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\DOI{10.5802/crmeca.340}
\datereceived{2025-06-01}
\daterevised{2025-10-19}
\dateaccepted{2025-11-04}
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\begin{noXML}

\CDRsetmeta{articletype}{research-article}

\shortrunauthors

\title{Influence of steel substrate behavior on the deformation and
cracking of Zn--Al--Mg coatings on galvanized steel sheets}

\alttitle{Influence du comportement du substrat en acier sur la
d\'{e}formation et la fissuration des rev\^{e}tements Zn--Al--Mg sur
les t\^{o}les d'acier galvanis\'{e}es}

\author{\firstname{Ahmed} \lastname{Zouari}\CDRorcid{0000-0002-5327-5425}\IsCorresp}
\address{Mines Paris PSL University, Centre des mat\'eriaux, CNRS UMR 7633, 21 All\'ee des Marronniers, 78000 Versailles, France}
\address{ArcelorMittal Maizi\`eres Research, voie Romaine, F-57280, Maizi\`eres les Metz, France}
\address{Safran Aircraft Engines Villaroche, Rond-Point Rene Ravaud, 77550 Moissy-Cramayel, France}
\email[A. Zouari]{Ahmed.Zouari@safrangroup.com}
\curraddr[A. Zouari]{Safran Aircraft Engines Villaroche, Rond-Point Rene Ravaud, 77550 Moissy-Cramayel, France}

\author{\firstname{Samuel} \lastname{Forest}\CDRorcid{0000-0002-8869-3942}\IsCorresp}
\addressSameAs{1}{Mines Paris PSL University, Centre des mat\'eriaux, CNRS UMR 7633, 21 All\'ee des Marronniers, 78000 Versailles, France}
\email[S. Forest]{Samuel.Forest@minesparis.psl.eu}

\author{\firstname{Sylvain} \lastname{D\'{e}pinoy}\CDRorcid{0000-0001-9706-4497}}
\addressSameAs{1}{Mines Paris PSL University, Centre des mat\'eriaux, CNRS UMR 7633, 21 All\'ee des Marronniers, 78000 Versailles, France}

\author{\firstname{Yazid}\nobreakauthor\lastname{Madi}\CDRorcid{0000-0002-3530-8668}}
\addressSameAs{1}{Mines Paris PSL University, Centre des mat\'eriaux, CNRS UMR 7633, 21 All\'ee des Marronniers, 78000 Versailles, France}

\author{\firstname{Jean-Michel} \lastname{Mataigne}}
\addressSameAs{2}{ArcelorMittal Maizi\`eres Research, voie Romaine, F-57280, Maizi\`eres les Metz, France}

\author{\firstname{Houssem Eddine} \lastname{Cha\"{i}eb}\CDRorcid{0000-0003-1185-4872}}
\address{ArcelorMittal Global R\&D Ghent, J.F. Kennedylaan 3, B-9060 Zelzate, Belgium}

\author{\firstname{Pascal}\nobreakauthor\lastname{Bertho}}
\addressSameAs{2}{ArcelorMittal Maizi\`eres Research, voie Romaine, F-57280, Maizi\`eres les Metz, France}

\author{\firstname{Aymen}\nobreakauthor\lastname{Bouzid}}
\addressSameAs{2}{ArcelorMittal Maizi\`eres Research, voie Romaine, F-57280, Maizi\`eres les Metz, France}

\author{\firstname{Aurelien} \lastname{Chopin}}
\addressSameAs{2}{ArcelorMittal Maizi\`eres Research, voie Romaine, F-57280, Maizi\`eres les Metz, France}

\author{\firstname{Coralie} \lastname{Jung}}
\addressSameAs{2}{ArcelorMittal Maizi\`eres Research, voie Romaine, F-57280, Maizi\`eres les Metz, France}

\keywords{\kwd{Zinc coatings}
\kwd{Crystallographic texture}
\kwd{L\"{u}ders banding}
\kwd{Coating cracking}
\kwd{Slip and twinning}
\kwd{Strain localization}
\kwd{Galvanized steel}}

\altkeywords{\kwd{Rev\^{e}tements de zinc}
\kwd{Texture cristallographique}
\kwd{Bandes de L\"{u}ders}
\kwd{Fissuration du rev\^{e}tement}
\kwd{Glissement et maclage}
\kwd{Localisation de la d\'{e}formation}
\kwd{Acier galvanis\'{e}}}

\begin{abstract} 
The objective of this study is twofold: (i) to comprehensively describe
the deformation modes of zinc coatings depending on their
crystallographic texture and (ii) to investigate the effect of
L\"{u}ders banding occurring in the steel substrate on the coating
deformation and cracking modes. Microscopic characterization and
mechanical tests were conducted on three types of galvanized steel: a
mild steel substrate and a high-strength low-alloy (HSLA) steel
substrate known to exhibit the L\"{u}ders banding phenomenon, and a
dual-phase steel substrate. The results reveal a direct correlation
between the coating texture, plastic deformation modes, and the
mechanical behavior of the respective substrates. Digital Image
Correlation (DIC) was employed to measure the strain fields and
characterize strain localization phenomena resulting from L\"{u}ders
banding. The strain field measured on the galvanized substrate
specimens showed that L\"{u}ders band propagation leads to accelerated
plastic deformation and cracking in the Zn--Al--Mg coating. Finite
element simulations were performed by considering the real coating
microstructure and incorporating the macroscopic behavior of the
substrate. The simulation results demonstrate that slip and twinning
activities of the coating grains are strongly influenced by the
underlying substrate behavior.
\end{abstract} 

\begin{altabstract} 
L'objectif de cette \'{e}tude est double : (i) d\'{e}crire de
mani\`{e}re exhaustive les modes de d\'{e}formation des rev\^{e}tements
de zinc en fonction de leur texture cristallographique et (ii)
\'{e}tudier l'effet des bandes de L\"{u}ders se d\'{e}veloppant dans le
substrat en acier sur les modes de d\'{e}formation et de fissuration du
rev\^{e}tement. Une caract\'{e}risation microscopique et des essais
m\'{e}caniques ont \'{e}t\'{e} r\'{e}alis\'{e}s sur trois types d'acier
galvanis\'{e} : un substrat en acier doux, un substrat en acier \`{a}
haute r\'{e}sistance et faiblement alli\'{e} (HSLA) connu pour
pr\'{e}senter le ph\'{e}nom\`{e}ne de bandes de L\"{u}ders, et un
substrat en acier \`{a} microstructure biphas\'{e}e (Dual-Phase). Les
r\'{e}sultats r\'{e}v\`{e}lent une corr\'{e}lation directe entre la
texture du rev\^{e}tement, les modes de d\'{e}formation plastique et le
comportement m\'{e}canique des substrats respectifs. La corr\'{e}lation
d'images num\'{e}riques (DIC) a \'{e}t\'{e} utilis\'{e}e pour mesurer
les champs de d\'{e}formation et caract\'{e}riser les
ph\'{e}nom\`{e}nes de localisation associ\'{e}s aux bandes de
L\"{u}ders. Les champs de d\'{e}formation mesur\'{e}s sur les
\'{e}prouvettes de substrat galvanis\'{e} montrent que la propagation
des bandes de L\"{u}ders entra\^{i}ne une acc\'{e}l\'{e}ration de la
d\'{e}formation plastique et favorise la fissuration dans le
rev\^{e}tement Zn--Al--Mg. Des simulations par \'{e}l\'{e}ments finis
ont \'{e}t\'{e} r\'{e}alis\'{e}es en consid\'{e}rant la microstructure
r\'{e}elle du rev\^{e}tement et en int\'{e}grant le comportement
macroscopique du substrat. Les r\'{e}sultats de la simulation
d\'{e}montrent que les m\'{e}canismes de glissement et de maclage au
sein des grains du rev\^{e}tement sont fortement influenc\'{e}s par le
comportement m\'{e}canique du substrat sous-jacent.
\end{altabstract} 

\maketitle

\end{noXML}

\section{Introduction}\label{sec1}
In recent years, significant efforts have been made to enhance the
properties of zinc coatings by introducing new chemical elements into
the zinc bath during the hot-dip galvanizing (HDG)  process~\cite{1}.
These elements aim at improving corrosion resistance by adding Al and
Mg to  zinc~\cite{2,3,4}. However, the incorporation of these elements
leads to substantial changes in the microstructure and crystallographic
texture of the coatings, which in turn directly impacts their
mechanical behavior and adhesion performance during forming and shaping
processes. 

