\makeatletter
\@ifundefined{HCode}                  
{\documentclass[screen,CRMECA,Unicode,biblatex,published]{cedram}
\addbibresource{crmeca20250545.bib}
\newenvironment{noXML}{}{}
\def\mathbi#1{\text{\textbf{\textit{#1}}}}
\csdef{Seqnsplit}{\\}
\newcounter{runlevel}
\usepackage{multirow} 
\def\thead{\noalign{\relax}\hline}
\def\tbody{\noalign{\relax}}
\def\endthead{\noalign{\relax}\hline}
\def\tabnote#1{\vskip4pt\parbox{.9\linewidth}{#1}}
\usepackage{hyperref}
\def\botline{\\\hline}
\def\0{\phantom{0}}
\let\MakeYrStrItalic\relax
\def\jobid{crmeca20250545}
%\graphicspath{{/tmp/\jobid_figs/web/}} 
\graphicspath{{./figures/}} 
\def\refinput#1{}
\def\back#1{}
\def\tsup#1{$^{{#1}}$}
\def\og{\guillemotleft} 
\def\fg{\guillemotright}
\def\bchi{\boldsymbol{\chi}}
\def\bLambda{\boldsymbol{\Lambda}}
\newenvironment{inftab}{}{}
\DOI{10.5802/crmeca.339}
\datereceived{2025-07-02}
\daterevised{2025-09-27}
\datererevised{2025-10-29}
\dateaccepted{2025-10-31}
\ItHasTeXPublished
}                                                                 
{
\documentclass[crmeca]{article}
\usepackage[T1]{fontenc} 
\def\CDRdoi{10.5802/crmeca.339}
\def\bibliographystyle#1{}
\def\bibliography#1{}
\let\refinput\input
\def\newline{\unskip\break}
\def\hypertarget#1#2{\label{#1}}
\def\CDRsupplementaryTwotypes#1#2{}
}

\usepackage{upgreek}

\dateposted{2025-12-09}
\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{research-article}

\title{An engineering methodology for predicting crack growth of
2024-T3 aluminium alloys under variable amplitude loading}

\alttitle{Une m\'{e}thodologie d'ing\'{e}nierie pour pr\'{e}dire la
croissance des fissures des alliages d'aluminium 2024-T3 sous charge
d'amplitude variable}

\author{\firstname{Brahim} \lastname{Bouaziz}\IsCorresp}
\address{Mechanical Laboratory of Sousse, National Engineering School
of Sousse, University of Sousse, BP 264, Erriadh, 4023 Sousse, Tunisia}
\email[B. Bouaziz]{bouaziz\_ibrahim@yahoo.fr}

\author{\firstname{Maher} \lastname{Eltaief}}
\addressSameAs{1}{Mechanical Laboratory of Sousse, National Engineering
School of Sousse, University of Sousse, BP 264, Erriadh, 4023 Sousse,
Tunisia}
\email[M. Eltaief]{eltaiefmaher@gmail.com}

\author{\firstname{Chokri} \lastname{Bouraoui}\CDRorcid{0009-0004-4639-2940}}
\addressSameAs{1}{Mechanical Laboratory of Sousse, National Engineering
School of Sousse, University of Sousse, BP 264, Erriadh, 4023 Sousse,
Tunisia}
\email[C. Bouraoui]{chokri.bouraoui2@gmail.com}

\begin{abstract}
The propagation of fatigue cracks can significantly reduce the lifespan
of components, potentially leading to failures. This study presents an
engineering approach that accounts for variable-amplitude loading while
incorporating residual stresses at the crack tip.

The objective is to evaluate the fatigue life of 2024-T3 aluminum
alloys under variable loading by transforming the problem into an
equivalent one with constant loading, eliminating complex
cycle-by-cycle calculations.

The proposed strategy is integrated into the two-parameter fatigue
crack growth model, which considers both the total maximum stress
intensity factor $K_{\max\_\mathrm{tot}}$ and the total stress
intensity factor range $\Delta K_{\mathrm{tot}}$. Initially, plastic
zone interaction effects are neglected to establish an equivalence
relationship between variable and constant loading.

The Willenborg model is then used to determine the equivalent stress,
ensuring a comparable crack growth rate. To refine the approach, a
finite element (FE) elasto-plastic analysis based on the Chaboche model
is conducted. This analysis estimates the residual stress distribution
at the crack tip, converted into a residual stress intensity factor
$K_{\mathrm{r}}$. This factor is integrated into a correction
coefficient equation to account for plastic zone interaction effects.

Three loading spectra were examined. Predicted results were compared
with experimental data, showing that the proposed model improves crack
growth life in 2024-T3 aluminum alloy structures subjected to
variable-amplitude loading.
\end{abstract}

\begin{altabstract}
La propagation des fissures de fatigue peut r\'{e}duire
consid\'{e}rablement la dur\'{e}e de vie des composants, ce qui peut
entra\^{i}ner des d\'{e}faillances. Cette \'{e}tude pr\'{e}sente une
approche d'ing\'{e}nierie qui tient compte des charges d'amplitude
variable tout en incorporant des contraintes r\'{e}siduelles \`{a}
l'extr\'{e}mit\'{e} de la fissure.

L'objectif est d'\'{e}valuer la r\'{e}sistance \`{a} la fatigue des
alliages d'aluminium 2024-T3 sous charge variable en transformant le
probl\`{e}me en un probl\`{e}me \'{e}quivalent \`{a} charge constante,
en \'{e}liminant les calculs complexes cycle par cycle.

La strat\'{e}gie propos\'{e}e est int\'{e}gr\'{e}e dans le mod\`{e}le
de croissance des fissures de fatigue \`{a} deux param\`{e}tres, qui
prend en compte \`{a} la fois le facteur d'intensit\'{e} de contrainte
maximal total $K_{\max\_\mathrm{tot}}$ et la plage de facteurs
d'intensit\'{e} de contrainte totale $\Delta K_{\mathrm{tot}}$. Initialement, les effets d'interaction de la zone plastique sont
n\'{e}glig\'{e}s afin d'\'{e}tablir une relation d'\'{e}quivalence
entre le chargement variable et le chargement constant.

