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\DOI{10.5802/crmeca.286}
\datereceived{2024-11-15}
\daterevised{2025-01-15}
\dateaccepted{2025-01-15}
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\dateposted{2025-02-06}
\begin{document}

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\newcommand{\ie}{i.e.}

\newcommand{\Ginc}{\mathcal{G}_{\mathrm{inc}}}
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\begin{noXML}

\CDRsetmeta{articletype}{research-article}

\title{Mechanisms of transverse cracking induced by adjacent ply matrix
cracks in composite laminates}

\alttitle{M\'{e}canismes de fissuration transverse induits par des
fissures matricielles dans des plis adjacents de stratifi\'{e}s
composites}

\author{\firstname{Nicolas} \lastname{Carr\`{e}re}\CDRorcid{0000-0003-3921-6151}}
\address{ENSTA Bretagne, CNRS, IRDL, UMR 6027, F-29806 Brest, France}
\email[N. Carr\`{e}re]{nicolas.carrere@ensta-bretagne.fr}

\author{\firstname{Aur\'{e}lien} \lastname{Doitrand}\IsCorresp\CDRorcid{0000-0003-1000-3507}}
\address{Universit\'e Lyon, INSA-Lyon, UCBL, CNRS, MATEIS, UMR5510, F-69621 Villeurbanne, France}
\email[A. Doitrand]{aurelien.doitrand@insa-lyon.fr}

\author{\firstname{Mathilde} \lastname{Conan}}
\addressSameAs{1}{ENSTA Bretagne, CNRS, IRDL, UMR 6027, F-29806 Brest, France}
\email[M. Conan]{mathilde.conan@ensta-bretagne.org}

\author{\firstname{Eric} \lastname{Martin}}
\address{Bordeaux-INP, Univ. Bordeaux, F-33400 Talence, France}
\email[E. Martin]{eric.martin@enseirb-matmeca.fr}

\begin{abstract} 
Cracking in composite laminates containing \tralicstex{\textit{θ}}{$\theta$}-plies adjacent to
90\textdegree{} plies is studied experimentally and numerically using the
coupled criterion and finite element calculations. Different damage
mechanisms are considered, namely transverse cracking in 90\textdegree{},
cracking in \tralicstex{\textit{θ}}{$\theta$}-plies, or debonding between adjacent misoriented
plies. The influence of the stacking sequence on the damage mechanism
sequence is investigated. Experimental observations of the composite
edge under tensile loading evidence \tralicstex{\textit{θ}}{$\theta$}-ply cracking (i) at an
imposed strain level much larger than first transverse cracking in
90\textdegree{} ply for a sufficiently large orientation mismatch between
adjacent plies or (ii) at a similar imposed strain level if the mismatch
angle between two adjacent plies is small. The latter phenomenon may be
mitigated by the presence of a 0\textdegree{} ply between the 90\textdegree{} and
the \tralicstex{\textit{θ}}{$\theta$}-plies.  

These conclusions are supported by numerical
simulation of the experimentally observed damage mechanisms, evidencing
a change in the damage mechanism sequence depending on the \tralicstex{\textit{θ}}{$\theta$}-ply
misorientation. The numerical simulations also highlight that debonding
between adjacent plies may occur as it becomes more favorable that
adjacent ply crack re-initiation for sufficiently large adjacent ply
mismatch angle.   
\end{abstract}

