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\DOI{10.5802/crmeca.264}
\datereceived{2024-04-24}
\daterevised{2024-07-09}
\dateaccepted{2024-07-12}
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\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{research-article}

\title{Study of V-notch fatigue strength based on affected depth
approach}

\alttitle{\'{E}tude de la r\'{e}sistance \`{a} la fatigue des entailles
en V bas\'{e}e sur l'approche de la profondeur affect\'{e}e}

\author{\firstname{Marwa} \lastname{Youssef}\CDRorcid{0000-0002-4902-2097}\IsCorresp}
\address{LGM, ENIM, Universit\'{e} de Monastir, Avenue Ibn Eljazzar,
Monastir 5019, Tunisia}
\email[M. Youssef]{marwayousseff@gmail.com}

\author{\firstname{Anouar} \lastname{Nasr}\CDRorcid{0000-0002-2152-9910}}
\addressSameAs{1}{LGM, ENIM, Universit\'{e} de Monastir, Avenue Ibn
Eljazzar, Monastir 5019, Tunisia}
\address{IPEIM, Universit\'{e} de Monastir, Avenue Ibn Eljazzar,
Monastir 5019, Tunisia}

\keywords{\kwd{High-cycle fatigue}\kwd{Affected depth}\kwd{Fatigue
Limit diagram}\kwd{V-notch}\kwd{Notch tip radius}}

\altkeywords{\kwd{Fatigue \`{a} grand nombre de cycle}\kwd{Profondeur
affect\'{e}e}\kwd{Diagramme de limite de fatigue}\kwd{Entaille en
V}\kwd{Rayon de l'extr\'{e}mit\'{e} de l'entaille}}

\begin{abstract} 
In this work, the Affected Depth (AD) approach is employed to estimate
the high cycle fatigue limit of AISI 416 steel weakened by notches.
This approach is based on a critical depth parameter identified from
the analysis of the stress distribution around the defect.
Elasto-plastic finite element calculations have been performed to
determine the stress field close to the notch. Where the notch is
modeled by a discontinuity of matter (void) characterised by: (i) a
notch depth and (ii) a notch tip radius. The Crossland equivalent
stress is employed to compute the stress surrounding the notch of a
V-notched material submitted to fully reversed torsion and tension
loadings. The experimental results of notched materials, used for model
validation, are obtained from the literature. In the first part of the
paper, Fatigue Limit diagram is simulated by assessing the fatigue
limits to validate the AD model for notched components. The prediction
of the fatigue strength is consistent with the experimental
investigation taken from the literature, and it provides interesting
results. In the second part of the paper, two points are treated:
(i) The effect of the notch acuity on the plastic strain and the stress
concentration. It allows testing the capability of the AD approach to
describe the fatigue behavior of the V-notched components. (ii) The
impact of the notch tip radius on the affected depth parameter of
notched low carbon steel.\vspace*{.25pc}
\end{abstract} 

\begin{altabstract} 
Dans ce travail, l'approche de la profondeur affect\'{e}e (AD) est
employ\'{e}e pour estimer la limite de fatigue \`{a} haut cycle de
l'acier AISI 416 affaibli par des entailles. Cette approche est
bas\'{e}e sur un param\`{e}tre de profondeur critique identifi\'{e}
\`{a} partir de l'analyse de la distribution des contraintes autour du
d\'{e}faut. Des calculs \'{e}lasto-plastiques par \'{e}l\'{e}ments
finis ont \'{e}t\'{e} effectu\'{e}s pour d\'{e}terminer le champ de
contraintes \`{a} proximit\'{e} de l'entaille. L'entaille est
mod\'{e}lis\'{e}e par une discontinuit\'{e} de mati\`{e}re (vide)
caract\'{e}ris\'{e}e par :\break (i) une profondeur d'entaille et (ii) un
rayon de la pointe de l'entaille. La contrainte \'{e}quivalente de
Crossland est utilis\'{e}e pour calculer la contrainte entourant
l'entaille d'un mat\'{e}riau entaill\'{e} en V soumis \`{a} des charges
de torsion et de tension totalement invers\'{e}es. Les r\'{e}sultats
exp\'{e}rimentaux des mat\'{e}riaux entaill\'{e}s, utilis\'{e}s pour la
validation du mod\`{e}le, sont tir\'{e}s de la litt\'{e}rature. Dans la
premi\`{e}re partie de l'article, le diagramme de limite de fatigue est
simul\'{e} en \'{e}valuant les limites de fatigue afin de valider le
mod\`{e}le AD pour les composants entaill\'{e}s. La pr\'{e}diction de
la r\'{e}sistance \`{a} la fatigue est coh\'{e}rente avec l'\'{e}tude
exp\'{e}rimentale tir\'{e}e de la litt\'{e}rature et fournit des
r\'{e}sultats int\'{e}ressants. Dans la deuxi\`{e}me partie de
l'article, deux points sont trait\'{e}s : (i) l'effet de l'acuit\'{e}
de l'entaille sur la d\'{e}formation plastique et la concentration de
contrainte. Cela permet de tester la capacit\'{e} de l'approche AD
\`{a} d\'{e}crire le comportement en fatigue des composants
entaill\'{e}s\break en V. (ii) L'impact de l'acuit\'{e} de l'entaille sur la
d\'{e}formation plastique et la concentration de
contrainte.\break (ii) L'impact du rayon de la pointe de l'entaille sur le
param\`{e}tre de profondeur affect\'{e}e de l'acier \`{a} faible teneur
en carbone entaill\'{e}.
\end{altabstract}