Previous research has primarily focused on investigating the impact of
microstructure and crystallographic orientation on the deformation and
cracking behaviors of Zn  grains~\cite{5,6,7,8,9,10,11,12}. These
studies have revealed that the morphology of zinc grains and of
eutectic phases can be considerably affected by the relatively low
variations of Al and  Mg~\cite{13}. Furthermore, inclusion of Mg in
zinc coatings induces the formation of a brittle binary eutectic phase
(i.e.\ Zn ${+}$ MgZn$_{2}$) which acts as the primary cause of
micro-cracking within Zn--Al--Mg  coatings~\cite{14}.

Plastic slip activity of Zinc alloy coatings has been investigated by
Parisot  et~al.~\cite{15} in two different coating microstructures and
also in bulk zinc using uniaxial tension, plane-strain tension, and
expansion loading. This study demonstrates that the primary deformation
mechanism is basal slip in bulk zinc and in the recrystallized
coatings. However, in temper rolled coatings or ``Skin Pass'' coatings,
and in non-temper rolling, the major modes of deformation are
identified as pyramidal $\uppi^2$  slip and mechanical twinning, which
are additional characteristic deformation modes in hcp metals and
alloys. A specific study regarding the deformation of Zn--Al--Mg
coatings was conducted by  Ahmadi et~al.~\cite{13} using an in-situ
uniaxial tensile test followed by Electron-Back-Scattered Diffraction
(EBSD) analyses and SEM observations. These authors showed that zinc
grains can develop five independent slip/twinning systems, namely
basal, prismatic, 2nd order pyramidal, 1st order pyramidal and
twinning. 

Park et~al.~\cite{8} examined the effect of the crystallographic
orientation of Zn grains on the cracking of two Zn--Al--Mg coatings by
combining mechanical deformation and post-mortem EBSD analyses. They
demonstrated that the primary zinc and eutectic zinc phases,
characterized by basal planes with the [0001] direction perpendicular
to the normal direction (ND) of the sheet, are particularly prone to
cracking when subjected to stress along the $c$-axis. Jaffrey 
et~al.~\cite{16} presented experimental evidence that non-basal grains
exhibited a greater occurrence of cleavage cracks compared to other
oriented grains. According to Parisot  et~al.~\cite{17}, the occurrence
of cleavage cracking in the coatings is influenced by the favored
orientations of the grains. They observed that the maximum normal
stress component required for cleavage to take place is highest in
grains that have their $c$-axis aligned along the tensile direction.

The existing research exploring the impact of the steel substrate on
the deformation and cracking behavior of Zn--Al--Mg alloy coating is
notably limited. A particular gap exists in understanding the strain
localization behavior of zinc grains, in the presence of L\"{u}ders
banding of the steel substrate. Only two notable studies, conducted by
Parisot  et~al. and Ahmadi et~al., have explored this 
aspect~\cite{13,17}. Parisot et~al. focused on Zn--Al coating on
interstitial-free steel, while Ahmadi et~al. investigated the effect of
HSLA steel with L\"{u}ders banding and IF steel on Zn--Al--Mg coating
cracking behavior. However, a comprehensive understanding of the
underlying substrate influence on coating deformation and cracking has
not been thoroughly scrutinized or clarified. How does the L\"{u}ders
band arising from the substrate propagate into the coating? Moreover,
there is a need for a better understanding of the impact of L\"{u}ders
band propagation on coating formability, taking into account real
microstructural factors. This crucial aspect has never been thoroughly
investigated through simulations that incorporate the actual
microstructure, allowing for a detailed examination of how L\"{u}ders
band influences the deformation behavior of the coating.

Therefore, it is crucial to better understand the interplay between the
coating, including its microstructure and crystallographic texture, and
the substrate through appropriate experimental and numerical
investigations. To address this research gap, the present study aims at
gaining a comprehensive understanding of the effect of steel substrate
on the formability of zinc coatings. This investigation takes into
account various factors such as the microstructure of the coating, the
crystallographic orientation of the phases and the influence exerted by
the underlying steel substrate. In pursuit of this objective, we have
examined three specific types of steel substrates with different
hardening behavior, namely Mild, HSLA, and Dual-Phase steels. These
substrates were coated with an identical Zn--Al--Mg alloy. The
materials were tested in tension and some tests were interrupted to
follow the propagation of the L\"{u}ders band through the coating and
identify the deformation and cracking modes. A three-dimensional
finite-element crystal-plasticity simulation of a zinc coating on a
galvanized steel sheet has been performed to assess the local impact of
substrate behavior and give access to mechanical and physical
quantities that are experimentally challenging to measure. These
quantities include the effect of L\"{u}ders banding on the coating
local strain field, the slip activities in each grain as well as the
strain gradient throughout the coating thickness. The simulation takes
into consideration the actual polycrystalline structure of the coating,
providing valuable insights into the material behavior that would
otherwise be difficult to obtain experimentally.

The microstructures and textures of the samples are described in 
Section~\ref{sec2}. Uniaxial and equi-biaxial mechanical tests coupled
with digital image correlation techniques to characterize strain
localization due to L\"{u}ders banding for the coated mild steel
(C\_Mild) and the coated HSLA steel have been conducted and are
presented in  Section~\ref{sec3}. In the same section, the results of
the mechanical tests are presented. The effect of the crystallographic
orientation and microstructure of Zn grains on plastic deformation
modes such as plastic slip, deformation twinning and cracking modes
have been investigated by combining mechanical deformation, post-mortem
Scanning Electron Microscopy (SEM) observations and EBSD analyses. 
Section~\ref{sec4} of the study introduces the numerical method
employed to model the propagation of L\"{u}ders banding within the
substrate and the coating and to simulate the actual microstructure of
the zinc coating. This method takes into account the individual
orientations of the grains. This  section also contains the numerical
results of these finite element simulations.

\section{Materials and method}\label{sec2}
The current study investigates three distinct steel substrates coated
with identical Zn--Al--Mg coatings, all processed on the same
galvanizing line. The first specimen (C\_Mild) is a galvanized
{Mild steel sheet}, the second (C\_HSLA) is a
galvanized {High-Strength Low-Alloy steel (HSLA) sheet}, and the
last (C\_Dual-Phase) is a galvanized {Dual-Phase steel sheet},
characterized by a heterogeneous microstructure composed of a soft
ferritic matrix and hard martensitic islands providing a balanced
combination of high strength and ductility. Both the Mild and the HSLA
steel substrates exhibit the L\"{u}ders phenomenon with different
L\"{u}ders plateau lengths at the beginning of the plastic flow. 
Table~\ref{tab1} provides details regarding the materials used in this
study, and the chemical compositions of steel substrates are given in 
Table~\ref{tab2}.

\begin{table}
\caption{\label{tab1}Characteristics of Zn--Al--Mg coated steel sheet
samples investigated in the study: the coating alloy is the same for
all three substrates}
\begin{tabular}{ccccc}  
\thead
Sample label & Coating & Steel type & 
Substrate thickness ($\upmu$m) & 
Coating thickness ($\upmu$m) \\ 
\endthead
C-Mild & \morerows{2}{Zn--Al--Mg} & Mild steel & 2100 & \morerows{2}{7} \\
C-HSLA & & HSLA & 1200 & \\
C-Dual-phase & & Dual-phase & 1200 &
\botline
\end{tabular}
\end{table}

\begin{table}
\caption{\label{tab2}Chemical composition (wt\%) of steel substrates}
\begin{tabular}{ccccccc}  
\thead
Element (wt\%) & C & Si & Mn & Al & Cr & Phase \\ 
\endthead
Mild steel & 0.05 & 0.02 & 0.3 & 0.04 & 0.05 & Ferritic \\ 
HSLA & 0.08 & 0.25 & 1.6 & 0.06 max & - & Ferritic \\ 
Dual-phase & 0.09 & 0.25 & 1.9 & 0.045 & 0.22 & Dual-phase 
\botline 
\end{tabular}
\end{table}

The average total coating weight for the Zn--Al--Mg coating is about 
90~g/m$^{2}$ which corresponds to a coating thickness of about 
7~$\upmu$m. The galvanizing process is as follows. Steel sheets,
produced from cold rolling, are transported to the galvanization line.
The coils undergo a surface cleaning process to remove rolling oils,
followed by a surface rinse, either chemically or electrochemically.
Then, the sheet undergoes a metallurgical annealing process in a
furnace at an annealing temperature up to  800~\textdegree C with a dry
nitrogen and hydrogen atmosphere (5 to 20~vol\% H2), with a controlled
dew point. The surface iron oxides are reduced during this step to
expose the metallic surface to the molten zinc alloy, ensuring good
adhesion between the sheet and the coating. Following the completion of
the two primary surface preparation stages, the temperature of the
sheet gradually decreases until it reaches the temperature of the zinc
bath, which is approximately  460~\textdegree C. At this point, the
sheet is immersed in the bath of molten zinc alloy for a brief
duration, resulting in the formation of a liquid zinc layer as it
emerges from the bath.