\begin{altabstract} 
La fissuration dans les stratifi\'{e}s composites, contenant des plis
orient\'{e}s \`{a} \tralicstex{\textit{θ}}{$\theta$}\textdegree{} de la direction de chargement
adjacents \`{a} des plis \`{a} 90\textdegree{}, est \'{e}tudi\'{e}e
exp\'{e}rimentalement et num\'{e}riquement \`{a} l'aide du crit\`{e}re
coupl\'{e} et de calculs par \'{e}l\'{e}ments finis. Diff\'{e}rents
m\'{e}canismes d'endommagement sont pris en compte : la fissuration
transverse dans le pli \`{a} 90\textdegree{}, la fissuration dans les plis
\`{a} \tralicstex{\textit{θ}}{$\theta$}\textdegree{}, ou la d\'{e}coh\'{e}sion entre des plis adjacents
d'orientation diff\'{e}rente. L'influence de la s\'{e}quence
d'empilement sur la s\'{e}quence des m\'{e}canismes d'endommagement est
analys\'{e}e. Les observations exp\'{e}rimentales effectu\'{e}es sur
des stratifi\'{e}s sollicit\'{e}s en traction mettent en \'{e}vidence
une fissuration dans les plis \`{a} \tralicstex{\textit{θ}}{$\theta$} : (i) \`{a} un niveau de
d\'{e}formation impos\'{e} beaucoup plus \'{e}lev\'{e} que celui de la
premi\`{e}re fissuration transverse dans les plis \`{a} 90\textdegree{} pour une
d\'{e}sorientation suffisamment grand entre les plis adjacents, ou (ii)
\`{a} un niveau de d\'{e}formation impos\'{e} similaire si l'angle de
d\'{e}sorientation entre deux plis adjacents est faible. Ce dernier
ph\'{e}nom\`{e}ne peut \^{e}tre att\'{e}nu\'{e} par la pr\'{e}sence d'un
pli \`{a} 0\textdegree{} entre le pli \`{a} 90\textdegree{} et les plis \`{a} 
\tralicstex{\textit{θ}}{$\theta$}.

Ces conclusions sont appuy\'{e}es par des simulations num\'{e}riques
des m\'{e}canismes d'endommagement observ\'{e}s exp\'{e}rimentalement,
mettant en \'{e}vidence un changement dans la s\'{e}quence des
m\'{e}canismes en fonction de l'angle de d\'{e}sorientation des plis
\`{a} \tralicstex{\textit{θ}}{$\theta$}. Les simulations num\'{e}riques montrent \'{e}galement
que la d\'{e}coh\'{e}sion entre plis adjacents peut se produire et
devenir plus favorable que le r\'{e}-amor\c{c}age des fissures dans les
plis adjacents lorsque l'angle de d\'{e}sorientation entre ces plis est
suffisamment grand.  
\end{altabstract}

\shortrunauthors

\keywords{\kwd{Finite fracture mechanics}
\kwd{Coupled criterion}
\kwd{Laminates}
\kwd{Crack re-initiation}}

\altkeywords{\kwd{M\'{e}canique de la rupture incr\'{e}mentale}
\kwd{Crit\`{e}re coupl\'{e}} 
\kwd{Stratifi\'{e}s} 
\kwd{R\'{e}-amor\c{c}age de fissures}}