\maketitle

\end{noXML}

\section{Introduction} \label{sec1}

Different geometric irregularities can exist in structures, especially
notches. Various studies have aimed to provide different approaches to
accessing the fatigue limit of notched components. Initially, Lazzarin
and Susmel~\cite{1} introduced a methodology for predicting the fatigue
limit of notched components under multiaxial-fatigue-loading
conditions. This method was based on a non-conventional bi-parametric
Wohler curve and a critical plane approach. The authors assumed that
the plane of the maximal shear stress amplitude would coincide with the
initiation plane of the microcrack. The Wohler curve plotted the
fatigue strength by considering two factors: the maximal shear stress
amplitude and the maximal normal stress amplitude calculated on the
same plane. Then, this method was applied to the low and medium regimes
to predict the fatigue life for notched components. Furthermore, the
authors provided a practical rule for calculating the fatigue limit
reduction factor~\cite{2}.

Years later, the fatigue behavior of notched metallic structures was
evaluated by applying the Theory of Critical Distances (TCD) suggested
by Taylor~\cite{3}. To calculate the life limit, different methods were
proposed: the point method, the line method, the area method, and the
volume method. The point method consisted of comparing the maximal
local stress at a single point to the smooth sample fatigue limit.
However, the line method and the area method used a comparison between
the average value of the maximal local stress along a line or over an
area and the smooth sample fatigue limit. Finally, the volume method
suggested that the maximal principal stress is calculated as an average
over a finite volume located at a certain distance from the notch tip.
In accordance with this, the author extended the TCD approach to
evaluate the fatigue limit for the cracked or notched material using
the theory of linear elastic fracture mechanics. The author used this
approach to provide interesting results in predicting the static
strength of notched brittle components under a mixed-mode loading where
the crack propagation direction can also be estimated in notched
components~\cite{4}.

In a similar context, an interesting and complementary work by Susmel
and Pelekis~\cite{5} presented a reformulated version of the TCD method
and the authors showed that it was accurately able to calculate the
static and dynamic fatigue limit of notched plain concrete without
requiring the use of complex nonlinear constitutive laws.

Furthermore, an energetic approach named the Strain Energy Density
(SED) has been suggested in the literature to evaluate the fatigue life
of a notched material under uniaxial tensile stress~\cite{6} and
multiaxial stress~\cite{7}. This approach was extended from the notch
tip~\cite{6} to the finite size volume near the V-notch tip~\cite{8}.
The concept was that the SED value averaged over a finite size volume
of material near to the tip of V-notch can be employed to predict the
fatigue life~\cite{9}. Later, Berto {et~al.}~\cite{10,11} demonstrated
that the SED approach could successfully predict the fracture behavior
of notched cast iron specimens and incompressible hyperelastic
materials. In line with this, Hu {et~al.}~\cite{12} developed a
comparison of TCD and SED approaches. This comparative study
demonstrated that the TCD approach, particularly the line method, was
more effective for predicting the fatigue limit of structures under
uniaxial loading. Then, the authors concluded that the SED approach was
suitable for predicting fatigue life for both uniaxial and multiaxial
loading. 