\subsection{Experimental protocol}\label{sec2.1}
The microstructure of the materials was investigated by means of Field
Emission Gun Scanning Electron Microscopy (FEG-SEM) using a JEOL
IT800SHL and a FEI Nova NanoSEM 450. The crystallographic orientation
of the grains in coatings was determined by Electron Backscatter
Diffraction (EBSD) using an Oxford camera. To this end, the coating
surface was ion polished during 20~min with 2~keV and 
85\textdegree\ tilt on a Hitachi IM4000${+}$ flat-milling machine. No
mechanical polishing has been applied to the surface to avoid
mechanical deformation of the coating. All the EBSD maps represent the
orientation of the coating Normal Direction with respect to the single
crystal axes. The red color indicates that the ND is parallel to the
[0001] crystal direction.

Uniaxial tests were conducted using a ZwickRoell uniaxial tensile
machine with a strain rate of  $10^{-4}~\mathrm{s}^{-1}$. Dog
bone samples were extracted from galvanized sheets using laser cutting,
with a gauge length of 60~mm  (see the sample geometry in
Figure~\ref{fig5}a). The strain field on the coating surface was
measured using the Digital Image Correlation (DIC) method. Before each
test, the surface of the Zn--Al--Mg coating is first painted with a
white background, and then a random speckle pattern is applied using
black spray paint, enabling accurate measurement of the strain field.
The system uses one camera with a resolution of 3048--3048 pixels, a
focal length of 50~mm and an aperture of f/22. The stand-off distance
is approximately 200~mm. The image scale is 2~$\upmu$m/pixel. The
acquisition rate is two images per second. The correlation analysis in
the software uses a subset size of 27 pixels with a step size of 7
pixels. The calculated engineering strain fields are treated with a
Gaussian spatial filter before analysing the results.

\subsection{Microstructure of the coatings}\label{sec2.2}
Back-scattered electron (BSE) observations of the microstructures of
the three specimens are presented in  Figure~\ref{fig1}. The coating is
predominantly composed of proeutectic zinc dendrites, accounting for
approximately 70\% of the total area. Two eutectic phases are present
in the remaining area: a coarse brittle binary eutectic Zn/MgZn2 (BE)
and a thin ternary eutectic Zn/Al/MgZn2 (TE). The size and morphology
of these dendrites are  {dependent} by the underlying substrate. On the
C\_Mild specimen, the coating dendrites are two to three times larger
than those on the C\_HSLA and C\_Dual-Phase, with an average size of
83~$\upmu$m. Moreover, C\_Mild dendrites exhibit distinct primary and
secondary arms, while SEM images of C\_HSLA and C\_Dual-Phase reveal
globular-shaped dendrites averaging  45~$\upmu$m and 30~$\upmu$m in
size, respectively. Particularly, the coating deposited on dual-phase
steel exhibits very small grains with diameters smaller than 
10~$\upmu$m, as depicted in  Figure~\ref{fig1}f.

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig1}BSE-SEM micrographs at two different scales
showing the different phases observed on the surface of (a,d)
C\_Mild, (b,e) C\_HSLA, and (c,f) C\_Dual-Phase.}
\end{figure}

In this part of the study, a qualitative description of the SEM
observations is provided to highlight the relationship between the
coating morphology and the nature of the substrate, rather than to
suggest a direct influence of substrate grain size on dendrite shape.

\subsection{Crystallographic texture}\label{sec2.3}
In addition to the dendrite morphology, the coating texture is
influenced by the underlying substrate. The harmonic textures by pole 
figure (PF) are illustrated in  Figure~\ref{fig2}. Texture analysis is
based only on the zinc phase (dendrite ${+}$ zinc eutectic) which gives
good indexation, except for the eutectic phase in some cases. This is
due to the fact that eutectic phases are very thin (especially in the
ternary eutectic). In the case of C\_Mild, the coating surface displays
both basal (0001) and prismatic  $(1\underline{1}00)$ or (1010)
orientations  (Figure~\ref{fig2}a). On the other hand, the coating
deposited on C\_HSLA and C\_Dual-Phase steel predominantly shows a high
basal texture (0001) as demonstrated by the Figures~\ref{fig2}b 
and~\ref{fig2}c. 

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig2}Direct pole figures in the RD-TD plane (left) and
ND IPF maps (right) of the coatings on (a) C\_Mild, (b) C\_HSLA and
(c) C\_Dual-Phase substrates.}
\end{figure}

To understand the relationship between grain morphology and the
underlying substrate, a comprehensive investigation was conducted using
scanning transmission electron microscopy (STEM) analysis with a Jeol
TEM instrument. The analysis focused on the interface between a
\mbox{Zn--Al--Mg} coating and Mild and Dual-Phase steel. During the
hot-dip galvanizing (HDG) process, a well-defined Al-rich inhibition
layer, denoted as FeAl$_3$Zn$_x$ or Fe$_2$Al$_5$Zn$_x$, was observed to
form at the interface between the steel substrate and the 
coating~\cite{18,19,20}. This layer exhibits a uniform thickness of
approximately 50~nm and plays a crucial role in preventing the
formation of brittle FeZn compounds while significantly enhancing the
overall adhesion of the coating.

Figure~\ref{fig3} presents the results of TEM analysis conducted on the
coating/substrate interface of the C\_Mild sample. In order to prepare
a lamella for the analysis, a focused ion beam (FIB) method was
employed, as depicted in  Figure~\ref{fig3}b. Moreover, 
Figure~\ref{fig3}c showcases the STEM energy-dispersive X-ray
spectroscopy (EDS) results obtained from a designated region
(highlighted in  Figure~\ref{fig3}a) encompassing the secondary arm
dendrite, binary eutectic phase, and ternary eutectic phase. The
chemical composition analysis of the coating (Zn, Al, and Mg) presented
in  Figure~\ref{fig3}c reveals that the binary eutectic phase, observed
on the surface, extends all the way below the secondary zinc dendrite.
Notably, an additional continuous layer consisting of aluminum has been
observed at the interface of the coating and substrate, positioned
beneath the binary eutectic and secondary arm dendrite phases. The
chemical composition of the intermetallic layer rich in Al has been
quantified utilizing the EDS technique by analyzing two regions. From
the EDS analysis results, these regions rich in aluminum and iron were
confirmed to be a FeAl$_{3}$Zn$_{x}$ intermetallic layer. A thickness
between 60 and 200~nm has been measured for this layer.

\begin{figure}
\includegraphics{fig03}
\caption{\label{fig3}Intermetallic layer TEM analysis for C\_Mild
specimen: (a)~deposited platinum on the location of the extracted STEM
lamella including secondary zinc arm dendrite, binary eutectic phase
and ternary eutectic phase, (b)~the FIB extracted lamella, (c)~chemical
composition of the Zn--Al--Mg coating and the Mild substrate, (d)~the
chemical composition of the intermetallic at the interface.}
\end{figure}

The TEM analysis was extended to the second specimen C\_Dual-Phase,
which exhibited a distinct coating microstructure characterized by
globular dendrites resembling that of the C\_HSLA specimen 
(Figure~\ref{fig1}b,c,e,f). This analysis revealed the presence of an
intermetallic layer similar to the first specimen, with a binary
eutectic phase extending below the zinc dendrite, as depicted in 
Figure~\ref{fig4}c. Additionally, at the coating/substrate interface, a
layer rich in manganese and silicon was observed. EDS local analysis,
focusing on areas enriched in Mn and Si, as presented in 
Figure~\ref{fig4}d, demonstrated that this layer primarily consists of
MnO and Mn$_2$SiO$_4$ oxides. This external oxide layer is commonly
known as selective oxidation and is frequently observed in steels
containing significant amounts of Mn and  Si~\cite{21}. Some of these
oxides are observed within the superficial layer of the substrate at
the grain boundaries. Notably, the local analysis specifically
targeting this oxide layer revealed that the formation of the
intermetallic layer FeAl3 was impeded in the presence of MnO and
Mn$_2$SiO$_4$ oxides.

\begin{figure}
\includegraphics{fig04}
\caption{\label{fig4}Intermetallic layer TEM analysis for C\_Dual-Phase
specimen, (a)~Deposited platinum on the location of the extracted STEM
lamella including zinc dendrite and binary eutectic phase, (b)~the FIB
extracted lamella, (c)~chemical composition of the Zn--Al--Mg coating
and the Dual-Phase substrate, (d)~the chemical composition of the
interface.}
\end{figure}

The presence of the oxide layer, composed primarily of MnO and
Mn$_2$SiO$_4$ oxides, at the interface between the steel substrate and
the Zn--Al--Mg coating can influence the microstructure of the coating.
This oxide layer may have an impact on grain morphology and size within
the coating. The interaction between the oxides and the coating
material could lead to alterations in the nucleation and growth of
grains, potentially resulting in changes to the overall microstructural
characteristics. Further studies are currently underway to explore in
greater detail the impact of these oxides on the germination and growth
of zinc grains.

