unlike
the OBE, this damage curve OFE takes into account the interactions
between high and low loadings.

Thus, by considering the load interaction effects, the damage
accumulation path is OA-AF-FE, and the corresponding equivalent state of
fatigue damage can be described as

The function represented by curve OBE has the same starting point and
ending point as that represented by curve OFE, and curve OFE can be
transformed by curve OBE. In general, this transformation relation is
power exponential. Peng\textsuperscript{16} modified Ye's model with the
proposed method by taking loading interactions into account, and in
comparison to Miner's rule and Ye's model, the modified model has a more
satisfactory forecasting result .\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image124/image124}

\end{center}
\end{figure}

FIGURE 3 Fatigue damage evolution under two-stage high-low loading

\textbf{3.2.A new residual life prediction model considering loading
interaction effects}

In order to consider the loading interaction effects, the fatigue
driving stress model is modified. From the evolution of the fatigue
driving stress in Figure 1 and the damage evolution in Figure 3, the
fatigue driving stress curve after an offset can be assumed to be EI, as
shown in Figure 4.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image125/image125}

\end{center}
\end{figure}

FIGURE 4 Fatigue driving stress evolution under two-stage high-low
loading

After translation and extension of the fatigue driving stress curve EI,
a fatigue driving stress curve F'G is finally obtained with a low
loading applied alone. In contrast to EI, the curve F'G accounts for the
loading interaction effects. Compared with EI, it can be expressed as:

The fatigue driving stress curve F'G is built based on curve FG, and the
following conditions should be satisfied:

\par\null

Under low-high loading

According to the equivalence of fatigue driving stress, the fatigue
driving stress generated by cycles under stress is equal to the fatigue
driving stress generated by cycles under stress according to curve F'G,
as shown in Figure 3. Therefore, considering the loading interaction
effects, the fatigue driving path is E F'-F'G, and the corresponding
equivalent state can be described as:

According to the studies\textsuperscript{20-23}, the greater the
difference between loading levels, the more significant the interaction
effects are. The loading interaction effects can be described by the
ratio of loading levels. Therefore, the interaction factor is defined in
this paper as the ratio of loading levels of two adjacent stages,.

In order to satisfy conditions (29) - (33), referring to the
Marco-Starkey model\textsuperscript{6} or Manson
model\textsuperscript{25}, we can obtain by adding the power exponent to
:

Then the equivalent driving stress can be expressed as:

By taking logarithms of both sides of equation (36), the equivalent
cyclic fraction under stress loading can be obtained:

If the member fails by fatigue under a secondary loading, the fatigue
driving stress will follow the path from until the critical driving
stress is reached.

\par\null

By substituting the equation into , the remaining life fraction of the
second stage of loading is obtained as

By fatigue driving stress equivalence, extending (39) to the multi-stage
loading case, the remaining life fraction at the ith stage is

\par\null

Where the equivalent life fraction is expressed as

Where is the loading interaction factor, expressed as the ratio of two
levels of stress:

In summary, equation (40) is the general form of the residual life
prediction model when considering the loading interaction effects. The
unknown parameters involved in the model can be determined only by the
S-N curve.

\textbf{4. Comparison of predicting results of different models}

Two-stage loading tests of Al2024 aluminum alloy, welded joints of
aluminum alloy body, and nodular cast iron (GS61) of high-speed EMUS and
four-stage loading test of aluminum alloy 6082T6 are used to verify the
validity of the proposed model.

\textbf{4.1.} \textbf{Two-stage loading}

\textbf{Results of alloy Al2024}

The material studied by Pavlou is aluminum alloy
Al2024\textsuperscript{26}. The life of loading stress 150Mpa is 430000
cycles, and the life of loading stress 200Mpa is 150000 cycles.

Table 1 gives the experimental remaining life fraction and the predicted
life fraction of Miner's rule, the K-R model, Zhu's model, Li's model,
and the new model.