Additionally, Lazzarin {et~al.}~\cite{13} suggested a Notch Stress
Intensity Factor (NSIFs) method to estimate the fatigue strength of the
notched component. The authors demonstrated that the NSIFs can honestly
characterise the initiation of cracks, near the V-notch by taking into
account the geometric parameter, the applied loads, and the material
properties~\cite{14}. Subsequently, some criteria have also been used
to assess fatigue behavior. First, a criterion for crack nucleation at
a notch was developed by Leguillon~\cite{15} while combining the
Griffith energy criteria~\cite{16} and the strength criteria. The
author demonstrated that both stress and energy criteria were necessary
conditions in the assessment of the fatigue behavior of notched
components. The above criterion was extended to notched materials and
bi-material structures, and comparisons with experiments show good
correlations~\cite{17}. Next, Carpinteri {et~al.}~\cite{18} proposed a
fracture criterion for quasi-brittle structures with sharp V-notches,
using Finite Fracture Mechanics (FFM). The authors incorporated both
energy aspects and stress requirements to lead more precise predictions
compared to classical approaches. More recently, Lazzarin
{et~al.}~\cite{19} proposed a comparison between these two Finite
Fracture Mechanics (FFM) criteria developed by Leguillon~\cite{17} and
by Carpinteri~\cite{18} and the averaged SED criterion. This analysis
was applied to the V-notched components under Mode I loading and was
based on the critical Notch Stress Intensity Factor. Then, the authors
concluded that the SED criterion and Carpinteri's criteria provided
interesting results in predicting the fatigue strength of notched
components. 

The present contribution aims at studying the fatigue strength of a
notched structure by using the Affected Depth (AD) approach put forward
by Nasr~\cite{20}. This method introduced a new parameter called
affected depth $a_{w}$, which denoted the depth of the tip of the
defect to the interior of the sample, where the Crossland criterion was
violated. This approach evaluated the fatigue limit of defective
material in different ways compared to the TCD approach. To access the
fatigue limit, the TCD method~\cite{3} considered the maximum principal
stress at a critical distance, while the AD model measured all
components of the stress tensor at the affected depth. Additionally,
unlike the TCD method, the AD model has been extended to multiaxial
loading by evaluating stress distribution in the plane perpendicular to
the maximum principal stress~\cite{20}.

The AD approach~\cite{21} was limited to spherical and elliptical
defects and it ignored notches. Through this study, we suggest proving
that the parameter affected depth can describe properly strength
fatigue of V-notched components under different loadings. First, we
propose two affected depth models for notched specimens under torsion
and tensile loads. The Finite Element (FE) calculations are performed
for AISI 416 steel and the predictions of fatigue strength are
consistent with the experimental investigation. Finally, numerical
simulation is carried out for AISI 416 steel and low carbon steel to
quantify the impact of the notch tip root on the fatigue strength and
the affected depth parameter.

\section{Material} \label{sec2}

In this study, the experimental fatigue results of V-notch AISI416
steel are used. The experimental tests, mechanical properties, and
endurance limits of these steel materials are obtained from previous
research conducted by Berto {et~al.}~\cite{22} (Table~\ref{tabI}).

\begin{table}[t!] %tab1
\caption{\label{tabI}Mechanical properties and endurance limits of AISI
416~\cite{22}}
\begin{tabular}{ccccccc}
\thead
$E$ (MPa) & $R_{p0.2}$ (MPa) & $R_{m}$ (MPa) & $\sigma_{D-1}$ (MPa) &
$\tau_{D-1}$ (MPa) & $R$ & $N$ (cycles)\\
\endthead
210,000 & 570 & 700 & 348.4 & 236.9 & $-1$ & $2 \times 10^{6}$
\botline
\end{tabular}
\end{table}

Cylindrical specimens are employed with V-notch (Figure~\ref{fig1})
characterised by a notch tip radius $r_n$, an opening angle $2\alpha$,
and different depths $p$ varying in the [500~$\rmmu$m--4000~$\rmmu$m]
range. These variations in the notch parameters allow for comprehensive
analysis of their impact on the fatigue behavior of the material. The
same specimen was used for both tension and torsion fatigue limit
testing.

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig1}Notch and specimen geometries.}
\end{figure}

\section{FE simulation} \label{sec3}

The presence of notches in materials subjected to cyclic loading
creates additional stress. To predict the stress distribution near the
V-notch, a FE model is employed using ABAQUS software~\cite{23}. Two
fatigue loads, tension and torsion, are simulated. For tension
loadings, a simplified model, taking into account the symmetries of
loadings and the geometry, is considered. Three planes of symmetry are
considered, and the simplified model is an eighth of the cylinder
having a length 10 times the width of the notch. However, for torsion
loading, a completed model is simulated (Figure~\ref{fig2}). An
elastic--plastic model is adopted to simulate the cyclic FE
calculations. The combined cyclic hardening behavior law is used to
model the mechanical behavior of the material. It combines the
isotropic and nonlinear kinematic hardening laws.