\section{Deformation modes of the coatings}\label{sec3}
In addition to the observed differences in coating microstructure and
crystallography, as discussed in  Sections~\ref{sec2.1}
and~\ref{sec2.2}, it is important to recognize that the mechanical
behavior of the underlying substrate can profoundly impact the
deformation characteristics of the Zn--Al--Mg coating. In this 
section, the deformation behavior is analysed in function of the
coating microstructure and crystallography, the steel behavior and the
loading mode.

\subsection{Macroscopic uniaxial tensile tests}\label{sec3.1}
Uniaxial tensile tests were performed on C\_Mild, C\_HSLA, and
C\_Dual-Phase coated steels at various strain levels to examine the
local plasticity behavior of the Zn--Al--Mg coating.  Figure~\ref{fig5}
illustrates the macroscopic behavior observed in these tests. Notably,
as shown in  Figure~\ref{fig5}, the plastic deformation responses of
the studied samples exhibit significant differences. It is important to
note that the relatively thicker substrates have a more prominent
influence on the observed global mechanical properties, predominantly
reflecting the plastic deformation tendencies of the substrates rather
than the coatings themselves. In the case of the coatings on Mild and
HSLA substrates, a discontinuous yielding behavior characterized by
L\"{u}ders peak and plateau stresses is observed, occurring after the
initial elastic regime. The length of the L\"{u}ders plateau is found
to be 8\% and 3.5\% for the C\_Mild and C\_HSLA steels, respectively.
Conversely, the samples with the Dual-Phase substrate demonstrate
continuous yielding behavior without the presence of L\"{u}ders bands.

\begin{figure}
\includegraphics{fig05}
\caption{\label{fig5}(a)~Sample geometry, (b)~engineering stress--strain
responses during uniaxial tensile tests of C\_Mild, C\_HSLA, and
C\_Dual-Phase steels.}
\end{figure}

DIC measurements conducted on the surfaces of C\_Mild and C\_HSLA
specimens, as depicted in  Figures~\ref{fig6}a and~\ref{fig6}b
respectively, reveal the presence of strain localization at the initial
stages of loading. At a macroscopic strain of 1.4\%, two
non-homogeneous bands with localized deformation spanning between 6\%
and 8\% appear at the edge of the gauge area in the C\_Mild specimen.
These bands gradually encompass half of the specimen after 4\% of
macroscopic strain and eventually propagate across the entire useful
cross-sectional area by 8\% macroscopic strain. In the case of the
C\_HSLA steel, a L\"{u}ders band with a magnitude of 3.5\% is measured
at a macroscopic strain of 1.4\%, initially occupying almost half of
the gauge length. As the macroscopic strain reaches 3.5\%, this band
expands to cover the entire gauge length. Before the band propagates to
the entire specimen, two distinct areas can be identified within the
gauge  section: a plastically Deformed Area (DA) and a plastically
Non-Deformed Area (NDE), as depicted in  
\mbox{Figures~\ref{fig6}a} 
and~\ref{fig6}b at a macroscopic strain of~0.014. In contrast, for
the last specimen, C\_Dual-Phase, which does not exhibit L\"{u}ders
effects on the tensile curve, no strain localization is observed at the
gauge length of the uniaxial tensile specimen, indicating a homogeneous
mechanical behavior  (Figure~\ref{fig6}c). In the remainder of the
article, all SEM analyses presented in the following  sections were
conducted in the DAs.

\begin{figure}
\includegraphics{fig06}
\caption{\label{fig6}Local  horizontal engineering strain
distributions along the gauge length of the specimens, obtained using
digital image correlation (DIC) during uniaxial tensile tests:
(a)~position of L\"{u}ders bands at different levels of macroscopic
applied strain for the C\_Mild steel specimen, (b)~position of
L\"{u}ders bands at different levels of macroscopic applied strain for
the C\_HSLA steel specimen, and (c)~strain field distribution at
different levels of macroscopic applied strain for the C\_Dual-Phase
steel specimen.}
\end{figure}

L\"{u}ders band formation is a material instability which can occur in
principle at any location of the steel sample. Stress concentration
points, such as the width transition zone of the sample, may act as
perturbations of the field and trigger the band initiation. This is
indeed the case in the experiments of Figure~\ref{fig6}. However, this
is not always the case in L\"{u}ders experiments.

\subsection{Deformation modes}\label{sec3.2}
Before studying the effect of L\"{u}ders banding on the deformation
behavior of the zinc coating, the principal deformation modes of
individual grains are investigated.  Figure~\ref{fig7} shows a SEM
micrograph and an IPF map of the same zone for the C\_Dual-Phase
specimen deformed at 0.07 strain level. The plastic deformation within
the zinc grains can be observed through several means. Firstly, slip
traces of different slip families within the grains can be visualized
in SEM images, as shown in  Figure~\ref{fig7}a. Additionally,
orientation gradients on the IPF (Inverse Pole Figure) map, as depicted
in  Figure~\ref{fig7}b, provide further evidence of plastic deformation
occurring within the zinc grains. Slip trace analysis reveals that
basal slip (0001) and 2nd order pyramidal slip, indicated by red and
blue lines on  Figure~\ref{fig7}a, respectively, are the dominant modes
of slip deformation within the zinc grains  (Figure~\ref{fig7}c).  

\begin{figure}
\vspace*{-2pt}
\includegraphics{fig07}
\vspace*{-2pt}
\caption{\label{fig7}(a)~SEM micrograph of C\_Dual\_phase specimen
after tensile loading at a strain value of 0.07; (b)~the corresponding
IPF map; (c)~possible slip trace directions in selected grains.}
\end{figure}

Furthermore, IPF maps of the uniaxial tests presented in 
Figure~\ref{fig8} demonstrate that for high enough applied strain,
twinning emerges as the primary deformation mode within some zinc
grains. This twinning mechanism plays a crucial role in redistributing
stress and facilitating the plastic deformation of the grains under
such conditions.

\begin{figure}
\vspace*{-2pt}
\includegraphics{fig08}
\vspace*{-2pt}
\caption{\label{fig8}ND IPF\_Z and GND maps of (a,b) C\_Mild, 
(c,d) C\_HSLA and (e,f) C\_Dual-Phase steel specimens after 0.014 of
major strain during uniaxial tensile tests.}
\end{figure}

To investigate the influence of L\"{u}ders bands on plasticity within
zinc grains, EBSD analyses were conducted on all specimens after
applying an uniaxial macroscopic strain of 0.014. For each specimen, an
area containing approximately 60 grains in the DA zone was analyzed. 
Figure~\ref{fig8} presents the IPF maps and the GND (Geometrically
Necessary Dislocation) of the deformed area. The GND maps were
calculated using the AZtecCrystal  software$^{\mbox{\textregistered}}$ from
Oxford, employing the ``Weighted Burgers Vector'' (WBV) technique.
Dislocation density was calculated utilizing the basal slip system
(0001) and the pyramidal slip system $(1\underline{1}01)$ and
$(1\underline{2}1\underline{2})$. In addition, the quantification of
the twinned area fraction within the selected grains was also performed
using the same AZtecCrystal software, based on the automatic
identification of twin orientation relationships.