TABLE 1 Experimental and model prediction results of Al2024\selectlanguage{english}
\begin{longtable}[]{@{}llllllllll@{}}
\toprule
\begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
Load level (Mpa)\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
Load level (Mpa)\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
Load sequence\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
Load sequence\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
Experimental\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
Experimental\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
Experimental\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
Experimental\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
Experimental\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
Experimental\strut
\end{minipage}\tabularnewline
\midrule
\endhead
\begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
200-150\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
200-150\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
High-Low\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
High-Low\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
30000\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
30000\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.2\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
228700\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
228700\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.67\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
60000\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
60000\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.4\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
101050\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
101050\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.24\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
90000\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
90000\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.6\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
76050\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
76050\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.18\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
150-200\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
150-200\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
Low-High\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
Low-High\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
86000\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
86000\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.2\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
144500\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
144500\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.96\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
172000\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
172000\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.4\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
133500\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
133500\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.89\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
258000\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
258000\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.6\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
81700\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
81700\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.54\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
Miner rule\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
K-R model\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
K-R model\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
Zhu model\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
Zhu model\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
Li model\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
Li model\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
Li model\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
Proposed model\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
Proposed model\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.8\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.73\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.73\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.71\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.71\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.70\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.70\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.70\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.63\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.63\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.6\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.55\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.55\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.53\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.53\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.56\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.56\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.56\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.45\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.45\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.4\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.37\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.37\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.36\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.36\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.38\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.38\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.38\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.29\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.29\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.8\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.87\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.87\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.85\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.85\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.92\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.92\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.92\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.93\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.93\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.6\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.65\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.65\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.64\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.64\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.65\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.65\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.65\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.76\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.76\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.4\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.44\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.44\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.43\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.43\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.42\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.42\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.42\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.53\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.53\strut
\end{minipage}\tabularnewline
\bottomrule
\end{longtable}

As can be seen in Figure 4, the sum of life fractions is 1 under Miner's
rule and the other models have sums of life fractions less than 1 under
high - low loading and greater than 1 under low - high loading, where
the models take into account load history effects.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image160/image160}

\end{center}
\end{figure}

FIGURE 5 Comparison of the experimental results of Al2024 with the
predicted results of the five models

Compared with Miner's Rule, the K-R model, Zhu's model, and Li's model,
the model proposed in this paper have better prediction results. As can
be seen from Figure 5, the predicted life of the model under high-low
loading falls within the 2x error band, and the predicted life under
low-high loading falls within the 1.5x error band.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image161/image161}

\end{center}
\end{figure}

FIGURE 6 Comparison of the predictive capability of five models for the
remaining life of Al2024

\textbf{Results of welded joints of aluminum alloy body}

The fatigue test of welded joints of the aluminum alloy body of a
high-speed EMU was analyzed as an example\textsuperscript{27}. When the
loading stress of aluminum alloy butt joints is 104, 89, and 74Mpa, the
fatigue life is 549300, 880500, and 1540100 cycles, respectively. The
fatigue life of aluminum alloy angle joints is 619800, 952300, and
1546100 cycles when the loading stress is 93, 83, and 73Mpa,
respectively.

Table 2 gives the experimental remaining life fraction and the predicted
life fraction of Miner's rule, the K-R model, Zhu's model, Li's model,
and the new model.