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig2}Load and boundary conditions for simulated models:
(a)~tension load; (b)~torsion load.}
\end{figure}

The FE parts are meshed using 10 nodes of the quadratic tetrahedral
solid element C3D10. The meshing process involves refining and
optimising the mesh around the noch. Under both tension and torsion
loading conditions, the convergence of the mesh was performed when the
maximum stress and strain at the defect tip converged to a stable value
with successive mesh refinements.

At the tip of the defect, the size of the tetrahedral elements is
measured in terms of the edge lengths of the elements in this area. The
smallest edge measures 0.558~$\rmmu$m, and the highest one is
1.178~$\rmmu$m. Figure~\ref{fig3} shows the refined mesh around the
V-notch sample subjected to tensile loadings. 

\begin{figure}
\includegraphics{fig03}
\caption{\label{fig3}Refined mesh in front of notch.}
\end{figure}

When a notched material is subjected to fatigue loadings, the crack
tends to extend perpendicular to the direction of the maximal principal
stress. For tension loading, the crack propagates within the plane
perpendicular to the specimen axis. For torsion loadings, the crack
propagates within the plane that forms a 45{\textdegree} angle with the
specimen axis~\cite{24,25}.

\section{Affected depth approach (background)} \label{sec4}

The AD approach, conceived by Nasr {et~al.}~\cite{20} was designed for
materials with surface defects under periodic loadings. The fatigue
limit of defective components was determined by applying the High-Cycle
Fatigue (HCF) criteria at a critical depth. In their study, the authors
introduced the concept of affected depth, denoted as $a_{w}$, which
represented the depth of the tip of the defect to the inside of the
sample where the Crossland criterion was violated.

The Crossland criterion is used, and it is defined by the equivalent
stress given by Equation~(\ref{eq1})~\cite{26}:
{\begin{equation} \label{eq1}
\sigma_{\mathrm{eqCr}}=\sqrt{J_{2.a}}+\alpha_{c}P_{\max}\leq \beta_{c}
\end{equation}}\unskip
where $\sqrt{J_{2,a}}$ is the amplitude of the square root of the
second invariant of the stress deviator obtained by a double
maximization over the loading period ($T$):

$P_{\mathrm{max}}$ is the maximal hydrostatic stress during a loading
cycle.

$\alpha_{c}$ and $\beta_{c}$ are the material parameters in the
Crossland criterion.

According to this approach, the defect shape has no impact on the
affected depth parameter as long as it is the same material and the
same loading ratio. By controlling the affected depth based on the load
amplitude, the fatigue limit can be estimated for materials with
different defect sizes using the following model
(Equation~(\ref{eq2})).
{\begin{equation} \label{eq2}
\sigma_{\mathrm{eqCr}}(a_{w})\leq \beta_{c}
\end{equation}}\unskip
where $\beta_{c}=\tau_{D-1}$ is the fully reversed torsion fatigue
limit of the defect-free material, and $a_{w}$ is the critical depth,
which is a parameter identified from one experimental fatigue limit of
the defective material by numerical simulation.

\section{Application of AD approach for notched material} \label{sec5}

In this work, the AD approach and the Crossland criterion will be used
to evaluate the notched fatigue limit of AISI 416 steel. According to
the principle of the affected depth approach, it is necessary to
perform iterative numerical simulations. For each size of the notch,
the fatigue limit could be calculated by controlling the affected
depth, and we repeat the numerical simulation by adjusting the value of
the applied load until we obtain the Crossland equivalent stress, which
is equal to the experimental limit fatigue.

The AD model is described in the flowchart provided in
Figure~\ref{fig4}.

\begin{figure}
\includegraphics{fig04}
{\vspace*{-.25pc}}
\caption{\label{fig4}Iterative calculation leading to predict fatigue
limit of notched component.}
{\vspace*{-.35pc}}
\end{figure}

The evaluation of the fatigue limit for notched structures involves
iterative steps. First, the identification of AD parameter through
numerical simulation depends on the load amplitude, the depth notch,
and the experimental limit fatigue. 

Second, the Crossland equivalent stress $\sigma_{\mathrm{eqCr}}(a_{w})$
is evaluated at the AD parameter $a_{w}$ and compared to $\beta_{c}$.
If the Crossland equivalent stress evaluated at the affected depth
exceeds $\beta_{c}$ ($\sigma_{\mathrm{eqCr}}(a_{w}) > \beta_{c}$), we
repeat the calculation using an applied loading $P$ less than the one
already applied. If the Crossland equivalent stress on the affected
depth is less than $\beta_{c}$ (\mbox{$\sigma_{\mathrm{eqCr}}(a_{w}) <
\beta_{c}$}), the calculation is repeated using an applied loading $P$
that exceeds the initially applied load. Finally, the notched fatigue
limit is obtained when the Crossland equivalent stress is{\break} equal
to~$\beta_{c}$.