In the deformed area, in  Figure~\ref{fig6}, the C\_Mild specimen
exhibits a significant amount of plastic strain, as evidenced by
numerous twinning bands within the zinc grains  (Figure~\ref{fig8}a).
Some of these twins have already invaded almost the entire grain,
indicating extensive plastic deformation. Note that the twin shear
value in zinc is 0.139. Moreover, several twinning systems can coexist
in the same grain. The newly oriented twinned grains can experience
further plastic deformation through slip activity, as evidenced by a
visible orientation gradient inside the twinned area shown in 
Figure~\ref{fig8}a. Additionally, twinning is highly prevalent in
grains oriented with their $c$-axis perpendicular to the sheet plane.
Twinned grains represent approximately 50\% of the observed area,
illustrating the significance of this deformation mechanism. 
Figure~\ref{fig8}b presents the dislocation density of the deformed
area of the C\_Mild specimen. Interestingly, the dislocation density of
grains with $c$-axis lying on the sheet surface is lower than that of
grains with other orientations. This is demonstrated in G1, G2, G3, and
G4 grains with $(1\underline{1}00)$ orientation. While not shown here,
no extensive plasticity was found in the Non-Deformed Area, confirming
that significant plasticity development only occurs due to the passage
of the L\"{u}ders bands.

In the case of the C\_HSLA specimen, similar deformation modes to those
seen in the C\_Mild specimen were observed. Twinning emerged as the
principal deformation mode, with approximately 65\% of twinned grains
in the deformed area after a macroscopic strain of 0.014, corresponding
to a local strain of 0.035 within the DA. At this strain level,
multiple twins formed within the zinc grains and began thickening. The
high percentage of twinning can be attributed to the crystallographic
orientation difference between the C\_Mild and C\_HSLA steel specimens.
As mentioned in  Section~\ref{sec2.2}, basal orientation is prevalent
in C\_HSLA steel, leading to an increased occurrence of twinning
compared to C\_Mild specimens, where twinning primarily occurred in
grains with $c$-axis parallel to the Normal Direction of the steel
surface (ND). However, the twin area fraction inside the grains in
C\_HSLA remains significantly lower than that in C\_Mild specimens,
despite its grain size being twice smaller. This difference can be
attributed to the local strain being twice higher in C\_Mild specimens
(8\% against 3.5\%). The GND map in  Figure~\ref{fig8}d illustrates a
high dislocation density in grains where twinning is prevalent.
Conversely, other grains do not exhibit significant plasticity,
irrespective of their crystallographic orientation. 

As for the final C\_Dual-Phase specimen depicted in 
Figure~\ref{fig8}e, out of the same number of investigated grains as
the previous two specimens, only 6\% began to twin, and the fraction of
twin area was very low despite the favourable grain orientation to
twinning. Moreover, the GND map presented in  Figure~\ref{fig8}f
demonstrates only a minor amount of plasticity inside the zinc grains.
This is due to the low applied macroscopic strain, 0.014, distributed
uniformly, in the absence of L\"{u}ders banding.

\subsection{Cracking modes}\label{sec3.3}
Cracking modes of the coatings have been observed for all materials at
several strain levels for uniaxial tests.  Figure~\ref{fig9} shows
cracking modes of C\_Mild, C\_HSLA and C\_Dual-Phase steel during
uniaxial tests at a macroscopic strain of 0.014, corresponding to the
areas shown in  Figure~\ref{fig8}.

\begin{figure}
\includegraphics{fig09}
\caption{\label{fig9}BSE-SEM images showing selected coating surface
areas presented in Figure~\ref{fig8} of (a,b) C\_Mild, 
(c,d) C\_HSLA, and (e,f) C\_Dual-Phase specimens deformed up to
0.014 of global strain.}
\end{figure}

In Figure~\ref{fig9}a, a significant number of cracks can be observed
traversing the zinc grains. To be specific, the grains designated as
G1, G2, G3, and G4, which are highlighted in both  
Figures~\ref{fig8}a
and~\ref{fig9}a, serve as typical examples that nearly all
grains with their $c$-axis oriented parallel to the sheet plane display
multiple instances of cleavage cracking. Some cleavage cracks are also
observed on the C\_HSLA specimen in grains with similar orientation (G5
and G6), as shown in
Figures~\ref{fig8}c and~\ref{fig9}c. Microcracks
in the brittle binary eutectic phase are also observed in both C\_Mild
and C\_HSLA specimens. Conversely, no cleavage cracking is evident in
the C\_Dual-Phase specimen, despite the presence of grains with $c$-axis
alignment parallel to the sheet plane. This discrepancy can be
attributed, firstly, to the textural differences in the coating of
C\_HSLA and C\_Dual-Phase steel, and secondly, to the lower strain
levels in comparison to the C\_Mild specimen. Notably, for the
C\_Dual-Phase specimen, the applied macroscopic strain remains low and
insufficient to induce any surface coating damage.

\section{Modeling and simulation}\label{sec4}
In this section, the three-dimensional mechanical behavior of zinc
coatings deposited on steel substrates is simulated by means of the
crystal plasticity finite element method. This 
\mbox{computational} endeavor
is aimed at gaining deeper insights into the complex interplay between
the coating and the substrate under various mechanical loading
scenarios. The simulation adopts an advanced crystal plasticity model
for the coating and an isotropic von Mises plasticity model for the
substrate by modeling the L\"{u}ders plateau and its effect on the
coating behavior. The computational approach initiated  in~\cite{15,17}
is extended here to the large deformation crystal plasticity framework
and applied to the three considered combinations of steel substrates
and coating systems.

\subsection{Simulation method}\label{sec4.1}
The approach referred to as the ``multicrystalline approach''
intricately considers the actual geometry and orientation of zinc
grains in the coating. Given the inherently complex multiphase
microstructure of the Zn--Al--Mg coating, it becomes imperative to
consider the various morphologies when generating the finite element
(FE) mesh. Additionally, the orientations of these grains play a
pivotal role in determining the mechanical response of the coatings, as
shown in  Section~\ref{sec3}, and must be incorporated into the
material model. This approach, initiated in  reference~\cite{22}, makes
use of the real coating microstructure obtained from EBSD analysis to
create a realistic mesh that encompasses both the microstructure and
the orientations of grains. 

The data obtained from the EBSD analysis is initially processed using
the MTEX  toolbox~\cite{23}, specifically designed for Matlab.
Subsequently, the MTEX2Gmsh tools, developed  by~\cite{24}, are
employed to generate 2D meshes from the EBSD data, utilizing the
features of the MTEX toolbox. The generated geometry file is further
refined using Gmsh  software~\cite{25} allowing control over mesh
density. An example of final mesh is given in  Figure~\ref{fig10}. In
this study, to simplify the calculations, we opted to include only the
zinc phase in the model, thus including the eutectic phases into the
neighboring grains according to the epitaxy orientation relationships. 

\begin{figure}
\vspace*{-2pt}
\includegraphics{fig10}
\vspace*{-2pt}
\caption{\label{fig10}Example of a finite element mesh of a tensile
specimen, (a)~free coating surface and (b)~coating/substrate
cross-section.}
\vspace*{-2pt}
\end{figure}

To extend this two-dimensional mesh into a three-dimensional
representation, a straightforward translation along the Normal
Direction (ND) is applied, since grain boundaries are perpendicular to
the sheet plane, with a good approximation. The coating is
monocrystalline throughout the thickness, as illustrated in 
Figures~\ref{fig2}b and~\ref{fig3}b. Each grain in the coating
has a specific label (ID number) which defines a set of elements which
a triplet of Euler angles is attributed  (Figure~\ref{fig10}a). 

The substrate is meshed by translating the same two-dimensional coating
mesh but with varying thickness. In the thickness direction, the
coating is segmented into six layers of regular elements while a
progressive meshing is used for the substrate, with elements identical
to the coating near the interface and coarser far from the interface 
(Figure~\ref{fig10}b). The final step is the conversion of the mesh to
a geometry file that is recognized by the finite element software
Z-set. 

In this simulation, we employed finite elements with quadratic
interpolation functions and reduced integration points to achieve
accurate results. Table~\ref{tab3} provides details regarding the number of
brick elements, integration points, and nodes for each specimen,
ensuring that mesh was properly refined for numerical precision of the
computed fields. Note that the fine meshes used in this work lead to a
large number of unknown degrees of freedom (dof), close to 3~million
dofs. The implicit resolution of the problem requires a large amount of
memory and was distributed over 10 processors. Long simulation times
(typically one week) were necessary to simulate the tensile tests. The
strain localization phenomenon associated with L\"{u}ders band
initiation and propagation leads to fine time steps in the simulation.
The Newton implicit method is used for both the resolution of global
static equilibrium and the resolution of the differential system of
constitutive equations at each Gauss point.

\begin{table}
\caption{\label{tab3}Mesh details for simulated specimens}
\begin{tabular}{ccc}
\thead
 & Nodes & Elements \\ 
\endthead
C\_Mild steel  & 987~103 & 380~160 \\ 
C\_HSLA steel & 779~501 & 298~064 \\ 
C\_Dual-phase steel & 737~367 & 283~030
\botline 
\end{tabular}
\end{table}

Continuity of the displacement vector (no crack nor slip) and of the
traction vector are assumed at the interface.  