TABLE 2 Experimental and model prediction results of welded joints\selectlanguage{english}
\begin{longtable}[]{@{}llllllllllll@{}}
\toprule
Type & Load sequence & Load sequence & Load level(Mpa) & Load
level(Mpa) & Load level(Mpa) & Experimental & Experimental &
Experimental & Experimental & Experimental & Experimental\tabularnewline
\midrule
\endhead
Butt joints & High-Low & High-Low & High-Low & 104-74 & 104-74 & 109900
& 109900 & 0.2 & 797600 & 797600 & 0.52\tabularnewline
& & & & 89-74 & 89-74 & 176100 & 176100 & 0.2 & 1029200 & 1029200 &
0.67\tabularnewline
& Low-High & Low-High & Low-High & 74-89 & 74-89 & 770100 & 770100 & 0.5
& 545600 & 545600 & 0.62\tabularnewline
& & & & 74-104 & 74-104 & 770100 & 770100 & 0.5 & 418900 & 418900 &
0.76\tabularnewline
Angle joints & High-Low & High-Low & High-Low & 93-73 & 93-73 & 309900 &
309900 & 0.5 & 587500 & 587500 & 0.38\tabularnewline
& & & & 83-73 & 83-73 & 476100 & 476100 & 0.5 & 681100 & 681100 &
0.44\tabularnewline
& Low-High & Low-High & Low-High & 73-83 & 73-83 & 509200 & 509200 &
0.33 & 708200 & 708200 & 0.74\tabularnewline
& & & & 73-93 & 73-93 & 773000 & 773000 & 0.5 & 426400 & 426400 &
0.69\tabularnewline
Miner rule & Miner rule & K-R model & K-R model & K-R model & Zhu model
& Zhu model & Li model & Li model & Li model & Proposed model & Proposed
model\tabularnewline
\bottomrule
\end{longtable}\selectlanguage{english}
\begin{longtable}[]{@{}lllll@{}}
\toprule
0.8 & 0.742 & 0.719 & 0.709 & 0.618\tabularnewline
0.8 & 0.768 & 0.762 & 0.750 & 0.704\tabularnewline
0.5 & 0.520 & 0.517 & 0.514 & 0.587\tabularnewline
0.5 & 0.539 & 0.527 & 0.527 & 0.663\tabularnewline
0.5 & 0.468 & 0.459 & 0.477 & 0.391\tabularnewline
0.5 & 0.483 & 0.481 & 0.488 & 0.440\tabularnewline
0.67 & 0.694 & 0.691 & 0.697 & 0.740\tabularnewline
0.5 & 0.534 & 0.527 & `0.524 & 0.622\tabularnewline
\bottomrule
\end{longtable}

As can be seen in Figure 7, the sum of life fractions is 1 under Miner's
rule and the other models have sums of life fractions less than 1 under
high - low loading and greater than 1 under low - high loading, where
the models take into account load history effects.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image164/image164}

\end{center}
\end{figure}

FIGURE 7 Comparison of the experimental results of welded joints with
the predicted results of the five models

Compared with Miner's Rule, the K-R model, Zhu's model, and Li's model,
the model proposed in this paper have better prediction results. As can
be seen from Figure 8, the predicted life of the model under high-low
loading falls within the 2x error band, and the predicted life under
low-high loading falls within the 1.5x error band.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image165/image165}

\end{center}
\end{figure}

FIGURE 8 Comparison of the predictive capability of five models for the
remaining life of welded joints

\textbf{Results ofnodular cast iron GS61}

Reference\textsuperscript{28} gives the fatigue loading test data of
nodular cast iron GS61. When the loading stress is 352, 320, and 303Mpa,
the fatigue life is 112866, 322580, and 588235 cycles, respectively.

Table 3 gives the experimental remaining life fraction and the predicted
life fraction of Miner's rule, the K-R model, Zhu's model, Li's model,
and the new model.

TABLE 3 Experimental and model prediction results of GS61\selectlanguage{english}
\begin{longtable}[]{@{}llllllllll@{}}
\toprule
\begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
Load sequence\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
Load level (Mpa)\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
Load level (Mpa)\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
Load level (Mpa)\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
Experimental\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
Experimental\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
Experimental\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
Experimental\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
Experimental\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
Experimental\strut
\end{minipage}\tabularnewline
\midrule
\endhead
\begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
High-Low\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
352-320\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
352-320\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
352-320\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
50000\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.443\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.443\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
130510\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
130510\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.405\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
352-303\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
352-303\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
352-303\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
50000\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.443\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.443\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
205040\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
205040\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.349\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
Low-High\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
320-352\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
320-352\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
320-352\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
110000\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.341\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.341\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
82830\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
82830\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.734\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
303-352\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
303-352\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
303-352\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
160000\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.272\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.272\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
116650\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
116650\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
1.030\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
Miner rule\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
Miner rule\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
K-R model\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
Zhu model\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
Zhu model\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
Zhu model\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
Li model\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
Li model\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
Proposed model\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
Proposed model\strut
\end{minipage}\tabularnewline
\bottomrule
\end{longtable}\selectlanguage{english}
\begin{longtable}[]{@{}lllll@{}}
\toprule
0.557 & 0.511 & 0.506 & 0.519 & 0.478\tabularnewline
0.557 & 0.488 & 0.477 & 0.500 & 0.438\tabularnewline
0.659 & 0.718 & 0.713 & 0.723 & 0.752\tabularnewline
0.728 & 0.831 & 0.816 & 0.865 & 0.874\tabularnewline
\bottomrule
\end{longtable}