\maketitle

\end{noXML}



\section{Introduction}\label{sec:introduction}

The growing utilization of composite materials in sectors demanding
both performance and safety is often guided by principles derived from
lessons learned. In specific industries, such as aeronautics, these
principles have been formalized into comprehensive best practice
guidelines outlined in the Military Handbook \cite{MIL-HDBK}. This
document outlines, among other aspects, the draping recommendations
that must be complied with when defining the lay-up of a laminated
composite structure. Among the draping recommendations details, we can
mention for example:
\begin{enumerate}[(1)]
\item Homogeneous stacking sequence are recommended for strength
controlled designs. The handbook reminds that \textit{the
interlaminar stress distributions are affected, around the periphery of
holes, by the variation of  ply orientations relative to a tangent to
the edge. Since it is difficult to optimize for a single lay-up in this
case, the best solution is to make the stacking sequence as homogeneous
as possible}.
\item Since transverse strength is highly dependent on ply thickness,
it is strongly recommended to \textit{minimize the grouping of plies
with identical orientations},
\item The mismatch in Poisson's ratio between adjacent plies, caused
by the anisotropy of the ply properties and the difference in
orientation between them, can generate out-of-plane stresses at the
interface of adjacent plies near the edges of a laminate. It is thus
recommended to perform analyses \textit{to predict free edge stresses
and delamination strain levels}.
\item It is recommanded that the laminate has at least four distinct
ply angles (e.g., 0\degr{}, ${\pm}\theta$\degr{}, 90\degr{}) with a
minimum of 10\% of the plies oriented at each angle.
\end{enumerate}
These recommendations are partly based on experimental observations.
Among numerous examples, and without claiming to be exhaustive, we can
highlight a few studies that illustrate the aforementioned
recommendations. Regarding recommendation \#2, the pioneering work
published in \cite{ParviziGarrett78} demonstrated that thickness has a
significant influence on the apparent strength of the ply. The impact
of blocked versus dispersed plies on strength (recommendation \#1) has
been illustrated, for example, in unnotched specimens
\cite{WISNOM200821} and in open-hole specimens \cite{WISNOM2009335}.
Delamination initiated from the edge and the effect of the stacking
sequence (recommendation \#3) have been studied in
\cite{LAGUNEGRAND20061315}. Finally, regarding the recommendation \#4,
the use of 0\degr{}, ${\pm}45$\degr{}, 90\degr{}  plies is a
well-established practice driven by the need to balance stiffness,
strength, processability, and damage tolerance. These orientations
offer a versatile and practical solution for a wide range of
engineering applications. It is essential to remember that the
recommendations provided in this handbook serve solely as guidelines
that should be verified based on the materials used and further
assessed in relation to new-generation materials and manufacturing
processes. Considering the high cost of testing and the numerous
optimization possibilities \cite{IRISARRI2009983}, using models to
estimate the influence of stacking on the laminate's strength is
crucial. 

{\advance\baselineskip by .2pt

Various models with differing levels of complexity have been proposed
in the literature to achieve this goal. Classical Laminate Theory (CLT)
is a simple and widely used  model for rapidly designing laminated
composite structures \cite{Pister1959,tsai1980}. Based on the elastic
properties of the individual laminae, the stacking sequence, and the
applied loading, this model enables the determination of the
laminate's overall elastic properties and the stress distribution
within each ply, while accounting for the effects of potential ply
failure \cite{Laurin2007}. However, due to its inherent assumptions,
this theory cannot account for the effects of discrete ply failure on
adjacent plies or the potential delamination between two plies, whether
caused by edge effects or induced by transverse failure. To effectively
describe some of these mechanisms, it is essential to capture the
discrete nature of the damage, such as transverse crack and
delamination \cite{Iarve2011}, particularly in terms of how a ply
failure affects the neighboring plies \cite{hallett2009,Journoud2020}.
This approach holds significant potential for accurately describing the
phenomena observed experimentally; however, it relies on sophisticated
finite element models combined with highly non-linear material
behaviors. Consequently, this often results in high computational
costs, limiting the applicability of these methods for parametric
studies or optimization. 