\subsection{Identification of critical AD parameter $a_{w}$}
\label{sec5.1}

In accordance with the AD approach, we propose to determine the AD
parameter, denoted as $a_{w}$, for AISI 416 steel using the Crossland
criterion. The affected depth is evaluated by measuring the distance
between the tip of the notch and the interior of the specimen subject
to its experimental notch fatigue limit.

To carry out this analysis, numerical simulation is performed on a
sample containing a \mbox{V-notch}. The specimen is subjected to its
experimental V-notch fatigue limit, which is equal to 85~MPa, under
tension loading with a load ratio $R_{\sigma} = -1$. The notch tip
radius is equal to 100~$\rmmu$m, the notch depth remains at a value of
2000~$\rmmu$m and the opening angle measures 90{\textdegree}. The
results of this numerical simulation are presented in
Figure~\ref{fig5}.

\begin{figure}
\includegraphics{fig05}
\caption{\label{fig5}Identification of critical depth for AISI steel
(fully reversed tension at 85~MPa, for V-notch ($p = 2000~\rmmu$m,
$r_{n} = 100~\rmmu$m)).}
\end{figure}

From Figure~\ref{fig5}, it can be observed that the equivalent stress
exceeds the maximal value in front of the notch, indicating localised
stress concentration. As the stress propagates away from the notch, it
gradually decreases until reaching an asymptotically constant value.
This behavior suggests that the presence of the notch influences the
stress distribution in the surrounding the tip of a sharp~V.

Furthermore, according to the AD model based on the Crossland
criterion, the estimated AD parameter $a_{w}$ is approximately
110~$\rmmu$m. This means that the affected region, where the Crossland
criterion is violated, extends to a depth of approximately 110~$\rmmu$m
from the tip of the notch into the specimen.

\subsection{V-notch fatigue limit estimation based on AD approach}
\label{sec5.2}

The simulation of Fatigue Limit diagram is performed based on the AD
approach by assessing the fatigue limits for different notch sizes. 

Table~\ref{tabII} presents the results of fatigue tests conducted on
AISI 416 steel under tension and torsion loadings. The specimens are
subjected to a total of $2 \times 10^{6}$ cycles~\cite{22}.

\begin{table}[t!] %tab2
\caption{\label{tabII}Summary of fatigue calibration curves for AISI
416 derived from experimental testing of notched specimens subjected to
axial loadings~\cite{22}}
\begin{tabular}{ccccccc}
\thead
\parbox[t]{.85cm}{\centering $p$ ($\rmmu$m)} &
\parbox[t]{.85cm}{\centering $r_{n}$ ($\rmmu$m)} &
\parbox[t]{.7cm}{\centering$2\alpha$ ({\textdegree})} &
\parbox[t]{3.4cm}{\centering Stress amplitude of tension loadings (MPa)}
& \parbox[t]{3.4cm}{\centering Stress amplitude of torsion loadings
(MPa)}\vspace*{2pt} & $R$ & \parbox[t]{1.2cm}{\centering $N$ (cycles)}\\
\endthead
4000 & 100\raisebox{10pt}{} & 90 & 38.2 & 47.5 & $-$1 & $2 \times 10^{6}$\\
2000 & 100 & 90 & 82 & 110 & $-$1 & $2 \times 10^{6}$\\
\0500 & 100 & 90 & 225 & 170 & $-$1 & $2 \times 10^{6}$
\botline
\end{tabular}
\end{table}

To evaluate the fatigue limit of V-notched parts, two loading types are
used. In this order, two Fatigue Limit diagrams are proposed for
modeling the influence of the V-notch subjected to torsion and tension
loadings. It is important to note that the notch tip radius remains
constant at 100~$\rmmu$m for different predicted loadings. In this
section, we consider the simulation of AISI 416 steel under the same
loading ratio so that the AD parameter $a_{w}$ will remain constant at
110~$\rmmu$m under tensile and torsion loadings.

