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\begin{document}

\title{Crack initiation and propagation mechanisms of an extruded dual-phase
Magnesium-Lithium alloy in very high cycle fatigue regime}



\author[1]{Xiangyu WANG}%
\author[1]{Chao HE}%
\author[1]{Xue LI}%
\author[1]{YongJie Liu}%
\author[1]{Qingyuan Wang}%
\author[1]{Hong ZHANG}%
\author[1]{Lang LI}%
\author[1]{Chong Wang}%
\affil[1]{Sichuan University}%


\vspace{-1em}



  
  \date{June 17, 2024}


\begingroup
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\selectlanguage{english}
\begin{abstract}
Ultra-light Mg-Li alloy is a promising alloy in aerospace since it is
known to the lightest structural alloy at present, but its fatigue
behaviors remain to be explored. This work focuses on very-high cycle
fatigue (VHCF) strength and small crack initiation behaviors of an
extruded dual-phase Mg-Li alloy (LZ91). The fatigue strength of the LZ91
alloy at 109 cycles is about 78 MPa, and the fatigue ratio is
approximately 0.46. Microstructure characterization demonstrates that
fatigue crack tends to initiate from the \selectlanguage{greek}β-\selectlanguage{english}Li phase-enriched region. The
\selectlanguage{greek}α-\selectlanguage{english}Mg phase, presenting \textless{}10-10 \textgreater{}  fiber texture
with the basal plane, deforms hardly along the extrusion direction and
acts as an enhanced phase compared with the \selectlanguage{greek}β-\selectlanguage{english}Li phase. The deformation
discrepancy localizes cyclic plasticity at the Li phase and finally
leads to the fatigue crack initiation.%
\end{abstract}%



\sloppy


\textbf{Crack initiation and propagation mechanisms of an extruded
dual-phase Magnesium-Lithium alloy in very high cycle fatigue regime}

Xiangyu WANG \textsuperscript{a}, Chao HE* \textsuperscript{a, b}, Xue
Li \textsuperscript{a}, Yongjie LIU\textsuperscript{a, b}, Qingyuan
WANG* \textsuperscript{a, c}, Lang LI \textsuperscript{a,b}, Hong
ZHANG\textsuperscript{a, b}, Chong WANG \textsuperscript{a, b}

a Key Laboratory of Deep Earth Science and Engineering, Ministry of
Education, College of Architecture and Environment, Sichuan University,
Chengdu 610065, China

b Failure Mechanics \& Engineering Disaster Prevention and Mitigation,
Key Laboratory of Sichuan Province, Sichuan University, Chengdu 610065,
China

c School of Architecture and Civil Engineering, Chengdu University,
Chengdu 610106, China

Corresponding authors:

Dr. Chao HE

Sichuan University

Email: hechao@scu.edu.cn

Prof. Qingyuan WANG

Sichuan University

Email: wangqy@scu.edu.cn

\textbf{Abstract:} Ultra-light Mg-Li alloy is a promising alloy in
aerospace since it is known to the lightest structural alloy at present,
but its fatigue behaviors remain to be explored. This work focuses on
very-high cycle fatigue (VHCF) strength and small crack initiation
behaviors of an extruded dual-phase Mg-Li alloy (LZ91). The fatigue
strength of the LZ91 alloy at 10\textsuperscript{9} cycles is about 78
MPa, and the fatigue ratio is approximately 0.46. Microstructure
characterization demonstrates that fatigue crack tends to initiate from
the \selectlanguage{greek}β-\selectlanguage{english}Li phase-enriched region. The \selectlanguage{greek}α-\selectlanguage{english}Mg phase, presenting
\textless{}10-10 \textgreater{}  fiber texture with the basal plane,
deforms hardly along the extrusion direction and acts as an enhanced
phase compared with the \selectlanguage{greek}β-\selectlanguage{english}Li phase. The deformation discrepancy
localizes cyclic plasticity at the Li phase and finally leads to the
fatigue crack initiation.

\textbf{Keywords:}Magnesium-lithium alloy; very-high cycle fatigue;
crack initiation and propagation; microstructural sensitivity.

\textbf{1.Introduction}

In recent decades, magnesium alloy is one of the lightest structural
metallic materials, the advantages of this alloy are its outstanding
specific strength, sound damping capabilities {[}1-3{]}. However, due to
its hexagonal closed-packed (HCP) lattice structure, magnesium alloy
possesses limited slip systems, low ductility, and unfavorable
formability at room temperature {[}4-6{]}. Recent studies have shown
that the formability and ductility of magnesium alloys would be highly
improved by adding an appropriate amount of Lithium (BCC, body-centered
cubic structure) {[}7, 8{]}. It is reported that Lithium addition would
significantly decrease the CRSS value of non-basal slip systems by
introducing more slip systems and reducing the c/a ratio {[}9{]}.
Besides, with the addition of lithium, the lightest metal (about 0.534
g/cm\textsuperscript{3}), the Mg-Li dual-phase alloy could significantly
reduce its density without losing too much mechanical strength {[}10,
11{]}. Therefore, Mg-Li alloy is widely considered as an ideal
structural material in the aerospace, automotive, and electronic
industry. Usually, the structural material is subjected to repeated
fatigue loadings during its service life, which may exceed more than
10\textsuperscript{7} cycles and reach the VHCF (very-high cycle
fatigue) regime {[}12-14{]}. However, knowledge of the VHCF properties
of Mg-Li is fairly limited, which prevents its wide application. Thus,
it is essential to fully investigate its VHCF behavior.

As we all know, small crack initiation behavior accounts for the
majority (about 90\%-95\%) of the total fatigue life {[}15-17{]} of the
material and it is closely related to the microstructure of the
material. For conventional magnesium alloy with a single phase of HCP
structure, it is widely reported that crack initiation is closely
related to the deformation texture. Crack is mainly induced by twinning
deformation in Mg alloys with strong deformation texture {[}18{]}, while
basal slip-induced crack is the predominant crack initiation mode in Mg
alloys with weak deformation texture {[}19-21{]}. In the case of
dual-phase Mg-Li alloy, the plastic deformation behavior of the material
becomes more complicated with the introduction of the \selectlanguage{greek}β-\selectlanguage{english}Li phase.
Currently, plastic deformation and fracture behavior have been examined
by some researchers {[}22, 23{]}. For instance, Guo et al. {[}23{]}
investigated the slip behavior of dual-phase Mg-Li alloy by tracing the
microstructure and texture evolution during the rolling procedure. It
proposed that \{001\}\textless{}110\textgreater{} was the dominant
texture component in the \selectlanguage{greek}β-\selectlanguage{english}Li phase and basal slip, pyramidal I and II
slip worked together to coordinate deformation strain in the \selectlanguage{greek}α-\selectlanguage{english}Mg phase.
On the other hand, Dong et al. {[}22{]} found that deformation and
fracture occurred mainly in the \selectlanguage{greek}β-\selectlanguage{english}Li phase, and \selectlanguage{greek}α-\selectlanguage{english}Mg phase participated
in fracture at high tensile speed. However, what remains unclear is how
the \selectlanguage{greek}α-\selectlanguage{english}Mg and \selectlanguage{greek}β-\selectlanguage{english}Li phase coordinate the deformation strain as well as the
fracture mechanism of dual-phase Mg- Li alloys in the regimes of
high-cycle and very high-cycle fatigue.