Le mod\`{e}le de Willenborg est ensuite utilis\'{e} pour d\'{e}terminer
la contrainte \'{e}quivalente, garantissant ainsi un taux de croissance
des fissures comparable. Pour affiner l'approche, une analyse
\'{e}lasto-plastique par \'{e}l\'{e}ments finis (EF) bas\'{e}e sur le
mod\`{e}le de Chaboche est r\'{e}alis\'{e}e. Cette analyse permet
d'estimer la distribution des contraintes r\'{e}siduelles \`{a}
l'extr\'{e}mit\'{e} de la fissure, convertie en facteur
d'intensit\'{e} des contraintes r\'{e}siduelles $K_{\mathrm{r}}$. Ce facteur
est int\'{e}gr\'{e} dans une \'{e}quation de coefficient de correction
pour tenir compte des effets d'interaction de la zone plastique.

Trois spectres de charge ont \'{e}t\'{e} examin\'{e}s. Les
r\'{e}sultats pr\'{e}dits ont \'{e}t\'{e} compar\'{e}s aux donn\'{e}es
exp\'{e}rimentales, montrant que le mod\`{e}le propos\'{e} am\'{e}liore
la concordance avec les r\'{e}sultats exp\'{e}rimentaux am\'{e}liore la
dur\'{e}e de vie de la croissance des fissures dans les structures en
alliage d'aluminium 2024-T3 soumises \`{a} une charge d'amplitude
variable.
\end{altabstract}

\shortrunauthors

\keywords{\kwd{Fatigue crack growth life}\kwd{Variable amplitude
loading}\kwd{Residual stress intensity factor}\kwd{Equivalent  stress
level}\kwd{Optimization}}

\altkeywords{\kwd{Dur\'{e}e de vie de la propagation des fissures de
fatigue}\kwd{Chargement \`{a} amplitude variable}\kwd{Facteur
d'intensit\'{e} de contrainte r\'{e}siduelle}\kwd{Niveau de stress
\'{e}quivalent}\kwd{Optimisation}}

\maketitle

{\vspace*{-12pt}}

\end{noXML}    

\newcommand{\diff}{\mathrm{d}}
\newcommand{\vecu}{\mathbf{u}}
\newcommand{\vecs}{\mathbf{s}}
\newcommand{\vecnull}{\mathbf{0}}
\newcommand{\vecp}{\mathbf{p}}
\newcommand{\vecK}{\mathbf{K}}
\newcommand{\vecC}{\mathbf{C}}
\newcommand{\vecX}{\mathbf{X}}
\newcommand{\vecx}{\mathbf{x}}

\section{Introduction}\label{sec1}

In most structural components used in the automotive or aerospace
industries, the presence of cracks or geometric discontinuities is
inevitable. These components are often subjected to cyclic loading of
variable amplitude. To reduce the risk of failure, fatigue life
prediction is a fundamental step in the design process. Considering
both the importance of safety and the current economic context in this
sector, research on extending the service life of these structures
remains highly relevant. Throughout their service life, structures are
subjected to random loads, which makes estimating crack growth and
consequently, fatigue life difficult and unreliable. Fatigue crack
growth (FCG) is influenced not only by the current load but also by the
loading \mbox{history~\cite{1,2,3},} which induces interaction effects in the
plastic zone, either accelerating or slowing down crack propagation
under realistic service conditions, sequence effects arising from the
order and magnitude of load surges and relaxations can modify the local
plastic zone and induce memory in the material, thereby altering crack
growth paths in ways that simple ${\Delta}K$ criteria cannot capture.
Numerous studies have been carried out in recent decades to predict
fatigue crack growth under variable loading. Some of these have focused
on calculating the size of the plastic zone and its effect on
propagation delay, as first proposed by Wheeler~\cite{3,4,5}. Others
focus on the concept of crack closure and its impact on propagation
behaviour, where Elber~\cite{6,7} was the first to introduce this
concept. Later, Noorozi and Glinka~\cite{8} proposed a two-parameter
model for the analysis of fatigue crack growth, based on the
elastic--plastic response ahead of the crack tip. The prediction of
fatigue life under the effect of a variable load is a major issue for
components and structures working under real service conditions. This
study aims to develop an engineering methodology that accounts for
various types of variable-amplitude loading, considering residual
stresses at the crack tip. In this context, the objective is to
estimate fatigue life under variable loading by replacing it with an
equivalent problem where the loading is constant, which is easier to
handle, while taking into account the effects of random loading
sequences~\cite{9}. In this paper, an equivalent stress transformation
approach is substituted for the two-parameter fatigue crack growth
model. This choice is physically justified: plastic interactions and
residual stresses generated at the crack tip modulate the load
gradients and, consequently, the driving force that governs
propagation. Firstly, all plasticised zone interaction effects are
neglected, allowing an equivalence relationship to be established
between variable loads and constant loading using the driving force
model~\cite{8,10}, which states that fatigue crack growth behaviour
combines both the total maximum stress intensity factor
$K_{\max\_\mathrm{tot}}$ and the magnitude of the total stress
intensity factor $\Delta K_{\mathrm{tot}}$.  Secondly, the Willenborg
model~\cite{11}, generalised by Meggiolaro and  Castro~\cite{12,14} and
Gallagher~\cite{13,14}, is used to determine the equivalent stress
$\Delta \sigma _{\mathrm{eq}}$ deduced from variable loading, which causes a
similar level of crack growth to that caused by constant loading.
However, the limitations of the Willenborg model~\cite{11} in
particular its inability to account for the influence of multiple
overloads and under loads and, therefore, to predict a lifetime greater
than that determined without the plastic--zone interaction~\cite{15}
effect justify the use of finite element (FE) analysis to incorporate
the effect of this zone. This makes it possible to consider the effect
of residual stresses induced by plastic deformation at the bottom of
the crack on the lifetime prediction. In this study, we propose an
innovative association between the driving force model and the
Willenborg model~\cite{11}, complemented by FE integration, to predict
the fatigue crack growth of structural components subjected to real
loads. The present work is organised as follows:
\begin{enumerate}[(i)]
\item Determination of equivalent loads at constant amplitudes without
accounting plastic deformation into account.
\item Determination of driving force modulus parameters from
experimental data.
\item FE modelling of residual stresses induced by plastic deformation
at the bottom of the crack.
\item Determination of corrected equivalent loads at constant
amplitudes, taking plastic deformation into account.
\end{enumerate}

\section{Procedure for calculating equivalent stresses}
\label{sec2}
\subsection{Equivalent stress without crack closure effect}
\label{ssec21}

Throughout its service life, each structural component undergoes actual
loading in the form of load spectra with varying magnitudes and stress
ratios. The objective, described in \mbox{references}~\cite{16,17},  is to
find a relationship that describes crack growth by applying a constant
load similar to that of the variable loads. This work modifies this
approach by using the finite element tool to take into account the
interactions of plastic zones and the effect of residual stresses on
crack growth prediction. A correction coefficient is implemented,
depending on the parameters of the driving force model, the maximum
stress intensity factor and the residual stress intensity factor, in
order to adjust the calculated equivalent stress without taking into
account the effect of plastic zone interaction.