As can be seen in Figure 9, the sum of life fractions is 1 under Miner's
rule and the other models have sums of life fractions less than 1 under
high - low loading and greater than 1 under low - high loading, where
the models take into account load history effects.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image166/image166}

\end{center}
\end{figure}

FIGURE 9 Comparison of the experimental results of GS61 with the
predicted results of the five models

Compared with Miner's Rule, the K-R model, Zhu's model, and Li's model,
the model proposed in this paper have better prediction results. As can
be seen from Figure 10, the predicted life of the model under high-low
loading falls within the 2x error band, and the predicted life under
low-high loading falls within the 1.5x error band.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image167/image167}

\end{center}
\end{figure}

FIGURE 10 Comparison of the predictive capability of five models for the
remaining life of GS61

\textbf{4.2.} \textbf{Multi-stage loading}

The four-stage loading test of aluminum alloy 6082T6 was carried
out\textsuperscript{29}. When the loading stress is 305, 280, 260, and
240Mpa, the fatigue life is 38000, 87612, 180660, and 394765 cycles,
respectively.

Table 4 gives the experimental remaining life fraction and the predicted
life fraction of Miner's rule, the K-R model, Zhu's model, Li's model,
and the new model.

Tables 4 and 5 give a comparison of the experimental results with
Miner's rule, the K-R model, Zhu's model, Li's model, and the new
model's prediction results

TABLE 4 Four-stage loading test results of aluminum alloy 6082T6\selectlanguage{english}
\begin{longtable}[]{@{}llllllllll@{}}
\toprule
\begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
Load sequence\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
Load level (Mpa)\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
Experimental\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
Experimental\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
Experimental\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
Experimental\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
Experimental\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
Experimental\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
Experimental\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
Experimental\strut
\end{minipage}\tabularnewline
\midrule
\endhead
\begin{minipage}[t]{0.10\columnwidth}\raggedright\strut
High-Low\strut
\end{minipage} & \begin{minipage}[t]{0.10\columnwidth}\raggedright\strut
305-280- 260-240\strut
\end{minipage} & \begin{minipage}[t]{0.10\columnwidth}\raggedright\strut
10950\strut
\end{minipage} & \begin{minipage}[t]{0.10\columnwidth}\raggedright\strut
0.288\strut
\end{minipage} & \begin{minipage}[t]{0.10\columnwidth}\raggedright\strut
19427\strut
\end{minipage} & \begin{minipage}[t]{0.10\columnwidth}\raggedright\strut
0.222\strut
\end{minipage} & \begin{minipage}[t]{0.10\columnwidth}\raggedright\strut
26258\strut
\end{minipage} & \begin{minipage}[t]{0.10\columnwidth}\raggedright\strut
0.145\strut
\end{minipage} & \begin{minipage}[t]{0.10\columnwidth}\raggedright\strut
52500\strut
\end{minipage} & \begin{minipage}[t]{0.10\columnwidth}\raggedright\strut
0.133\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
Low-High\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
240-260-280-305\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
103000\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.261\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
26258\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.145\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
19427\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.222\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
16800\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.442\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
Random\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
280-305-260-240\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
19427\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.222\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
10950\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.288\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
26258\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.145\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
43400\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.110\strut
\end{minipage}\tabularnewline
\bottomrule
\end{longtable}

TABLE 5 Comparison of experimental and model prediction results of
aluminum alloy 6082T6\selectlanguage{english}
\begin{longtable}[]{@{}llllllllllllllll@{}}
\toprule
Experimental & Experimental & Miner & Miner & Miner & K-R & K-R & K-R &
Zhu & Zhu & Zhu & Li & Li & Li & Purposed & Purposed\tabularnewline
\midrule
\endhead
52500 & 0.133 & 136098 & 0.344 & 98760 & 98760 & 0.250 & 97124 & 97124 &
0.246 & 92458 & 92458 & 0.234 & 88326 & 88326 & 0.223\tabularnewline
16800 & 0.442 & 14136 & 0.372 & 18886 & 18886 & 0.497 & 18578 & 18578 &
0.489 & 20903 & 20903 & 0.550 & 19941 & 19941 & 0.524\tabularnewline
43400 & 0.110 & 136098 & 0.345 & 124350 & 124350 & 0.315 & 123070 &
123070 & 0.312 & 110801 & 110801 & 0.281 & 109770 & 109770 &
0.278\tabularnewline
\bottomrule
\end{longtable}