Therefore, the purpose of this research has been to study the behavior
of dual-phase Mg-Li alloy LZ91 in the VHCF regime. The relation between
fatigue crack initiation and propagation and the microstructure of the
material was carefully investigated through microstructural
characterization and quantitative statistics. This research may increase
understanding of the formation mechanism of the fatigue crack of LZ91
alloy and potentially provide a way to improve its fatigue properties.

\textbf{2. Material and experimental procedure}

\textbf{2.1 Materials}

The material utilized in the present work is an as-extruded dual-phase
magnesium-lithium alloy LZ91. The material was prepared by melting high
purity element magnesium, lithium, and zinc in a vacuum induction
melting furnace under a protective gas of high purity argon gas,
followed by stirring and a setting of 3 min, then they were poured into
the cylindrical billets with a temperature of 730 . Then it was held at
250  for 1 h and to obtain an as-cast ingot. After that, the obtained
as-cast ingot was extruded at the temperature of 200  with an extruding
speed of 2 m/min. Finally, an extruded rod with dimensions of \selectlanguage{greek}Φ\selectlanguage{english}110 mm \selectlanguage{ngerman}×
400 mm was obtained. The chemical compositions of the LZ91 alloy are
displayed in \textbf{Table 1} , and the mechanical properties are listed
in \textbf{Table 2} , respectively.

In order to observe the microstructure of the LZ91 alloy, the specimen
surface was ground with 800, 1200, 2000 grits silicon carbide paper,
respectively, then it was an electrolytic etched perchloric acid-ethanol
solution (perchlorate acid: ethanol = 1: 9) for 90 seconds. The
metallography of the cross-section and side surface of the specimen was
presented in \textbf{Figure 1} , which was characterized by an optical
microscope (OM, OLYMPUS GX53). It can be seen clearly that the
microstructure of LZ91 alloy mainly consists of a brighter \selectlanguage{greek}α-\selectlanguage{english}Mg phase
and a relatively darker \selectlanguage{greek}β-\selectlanguage{english}Li phase. Moreover, it should be noted that
there is a significant difference between the microstructure of the
cross-section and the side surface of the specimen. The \selectlanguage{greek}α-\selectlanguage{english}Mg phase
in\textbf{Figure 1a} distributes in \selectlanguage{greek}β-\selectlanguage{english}Li phases with irregular shapes,
while \textbf{Figure 1b} shows that the \selectlanguage{greek}α-\selectlanguage{english}Mg and \selectlanguage{greek}β-\selectlanguage{english}Li phases were both
elongated along axial direction due to the radial extrusion.

\textbf{2.2 Experimental procedure}

\textbf{2.2.1 Nano-indentation}

Nano-indentation was applied to characterize the hardness values of the
\selectlanguage{greek}α-\selectlanguage{english}Mg phase and \selectlanguage{greek}β-\selectlanguage{english}Li phase. The nano-indentation test sample was cut by a
precision cutting machine, then it was mechanically ground. After that,
the electrolytic etching was employed to present the microstructure and
distinguish the \selectlanguage{greek}α-\selectlanguage{english}Mg phase and \selectlanguage{greek}β-\selectlanguage{english}Li phase of the material, and eliminate
the residual stress as well. The nano-indentation tests were performed
in KLA iNano, the maximum load and target depth were set as 20000 \selectlanguage{greek}μ\selectlanguage{english}N and
1000 nm to make indentations inner \selectlanguage{greek}α-\selectlanguage{english}Mg phase and \selectlanguage{greek}β-\selectlanguage{english}Li phase.

\textbf{2.2.2 Fatigue testing}

Data on long-life fatigue fracture of LZ91 were obtained by the fatigue
tests, which were conducted on an ultrasonic fatigue test system
(Shimadzu USF-2000) at room temperature in the air. The resonance
interval was 100 ms per 300 ms, and the stress ratio R = -1, up to a
limit of 10\textsuperscript{9} cycles except for failure. Fatigue test
specimens were performed on hourglass-shaped specimens, and the length
of the specimen was designed to give it a resonance of 20 kHz. The
detailed specimen configurations is shown in \textbf{Figure 3} . Before
fatigue testing, surfaces of specimens were polished in advance by
different grades of abrasive papers and cloths to eliminate the effect
of surface roughness on fatigue behavior. Temperature increment caused
by cyclic loadings used in this work was measured to be no more than 5 
during fatigue tests by an infrared camera (NEC-Avio R500EX). Besides,
compressed cold air was adopted to cool the specimens during fatigue
tests.

\textbf{2.2.3 Microstructure observation methods}

After fatigue tests, the microstructure of the material was
characterized by a scanning electron microscope (SEM, JEOL 7800F). The
detection mode was SEI (secondary electronic imaging) mode, and the
acceleration voltage is set to 15 kV, respectively. The SEM observation
area was concentrated on the fracture surface and specimen surface.
Especially, we carried out higher magnifications for the regions near
the crack initiation sites, which is focused on the crack initiation
behavior of the material. On the other hand, the EBSD (electron
backscatter diffraction) technique was applied to determine the grain
orientation of the material. The area detected is the cross-section of
the material, and the diffraction direction is perpendicular to the
cross-section.

\textbf{3. Experimental results}

3.1 Texture characterization

The microstructure of the material was characterized by the EBSD
technique before fatigue tests. \textbf{Figure 2} displays the EBSD
information of the cross-section of the material, including the grain
orientation map, the pole figure, and the inverse pole figure. The
striped phase in the grain orientation map in different colors is the
\selectlanguage{greek}α-\selectlanguage{english}Mg phase. It should be noted that the \selectlanguage{greek}β-\selectlanguage{english}Li phase can not be resolved
due to its high metal activity and high oxidation, thus it acts as a
black matrix in the grain orientation map. The EBSD results, especially
in the pole figure, show that the \selectlanguage{greek}α-\selectlanguage{english}Mg phase exhibits a strong texture
along the axial direction of the material, the grain orientation of most
grains is around \{10-10\} and \{11-20\} (the cylindrical directions of
the hexagonal close-packed structure). In other words, the basal plane
of most grains is parallel or nearly parallel to the axial direction of
the material. It is typical for the magnesium and its alloys to exhibit
a texture after extrusion, which mainly results from recrystallization
and a unidirectional deformation during extrusion {[}24-26{]}.