The two-parameter driving force model is used to estimate fatigue crack
growth, incorporating both the total maximum stress intensity factor
$K_{\max,\mathrm{tot}}$ and the total stress intensity factor range
$\Delta K_{\mathrm{tot}}$. The equation below describes the crack
growth rate:
{\begin{equation}\label{eq1}
\frac{\mathrm{d}a}{\mathrm{d}N} = C{K}_{\max}^{p} \Delta K^{q},
\end{equation}}\unskip
where $C,p,q$ are the calibration parameters of the fatigue crack
growth data.

In the case of linear damage, and when a crack is subjected to a cyclic
load spectrum of variable amplitude such as $\Delta \sigma _{i}$ for
$N_{i}$ cycles, the crack length during this spectrum is $\Delta a$
which can be calculated from crack growth rates such as~\cite{18}:
{\begin{equation}\label{eq2}
\Delta a = 
\left(\frac{\mathrm{d}a}{\mathrm{d}N}\right)_{1} \cdot N_{1} + 
\left(\frac{\mathrm{d}a}{\mathrm{d}N}\right)_{2} \cdot N_{2} + \cdots +
\left(\frac{\mathrm{d}a}{\mathrm{d}N}\right)_{n} \cdot N_{n}.
\end{equation}}\unskip

From the formulation in Equation~(\ref{eq1}), Equation~(\ref{eq2}) can
be rewritten as follows
{\begin{equation}\label{eq3}
\Delta a = C(
{K}_{1\max}^{p}{\Delta K}_{1}^{q}N_{1} + 
{K}_{2\max}^{p}{\Delta K}_{2}^{q}N_{2} + \cdots +
{K}_{n\max}^{p}{\Delta K}_{n}^{q}N_{n}).
\end{equation}}\unskip

Since the crack closure effect is negligible, the relationship between
the maximum stress intensity factor $K_{\max}$ and the stress intensity
factor range $\Delta K$ is given by~\cite{18}:
{\begin{equation}\label{eq4}
K_{\max} = \dfrac{\Delta K}{1-R},
\end{equation}}\unskip
where $R$ is the stress ratio, defined as ${\sigma _{\min}}/{\sigma
_{\max}}$.

We can rewrite the expression for crack length as a function
of $\Delta K$ as follows:
{\begin{eqnarray}
\Delta a 
&
=
&
C\left(\left(\frac
{\Delta K_{1}}{1-R_{1}}\right)^{p}
{\Delta K}_{1}^{q}N_{1} + \left(\frac
{\Delta K_{2}}{1-R_{2}}\right)^{p}
{\Delta K}_{2}^{q}N_{2} + \cdots + \left(\frac
{\Delta K_{n}}{1-R_{n}}\right)^{p}
{\Delta K}_{n}^{q}N_{n}\right) 
\nonumber \\&
=
&
C\left(\frac{
{\Delta K}_{1}^{p+q}}{(1-R_{1})^{p}}N_{1} + \frac{
{\Delta K}_{2}^{p+q}}{(1-R_{2})^{p}}N_{2} + \cdots + \frac{
{\Delta K}_{n}^{p+q}}{(1-R_{n})^{p}}N_{n}\right).
\label{eq5}
\end{eqnarray}}\unskip
The range of stress intensity factors $\Delta K$ is expressed as a
function of stress level and crack length:
{\begin{equation}\label{eq6}
\Delta K = Y \Delta \sigma \sqrt{\uppi a},
\end{equation}}\unskip
where $Y$ represents a geometric correction factor.

Thus, the total crack length can be rewritten as:
{\begin{equation}\label{eq7}
\Delta a = C \cdot Y^{p+q} (\uppi a)^{(p+q)/2} \cdot \left(
\frac{{\Delta \sigma }_{1}^{p+q}}{(1-R_{1})^{p}}N_{1} + 
\frac{{\Delta \sigma }_{2}^{p+q}}{(1-R_{2})^{p}}N_{2} + \cdots +
\frac{{\Delta \sigma }_{n}^{p+q}}{(1-R_{n})^{p}}N_{n}\right).
\end{equation}}\unskip

From Equation~(\ref{eq7}), it is possible to calculate the growth of
the equivalent crack from a constant amplitude loading: 
{\begin{equation}\label{eq8}
\Delta a = C \cdot Y^{p+q}(\uppi a)^{(p+q)/2} \cdot 
\left(\frac{{\Delta \sigma}_{\mathrm{eq}}^{p+q}}{(1-R_{\mathrm{eq}})^{p}}
N_{\mathrm{total}}\right).
\end{equation}}\unskip
Equality between Equations~(\ref{eq7}) and (\ref{eq8}) allows us to
calculate $\Delta \sigma _{\mathrm{eq}}$:
{\begin{equation}\label{eq9}
\Delta \sigma _{\mathrm{eq}} = 
\left(\sum _{i=1}^{n} \frac{N_{i}}{N_{\mathrm{total}}}
\frac{(1-R_{\mathrm{eq}})^{p}}{(1-R_{i})^{p}}
{\Delta \sigma }_{i}^{p+q}\right)^{1/(p+q)}.
\end{equation}}\unskip
Since the equivalent constant load is assumed to vary between $0$ and
$\sigma_{\max}$, hence $R_{\mathrm{eq}}=0$, the equivalent stress is
given by:
{\begin{equation}\label{eq10}
\Delta \sigma _{\mathrm{eq}} = \left(\sum _{i=1}^{n}
\frac{N_{i}}{N_{\mathrm{total}}}
\frac{{\Delta \sigma }_{i}^{p+q}}{(1-R_{i})^{p}}\right)^{1/(p+q)}.
\end{equation}}\unskip
By combining the two-parameter model with the Willenborg model, the
crack growth rate can be described in this form~\cite{19}:
{\begin{equation}\label{eq11}
\frac{\mathrm{d}a}{\mathrm{d}N} = C{K}_{\max}^{p}{\Delta K}_{\mathrm{eff}}^{q}
= \frac{C}{(1-R)^{p}}{\Delta K}_{\mathrm{eff}}^{p+q} = 
C_{T}{\Delta K}_{\mathrm{eff}}^{p+q},
\end{equation}}\unskip
where $R$ is the stress ratio for constant loading. Several studies
carried out on aluminium alloys show that $p=2$~\cite{20,21}. In
Equation~(\ref{eq11}), it can be seen that there are two unknowns to be
determined: the coefficient $C_{T}$ and $q$, which will be deduced from
experimental data on the crack growth rate curve
$({\mathrm{d}a}/{\mathrm{d}N})$ as a function of $\Delta K$ under
constant load $(R = \mathrm{constante})$. 