The following boundary conditions have been applied to simulate tension
in rolling direction~1 (RD) and to allow for the formation of
L\"{u}ders banding  (Figure~\ref{fig10}a). A vanishing displacement
$U_1=0$ is applied to all nodes of the surface  $x_1=0$ (substrate and
coating, left side in  Figure~\ref{fig10}). The displacement component
$U_1$ is prescribed to the nodes of the surface  $x_1=L$, $L$ 
being the length of the meshed plate (right side in 
Figure~\ref{fig10}a), with a constant displacement rate corresponding
to the loading rate of the experiment. All the lateral surfaces normal
to TD and ND (including the coating free surface and the bottom steel
surface) are left free of forces. In that way, the L\"{u}ders band can
propagate without boundary kinematical constraints, except at the
specimen's ends which remain flat due to the prescription of the $U_1$
displacement component. Appropriate additional displacements are
applied to fix the rigid body motion (not detailed here).

The microstructure of the coating, which was studied experimentally
through uniaxial tests, is now simulated in this  section, with the
results detailing the local deformation behavior.  Figure~\ref{fig11}
presents the EBSD maps of each specimen, with grain numbers assigned
after preparation using the MTEX toolbox. Notably, the C\_Dual-Phase
specimen exhibits a significantly higher grain density compared to the
C\_HSLA specimen. This difference arises from the presence of very
small grains, as depicted in  Figure~\ref{fig1}c. To ensure
consistency, a larger surface area was considered for~the C\_Mild
specimen to achieve a similar number of grains compared to the C\_HSLA
specimen.

\begin{figure}
\includegraphics{fig11}
\caption{\label{fig11}EBSD maps with the associate finite elements mesh
for the (a,b) C\_Mild steel specimen, (c,d) C\_HSLA steel
specimen and (e,f) C\_Dual-Phase steel specimen: grain numbers and
orientations on the left and finite element mesh of the coating 
surface.}
\end{figure}

\subsection{Steel sheet behavior}\label{sec4.2}
The mechanical behavior of the steel sheets used in this study has been
identified from the tensile tests performed in the rolling direction
(RD), as presented in  Section~\ref{sec3}. The plasticity law of the
steel sheets is based on a Mises yield criterion, as expressed 
in Equation~(\ref{eq1}). An isotropic nonlinear hardening model has been adopted
to describe the evolution of the yield stress as a function of the
cumulative plastic strain, denoted as $p$. In this work, the materials
are considered isotropic, and the constitutive model does not account
for potential anisotropy that could result from the manufacturing
process, particularly from the rolling operations. However, the DIC
measurements provide information on the Lankford coefficient for the
lateral strain during tension along RD, which was found compatible with
isotropy.

For the Dual-Phase steel substrate the hardening law is given 
by Equation~(\ref{eq2}) while the hardening law for Mild and HSLA steel
substrates is given by Equation~(\ref{eq3}). In the hardening laws, $R_{0}$
represents the initial yield stress, while $Q_{1}$, $Q_{2}$, $Q_{3}$,
$b_{1}$, $b_{2}$, and $b_{3}$ are material parameters characterizing
the exponential functions describing the hardening behavior. The
equation incorporates a classic Voce strain hardening behavior in its
initial  part~\cite{26,27}. To account for the L\"{u}ders effect and
associated strain localization observed during the experimental tests,
\mbox{attributed} to 
dislocation anchoring and release, we introduce an
additional softening term with a negative 
\mbox{coefficient~$Q_{2}$.} The
third additional hardening $Q_{3}$ term is employed to smooth out the
initial peak in the local material behavior and enhance convergence in
finite element calculations.  Figure~\ref{fig12} gives the comparison
between the experiments and the simulation of steel sheets in rolling
direction. 
{\begin{eqnarray}
& \displaystyle
f = (\tfrac{1}{2}[(\sigma_{11}-\sigma_{22})^{2}+
(\sigma_{2}-\sigma_{33})^{2}+
(\sigma_{33}-\sigma_{11})^{2}+
6(\sigma_{23}^{2}+
\sigma_{31}^{2}+
\sigma_{12}^{2})])^{0.5}-R 
\label{eq1}\Seqnsplit 
& \displaystyle
R = R_{0}+Q_{1}(1-\exp(-b_{1}p))+
Q_{2}(1-\exp(-b_{2}p))
\label{eq2}\Seqnsplit
& \displaystyle
R = R_{0}+Q_{1}(1-\exp(-b_{1}p))+
Q_{2}(1-\exp(-b_{2}p))+
Q_{3}(1-\exp(-b_{3}p))
\label{eq3}
\end{eqnarray}}\unskip

\begin{figure}
\includegraphics{fig12}
\caption{\label{fig12}Global mechanical responses of experiment and
coating/substrate FE simulation for C\_Mild, C\_HSLA and C\_Dual-Phase
steel specimens during uniaxial tensile tests.}
\end{figure}

The hardening parameters identified for the Mild steel substrate are
reported below. These values were obtained from the tensile tests in
the rolling direction and fitted  using Equation~(\ref{eq3}). The
identified parameters are: $R_{0}=310$~MPa,  $Q_{1}=140$~MPa, 
$b_{1}=2.6$, $Q_{2}=-65$~MPa,  $b_{2}=110$ and  $Q_{3}=150$~MPa  with
$b_{3}=1.5$.

\subsection{Coating behavior}\label{sec4.3}
To simulate the mechanical behavior of the zinc coating, a crystal
plasticity finite deformation model with the multiplicative
decomposition of the deformation gradient has been employed. The
multiplicative decomposition of the deformation gradient into an
elastic part $F^{e}$ and an inelastic part $F^{p}$ is written 
in Equation~(\ref{eq4}).

We define the velocity gradient tensor  $L$ as the sum of the
contributions of all activated slip  systems Equation~(\ref{eq5}) with 
$\dot{\gamma}^{s}$ being the slip rate for the slip system $s$. It is
recalled that the slip direction and normal to the slip plane are
denoted by $m^{s}$ and $n^{s}$, respectively. The total number of
considered slip systems is $N_{s}$. The criterion for the activation of
slip system $s$ is the Schmid law given in Equation~(\ref{eq6}). The
viscoplastic Norton-type flow law is also given  by Equation~(\ref{eq6}). A
cumulative slip variable is introduced  by Equation~(\ref{eq7}).

The Critical Resolved Shear Stress (CRSS) is given  by Equation~(\ref{eq8})
where $M$ is the Mandel stress tensor defined  by Equation~(\ref{eq9}) with
$\Pi^{e}$ is the second Piola--Kirchhoff stress tensor with respect to
the intermediate configuration. The isotropic strain hardening law is
given  by Equation~(\ref{eq11}). It involves an interaction matrix coupling the
slip systems of various families.
{\begin{eqnarray}
& \displaystyle
\underset{\sim}{F} =
\underset{\sim}{F}^{e}\underset{\sim}{F}^{p} 
\label{eq4} \Seqnsplit 
& \displaystyle
\underset{\sim}{\overline{L}}^{p} =
\underset{\sim}{F}^{p}
\underset{\sim}{F}^{p-1}=
\sum_{s}^{N_s}
\dot{\gamma}^{s}
(m^{s}\otimes n^{s}) 
& \displaystyle
\label{eq5}\Seqnsplit 
& \displaystyle
\dot{\gamma}^{s} =
\mathrm{sign}\;
(\tau^{s})
\left\langle \frac{|\tau^{s}| -\tau_{0}^{s}}{K^{s}}
\right\rangle^{n^s}
\label{eq6}\Seqnsplit 
& \displaystyle
\dot{\gamma}_{\mathrm{cum}}^{s} =
| \dot{\gamma}^{s}| 
\label{eq7}\Seqnsplit 
& \displaystyle
\tau^{s} = \underset{\sim}{M}\colon 
m^{s}\otimes n^{s}
\label{eq8}\Seqnsplit 
& \displaystyle
\underset{\sim}{M} = J_{e}
\underset{\sim}{F}^{eT}\cdot
\underset{\sim}{\sigma}\cdot
\underset{\sim}{F}^{e-T}=
\underset{\sim}{C}^{e}\cdot 
\underset{\sim}{\Pi}^{e}
\label{eq9}\Seqnsplit
& \displaystyle
\underset{\sim}{C}^{e} =
\underset{\sim}{F}^{eT}
\underset{\sim}{F}^{e}
\label{eq10}\Seqnsplit 
& \displaystyle
\tau_{c}^{s} =
\tau_{0c}^{s}+
\sum_{r=1}^{N}
Q^{s}h_{rs}
(1-\exp(-b_{r}\gamma_{\mathrm{cum}}^{r}))
\label{eq11}
\end{eqnarray}}\unskip
The material parameters  $\tau_{0c}^{s}$,  $Q^{s}$, $h_{rs}$, $K^{s}$
and $n^{s}$ were identified  by~\cite{28,15} for the basal, pyramidal
P1 and P2, and prismatic slip-system families of single-crystalline
zinc. Twinning is modeled as a pseudo slip system and does not consider
the lattice orientation change in the twinned phase. The reader is
referred  to~\cite{15} for a detailed description of the interaction
matrix and identification procedure. In particular, the viscosity
parameters  $K^{s}$ and $n^{s}$ account for the rate-dependent behavior
of zinc at room temperature.