\textbf{3.2 Nano-indentation results}

The mechanical properties of \selectlanguage{greek}α-\selectlanguage{english}Mg and \selectlanguage{greek}β-\selectlanguage{english}Li phase obtained by the
nano-indentation test are listed in \textbf{Table 3} , and the
loading-displacement curves are shown in \textbf{Figure 4} ,
respectively. By comparing the mechanical properties of these two
phases, it can be found that the hardness and elastic modulus values of
the \selectlanguage{greek}α-\selectlanguage{english}Mg phase and the \selectlanguage{greek}β-\selectlanguage{english}Li phase are fairly close, about 75 HV and 50.5
GPa, respectively. Notably, however, the loading-depth curve of the \selectlanguage{greek}α-\selectlanguage{english}Mg
phase varies greatly from the one of \selectlanguage{greek}β-\selectlanguage{english}Li phase. It can be seen that the
depth of the indentation in the \selectlanguage{greek}β-\selectlanguage{english}Li phase increased smoothly with the
increasing of the loading stress applied by the probe. As shown
in\textbf{Figure 4a} , however, the loading-depth curve of the \selectlanguage{greek}α-\selectlanguage{english}Mg
phase is quite discontinuous and presents a ladder-like pattern. Similar
results were obtained by some previous nano-indentation experiments of
Mg-Li alloy done by V. Kumar et al. and W. Wang et al. {[}27-29{]}, and
this phenomenon was named as ``pop-in'' effect. Guo et al. {[}30{]}
carried out nano-indentation teste on magnesium grains of \{10-10\} and
\{11-20\} crystal planes. It was concluded that the ``pop-in'' effect in
loading-depth curves is ascribed to the initiation of \{10-12\}
extension twinning. This can be indirectly supported by the EBSD results
of the material. As stated in section 3.1, the axial grain orientation
of \selectlanguage{greek}α-\selectlanguage{english}Mg grains is mostly concentrated in \{10-10\} and \{11-20\}
cylindrical directions. Therefore, when the \selectlanguage{greek}α-\selectlanguage{english}Mg phase is subjected to
an axial loading applied by the probe, the twinning deformation will be
activated discontinuously when the shearing stress reaches the critical
value, as presented in the schematic diagram in \textbf{Figure
4a\textsubscript{1}} . On the other hand, due to the BCC (body-centered
cubic) structure, the \selectlanguage{greek}β-\selectlanguage{english}Li phase possesses 48 slip deformation systems,
which is far more than that of the HCP structure. In other words, there
are more slip systems for \selectlanguage{greek}β-\selectlanguage{english}Li grains to coordinate the plastic
deformation when they are subjected to axial loadings. For these
reasons, the displacement increases more continuously in the
loading-displacement curve of the \selectlanguage{greek}β-\selectlanguage{english}Li phase as compared with that of
the \selectlanguage{greek}α-\selectlanguage{english}Mg phase.

\textbf{3.3 Very high-cycle fatigue}

The S-N data associated with stress amplitudes \selectlanguage{greek}σ\selectlanguage{english}\textsubscript{a} and
fatigue life N\textsubscript{f} of LZ91 alloy under asymmetric axial
loading with a stress ratio of R = -1 are shown in \textbf{Figure 5} .
The S-N data of some other Mg-Li alloys and magnesium alloys were also
included for comparison {[}19, 31, 32{]}. ``Run-out specimens''
represents that the fatigue life of the specimen reached a limit of
1\selectlanguage{ngerman}×10\textsuperscript{7} (for LAZ832-0.5Y alloy) and
1×10\textsuperscript{7} (for the rest three alloys). The nonlinear
fitting curves of these fatigue data were obtained by Basquin's
equation, which were plotted by dash lines.

It can be seen that the S-N curve of LZ91 alloy exhibits a continuous
decreasing trend, even in the range of
10\textsuperscript{7}-10\textsuperscript{9} cycles. By comparing the S-N
curve of LZ91 alloy and LAZ832-0.5Y alloy, it can be found that the
curve of LZ91 alloy decreases much more slowly. Meanwhile, the fatigue
performance of the LZ91 alloy is much better than that of the
LAZ832-0.5Y alloy when the fatigue life exceeds
10\textsuperscript{5}cycles. As compared to conventional magnesium
alloys {[}19, 32{]}, whose fatigue tests were also conducted at R=-1 and
a loading frequency of \textasciitilde{}20 kHz, the fatigue resistance
of the LZ91 alloy is obviously lower, about 20-35 MPa. However, as the
fatigue life approaching 10\textsuperscript{9} cycles, the fatigue
strength gap between them is narrowing, less than 20 MPa at
10\textsuperscript{9}cycles. Due to the significant difference in
mechanical properties between LZ91 and these two alloys, the lag of LZ91
in fatigue strength seems inevitable. After that, in order to compare
the fatigue performance of these alloys, we calculated their fatigue
ratios (the ratio of fatigue strength to the ultimate tensile strength),
and the results are displayed in \textbf{Table 4} . Clearly, the fatigue
ratio of the LZ91 alloy, about 0.46, is significantly higher than that
of the other three alloys (0.29, 0.27, 0.30 for LAZ832-0.5Y alloy, ZK60,
AZ31, respectively), which indicates that the LZ91 alloy exhibits
relatively good fatigue performance.

\textbf{3.4 Observation on fracture surfaces}

After ultrasonic fatigue tests, the scanning electron microscope was
applied to characterize the fracture morphologies of failed
specimens.\textbf{Figure 6} shows typical SEM images of two
representatives failed specimens at different magnifications. It can be
clearly seen that fatigue crack nucleated and initiated from the
specimen surface and then continue to expand, which resulting in the
ultimate fatigue failure, as presented in \textbf{Figures 6a} and
\textbf{6e} . Meanwhile, the crack growth direction can be determined
according to the radial ridge of the material fracture.

\textbf{Figures 6b} and \textbf{6f} display the enlarged images of the
regions near the crack initiation sites of the two failed specimens,
respectively. Obviously, along the growth direction of the crack, the
fracture morphology varies greatly with the crack grows deeper, which is
quite analogous to that of high-strength steels and titanium alloys
{[}17, 33, 34{]}. Firstly, the regions near the crack initiation sites,
with a distance of about 100 \selectlanguage{greek}μ\selectlanguage{english}m, present a relatively smooth morphology.
As the region went further away from the crack initiation sites, the
fracture morphology became rougher than Zone I. Meanwhile, lots of
radial ridges can be seen in Zone II. As the distance goes further, no
characteristic morphology but radial streaks can be observed in the
regions far away from the crack initiation sites. In summary, according
to the difference in morphology, we utilized dash lines to divide the
fracture surface into three distinct regions: Zone I, Zone II, and Zone
III. These regions corresponding to three stages of crack initiation and
propagation, Zone (I): crack initiation region; Zone (II): short crack
propagation region; (III): long-crack propagation region.