\subsubsection{Prediction of V-notch fatigue limit under tension
loadings $(R_{\sigma} = -1)$} \label{sec5.2.1}

Figure~\ref{fig6} demonstrates the variation in the calculated V-notch
fatigue limits versus the V-notch depth for AISI 416 steel subjected to
a fully reversed tension loading. 

\begin{figure}
\includegraphics{fig06}
\caption{\label{fig6}Calculated fatigue limits, using AD model for
V-notch AISI 416 steel submitted to fully reversed tension, compared to
experimental results.}
\end{figure}

\subsubsection{Prediction of V-notch fatigue limit under torsion
loadings $(R_{\sigma} = -1$)} \label{sec5.2.2}

Figure~\ref{fig7} depicts the variation in the calculated V-notch
fatigue limits versus the V-notch depth for AISI 416 steel subjected to
a fully reversed torsion loading.

\begin{figure}
\includegraphics{fig07}
\caption{\label{fig7}Calculated fatigue limits, using AD model for
V-notch AISI416 steel submitted to fully reversed torsion, compared to
experimental results.}
\end{figure}

This developed AD approach is applied to predict the Fatigue Limit
diagrams for two types of loadings by considering the V-notch depth.
Iterative FE calculations (Figure~\ref{fig4}) are carried out for the
AISI416 steel. Figures~\ref{fig6} and~\ref{fig7} show a comparison
between the experimental results and calculated V-notch fatigue limits.
The predicted Fatigue Limit diagrams are reliable with the experimental
fatigue results for the two tested loadings. Thus, the developed AD
approach describes well experimental tests and takes into consideration
the V-notch and its depth.

The results presented in Figures~\ref{fig6} and~\ref{fig7} highlight
the following points:
\begin{itemize}
\item For both loading types, the fatigue limits decline with the
V-notch depth and tend towards an asymptotic value.
\item For tensional loadings, the Fatigue Limit diagram is divided into
two different zones: (i)~A first zone is dedicated to the notch with a
depth less than 2000~$\rmmu$m. In this zone, the notch depth has a
significant effect on the HCF limit. Indeed, the fatigue limit
decreases with increasing V-notch depth. In addition, the fatigue limit
of the 2000~$\rmmu$m V-notch depth is about of 20\% of the fatigue
limit of notch-free material. (ii)~A second zone covers a larger notch
where the depth goes from 2000~$\rmmu$m to 4000~$\rmmu$m. By increasing
the depth notch, the HCF limit remains almost constant and tends
towards an asymptotic value.
\item For torsion loadings, the variation in the fatigue limit with the
notch depth is not the same as under tension loadings and it appears to
be linear. For a V-notch depth equal to 2000~$\rmmu$m, the fatigue
limit is about 50\% of the fatigue limit of the notch-free material
(20\% for tension loadings). It can be therefore concluded that the
circumferential \mbox{V-notch} has an important effect on the fatigue
strength and is much more significant on tension loadings.
\item Fatigue Limit's diagram applied for specimens containing the
V-notch loaded in tension affirms that the AD models based on the
Crossland criterion lead to very encouraging results when compared to
the experimental results~\cite{22}.
\item The comparison with the experimental results in~\cite{22}
demonstrates that the AD model can efficiently assess the effect of the
V-notch geometry on the fatigue strength.
\end{itemize}

\section{Effect of notch tip radius} \label{sec6}

To assess the influence of the notch tip radius on the fatigue life,
simulation is conducted on V-notched cylindrical specimens made of AISI
416 steel, where the V-notch depth is held constant at 500~$\rmmu$m,
whereas the notch tip radius $r_{n}$ varied between 100, 200, and
300~$\rmmu$m. These specimens are subjected to tensile loadings at
stress levels of 220~MPa.

The notch radius $r_{n}$ and the origin of the coordinate system
$r_{0}$ are related by the following equation (Equations~(\ref{eq3}),
(\ref{eq4})) based on trigonometric considerations (Figure~\ref{fig8}):
{\begin{equation} \label{eq3}
r_{n}=\frac{q\cdot r_{0}}{q-1}
\end{equation}}\unskip
where
{\begin{equation} \label{eq4}
q=\frac{2\uppi - 2\alpha}{\uppi}.
\end{equation}}\unskip

\begin{figure}
\includegraphics{fig08}
\caption{\label{fig8}Geometry of V-notch tip.}
{\vspace*{.6pc}}
\end{figure}

Figure~\ref{fig9} depicts a comparison of plastic deformations of a
V-notch specimen with different notch tip radii and constant notch
depth loaded at the same tensile loading.

\begin{figure}
\includegraphics{fig09}
\caption{\label{fig9}Comparison of plastic strains for different notch
tip radii with same tensile loading and constant depth $p = 500~\rmmu$m
for V-notch low carbon steel.}
\end{figure}

Figure~\ref{fig9} shows that when the range of sharpness is expanded
from 100~$\rmmu$m to 300~$\rmmu$m, a decrease in the equivalent plastic
deformation is seen at the bottom of the notch, as well as on the
surface of the specimen subjected to the same tensile loading.