In the following results, the plastic shear strain is assumed to occur
solely on the 15 HCP slip systems, 3 basal systems $\{0001\}\langle
11\underline{2}\rangle$,  6 pyramidal  $\uppi_{2}$ systems 
$\{1122\}\langle 11\underline{2}3\rangle$ and 6 twinning system
$\{10\underline{1}2\}\langle10\underline{11}\rangle$ characterized by
the slip directions  $m^{s}$ and the normal to the slip planes 
$n^{s}$. This choice was based on experimental observation of slip
systems shown in  Figure~\ref{fig7} and EBSD analysis presented in 
Figure~\ref{fig8}. 

\subsection{Numerical results and comparison with 
experiments}\label{sec4.4}
To offer a comprehensive insight into the mechanical behavior, we
present the distribution of slip activity in each grain within the
coating and strain component  $F_{11}$ along the tensile direction
within both the coating and the steel substrates.  
Figures~\ref{fig13}, 
\ref{fig14} and~\ref{fig15} provide a visual
representation of the strain distribution at various loading stages and
across cross-sectional layers for the C\_Mild, C\_HSLA and
C\_Dual-Phase steel specimen, respectively.  The component F11-1
coincides with the horizontal engineering strain component plotted in 
Figure~\ref{fig6}. They illustrate the propagation of the L\"{u}ders
band through the substrate and its interaction with the coating.

\begin{figure}
\includegraphics{fig13}
\caption{\label{fig13}Distribution of the strain component
$F_{11}$ along the tensile axis for the C\_Mild steel
specimen at overall strain levels 2\% (left) and 4\% (right): 
(a,b)~at the substrate surface (ND ${=}$ 0), (c,d) at the interface and 
(e,f) at the coating surface (ND ${=}$ 1).}
\end{figure}

\begin{figure}
\vspace*{-2pt}
\includegraphics{fig14}
\vspace*{-2pt}
\caption{\label{fig14}Distribution of the strain component
$F_{11}$ along the tensile axis for the C\_HSLA steel
specimen at overall strain levels 2\% (left) and 4\% (right): 
(a,b) the substrate surface (ND ${=}$ 0), (c,d) at the interface and 
(e,f) at the coating surface (ND ${=}$ 1).}
\vspace*{-2pt}
\end{figure}

\begin{figure}
\includegraphics{fig15}
\caption{\label{fig15}Distribution of the strain component
$F_{11}$ along the tensile axis for the C\_Dual-Phase
steel specimen at overall strain levels 2\% (left) and 4\% (right):
(a,b) the substrate surface (ND ${=}$ 0), (c,d) at the interface
and (e,f) at the coating surface (ND ${=}$ 1).}
\end{figure}

The fields are represented at three different locations in 
Figures~\ref{fig13}--\ref{fig15}. The surface DN ${=}$ 0 represents the
layer of Gauss integration points at the bottom of the substrate in 
Figure~\ref{fig5} and therefore characterizes the bulk steel response.
The name ``interface'' in the captions refers to the first layer of Gauss
integration points just above the interface, on the coating side. The
surface DN ${=}$ 1 denotes the first layer of Gauss integration points
just below the free surface of the coating. Information is taken at the
Gauss points because node values for stress and strains are generally
not continuous at grain boundaries and at the substrate/coating 
interface.   

In the simulations of the L\"{u}ders behavior, the width transition
zones in tensile specimens play a similar role in band triggering as in
the experiment, see the macroscopic simulations  in~\cite{26}. However,
in the present simulations of polycrystalline coatings on a substrate,
the width transition zones were not included in the finite element mesh
because only a small part of the gauge length could be considered. The
bands are observed to start from the left end of the mesh in 
Figure~\ref{fig13}, probably due to stress concentrations induced by
the boundary conditions (imposed axial displacement) in the coating
grains and transmitted to the substrate.

Figures~\ref{fig13}a and~\ref{fig13}b show the local strain 
$F_{11}$ at the bottom steel surface (ND ${=}$ 0) of the Mild steel.
These pictures capture the state of $F_{11}$ at two 2\% and 4\% of
applied macroscopic strain. The corresponding local strains at the
bottom steel surfaces of the C\_HSLA and C\_Dual-Phase steel specimens
are presented in Figures~\ref{fig14}a,b, 
and~\ref{fig15}a,b, respectively.

Upon reaching a macroscopic strain of 2\%, the Dual-Phase steel
exhibits a uniform strain field, matching the magnitude of the applied
macroscopic strain. In contrast, both Mild and HSLA steel show the
initiation of a L\"{u}ders band on the left side, traversing a portion
of the plate and leading to the development of heterogeneous strain.
Furthermore, the local strain in the Mild and HSLA steels is
respectively four times and two times greater than that observed in
Dual-Phase steel at the same macroscopic strain. This local plastic
strain correlates in magnitude to the length of the L\"{u}ders plateau.
For a macroscopic strain of 4\%, the strain field within the Dual-Phase
steel specimen remains uniform on the bottom surface, with the
magnitude aligning precisely with the applied strain, as shown in 
Figure~\ref{fig15}b. In contrast, for the HSLA steel, the L\"{u}ders
band propagated along the entire specimen  (Figure~\ref{fig14}b). In
the case of the Mild steel, shown in  Figure~\ref{fig13}b, it spans
only half of the specimen's length. This difference is due to the
length of the L\"{u}ders plateau, which is twice as long in Mild steel
as in HSLA steel.

The distribution of the strain component $F_{11}=1+\varepsilon_{11}$
along the tensile axis at the coating  surface (ND ${=}$ 1), at various
loading stages, is presented in  
Figures~\ref{fig13}e,f, 
\ref{fig14}e,f and~\ref{fig15}e,f, corresponding to the
C\_Mild, C\_HSLA, and C\_Dual-Phase steel specimens, respectively.
These strain maps are complemented by maps of plastic activity in the
coating in  
Figures~\ref{fig16}--\ref{fig18} where plastic activity is
represented by the sum of cumulative plastic slip amounts for each
system family (basal, pyramidal and twinning). In the case of the
C\_Dual-Phase specimen, as depicted in Figure~\ref{fig15}e, following
the application of macroscopic strains of 2\% and 4\%, the local strain
$F_{11}$ exhibits remarkable uniformity within each grain. This
uniformity is primarily attributed to the strain imposed by the
underlying substrate. It is worth noting that certain grains within
this specimen do not undergo plastic deformation and thus remain in an
elastic state. Within the deformed grains, however, there exists a
significant heterogeneity in plastic strain from one grain to another,
see  
Figures~\ref{fig18}a  and~\ref{fig18}c. This variation is
particularly pronounced at grain boundaries, where peaks of slip
activity are concentrated. This phenomenon is the result of the
plastic-strain incompatibility between neighboring grains and is
further influenced by the strong elastic anisotropy of zinc, especially
during the initial stages of plastic deformation.

\begin{figure}
\includegraphics{fig16}
\caption{\label{fig16}Slip activities at the surface of the C\_Mild
steel Specimen at 4\% of macroscopic strain, (a)~basal slip, 
(b)~pyramidal slip and (c)~twinning.}
\end{figure}

\begin{figure}
\includegraphics{fig17}
\caption{\label{fig17}Slip activities at the surface of the C\_HSLA
steel Specimen at 4\% of macroscopic strain, (a) basal slip, 
(b) pyramidal slip and (c) twinning.}
\end{figure}

\begin{figure}
\includegraphics{fig18}
\caption{\label{fig18}Slip activities at the surface of the
C\_Dual-Phase steel Specimen at 4\% of macroscopic strain, (a)
basal slip, (b) pyramidal slip and (c) twinning.}
\end{figure}