High-magnification images of Zone I and Zone II are presented
in\textbf{Figures 6c} and \textbf{6g} . It is obvious that there are
much more striped crystallographic profiles in Zone II comparing to Zone
I. It can be seen more clearly in \textbf{Figures 6d} and \textbf{6h} .
By comparing the striped profiles with the \selectlanguage{greek}α-\selectlanguage{english}Mg grains in the
metallographic diagram of the material, it was found that they are
highly similar in appearance as well as area proportion. Namely, these
striped crystallographic profiles result from the fracture of the \selectlanguage{greek}α-\selectlanguage{english}Mg
grains near the crack initiation site. It has been reported in some
previous researches that there exist strong and weak phases in duplex
stainless steel {[}35-38{]}. The crack initiation and propagation
behaviors are closely related to the coordinated plastic deformation of
the two phases (\selectlanguage{greek}γ-\selectlanguage{english}austenite phase with faced centered cubic structure
and \selectlanguage{greek}α-\selectlanguage{english}ferrite phase with body centered cubic structure). On one hand,
when the hardness of the two phases differs greatly, the crack tends to
initiate in the weaker phase of the material. In this case, the phase
with much higher hardness is a hard phase in the material. For instance,
M.W. Tofique et al {[}35{]}. and R. Strubbia et al. {[}37{]} found that
since the hardness of ferrite grains is much lower than that of
austenite grains, the plastic damage accumulation mainly in the ferrite
grains. On the other hand, the crack nucleation will be highly affected
by the microstructure (grain orientation, grain boundary, etc.) of the
material when the hardness of these two phases differs slightly {[}35,
38{]}. As a structural alloy with a dual-phase structure, the LZ91 alloy
may possess hard and soft phases as well. Therefore, we investigated the
microstructure of two phases of the LZ91 alloy in the discussion section
and attempted to associate it with crack initiation and propagation.

\textbf{3.5 Morphology of specimen surfaces}

\textbf{Figure 7} presents an oblique view of the specimen surface near
the crack initiation site, the \selectlanguage{greek}α-\selectlanguage{english}Mg grains were indicated by white
arrows. Note that the crack exhibit different characteristics when
passing through the \selectlanguage{greek}α-\selectlanguage{english}Mg phase and \selectlanguage{greek}β-\selectlanguage{english}Li phase. The fatigue crack passes
directly through the \selectlanguage{greek}β-\selectlanguage{english}Li phase in the process of crack growth. On the
contrary, the crack presents a `zigzag' pattern when passing through the
\selectlanguage{greek}α-\selectlanguage{english}Mg grains, as enclosed by yellow dash circles. An enlarged optical
image of a second crack near the crack initiation site is presented
in\textbf{Figure 8} . Clearly, there are numerous turning points in the
crack path, especially when the crack passes through \selectlanguage{greek}α-\selectlanguage{english}Mg grains, making
the crack appear pretty tortuous. This is quite analogous to the fatigue
behavior of duplex stainless steel {[}38{]}. It was reported that the
fatigue crack growth was affected by the slip directions of the
\selectlanguage{greek}γ-\selectlanguage{english}austenite grains, which is a relatively hard phase in the material.
The crack path appears quite tortuous results from the influence of the
local microstructure. Therefore, it can be inferred that the \selectlanguage{greek}α-\selectlanguage{english}Mg phase
may also act as a hard phase (i.e. strengthening phase) in the material.
The microstructure (grain orientation or grain boundary) of the \selectlanguage{greek}α-\selectlanguage{english}Mg
phase would hinder the initiation and early propagation of fatigue crack
to a certain extent. In the following discussion section, we attempt to
quantitatively analyze and figure out how the local microstructure has a
macro effect on the crack behavior of the material.

\textbf{4. Discussion}

As have been highly stressed in extensive previous studies {[}15-17{]},
small crack initiation and propagation account for the majority of the
total fatigue life of metal materials. In other words, crack initiation
and early propagation play an essential role in the fatigue behaviors of
metal materials. Therefore, we focus on the small crack initiation and
propagation behaviors of LZ91 alloy as well as the influencing factors
in the following discussion section.

\textbf{4.1 Fracture morphology analysis based on fracture mechanics
calculation}

To further investigate the nature of fatigue crack initiation and
propagation behavior, quantitative fracture analysis was performed with
the help of fracture mechanics. The stress intensity factor \selectlanguage{greek}Δ\selectlanguage{english}K of the
fracture surface of failed specimens was calculated. For surface crack
initiation in the present work, Zone I, II can be approximately regarded
as a semi-elliptic. The areas of Zone I and Zone II were measured and
their square root values\(\ \sqrt{}\text{area}\) , which was obtained by
projecting on the plane perpendicular to the loading axial, were used to
calculate the \selectlanguage{greek}Δ\selectlanguage{english}K values,given by the equation for surface cracks {[}16,
39{]}:

\begin{equation}
\Delta K=0.65\sigma\sqrt{\pi\cdot\sqrt{ar\mathbb{e}a}}\nonumber \\
 \end{equation}

Where \selectlanguage{greek}σ \selectlanguage{english}is the maximum stress amplitude of cyclic stress for R=-1, and
0.65 is the calculated parameter for surface crack
initiation.\textbf{Figure 9} demonstrates the calculated results of \selectlanguage{greek}Δ\selectlanguage{english}K
of Zone I and Zone II together with the values of the applied stress
amplitude and the fatigue life. Obviously, the values of \selectlanguage{greek}Δ\selectlanguage{english}K
\textsubscript{Zone I} and \selectlanguage{greek}Δ\selectlanguage{english}K \textsubscript{Zone II} maintain nearly
constant levels regardless of the fatigue strength and the fatigue life.
The average \selectlanguage{greek}Δ\selectlanguage{english}K \textsubscript{Zone I}value was evaluated to 1.44
MPa\selectlanguage{ngerman}·m\textsuperscript{1/2} and the corresponding value of \selectlanguage{greek}Δ\selectlanguage{english}K
\textsubscript{Zone II} was a little higher, about 2.40
MPa\selectlanguage{ngerman}·m\textsuperscript{1/2}, respectively. Compared with some previous
investigations{[}40, 41{]}, the \selectlanguage{greek}Δ\selectlanguage{english}K \textsubscript{Zone II} is slightly
lower than that of some conventional magnesium alloys, which is usually
considered as a threshold of stable crack propagation. But it is still
within the range of the reported crack growth thresholds of magnesium
alloys, about 1.2-5.0 MPa m\textsuperscript{1/2} {[}40-42{]}. Therefore,
the \selectlanguage{greek}Δ\selectlanguage{english}K value of Zone II can be regarded as the crack propagation
threshold of the material, which corresponds to the threshold for stable
crack propagation.