To develop this result, and according to the numerical simulation, we
calculate the plasticised area for each radius of acuity. For a radius
equal to 100~$\rmmu$m, we find a plasticised area equal to 10.55\%
relative to the total surface area. However, for a radius equal to
300~$\rmmu$m, the plasticised area reaches 7.82\%. The sharpness of the
V-notch influences the stress distribution and concentration and it can
lead to an important plastic deformation surrounding the tip of the
notch (Figure~\ref{fig10}).

\begin{figure}
\includegraphics{fig10}
\caption{\label{fig10}Plasticized area on V-notched sample.}
\end{figure}

Therefore, in order to examine the impact of the notch tip radius more
effectively, it is essential to assess the stress concentration caused
by different notch root radii of the V-notch on fatigue strength.
Therefore, we conduct a numerical simulation to determine the
elastic--plastic stress concentration factor Kep at high-load
locations.

It was observed that for $r_n$ equal to 300~$\rmmu$m, Kep is 3.12 along
the entire circumference of the V-notch cavity on the plane
perpendicular to the direction of tension. When $r_{n}$ is
200~$\rmmu$m, Kep reaches a value of 3.35 at the maximal loaded point
(the tip of the notch). Finally, the \mbox{V-notch} exhibits the
highest elastic--plastic stress concentration factor when $r_{n}$ is
100~$\rmmu$m, with Kep approximately 4.30 at the maximal loaded point
(Table~\ref{tabIII}).

\begin{table}[t!] %tab3
\caption{\label{tabIII}Calcul of plasticized area and elastic--plastic
stress concentration factor for different notch tip radius}
\tabcolsep=2.7pt
\begin{tabular}{ccccc}
\thead
\parbox[t]{2.2cm}{\centering Notch tip radius $r_{n}$ ($\rmmu$m)} &
\parbox[t]{2cm}{\centering Plasticized zone (mm$^2$)} &
\parbox[t]{2cm}{\centering \% Plasticized zone (${d}_{p}/{d}_{n}$)} &
\parbox[t]{2.8cm}{\centering Equivalent plastic strain max (MPa)} &
\parbox[t]{3.7cm}{\centering Elastic--plastic stress concentration factor Kep}\vspace*{2pt}\\
\endthead
100 & 3.36 & 10.55 & 0.028 & 4.30\\
200 & 3 & \09.43 & 0.013 & 3.35\\
300 & 2.49 & \07.82 & 0.006 & 3.12
\botline
\end{tabular}
\end{table}

When the radius of the root notch decreases, the stress field becomes
more concentrated, and the elastic--plastic stress concentration factor
increases. The smaller radius causes the stress to be more localised
and intensified at the notch root, resulting in a higher stress
concentration factor.

It can be therefore concluded that the prediction of the fatigue
behavior not only depends on the depth notch but that the notch root
has an important effect and can greatly impact AISI416 tension fatigue
limits. 

\section{Impact of notch tip on AD parameter} \label{sec7}

To investigate the effect of the notch tip of the AD parameter,
simulation is carried out on \mbox{V-notched} cylindrical specimens
made of low carbon steel under fully reversed tensile loadings.

By analysing the experimental results~\cite{27} and correlating them
with the different notch tip radii, the impact of the notch root radius
on the fatigue behavior of low carbon steel can be estimated.

In the simulation, the V-notch depth is held constant at 2540~$\rmmu$m,
while the notch tip radius varies between 200~$\rmmu$m and
400~$\rmmu$m. These specimens are subjected to tensile loadings at
stress levels of 91.6~MPa and 93.6~MPa, respectively, representing
their experimental fatigue limits.

Based on the AD model, we propose to determine the AD parameter $a_{w}$
of low carbon steel as the depth of the tip of the notch into the
interior of the specimen, which is submitted to its experimental
notched fatigue limit, where the fatigue equivalent stress is greater
than $\beta_{c} = 187$~MPa.