The Willenberg model~\cite{11} used predicts a maximum delay just after
the overload, which does not allow the influence of multiple overloads
and under loads to be predicted correctly, as the lifetime is on
average equivalent to that determined without the effect of plastic
zone interaction~\cite{15}.

Figure~\ref{fig1} illustrates the procedure for converting a variable
load into an equivalent constant load at $R = 0$.

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig1}Equivalent stress.}
\end{figure}

\subsection{Effect of crack closure on equivalent stress}
\label{ssec22}

In this section, the two-parameter driving force model is modified to
account for the effect of crack closure on crack propagation~\cite{8}.

Residual stresses play a crucial role in crack growth prediction. To
integrate them into the driving force model, weight functions are used
to convert the residual stress distribution into a residual stress
intensity factor $K_{\mathrm{r}}$~\cite{22}, which is added only to the maximum
stress intensity factor since residual stresses do not influence the
minimum stress intensity factor~\cite{23}. The range of total stress
intensity factors and the total maximum stress intensity factor can be
defined as follows~\cite{24}:
{\begin{equation}\label{eq12}
\begin{cases} 
K_{\mathrm{max\_tot}} = K_{\mathrm{max\_app}} + K_{\mathrm{r}} \\
\Delta K_{\mathrm{tot}} = \Delta K_{\mathrm{app}}.
\end{cases}
\end{equation}}\unskip
The crack growth rate, as indicated in Equation~(\ref{eq1}), is
modified:
{\begin{equation}\label{eq13}
\left(\frac{\mathrm{d}a}{\mathrm{d}N}\right)_{\mathrm{corrected}} = 
C(K_{\max} + K_{\mathrm{r}})^{p} (\Delta K_{\mathrm{app}})^{q}.
\end{equation}}\unskip
The link between Equations~(\ref{eq13}) and~(\ref{eq1}) allows
us to define a coefficient such that:
{\begin{equation}\label{eq14}
\frac{\left(\frac{\mathrm{d}a}{\mathrm{d}N}\right)_{\mathrm{corrected}}}
{\frac{\mathrm{d}a}{\mathrm{d}N}} = \frac{(K_{\max} + K_{\mathrm{r}})^{p}
(\Delta K_{\mathrm{app}})^{q}}
{{K}_{\max}^{p}{\Delta K}_{\mathrm{app}}^{q}} = 
\left(\frac{K_{\max} + K_{\mathrm{r}}}{K_{\max}}\right)^{p} = 
\left(1 + \frac{K_{\mathrm{r}}}{K_{\max}}\right)^{p}.
\end{equation}}\unskip
Thus, the effect of crack closure is deduced by a correction
coefficient that depends on the stress intensity factor at each cycle,
the residual stress intensity factor and the material coefficients $p$
and $q$.

The corrected crack growth rate is given by:
{\begin{equation}\label{eq15}
\left(\frac{\mathrm{d}a}{\mathrm{d}N}\right)_{\mathrm{corrected}} = 
\left(1 + \frac{K_{\mathrm{r}}}{K_{\max}}\right)^{p}
\frac{\mathrm{d}a}{\mathrm{d}N}.
\end{equation}}\unskip
Referring to Equations~(\ref{eq4}) and~(\ref{eq6}), the crack growth
rate is expressed as follows:
{\begin{equation}\label{eq16}
\frac{\mathrm{d}a}{\mathrm{d}N} = 
C{K}_{\max}^{p} \Delta K^{q} = 
C\frac{\Delta K^{p+q}}{(1-R)^{p}} = 
\frac{C}{(1-R)^{p}} (Y\Delta \sigma _{\mathrm{eq}} \sqrt{\uppi a})^{p+q}.
\end{equation}}\unskip
By combining Equations~(\ref{eq15}) and~(\ref{eq16}), the
corrected crack growth rate is expressed as follows:
{\begin{eqnarray}
\left(\frac{\mathrm{d}a}{\mathrm{d}N}\right)_{\mathrm{corrected}} &=&
\frac{C}{(1-R)^{p}}(Y\Delta \sigma _{\mathrm{eq\_corrected}}
\sqrt{\uppi a})^{p+q} \nonumber \\
&=& \left(1 + \frac{K_{\mathrm{r}}}{K_{\max}}\right)^{p}
\frac{C}{(1-R)^{p}} (Y\Delta \sigma _{\mathrm{eq}}
\sqrt{\uppi a})^{p+q}. \label{eq17}
\end{eqnarray}}\unskip
Incorporating the residual stresses generated by plastic deformation,
the corrected equivalent stress is determined from the earlier
equivalent-stress expression in Equation~(\ref{eq10}):
{\begin{equation}\label{eq18}
\Delta \sigma _{\mathrm{eq\_corrected}} = 
\left(1 + \frac{K_{\mathrm{r}}}{K_{\max}}\right)^{\frac{p}{p+q}}
\Delta \sigma _{\mathrm{eq}} = \mu \Delta \sigma _{\mathrm{eq}},
\end{equation}}\unskip
where $\mu$ is defined as:
{\begin{equation}\label{eq19}
\mu = \left(1 + \frac{K_{\mathrm{r}}}{K_{\max}}\right)^{\frac{p}{p+q}}.
\end{equation}}\unskip
A correction is then made to $\Delta \sigma _{\mathrm{eq}}$ found in
Equation~(\ref{eq10}), corresponding to equivalent amplitude loading at
$R=0$, as illustrated in the Figure~\ref{fig2}.

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig2}Corrected equivalent load.}
\end{figure}

\section{Procedure for calculating correction coefficient $\mu$}
\label{sec3}

This coefficient is used to introduce the effect of plastic zone
interactions on the equivalent constant stress level and it depends on
the properties of the material, the maximum stress intensity factor and
the residual stress intensity factor. The transition from an equivalent
constant stress level in Equation~(\ref{eq10}) to a corrected equivalent
stress level in Equation~(\ref{eq18}) is adjusted by this coefficient in
Equation~(\ref{eq19}).

This coefficient is computed cycle by cycle, allowing for an accurate
estimation of its impact on service fatigue life.