Therefore, in the present case, the fatigue failure process of the LZ91
alloy can be divided into three main stages according to the differences
in the \selectlanguage{greek}Δ\selectlanguage{english}K values and the characteristics of the fracture surface. As
demonstrated in Section 3.3, the crack initiation region (Zone I)
presents a relatively special morphology, which is similar to the rough
area (RA) and facet morphology of some structural alloys {[}43-45{]}. In
the titanium alloys {[}44{]}, the facets were formed due to the
inconsistent plastic strain of the two phases in the crack initiation
region. Based on this, it can be preliminarily deduced that due to the
difference in the plastic deformability of the two phases in the
material, the plastic strain was highly concentrated in one of them.
Subsequently, the local deformation inhomogeneity of the two phases
would lead to the nucleation and initiation of micro-cracks. As the
micro-crack grows, the size of the plastic zone at the crack tip gets
larger, as well as the stress intensity factor. This would therefore
result in a higher level of stress in a local region near the
micro-cracks, as compared with neighboring regions. As a result,
numerous micro-cracks were initiated, propagated, and then coalesced in
Zone II. Namely, the formation of Zone II corresponds to a multi-crack
propagation stage in the process of fatigue failure. This is the reason
why there are a large number of radial ridges in Zone II, which make it
relatively rough. Afterward, when the stress intensity reaches the crack
propagation threshold value, namely, the \selectlanguage{greek}Δ\selectlanguage{english}K value of the edge of Zone
II, the crack propagation pattern changes again. In some researches of
high-strength steels, it was proposed that {[}46, 47{]} the
\selectlanguage{greek}Δ\selectlanguage{english}K\textsubscript{RA} (\selectlanguage{greek}Δ\selectlanguage{english}K \textsubscript{Zone II} in the present case)
value corresponds to the relevant threshold for long crack growth.
Therefore, the growth pattern of the crack in Zone III switched from the
multiple short crack growth to the steady single long-crack growth that
follows Paris law.

\selectlanguage{greek}\textbf{4.2 Influence of local proportion of α-Mg grain on crack
initiation behavior}\selectlanguage{english}

As a dual-phase Mg-Li alloy, there may also exist hard and soft phases
in the LZ91 alloy, just similar to the duplex stainless steels that have
been studied before. The plastic deformation will preferentially
accumulate in the soft phase during the subsequent micro-crack
initiation and early propagation {[}35, 36{]}. Therefore, we calculated
the proportion of \selectlanguage{greek}α-\selectlanguage{english}Mg and \selectlanguage{greek}β-\selectlanguage{english}Li phases in the region near the crack
initiation sites. In this way, we tried to explore whether the
difference in local microstructure will affect the fatigue performance
of the material. As presented in \textbf{Figure 10} , we calculated the
area of \selectlanguage{greek}α-\selectlanguage{english}Mg grains in the region near the crack initiation site in each
fracture topography. Note that the area of each calculated region is the
same, and the area of the calculated region is determined by the largest
area of Zone I (specimen No. 5). In this way, the proportion of \selectlanguage{greek}α-\selectlanguage{english}Mg
grains near the crack initiation site can be quantified more
intuitively. The conversion of the area of \selectlanguage{greek}α-\selectlanguage{english}Mg and \selectlanguage{greek}β-\selectlanguage{english}Li grains was done
by the mathematical software. The statistical results of the area of \selectlanguage{greek}α
\selectlanguage{english}and \selectlanguage{greek}β \selectlanguage{english}phases in the calculated region are listed in \textbf{Table 5} ,
and the \selectlanguage{greek}α-\selectlanguage{english}Mg ratio-Fatigue life diagram is presented in \textbf{Fig 10b}
, respectively. It is obvious that there is a simple linear relationship
between the fatigue life of the material and the proportion of \selectlanguage{greek}α-\selectlanguage{english}Mg
phase in the region near the crack initiation site. Namely, the lower
the proportion of \selectlanguage{greek}α-\selectlanguage{english}Mg phase in the region is, the longer the fatigue
life will be. Besides, it is worth noting that the proportion of \selectlanguage{greek}α-\selectlanguage{english}Mg
phase in most specimens is below the average value of the material.
Based on the discussion above, it can be preliminarily inferred that the
\selectlanguage{greek}α \selectlanguage{english}phase is a strengthening phase in the material as compared with the \selectlanguage{greek}β
\selectlanguage{english}phase. In other words, fatigue crack is more likely to nucleate and grow
in a local region, where the proportion of \selectlanguage{greek}α-\selectlanguage{english}Mg grains is relatively
low. As demonstrated in Section 3.2, however, the hardness and the
modulus of \selectlanguage{greek}α-\selectlanguage{english}Mg and \selectlanguage{greek}β-\selectlanguage{english}Li phase vary barely. As pointed above, the
crystallographic structure of the two phases is different, in which the
\selectlanguage{greek}α-\selectlanguage{english}Mg phase is the HCP, and the \selectlanguage{greek}β-\selectlanguage{english}Li phase is the BCC structure,
respectively. According to previous studies, the texture of magnesium
alloy will significantly affect the fatigue behavior of the material.
Therefore, in the present work, the most likely cause for this may be
the enhancing effect of the texture and the grain orientation of \selectlanguage{greek}α-\selectlanguage{english}Mg
grains.

\selectlanguage{greek}\textbf{4.3 Effect of the grain orientation of α-Mg grains on fatigue
behavior}\selectlanguage{english}