In the case of the C\_Mild and C\_HSLA specimens, applying the same
macroscopic strain leads to substantial strain localization, primarily
attributed to the presence of L\"{u}ders banding. Notably, the average
strain within the C\_Mild and C\_HSLA steel specimens is four times and
two times greater, respectively, than that observed within the grains
of the C\_Dual-Phase steel specimen. Within the deformed grains, the
macroscopic strain  $F_{11}$ exhibits a strain gradient, which can be
attributed to the heterogeneous strain in the underlying substrate. The
local strain within each grain displays heterogeneity, varying from one
grain to another. Nonetheless, the average strain remains closely
aligned with the local strain observed in the deformed region of the
underlying substrate. In  
Figures~\ref{fig13} and~\ref{fig14}, the
progress of the L\"{u}ders band is clearly visible, separating the
samples in two parts, the plastically deformed and undeformed zones.
The strain field is rather smooth inside the plastic zones of the
substrate. This is in contrast to the coating surface where additional
strain heterogeneity is induced by the anisotropic response of the zinc
grains. It is remarkable that enhanced strain levels beyond 15\% are
observed in some grains showing the strong impact of L\"{u}ders banding
on the coating's behavior.

At the interface between the coating and the substrate, it is evident
that local strain undergoes significant influence from the strain
localization occurring at the zinc grain boundaries, which are notably
observed at the surface (ND ${=}$ 1). This influence becomes particularly
pronounced when examining the distinct strain discontinuity closely
linked to the precise positioning of zinc grain boundaries. This
consistent observation is visually depicted in  
Figures~\ref{fig13}d,~\ref{fig14}d and~\ref{fig15}d. 
Upon comparing the local
strain values at the surface and those at the interface, a noticeable
strain gradient emerges, signifying a reduction in strain localization
compared to what is observed at the coating surface. Nevertheless, it
is important to note that even at the interface, the intensity of
strain localization remains significant in the cases of C\_Mild and
C\_HSLA specimens, primarily due to the presence and influence of
L\"{u}ders banding.

The slip activities in zinc grains are presented in  
Figure~\ref{fig16}
for C\_Mild,  Figure~\ref{fig17} for C\_HSLA and  Figure~\ref{fig18}
for C\_Dual-Phase steel specimen. The maps show the cumulative plastic
strain computed separately for each system family (it means that, for
the computation of these variables, the sum is taken over all the slip
systems belonging to one chosen family only). 

For the C\_Mild steel specimen subjected to a macroscopic strain of
4\%, nearly all zinc grains exhibit deformation predominantly through
basal slip as depicted in  Figure~\ref{fig16}a. Furthermore, in
addition to basal sliding, some grains also undergo pyramidal slip and
twinning, with a significantly lower cumulative plastic strain compared
to basal slip within the same grain as presented in 
Figures~\ref{fig16}b and~\ref{fig16}c. The significant
activation of basal slip at the free surface is in accordance with the
experimental observations of  Figure~\ref{fig7}. 

In the case of the C\_HSLA steel specimen, basal slip is significantly
activated, akin to the behavior observed in the C\_Mild steel specimen,
as illustrated in  Figure~\ref{fig17}a. Interestingly, within the
deformed region, certain grains exhibit little to no basal deformation,
such as grains 
\mbox{numbered} 5, 13, 14, 28, and 34. These grains are
primarily characterized by c axes perpendicular to the loading
direction, resulting in almost vanishing Schmid factor. Instead, 
Figure~\ref{fig17}c demonstrates that these grains undergo deformation
through twinning. Additionally, minor plasticity is attributed to
pyramidal slip in this specimen. Similar slip activity is observed in
the \mbox{C\_Dual-Phase} steel specimen, with notable basal slip and twinning,
and minimal activation of pyramidal slip as presented in 
Figure~\ref{fig18}. Similarly, in the case of the C\_HSLA steel
specimen, grains within the dendritic structure that do not deform
through basal slip but through twinning exhibit a $c$-axis orientation
perpendicular to ND. Examples include grains 29, 37, 49, 64, 96, and
147. Such behavior has been experimentally observed during uniaxial
testing within the deformed region as depicted in  Figure~\ref{fig18}.

In summary, the simulation results underscore a pronounced dependence
of deformation modes on the coating texture. The numerical findings
corroborate that the coating primarily undergoes deformation through
basal slip and twinning for grain with $c$-axis paralleled to the ND,
aligning with observations from uniaxial tests. Additionally, a
noticeable degree of pyramidal slip was recorded in both simulations
and experimental observations. Notably, the simulations reveal that the
substrate behavior in the presence of L\"{u}ders bands significantly
influences the acceleration of deformation mechanisms, leading to
localized deformation at the coating-substrate interface at early
stages of macroscopic deformation. 

\section{Conclusion}\label{sec5}
In this study, a comprehensive investigation of the mechanical
interaction between Zn--Al--Mg-coatings and the steel substrate, aiming
to unravel the intricate mechanisms of plastic deformation and
cracking. Through a meticulous blend of experimental tests,
sophisticated simulations, and crystallographic analyses, the research
provides invaluable insights into the complex deformation phenomena and
strain localization mechanisms inherent in Zn--Al--Mg-coated steel
specimens. 

A primary focus of this study is the profound influence of
crystallographic orientations on deformation patterns within both the
coating and substrate. The incorporation of a crystal plasticity model
enables a comprehensive understanding of how grain orientations dictate
slip and twinning activity, leading to strong variations in plastic
strain distribution. Noteworthy is the association of specific
crystallographic orientations, such as (0001) parallel to the ND, with
twinning.

The examination of strain distribution at various loading stages
provides important insights, showcasing uniform strain fields in
Dual-Phase steel and heterogeneous strain in Mild and HSLA steel
specimens, attributed to the presence of L\"{u}ders banding.
Particularly noteworthy is the heightened intensity of strain
localization observed in the C\_Mild and C\_HSLA specimens, extending
even beyond the interface between the coating and substrate.

Additionally, the investigation extends to the simulation of L\"{u}ders
bands, shedding light on their impact on strain localization within the
coating and at the coating-substrate interface. The study emphasizes
the pivotal role played by the length of the L\"{u}ders plateau in
different steel substrates, influencing the extent of strain
localization and shaping the overall deformation behavior of the coated
steel specimens.

Furthermore, the investigation into slip activities within the zinc
grains revealed that basal slip is a dominant mechanism, particularly
at the free surface. Twinning, as an additional deformation mechanism,
was consistently observed in all three specimens, confirming
experimental findings and emphasizing their significance in strain
accommodation.

The knowledge obtained from this study holds significant implications
for the optimization of Zn--Al--Mg coatings in industrial applications.
Understanding the crystallographic aspects, strain localization, and
deformation mechanisms can guide the development of predictive models
to enhance the performance and longevity of these coatings.

Several simplifications in the modelling and simulation approach should
be underlined. A more realistic description of the interaction of the
coating grains with the substrate should include the explicit meshing
of the steel microstructure, including crystal plasticity in the steel
grains. This would allow for a more precise prediction of
coating/substrate interface stresses and associated interface
resistance. Due to the smaller steel grain size, this effect has been
neglected in the present work. Twinning is an important deformation
mode in zinc grains. The pseudo-slip model for twinning used in the
present work does not account for the formation and propagation of
fully formed twins. This limitation has been overcome in the recent 
paper~\cite{29} where a localized twinning model was introduced. This
leads to significantly higher local stress concentration close to the
interface and to strong interaction between plasticity and twinning.
The  work~\cite{29} also contains a detailed comparison of the
experimental and simulated strain and orientation fields, based on
meshing of the real microstructure. An application to the interaction
of L\"{u}ders bands and twin formation and propagation is in progress.
Another missing aspect is the consideration of brittle eutectic phases,
which remains open in the current modelling of zinc coatings. The next
steps {will be} dedicated to a closer identification of damage mode
including  cleavage~\cite{30} and spalling, not observed in the present
materials in spite of high local strain values during the passage of
L\"{u}ders bans. This research contributes to the broader field of
advanced coatings and materials science, offering a foundation for
further studies in this area.

\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.

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