\subsection{Calculation of the equivalent residual stress intensity
factor $K_{\mathrm{r}}$}\label{ssec31}

To estimate the equivalent residual stress distribution $\sigma _{\mathrm{r}}$ 
created upstream of the crack tip, the FE method is used. At the end of
unloading, the residual stress distribution is extracted, which will
remain constant during constant amplitude loading. A simulation on
ABAQUS~\cite{25}, close to the crack tip, was carried out, where the
behaviour close to the crack tip is based on the Chaboche 
model~\cite{26}. This model is capable of describing the cyclic
response of plastic deformation upstream of the crack under constant
and variable loading. Elasto-plastic finite element analysis is used to
determine the residual stress distribution $\sigma _{\mathrm{r}}$ ahead of the
crack tip under constant cyclic loading of maximum value $\Delta
\sigma _{\mathrm{eq}}$ and load ratio $R=0$. Then, the residual stress
intensity factor $K_{\mathrm{r}}$ is deduced using the following
relationship~\cite{22}:
{\begin{equation}\label{eq20}
K_{\mathrm{r}} =  \int \sigma _{\mathrm{r}}m(x,a)\,\mathrm{d}x,
\end{equation}}\unskip
where $x$ represents the position of the point along the crack front
and $m(x,a)$ is the weight function, given by~\cite{27}:
{\begin{equation}\label{eq21}
m(x,a) = \frac{2P}{\sqrt{2\uppi (a-x)}},
\end{equation}}\unskip
where $P$ represents the load applied.

\subsection{Calculation of the maximum stress intensity factor
$K_{\max}$}\label{ssec32}

The prediction of crack increment evolution is influenced at each
loading cycle by various parameters including material properties,
loading type, specimen geometry, location and initial crack size
$a_{0}$. Each variation in these parameters leads to the evolution of
the crack, controlled by the stress intensity range and the maximum
stress intensity factor. Indeed, for loading ``$i$'', the maximum
stress intensity factor is expressed as follows:
{\begin{equation}\label{eq22}
K_{\max,i} = Y \Delta \sigma _{\mathrm{eq}} \sqrt{\uppi a_{i}},
\end{equation}}\unskip
where $Y$ is the geometric factor that depends on both the geometry 
considered and the loading mode. The value of $Y$ can be estimated 
either through extensive stress analyses using finite element methods
or by employing available ready-made solutions~\cite{35}.

The most well-known and used $Y$ factor solutions for middle-tension
Figure~\ref{fig3} specimens under tension loading are:
{\begin{equation}\label{eq23}
Y = \dfrac{1-0\cdot 5\left(\frac{a}{b}\right) + 
0\cdot 32\left(\frac{a}{b}\right)^{2}}{\sqrt{1-\frac{a}{b}}},
\end{equation}}\unskip
where $a$ is a crack length and $b$ is a specimen width.

\begin{figure}
\includegraphics{fig03}
\caption{\label{fig3}Example of middle-tension specimen.}
\end{figure}

\subsection{Calculation of calibration parameters $C_{T},p,q$}
\label{ssec33}

To establish these parameters, we utilize experimental data for the
crack growth rate curve $({\mathrm{d}a}/{\mathrm{d}N})$ in relation to
the range of the stress intensity factor $(\Delta K)$ under a
constant load. The curve depicting the relation ship between the crack
growth rate $({\mathrm{d}a}/{\mathrm{d}N})$ and the $(\Delta K)$
range is employed to fine-tune the parameters $p,q$, and $C_{T}$.
Multiple studies on aluminum alloys (Al-2024-T3) have shown that $p=2$
\cite{20,21}, this fixed exponent serves as a reference for calibrating
additional coefficients $C_{T}$ and $q$. These parameters are then
incorporated into the equation for calculating the equivalent load.

\subsection{Optimisation of the correction coefficient $\mu$}
\label{ssec34}

From Equation~(\ref{eq22}), for the maximum stress intensity factor for
each cycle, one observes that the maximum stress intensity factor
varies from cycle to cycle, which drives a variation in the correction
coefficient $\mu$ that is also variable. The aim is to have, in a
practical framework, a correction coefficient~$\mu$ that enables
mapping a variable real load to a stabilized equivalent stress, thereby
facilitating life estimation.

According to Equation~(\ref{eq19}), the correction factor is introduced
as:
{\begin{equation}\label{eq24}
\mu _{i} = \left(1 + \frac{K_{\mathrm{r}}}{K_{\max,i}}\right)^{\frac{p}{p+q}}
\end{equation}}\unskip
which implies that the correction coefficient $\mu$ depends on the
material properties $p$ and $q$, on the maximum stress intensity factor
$K_{\max,i}$ for each cycle and on the residual stress intensity factor
$K_{\mathrm{r}}$. In practice, the variation of $K_{\max,i}$ from cycle
to cycle induces a variation in the correction coefficient $\mu _{i}$.

In typical load sequences (e.g., recurrent spectra or block loading),
variations in~correction coefficient $\mu$ can be considered by its
average value of the cycle mean and the distribution of cycles do not
dramatically alter the distribution of $K_{\max,i}$. In this context,
fixing~the correction coefficient $\mu$ to a representative mean value
computed from a representative set of cycles can provide an effective
estimate of the corrected equivalent stress without requiring
continuous recalibration. But if the load sequence varies strongly
(e.g., frequent alternating peaks and troughs, or abrupt transitions
between loading regimes), the correction coefficient $\mu$ may become
sensitive to the sequence and require periodic recalibration.

In this study, the approach adopted initially is to use a constant
value of the correction coefficient $\mu$ to simplify the estimation of
the corrected equivalent stress defined by Equation~(\ref{eq18}):
$\Delta \sigma _{\mathrm{eq\_corrected}} = 
\mu \Delta \sigma _{\mathrm{eq}}$, where
$\Delta \sigma _{\mathrm{eq}} = f(\Delta \sigma)$, Equation~(\ref{eq10}),
as described previously in the model framework, with $\Delta \sigma$ 
representing the actual variable load and $\Delta \sigma
_{\mathrm{eq}}$ the constant equivalent load.

According to Equation~(\ref{eq22}), the distribution of $K_{\max,i}$ is
not strongly affected by the cycle distribution, as $\Delta \sigma
_{\mathrm{eq}}$ remains constant, therefore, the correction coefficient
$\mu$ may vary modestly around a mean value. Fixing the correction
coefficient $\mu$ to a representative mean can thus provide an
effective estimate of the corrected equivalent stress.