It is demonstrated in the previous section that there is a certain
relationship between the local proportion of \selectlanguage{greek}α-\selectlanguage{english}Mg in the region near the
crack initiation site and the fatigue life of the material. Since the
hardness and elastic modulus of the two phases vary slightly, the
microstructure of the material may be the governing affecting factor of
the fatigue properties. As displayed in \textbf{Figure 2} , the grain
orientation of most \selectlanguage{greek}α-\selectlanguage{english}Mg grains is around \{10-10\} and \{11-20\}
cylindrical direction, namely, the basal plane of them is almost
parallel to the loading direction during fatigue tests. It is well known
that the magnesium possesses only 12 slip systems {[}48-50{]}, with
three basal slip systems \{0001\} \textless{}-12-20\textgreater{}, three
prismatic slip systems \{10-10\} \textless{}-12-20\textgreater{}, and
six pyramidal slip systems \{10-10\} \textless{}11-23\textgreater{}. It
has been reported that the initial CRSS (critical resolved shear stress)
value of basal slip systems (about 0.45\textasciitilde{}0.81 MPa) and
\{10-12\} tensile twinning deformation (about 2.0\textasciitilde{}2.8
MPa) is far below that of non-basal slip systems (about 39.2 MPa of
prismatic slip and 45\textasciitilde{}91 MPa of pyramidal slip) and
compressive twinning deformation (about 76-153 MPa) at room temperature.
Thus, the basal slip systems and tensile twinning deformation are most
easily activated during the process of deformation. However, due to the
fact that the basal plane of the \selectlanguage{greek}α \selectlanguage{english}phase is nearly perpendicular to the
loading direction, the activation of basal slip is highly hindered.
Meanwhile, the loading-depth curve of the \selectlanguage{greek}α-\selectlanguage{english}Mg grain obtained by the
nano-indentation test demonstrates that the \selectlanguage{greek}α-\selectlanguage{english}Mg and \selectlanguage{greek}β-\selectlanguage{english}Li phases exhibit
distinct characteristics. Clearly, the loading-depth curve of the \selectlanguage{greek}α-\selectlanguage{english}Mg
phase exhibits a ``pop-in'' effect comparing to the \selectlanguage{greek}β-\selectlanguage{english}Li phase, which is
proved to be caused by the twinning deformation of \selectlanguage{greek}α-\selectlanguage{english}Mg grains under
compressive loading. Therefore, it can be preliminary deduced that the
twinning deformation is the predominating deformation pattern of \selectlanguage{greek}α-\selectlanguage{english}Mg
grains when subjected to axial loading.

As been revealed in Section 4.2, there is a linear relationship between
the fatigue life and the proportion of \selectlanguage{greek}α-\selectlanguage{english}Mg phase in the region near the
crack initiation site. Namely, a higher local proportion of \selectlanguage{greek}α-\selectlanguage{english}Mg phase,
which acts as a hard phase, seems to increase the fatigue life of the
material to a certain extent. Two main reasons, which make the \selectlanguage{greek}α-\selectlanguage{english}Mg
phase the hard phase in the material, can be concluded from the above
discussion. Firstly, the slip system of the HCP structure is less than
the BCC structure, thus the local coordinate deformability of the \selectlanguage{greek}β-\selectlanguage{english}Li
phase is much better than that of the \selectlanguage{greek}α-\selectlanguage{english}Mg phase. Secondly, the
orientation of the \selectlanguage{greek}α-\selectlanguage{english}Mg phase is mainly around the \{10-10\} and
\{11-20\} cylindrical direction, which makes the basal slip of the \selectlanguage{greek}α-\selectlanguage{english}Mg
phase difficult to be activated. Namely, the \selectlanguage{greek}α-\selectlanguage{english}Mg phase becomes the hard
phase in the material, whose deformability is much lower than the \selectlanguage{greek}β-\selectlanguage{english}Li
phase. As a result, the \selectlanguage{greek}β-\selectlanguage{english}Li phase takes the major part of the plastic
deformation during the fatigue testing, and the \selectlanguage{greek}α-\selectlanguage{english}Mg grains become a
strengthening phase that is difficult to deform. Therefore, the fatigue
crack tends to nucleate and initiate from the region where the \selectlanguage{greek}α-\selectlanguage{english}Mg
phase is relatively sparse.

\textbf{4.4 A proposed model of the fatigue behavior of LZ91 dual-phase
Mg-Li alloy}

As discussed above, small crack initiation and propagation are highly
affected by the microstructure of the material. \textbf{Figure 11} thus
presents the schematic illustration of crack initiation and early
propagation process. The stripe phase in white is the \selectlanguage{greek}α-\selectlanguage{english}Mg phase, and
the matrix in blue is the \selectlanguage{greek}β-\selectlanguage{english}Li phase, respectively. Severe plastic
deformation tends to occur in the `soft' phase, i.e., the \selectlanguage{greek}β-\selectlanguage{english}Li phase
during cyclic loading, as shown in Figure 11a. As crack grows and has to
pass through \selectlanguage{greek}α-\selectlanguage{english}Mg grains, the crack propagation will be inevitably
hindered by \selectlanguage{greek}α-\selectlanguage{english}Mg grain due to the hard grain orientations. As a result,
there are some turning points (marked by black arrows) in the crack path
when it passes through \selectlanguage{greek}α-\selectlanguage{english}Mg grains, which make it appear more tortuous.
Eventually, crack continues to propagate, resulting in the final failure
of the material.

In the present investigation, the experimental results and
microstructural analysis of the crack initiation will fill the gap of
our understanding of the VHCF behavior of the dual-phase Mg-Li alloy.
More importantly, they would provide more valuable advice to improve the
anti-fatigue ability of the Mg-Li alloys.

\textbf{5. Conclusions}

In the present paper, the VHCF behavior of a dual-phase Mg-Li alloy
(LZ91) was studied. The underlying mechanisms of the crack initiation
and early propagation have been fully investigated. The following
conclusions can be drawn from the discussions:

\begin{enumerate}
\tightlist
\item
  The S-N curve of the LZ91 alloy decreases continuously without a
  fatigue limit. The fatigue strength is evaluated to 78 MPa at
  10\textsuperscript{9} cycles, and the fatigue ratio of the material is
  calculated to 0.46, respectively.
\item
  The \selectlanguage{greek}α-\selectlanguage{english}Mg grains in the material exhibit strong textures due to the
  extrusion, in which the basal planes \selectlanguage{greek}α-\selectlanguage{english}Mg grains are almost parallel
  to the loading direction. The twinning deformation is the predominant
  pattern of \selectlanguage{greek}α-\selectlanguage{english}Mg grains since the basal slip is highly hindered during
  fatigue tests, making the \selectlanguage{greek}α-\selectlanguage{english}Mg the strengthening phase in the
  material.
\item
  The fractography of the material presents a noticeable microstructural
  sensitivity. The distinct morphologies of Zone I, II, and III resulted
  from the differences in the size of the plastic zone at the crack tip.
\item
  The fatigue crack initiation is significantly affected by the local
  microstructure of the material. The proportion of the \selectlanguage{greek}α-\selectlanguage{english}Mg phase of
  the region near the crack initiation region is lower than the average
  level. Cracks tend to nucleate and initiate from the local region with
  fewer \selectlanguage{greek}α-\selectlanguage{english}Mg grains, and the fatigue life is linearly positively
  correlated with the proportion of \selectlanguage{greek}α-\selectlanguage{english}Mg phase in the region.
\end{enumerate}

\textbf{Acknowledgment}

The authors gratefully acknowledge the financial support from the
National Natural Science Foundation of China (Nos. 12072212 and
11832007), the National Key Research and Development Program of China
(No. 2018YFE0307104) and the Applied Basic Research Programs of Sichuan
Province (No.21YYJC2743).