Figure~\ref{fig11} depicts a comparison of the equivalent stress of a
V-notch low carbon steel specimen with two different notch tips, loaded
at their respective fatigue limits and with a constant notch depth
equal to 2540~$\rmmu$m.

\begin{figure}
\includegraphics{fig11}
\caption{\label{fig11}Identification of AD parameter for low carbon
steel with same V-notch ($p = 2540~\rmmu$m) (fully reversed tension at
91.6~MPa for notch tip radius $r_{n}=200~\rmmu$m, and fully reversed
tension at 93.6~MPa for notch tip radius $r_{n} = 400~\rmmu$m).}
\end{figure}

It is observed that the AD varies depending on the V-notch acuity. For
a notch radius equal to 200~$\rmmu$m, the equivalent stress reaches the
$\beta_{c}$ value at a depth equal to $a_{w}=645~\rmmu$m. However, for
a radius equal to 400~$\rmmu$m, the affected depth increases and
reaches $760~\rmmu$m.

The increase in the notch tip radius induces a rise in the stress
distribution near the V-notch. Eventually, it accelerates the crack
initiation and propagation, which generates a higher AD hence
influencing the fatigue strength of the notched component.

Previous studies have shown that the defect shape influences the stress
state in the tip of the defect and it has no influence on the affected
depth parameter as long as it is the same material and the same loading
ratio~\cite{20}. However, our study has demonstrated the dependence
between the radius of the V-notch and the AD parameter.

To sum up, the fatigue limit is affected by the geometry of the
V-notch, especially by the notch tip radius.

\section{Conclusion}

The prediction of the fatigue limit for components weakened by
V-notches has been presented in the paper, and it has been performed
using the affected depth approach. The AD method has been extended to
illustrate the morphology influence of the geometry of the notch, and a
V-notch void has been considered in this paper. FE models have been
generated to determine the stress distribution at the notch tip under
torsion and tension loadings at load ratio $R_{\sigma} = -1$ for AISI
416 steel. Several experimental data taken from the literature have
been used for the validation, and a good agreement of the affected
depth model has been found.

It was observed that the affected depth parameter is suitable for
describing the fatigue behavior of V-notched components and may be
useful in different loadings. Then the influence of the notch tip
radius has been examined in detail in the second part of this
contribution. It can be deduced that the notch root is a significant
factor in the prediction of the fatigue life of the \mbox{V-notched}
component. Moreover, in order to assess the stress concentration
generated by \mbox{different} notch root radii of the V-notch, a
numerical simulation has been carried out to determine the
elastic--plastic stress concentration factor. 

Therefore, it would be of great interest to take into account the notch
root acuity in fatigue assessment for V-notched components.
Furthermore, to explore the effect of the notch tip of the affected
depth, a simulation has been conducted on V-notched cylindrical
specimens made of low carbon steel under fully reversed tensile
loadings. The results indicate that reducing the tip radius of the
V-notch creates a higher stress concentration area near the notch, and
it leads to an increase in the AD parameter. 

The perspective of this work is to confirm the applicability of the AD
approach for notched components tested under complex multiaxial
loadings with variable loading conditions.
{\vspace*{-.2pc}}

\section*{Nomenclature}

\begin{inftab}
\fontsize{10}{12.5}\selectfont
\begin{tabular}{ll}
\tbody
$a_{w}$ & Affected depth at fatigue limit ($\rmmu$m)\\
HCF & High Cycle Fatigue\\
$E$ & Young's modulus (MPa)\\
$R_{p0.2}$ & 0.2\% monotonous yield stress (MPa)\\
$R_{m}$ & Ultimate tensile strength (MPa)\\
$\nu$ & Poisson's ratio\\
$\sigma_{D-1}$ & Fatigue limit under fully reversed tension (MPa)\\
$\tau_{D-1}$ & Fatigue limit under fully reversed torsion (MPa)\\
$R_{\sigma}$ & Load ratio: $R_{\sigma}={\sigma_{\mathrm{min}}}/
{\sigma_{\mathrm{max}}}$\\
$N$ & Number for cycles to failure (cycles)\\
$r_n$ & Notch tip radius ($\rmmu$m)\\
$2\alpha$ & Notch opening angle ({\textdegree})\\
$p$ & V-notch depth ($\rmmu$m)\\
$\sigma_{\mathrm{eqCr}}$ & Crossland equivalent stress (MPa)\\
$\beta_{c}$ & Material parameter in Crossland criterion (MPa)\\
$\alpha_{c}$ & Crossland criterion parameter\\
$P$ & Applied loading (MPa)\\
$r_0$ & Origin of coordinate system ($\rmmu$m)\\
Kep & Elastic--plastic stress concentration factor 
\end{tabular}
\end{inftab}
{\vspace*{-.1pc}}

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.

{\vspace*{-.2pc}}

\section*{Funding}

The authors declare that no funds, grants, or other support were
received during the preparation of this manuscript. The research is not
supported.

{\vspace*{-.25pc}}

\back{}

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\end{document}