\section{Numerical application}\label{sec4}
\subsection{A plate with a central crack}\label{ssec41}

The analysis was carried out on a sample (Figure~\ref{fig4}) of 2024-T3
aluminium alloy with a central crack of initial size $(2a)$ equal to
12.7~mm, a length $L$ of 610~mm, a width $w$ of 229~mm and a thickness
$t$ of 4.1~mm.

\begin{figure}
\includegraphics{fig04}
\caption{\label{fig4}Specimen dimension.}
\end{figure}

\subsection{Calculation of parameters $C_{T},q$}
\label{ssec42}

To determine the equivalent load according to Equation~(\ref{eq10}), it
is necessary to find the unknown $q$ (the value of $p$ being equal to
2). This is done using Equation~(\ref{eq11}), which has two unknowns
($C_{T}$ and $q$). In order to determine these two parameters,
experimental data of the crack growth rate curve
${\mathrm{d}a}/{\mathrm{d}N}$ as a function of the stress intensity
factor range $\Delta K$ under constant load is used. Then, the
following curve (Figure~\ref{fig5}) represents the variation of the
crack growth rate ${\mathrm{d}a}/{\mathrm{d}N}$ as a function of the
stress intensity range $\Delta K$ is used to fit the parameters $p,q$
and $C_{T}$.

\begin{figure}
\includegraphics{fig05}
\caption{\label{fig5}$({\mathrm{d}a}/{\mathrm{d}N}) - \Delta K$ of
2024-T3 aluminium alloys for $R = 0.4$~\cite{28}.}
\end{figure}

For an aluminium alloy, we take $p=2$, which makes it possible to
determine $C_{T}=1.2037\times 10^{-11}$ and $q=1.7877$. 

\subsection{Calculation of the equivalent stress 
$\Delta\sigma_{\mathrm{eq}}$}\label{ssec43}

We then use the loading block shown in Figure~\ref{fig6}, according to
references~\cite{29}.

\begin{figure}
\includegraphics{fig06}
\caption{\label{fig6}Loading sequence.}
\end{figure}

According to Equation~(\ref{eq10}), the equivalent load associated with
each loading sequence is expressed in the following form:
{\begin{equation}\label{eq25}
\Delta \sigma _{\mathrm{eq}} = \left(\sum _{i=1}^{3}
\frac{N_{i}}{N_{\mathrm{total}}} \frac{{\Delta \sigma}_{i}^{p+q}}
{(1-R_{i})^{p}}\right)^{1/(p+q)},
\end{equation}}\unskip
where the load ratio $R=0$ leads to $\Delta \sigma _{\mathrm{eq}} =
\sigma _{\max}$.

\subsection{Calculation of the residual stresses $\sigma_{\mathrm{r}}$}
\label{ssec44}

The residual stress generated ahead the crack tip, extracted at the end
of unloading phase, are estimated using FE analysis based on the
Chaboche model~\cite{26}, this model can describe cyclic plasticity
with high precision by accounting for the Bauschinger effect, mean
stress relaxation, and cyclic hardening during simulations. In our
study, the nonlinear kinematic hardening model is integrated into
ABAQUS to evaluate the influence of a variable-amplitude loading on the
cyclic plastic behavior at the crack tip of the 2024-T3 aluminum alloy.

The Von Mises criterion applied in this study is defined by the
following expression:
{\begin{equation}\label{eq26} 
f(\overline{\overline{\sigma}}, \overline{\overline{X}}) = 
J_{2}(\overline{\overline{\sigma}} -\overline{\overline{X}}) - 
\sigma _{y},
\end{equation}}\unskip
where $f$ is the elasticity surface,
$J_{2}(\overline{\overline{\sigma}} - \overline{\overline{X}})$
indicates the second invariant of the stress deviator tensor and
$\sigma_{y}$ is the elastic limit.

The non-linear hardening component $\overline{\overline{X}}$ is
described by the following equation:
{\begin{equation}\label{eq27}
\mathrm{d}\overline{\overline{X}} = \textstyle\frac{2}{3}
C_{i}\overline{\overline{\dot{\varepsilon}}}^{p}-\gamma _{i}
\overline{\overline{X}}_{{p^{.}}},
\end{equation}}\unskip
where $\overline{\overline{\dot{\varepsilon}}}^{p}$ is the plastic
strain tensor and $p^{.}$ the cumulative plastic deformation rate,
which are defined as follows:
{\begin{eqnarray}
p^{.} &=& \sqrt{\frac{2}{3}\overline{\overline{\dot{\varepsilon}}}^{p}\colon 
\overline{\overline{\dot{\varepsilon}}}^{p}} \label{eq28} \Seqnsplit
\overline{\overline{\dot{\varepsilon}}}^{p} &=&
\underline{\lambda }\frac{\delta f}{\delta \sigma}. \label{eq29}
\end{eqnarray}}\unskip

In order to correctly predict the kinematic effect, several parameters
are introduced, in which $C_{i}$ and $\gamma_{i}$ represent the
characteristic parameters of each material.

This residual stress distribution remains unchanged under constant
amplitude loading. The mechanical and cyclic fatigue properties of the
Al2024-T3 alloy are summarised in the Table~\ref{tab1}, where $C_{i}$
and $\gamma_{i}$ represent the Chaboche kinematic hardening parameters.

%tab1
\begin{table}
\caption{\label{tab1}Summary of cyclic fatigue and mechanical
properties of Al2024-T3 alloy~\cite{31}}
\begin{tabular}{ccccccccc}
\thead
$E$ (MPa) & $\nu$ & $\sigma_{y}$ (MPa) & $C_{1}$ (MPa) & $C_{2}$ (MPa)
& $C_{3}$ (MPa) & $\gamma_{1}$ & $\gamma_{2}$ & $\gamma_{3}$ \\
\endthead
73~000 & 0.33 & 289 & 2169.7 & 7471.9 & 65.761 & 11.74 & 90.05 & 555.09
\botline
\end{tabular}
\tabnote{$E$ is the elastic modulus, $\nu$ is the Poisson's ratio, 
$\sigma_{y}$ is the yield stress, $C_{1}$, $C_{2}$ and $C_{3}$ are the
kinematic stress parameter for the back stress and $\gamma_{1}$,
$\gamma_{2}$ and $\gamma_{3}$ are the corresponding kinematic
saturation rate parameter.}
\end{table}

Figures~\ref{fig7} and~\ref{fig8} illustrate the finite element model,
as well as the boundary conditions and the type of mesh used.

\begin{figure}
\includegraphics{fig07}
\caption{\label{fig7}Boundary conditions used in the simulation.}
\end{figure}

\begin{figure}
\includegraphics{fig08}
\caption{\label{fig8}Type of finite element mesh.}
\end{figure}

To enhance result accuracy and better predict the crack tip response to
plastic deformation in the FE analysis, a highly refined mesh was
applied around the $CPS4R$ crack region, with an element size of $0.05$
\cite{3,24}. This FE analysis results in a residual stress
distribution, as shown in Figure~\ref{fig9}.

\begin{figure}
\includegraphics{fig09}
\caption{\label{fig9}Stress distribution.}
\end{figure}

Using the FE method, the residual stress is estimated based on the
$S_{22}$ stress distribution. It is observed that the compressive
stress distribution increases in absolute value near the crack tip,
which is confirmed by the results of several researchers~\cite{33}.