\textbf{Author Contribution Statement}

\textbf{Xiangyu WANG} : Data curation, Methodology, Software, Writing --
original draft. \textbf{Chao HE} : Project administration, Methodology,
Software, Writing -review & editing. \textbf{Xue LI} : Resources,
Writing -- original draft. \textbf{Yongjie LIU} : Formal analysis,
Supervision. \textbf{Qingyuan Wang} : Conceptualization, Funding
acquisition. \textbf{Lang Li} : Investigation,
Visualization.\textbf{Hong Zhang} : Investigation, Visualization.
\textbf{Chong Wang} : Formal analysis, Supervision.

\textbf{References}

{[}1{]} Z. Yang, J.P. Li, J.X. Zhang, G.W. Lorimer, J. Robson, Review on
research and development of magnesium alloys, Acta metallurgica sinica
(English letters) 21(5) (2009) 313-328.

{[}2{]} B.L. Mordike, T. Ebert, Magnesium:
properties---applications---potential, Materials Science and
Engineering: A 302(1) (2001) 37-45.

{[}3{]} V. Kumar, R. Shekhar, R. Balasubramaniam, K. Balani,
Microstructure evolution and texture development in thermomechanically
processed Mg--Li--Al based alloys, Materials Science and Engineering: A
547 (2012) 38-50.

{[}4{]} Z. Wu, R. Ahmad, B. Yin, S. Sandl\selectlanguage{ngerman}öbes, W.A. Curtin, Mechanistic
origin and prediction of enhanced ductility in magnesium alloys, Science
359(6374) (2018) 447-452.

{[}5{]} K. Srivastava, J.A. El-Awady, Deformation of magnesium during
c-axis compression at low temperatures, Acta Materialia 133 (2017)
282-292.

{[}6{]} M.R. Barnett, M.D. Nave, C.J. Bettles, Deformation
microstructures and textures of some cold rolled Mg alloys, Materials
Science and Engineering: A 386(1-2) (2004) 205-211.

{[}7{]} R.-z. Wu, Y.-d. Yan, G.-x. Wang, L.E. Murr, W. Han, Z.-w. Zhang,
M.-l. Zhang, Recent progress in magnesium--lithium alloys, International
Materials Reviews 60(2) (2015) 65-100.

{[}8{]} Z. Trojanová, Z. Drozd, S. Kúdela, Z. Száraz, P. Luká\selectlanguage{polish}č,
\selectlanguage{english}Strengthening in Mg--Li matrix composites, Composites Science and
Technology 67(9) (2007) 1965-1973.

{[}9{]} J. Hirsch, T. Al-Samman, Superior light metals by texture
engineering: Optimized aluminum and magnesium alloys for automotive
applications, Acta Materialia 61(3) (2013) 818-843.

{[}10{]} Y. Zou, L. Zhang, Y. Li, H. Wang, J. Liu, P.K. Liaw, H. Bei, Z.
Zhang, Improvement of mechanical behaviors of a superlight Mg-Li base
alloy by duplex phases and fine precipitates, Journal of Alloys and
Compounds 735 (2018) 2625-2633.

{[}11{]} H.-Y. Wu, J.-C. Yan, H.-H. Tsai, C.-H. Chiu, G.-Z. Zhou, C.-F.
Lin, Tensile flow and strain-hardening behaviors of dual-phase
Mg--Li--Zn alloy thin sheets, Materials Science and Engineering: A
527(27-28) (2010) 7197-7203.

{[}12{]} C. Bathias, P.C. Paris, Gigacycle Fatigue in Mechanical
Practice. Dekker, (2005).

{[}13{]} Q.Y. Wang, J.Y. Berard, A. Dubarre, G. Baudry, S. Rathery, C.
Bathias, Gigacycle fatigue of ferrous alloys, Fatigue \& Fracture of
Engineering Materials \& Structures 22(8) (1999) 667-672.

{[}14{]} S. Stanzl-Tschegg, H. Mughrabi, B. Schoenbauer, Life time and
cyclic slip of copper in the VHCF regime, International journal of
fatigue 29(9-11) (2007) 2050-2059.

{[}15{]} M.D. Sangid, The physics of fatigue crack initiation,
International journal of fatigue 57 (2013) 58-72.

{[}16{]} K. Yang, C. He, Q. Huang, Z.Y. Huang, C. Wang, Q. Wang, Y.J.
Liu, B. Zhong, Very high cycle fatigue behaviors of a turbine engine
blade alloy at various stress ratios, International Journal of Fatigue
99 (2017) 35-43.

{[}17{]} Y. Hong, Z. Lei, C. Sun, A. Zhao, Propensities of crack
interior initiation and early growth for very-high-cycle fatigue of high
strength steels, International Journal of Fatigue 58 (2014) 144-151.

{[}18{]} Y. Murakami, Metal fatigue: effects of small defects and
nonmetallic inclusions, Academic Press2019.

{[}19{]} F. Yang, S.M. Yin, S.X. Li, Z.F. Zhang, Crack initiation
mechanism of extruded AZ31 magnesium alloy in the very high cycle
fatigue regime, Materials Science and Engineering: A 491(1-2) (2008)
131-136.

{[}20{]} K. Tokaji, M. Kamakura, Y. Ishiizumi, N. Hasegawa, Fatigue
behaviour and fracture mechanism of a rolled AZ31 magnesium alloy,
International Journal of Fatigue 26(11) (2004) 1217-1224.

{[}21{]} J. Pan, P. Fu, L. Peng, B. Hu, H. Zhang, A.A. Luo, Basal slip
dominant fatigue damage behavior in a cast Mg-8Gd-3Y-Zr alloy,
International Journal of Fatigue 118 (2019) 104-116.

{[}22{]} D. Hanwu, W. Limin, L. Ke, W. Lidong, J. Bin, P. Fusheng,
Microstructure and deformation behaviors of two Mg--Li dual-phase alloys
with an increasing tensile speed, Materials \& Design 90 (2016) 157-164.

{[}23{]} F. Guo, L. Liu, Y. Ma, L. Jiang, Y. Zhang, D. Zhang, F. Pan,
Slip behavior and its effect on rolling texture development in a
dual-phase Mg--Li alloy, Journal of Alloys and Compounds 813 (2020)
152117.

{[}24{]} M. Shahzad, L. Wagner, Influence of extrusion parameters on
microstructure and texture developments, and their effects on mechanical
properties of the magnesium alloy AZ80, Materials Science and
Engineering: A 506(1-2) (2009) 141-147.

{[}25{]} J. Bohlen, S. Yi, D. Letzig, K.U. Kainer, Effect of rare earth
elements on the microstructure and texture development in
magnesium--manganese alloys during extrusion, Materials Science and
Engineering: A 527(26) (2010) 7092-7098.