It can be seen from Figure 10 that the proposed model has better life
prediction ability under high-low loading and random loading. The
predictive capability of the proposed model is similar to that of the
K-R model, Zhu's model, and Li's model under low-high loading but better
than that of the Miner model.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image183/image183}

\end{center}
\end{figure}

FIGURE 11 Comparison of the predictive capability of five models for the
remaining life of 6082T6

\textbf{5. Discussions}

Based on the results of the above four tests, it is shown that the
proposed model has improved the accuracy of predicting the remaining
life of fatigue loading to a considerable extent compared to the fatigue
driving stress model (K-R model). Miner's rule has a simple application
in the form that fatigue damage at different levels can be summed
linearly, but fatigue failure in many metallic materials often exhibits
highly nonlinear damage behavior. The fatigue driving force model (K-R
model) improves on the inherent flaws of the linear damage accumulation
rule. It is a nonlinear model that requires only the S-N curve of the
material. In addition, it does not require many material property
parameters and takes into account the effects of the loading sequence.
The K-R model has improved predictions compared to Miner's rule by
having a sum of damage values greater than 1 in the low-high loading
sequence and less than 1 in the high-low loading sequence. However, it
was found that loading interactions are also a part of the load history
and the K-R model does not consider loading interactions. Zhu's model
considers loading interactions by adding the ratio of the two loading
levels, while Li's model considers load interactions by adding a
pre-circulation factor. However, it was found that the improvement in
the predictive capability of the two modified K-R models relative to the
original model was limited, probably due to the complexity of the
inherent variable amplitude loading damage evolution, and the two
modified models did not provide an explanation in terms of fatigue
driving stress evolution. In this paper, based on the driving stress
evolution, a new driving stress evolution curve is obtained by taking
into account the load interactions, so that the original evolution curve
is shifted. Then, the fatigue driving stress equivalence was carried out
to obtain a new fatigue residual life prediction model. In particular,
compared with Miner's rule, K-R model, Zhu's model, and Li's model, the
proposed model has similar prediction ability under multi-stage loading,
but for two-stage loading, the proposed model has significantly higher
prediction accuracy.

\textbf{6. Conclusions}

The main purpose of this paper was to improve the life prediction
ability by studying the fatigue driving stress model, considering the
loading interactions and adding the interaction factor to the fatigue
driving stress model. The main work was summarized as follows:

A new fatigue residual life prediction model was proposed by analyzing
the fatigue driving force evolution, based on making the original
evolution curve shift. The effects of the loading interaction were
considered by introducing the ratio of two stresses into the fatigue
driving stress equivalence equation.

The validity of the new model was verified by using experimental data
from two-stage and four-stage loadings. Compared with Miner's rule, the
K-R model, Zhu's model, and Li's model, the proposed model gives more
satisfactory predictions and shows the ability to combine loading
sequences and loading interactions.

\textbf{Highlights}

\begin{enumerate}
\tightlist
\item
  A novel evolution curve for fatigue driving stress was created by
  including the loading interaction factor.
\item
  A new fatigue residual life prediction model was proposed.
\item
  Proposed model can achieve higher reliability .
\end{enumerate}

\textbf{Credit authorship contribution statement}

\textbf{Kaiwen Wang} : Investigation, Methodology, Validation,
Visualization,Writing - original draft ,Writing -review \& editing,
Supervision, Project administration. \textbf{Xu Zhao} : Writing -review
\& editing, Investigation, Methodology, Data curation. \textbf{Xuming
Niu} : Investigation, Methodology, Writing -review \& editing, Data
curation. \textbf{Yingdong Song} : Validation, Project administration,
Writing -- review \& editing, Supervision. \textbf{Zhigang Sun} :
Supervision, Project administration, Writing -- review \& editing, Data
curation.

\textbf{Acknowledgments}

The authors would like to acknowledge Engineer Wang from Shijinshi
Company for constructive suggests.

\textbf{Conflict of interest}

The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to
influence the work reported in this paper.

\textbf{Data availability statement}

The raw/processed data required to reproduce these findings cannot be
shared at this time as the data also forms part of an ongoing study.

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