To take account of the effect of the loading sequence, the residual
stresses are estimated from the loading sequences (a and b) shown in
Figure~\ref{fig6} and then compared to the residual stresses determined
from the equivalent load calculated from Equation~(\ref{eq25}).

Equations~(\ref{eq20}) and~(\ref{eq21}) are used to calculate the 
residual stress intensity factor $K_{\mathrm{r}}$.

\subsection{Calculation of the maximum stress intensity factor
$K_{\max}$}\label{ssec45}

The aim now is to estimate the correction coefficient $\mu$ and,
subsequently, the value of the corrected equivalent stress from
Equation~(\ref{eq18}).

The correction coefficient is given in Equation~(\ref{eq24}). In this
equation; the coefficients $p,q$ and $K_{\mathrm{r}}$ are constant, while the
maximum stress intensity factor $K_{\max}$ is variable and depends on
the crack length $a$ according to Equation~(\ref{eq22}).

The variation of $\Delta \sigma _{\mathrm{eq}}$ is calculated from
Equation~(\ref{eq10}). In this study, the applied load is $\Delta
\sigma _{\mathrm{eq}} = \sigma _{\max} = 62,63$ MPa. The variation of
$\mu$ as a function of $K_{\max}$ is shown in the following figure for
the two loading sequences (a and b) illustrated in Figure~\ref{fig10}
and is compared with that deduced from the equivalent load.

\begin{figure}
\includegraphics{fig10}
\caption{\label{fig10}Variation of $\mu$ as a function of $K_{\max}$.}
\end{figure}

\subsection{Optimisation of the correction coefficient $\mu$}
\label{ssec46}

Combining Equation~(\ref{eq18}) with Equation~(\ref{eq24}), we obtain:
{\begin{equation}\label{eq30}
\Delta \sigma _{\mathrm{eq\_corrected},i} = 
\mu _{i} \Delta \sigma _{\mathrm{eq}}.
\end{equation}}\unskip
We therefore look for determining $\mu_{i}$ allowing us to obtain
$\Delta \sigma _{\mathrm{eq\_corrected},i}$ to predict  a
lifetime as close as possible to the experimental results.
Table~\ref{tab2} shows some calculation points taken from the
correction coefficient curve (Figure~\ref{fig11}), where the residual
stress is deduced from the equivalent load.

\begin{figure}
\includegraphics{fig11}
\caption{\label{fig11}Variation of $\mu$ as a function of $K_{\max}$
(equivalent residual stress).}
\end{figure}

%tab2
\begin{table}
\caption{\label{tab2}Effect of correction coefficient $\mu$ on
equivalent load}
\begin{tabular}{ccc}
\thead
Point & $\mu$ & $\Delta \sigma_{\mathrm{eq\_corrected}}$ \\
\endthead
A & 0.9655   & 60.4776 \\
B & 0.97\0\0 & 60.7594 \\
C & 0.98\0\0 & 61.3858 \\
D & 0.985\0  & 61.699\0  \\
E & 0.99\0\0 & 62.0122 \\
F & 0.9904   & 62.0373
\botline
\end{tabular}
\end{table}

Figure~\ref{fig12} shows the effect of the correction coefficient on
the variation curve of the crack length $a$ (mm) as a function of the
number of cycles $N$ (cycles) for the two loading sequences (a and b).

\begin{figure}
\vspace*{-4pt}
\includegraphics{fig12}
\vspace*{-2pt}
\caption{\label{fig12}Effect of correction coefficient on crack
evolution.}
\vspace*{-2pt}
\end{figure}

The values of the correction coefficients obtained at different points
will be multiplied by $\Delta \sigma _{\mathrm{eq}}$ to determine
$\Delta \sigma _{\mathrm{eq\_corrected}}$. Thus, for each
cyclic loading spectrum corresponds an equivalent loading $\Delta
\sigma _{\mathrm{eq}}$ multiplied by a selected correction coefficient 
$\mu _{i}$, leading to the $\Delta \sigma _{\mathrm{eq\_corrected}}$ corresponding to a curve dipicting the evolution
crack length as a function of the number of cycles. It is observed that
the lower this factor, the closer the curve is to the experimental
results.

We can conclude that a minimal correction coefficient, corresponding to
the first iteration $(i=1)$, where the maximum stress intensity
$K_{\max}$ and crack length are minimal $(a=a_{0})$, is the optimal
coefficient (corresponding to point A in Figure~\ref{fig11}), as it
offers a variation in crack length as a function of the number of
cycles that is most consistent with the experimental curve.

To take account of the effect of the loading sequence, we consider the
minimum values of the correction coefficients for each loading sequence
(according to Figure~\ref{fig13}) and compare them with the results from
point~A.

\begin{figure}
\includegraphics{fig13}
\vspace*{-2pt}
\caption{\label{fig13}Variation of $\mu$ as a function of $K_{\max}$
for three loading sequences.}
\vspace*{-2pt}
\end{figure}

We obtain the results on the variation of the crack size as a function
of the number of cycles for the selected points, as follows
(Figure~\ref{fig14}).

\begin{figure}
\includegraphics{fig14}
\caption{\label{fig14}Effect of the correction coefficient on the
evolution of the crack for different points.} 
\end{figure}

It can be seen that the correction coefficient corresponding to point
A, where residual stresses are deduced from constant loads (equivalent
load), produces a variation in crack size as a function of the number
of cycles that is closest to the experimental results. In comparison,
the curve produced by the coefficient corresponding to point A1, where
the residual stresses are deduced from the variable loads (loading
sequence a), differs, as does that produced by the coefficient
corresponding to point A2, where the residual stresses are based on the
variable loads (loading sequence~b).

This means that the loading sequence is not taken into account in the
following calculation methodology.