{[}26{]} S.B. Yi, H.G. Brokmeier, J. Bohlen, D. Letzig, K.U. Kainer,
Neutron diffraction study on the texture development during extrusion of
magnesium alloy AZ31, Physica B: Condensed Matter 350(1-3) (2004)
E507-E509.

{[}27{]} V. Kumar, A. Gupta, D. Lahiri, K. Balani, Serrated yielding
during nanoindentation of thermomechanically processed novel
Mg--9Li--7Al--1Sn and Mg--9Li--5Al--3Sn--1Zn alloys, Journal of Physics
D: Applied Physics 46(14) (2013) 145304.

{[}28{]} W. Wang, C.B. Jiang, K. Lu, Deformation behavior of Ni3Al
single crystals during nanoindentation, Acta materialia 51(20) (2003)
6169-6180.

{[}29{]} V. Kumar, K. Philippe, R. Shekhar, K. Balani, Processing and
Nano-mechanical Characterization of Mg-Li-Al Based Alloys, Procedia
Materials Science 5 (2014) 585-591.

{[}30{]} T. Guo, F. Siska, J. Cheng, M. Barnett, Initiation of basal
slip and tensile twinning in magnesium alloys during nanoindentation,
Journal of Alloys and Compounds 731 (2018) 620-630.

{[}31{]} X. Peng, X. Liang, W. Liu, G. Wu, H. Ji, X. Tong, L. Zhang, W.
Ding, High-cycle fatigue behavior of Mg-8Li-3Al-2Zn-0.5 Y alloy under
different states, Journal of Magnesium and Alloys (2020).

{[}32{]} C. He, Y. Liu, J. Li, K. Yang, Q. Wang, Q. Chen,
Very-high-cycle fatigue crack initiation and propagation behaviours of
magnesium alloy ZK60, Materials Science and Technology 34(6) (2018)
639-647.

{[}33{]} H. Liu, H. Wang, Z. Huang, Q. Wang, Q. Chen, Comparative study
of very high cycle tensile and torsional fatigue in TC17 titanium alloy,
International Journal of Fatigue 139 (2020).

{[}34{]} F. Liu, C. He, Y. Chen, H. Zhang, Q. Wang, Y. Liu, Effects of
defects on tensile and fatigue behaviors of selective laser melted
titanium alloy in very high cycle regime, International Journal of
Fatigue 140 (2020).

{[}35{]} M.W. Tofique, J. Bergstr\selectlanguage{ngerman}öm, K. Svensson, S. Johansson, R.L.
Peng, ECCI/EBSD and TEM analysis of plastic fatigue damage accumulation
responsible for fatigue crack initiation and propagation in VHCF of
duplex stainless steels, International Journal of Fatigue 100 (2017)
251-262.

{[}36{]} I.H. Lo, W.-T. Tsai, Effect of selective dissolution on fatigue
crack initiation in 2205 duplex stainless steel, Corrosion Science 49(4)
(2007) 1847-1861.

{[}37{]} R. Strubbia, S. Hereñú, I. Alvarez-Armas, U. Krupp, Short
fatigue cracks nucleation and growth in lean duplex stainless steel LDX
2101, Materials Science and Engineering: A 615 (2014) 169-174.

{[}38{]} J.K. Sahu, U. Krupp, H.J. Christ, Fatigue crack initiation
behavior in embrittled austenitic--ferritic stainless steel,
International Journal of Fatigue 45 (2012) 8-14.

{[}39{]} Y. Murakami, S. Kodama, S. Konuma, Quantitative evaluation of
effects of non-metallic inclusions on fatigue strength of high strength
steels. I: Basic fatigue mechanism and evaluation of correlation between
the fatigue fracture stress and the size and location of non-metallic
inclusions, International Journal of Fatigue 11(5) (1989) 291-298.

{[}40{]} C. He, Y. Liu, J. Li, K. Yang, Q. Wang, Q. Chen,
Very-high-cycle fatigue crack initiation and propagation behaviours of
magnesium alloy ZK60, Materials Science and Technology 34(6) (2017)
639-647.

{[}41{]} X.H. Shao, H.Q. Liu, H.J. Yang, C. He, N. Su, Y.J. Wu, Q. Chen,
X.L. Ma, Enhanced very high cycle fatigue resistance of solution treated
Mg--10Gd--3Y--1Zn--0.5Zr magnesium alloy containing long-period stacking
ordered phase, Materialia 11 (2020).

{[}42{]} U. Karr, B.M. Schönbauer, H. Mayer, Near\selectlanguage{english}-threshold fatigue
crack growth properties of wrought magnesium alloy AZ61 in ambient air,
dry air, and vacuum, Fatigue \& Fracture of Engineering Materials \&
Structures 41(9) (2018) 1938-1947.

{[}43{]} X. Pan, H. Su, C. Sun, Y. Hong, The behavior of crack
initiation and early growth in high-cycle and very-high-cycle fatigue
regimes for a titanium alloy, International Journal of Fatigue 115
(2018) 67-78.

{[}44{]} W. Li, H. Zhao, A. Nehila, Z. Zhang, T. Sakai, Very high cycle
fatigue of TC4 titanium alloy under variable stress ratio: Failure
mechanism and life prediction, International Journal of Fatigue 104
(2017) 342-354.

{[}45{]} K. Yang, Q. Huang, Q. Wang, Q. Chen, Competing crack initiation
behaviors of a laser additively manufactured nickel-based superalloy in
high and very high cycle fatigue regimes, International Journal of
Fatigue 136 (2020).

{[}46{]} X. Pan, Y. Hong, High\selectlanguage{english}-cycle and very\selectlanguage{english}-high\selectlanguage{english}-cycle fatigue
behaviour of a titanium alloy with equiaxed microstructure under
different mean stresses, Fatigue \& Fracture of Engineering Materials \&
Structures 42(9) (2019) 1950-1964.

{[}47{]} Y. Hong, C. Sun, The nature and the mechanism of crack
initiation and early growth for very-high-cycle fatigue of metallic
materials--An overview, Theoretical and Applied Fracture Mechanics 92
(2017) 331-350.

{[}48{]} W.B. Hutchinson, M.R. Barnett, Effective values of critical
resolved shear stress for slip in polycrystalline magnesium and other
hcp metals, Scripta Materialia 63(7) (2010) 737-740.

{[}49{]} T. Hama, H. Takuda, Crystal-plasticity finite-element analysis
of inelastic behavior during unloading in a magnesium alloy sheet,
International Journal of Plasticity 27(7) (2011) 1072-1092.

{[}50{]} T. Hama, N. Kitamura, H. Takuda, Effect of twinning and
detwinning on inelastic behavior during unloading in a magnesium alloy
sheet, Materials Science and Engineering: A 583 (2013) 232-241.

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