\vspace*{-2pt}

\section{Flowchart for determining the crack growth increment}
\label{sec5}

The following flowchart illustrates the method for determining the
fatigue cycle growth increment in relation to a fatigue crack growth
increment (Figure~\ref{fig15}).

\begin{figure}
\vspace*{2pt}
\includegraphics{fig15}
\vspace*{2pt}
\caption{\label{fig15}Flowchart for determining the crack growth
increment.}
\end{figure}

\section{Application of the model for different load sequences}
\label{sec6}

To validate the computational model, three loading sequences are
studied~\cite{29,34}. The first sequence is shown in
(Figure~\ref{fig16}a), where the maximum load is constant and equal to
83~MPa, with a variation of $\Delta \sigma$ from 75~MPa to 55~MPa and
14~MPa. The (Figure~\ref{fig16}c) shows the second loading sequence
(low-high) with three loading blocks, where the maximum stress is
increased from 69~MPa to 83~MPa and then to 96~MPa, giving a constant
$\Delta \sigma$ equal to 49~MPa. In (Figure~\ref{fig16}e), we have a
loading sequence of four loading blocks (top--bottom) with a constant
$\Delta\sigma$ equal to 49~MPa.

\begin{figure}
\vspace*{-2pt}
\includegraphics{fig16}
\vspace*{-2pt}
\caption{\label{fig16}(b,d,f) Comparison of model results with
experimental results for 2024-T3 aluminium~\cite{29} and 
model~\cite{17}, using three types of variable loading spectra applied
(a,c,e).} 
\end{figure}

For each loading sequence:
\begin{enumerate}[(i)]
\item The equivalent stress $\Delta \sigma _{\mathrm{eq}}$ is
calculated.
\item Using the Chaboche model, the residual stresses close to the
crack are assessed.
\item The residual stress intensity factor is determined.
\item The correction coefficient is determined from which the corrected
equivalent stress $\Delta \sigma _{\mathrm{eq\_corrected}}$ is
calculated.
\end{enumerate}

We move from studying a variable cyclic charge spectrum problem to a
constant charge spectrum problem.

Comparison of the results in Figure~\ref{fig16}(b,d,f) shows that the
experimental results are in agreement with the data from the fatigue
life prediction model. Indeed, the crack propagation prediction curve
reproduces, in most cases, the experimental data~\cite{29} than the
one proposed by reference~\cite{17}, who design and evaluate effective
probabilistic methods allowing real-time prognosis of fatigue damage in
structures, taking into account uncertainties related to loading,
material properties and in situ diagnostics, based on the idea of
finding an equivalent crack growth process under constant amplitude
loading, which products a similar crack kinetics with that of the true
random loading case. This constant amplitude loading is modified in
this section by integrating the effect of residual stress determined by
the FE method and which are then introduced into the calculation of the
equivalent loading. Corresponding to Figure~\ref{fig9},  it is observed
that the compressive stress distribution increases in absolute value
near the crack tip, which is confirmed by the results of several
researchers~\cite{33}. Since in our model, the evaluation of the
applied equivalent stress is more reliable. A notable difference
between the calculation model described in reference~\cite{17} and the
experimental data occurs in the case of the second and third spectrum,
which is corrected in our model by taking into account the effects of
the residual stress determined by the FE method and which is then
introduced into the calculation of the equivalent loading.

\section{Conclusion}\label{sec7}

\vspace*{-2pt}

This study aims to provide a simple and effective approach to
predicting fatigue life under variable loading by transforming variable
load blocks into constant loads, thereby simplifying evaluation while
preserving the essential mechanical effects. The influence of plastic
deformations at the crack tip is accounted for with a correction
coefficient that adjusts crack propagation predictions across different
loading sequences. This correction improves agreement with experimental
results. The following conclusions can be drawn:
\begin{enumerate}[(i)]
\item The stress-equivalent transformation approach avoids
cycle-by-cycle calculations and reduces computational cost.
\item The correction coefficient has a stronger effect for low-to-high
loading than for high-to-low, highlighting the importance of load order
and loading dynamics on residual damage.
\item The model shows better agreement with experimental data when
corrections due to plastic variations and residual damage are included,
validating the usefulness of the proposed approach for diverse loading
types.
\item Residual compressive stresses generally slow crack growth, but
they exhibit substantial dispersion and large deformations; these
aspects are incorporated into the model through the correction
coefficient and the consideration of mean loading.
\end{enumerate}

\vspace*{-6pt}

\section*{Nomenclature}

\vspace*{-2pt}

\begin{inftab}
\fontsize{10}{11.8}\selectfont
\begin{tabular}{ll}
\tbody
$K_{\max}$ & Maximum stress intensity factor \\ 
$K_{\max,i}$ & Maximum stress intensity factor in the $i${th} cycle \\ 
$K_{\max\_\mathrm{tot}}$ & Total maximum stress intensity factor \\ 
$K_{\max\_\mathrm{app}}$ & Maximum applied stress intensity factor \\ 
$K_{\mathrm{r}}$ & Residual stress intensity factor \\ 
$\Delta K$ & Stress intensity factor range \\ 
$\Delta K_{i}$ & Stress intensity factor range in the $i${th} cycle \\ 
$\Delta K_{\mathrm{tot}}$ & Total stress intensity factor range \\ 
$\Delta K_{\mathrm{app}}$ & Applied stress intensity factor range \\ 
$\Delta K_{\mathrm{eff}}$ & Effective stress intensity factor range \\ 
$\Delta \sigma$ & Stress level \\ 
$\Delta \sigma_{i}$ & Stress level in the $i${th} cycle \\ 
$\Delta \sigma_{\mathrm{eq}}$ & Equivalent stress level \\ 
$\Delta \sigma_{\mathrm{eq\_corrected}}$ & Corrected equivalent stress level \\ 
$a$ & Crack length \\ 
$a_{i}$ & Crack length in the $i${th} cycle \\ 
$b$ & Width of specimen \\ 
$a_{0}$, $a_{\mathrm{c}}$ & Initial and critical crack lengths, respectively \\ 
$\Delta a$ & Increment of crack length \\ 
$Y$ & Geometric correction factor \\ 
$R$ & Stress ratio \\ 
$R_{i}$ & Stress ratio in the $i${th} cycle \\ 
$\sigma_{\min}$ and $\sigma_{\max}$ & Minimum and maximum stress, respectively \\ 
$N_{\mathrm{tot}}$ & Total number of loading cycles \\ 
$N_{i}$ & Number of loading cycles in the $i${th} cycle \\ 
$C$, $C_{T}$, $p$ and $q$ & Material parameters \\ 
$\mu$ & Correction coefficient \\ 
$\mu_{i}$ & Correction coefficient in the $i${th} cycle \\ 
$m$ & Weight function \\ 
$P$ & Applied load
\end{tabular}
\end{inftab}

\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.

\back{}

\printbibliography
\refinput{crmeca20250545-reference.tex}

\end{document}