The coupled criterion (CC), introduced by Leguillon in
\cite{leguillon2001CC,Leguillon2002}, provides an efficient approach to
address this challenge while preserving a discrete representation of
the damage. It requires only a few elastic calculations to compute the
stress field in the plane of the potential crack and the energy
released by the crack initiation. It has been applied on composite
materials in order to study several of the aforementioned 
issues~\cite{Martin2016chapter}. The
CC was used to study crack initiation in open hole composite plates
\cite{Camanho2012,Martin2012,Martin2015,Felger2017} or in fastened
joints composites \cite{Catalanotti2013} (recommendation \#1 underlined
at the beginning of this introduction and taken from the Military
Handbook). Regarding composite laminates, the focus was mainly made on
transverse cracking in 90\degr{} plies (\ie~when the loading direction is
perpendicular to the fiber direction) and the effect of the number of
grouped plies on the apparent strength (recommandation \#2). This topic
was studied in 2D
\cite{Leguillon2002,Carrere2016,Kashtalyan2018,Doitrand2022EJMAS} or
3D \cite{Garcia2016,Vereecke2024}, also considering manufacturing
thermal residual stresses \cite{Garcia2018,Garcia2019}. The CC has also
been employed to investigate the onset of free-edge delamination in
angle-ply laminates \cite{martin2010} (recommendation \#3).
Additionally, it has been applied to examine other types of composites.
For instance, the CC has been utilized to study 3D transverse
cracking and debonding in polymer matrix and glass fiber plain weave
composites \cite{Doitrand2017EFM,Doitrand2017CPA}, as well as 2D
damage initiation and propagation in a ceramic matrix woven composite
\cite{Li2018}. Furthermore, the failure mechanisms at the microscopic
scale have been explored, including the debonding of a particle
\cite{Gentieu2018,Martin2020} and the fiber-matrix interface
\cite{Mantic2009,Mantic2012,Garcia2014,MunozReja2016,Girard2024_2D3D}
as well as the influence of the distance and angle between two neighbor
fibers on interface debonding initiation \cite{Girard2024_twofibers}.

A main difficulty in applying the CC in a 3D problem is determining
the shape of the potential crack surface within a 3D geometry. A
first way to implement the CC in 3D consists in assuming a crack path
and applying stress and energy criteria corresponding to increasingly
large cracks following this path. This strategy was adopted to study
transverse cracking in laminates based on rectangular crack shapes
\cite{Garcia2016,Garcia2018,Hamam2022} as well as partially elliptical
crack shapes to simulate crack initiation at a V-notch
\cite{Mittelman2014,Mittelman2015,Yosibash2016}, tunneling crack in
layered ceramics \cite{Papsik2023JECS} or free edge delamination in
angle ply laminates \cite{Burhan2024}. Parameterized crack shapes may
not be the optimal crack shapes, \ie~the ones for which the initiation
loading is the minimum. A tentative to get crack shapes closer to the
optimal ones consists in defining the crack fronts based on the stress
isocontours \cite{LEGUILLON20143D,Doitrand2018IJSS,Doitrand2018IJAA}.
This approach yields crack shapes that are similar to those observed
experimentally in some configurations
\cite{Doitrand2018IJF,Doitrand2019EJM} and enables the possible
initiation crack shapes to be described by a single parameter, \ie~the
crack surface or equivalently, the associated stress isocontour level.
It results in  an efficient way to avoid testing all possible crack
shapes and was implemented for various 3D configurations  
\cite{Sevecek2019,Doitrand2020JMPS,Doitrand2020TAFMEC,Carrere2021}. 

}

It has been shown here that the CC is a pertinent tool to be used
during an optimization process taking into account the recommendation
of the Military Handbook. Current manufacturing methods, such as
automated fiber placement, due to their precision and the intricate
deposition paths they facilitate, should pave the way for increasingly
complex and optimized stacking sequence such as Bouligand laminates
which exhibits lower damage than classical laminates
\cite{korbelin2021}. However, due to the lack of data, the absence of
design rules, and conservatism arising from traditional practices, many
industries continue to rely solely on 0\degr{}, ${\pm}45$\degr{}, and
90\degr{} plies. Indeed, the use of varied ply orientations can lead
to more complex damage scenarios. Literature has shown that the
stacking sequence can significantly affect both the onset and
progression of damage, not only in the initially failed ply but also in
adjacent plies \cite{yokozeki2005,kobayashi1999}.

The objective of this work is to study crack initiation in
$\theta$-plies that are adjacent to a 90\degr{} ply in a composite
laminate. Experiments carried out on laminates having both 90\degr{} 
and $\theta$\degr{}-plies are first presented in
Section~\ref{sec:Experiments}. The aim of this section is to highlight
the effect of different parameters (ply thicknesses, angle differences
between 90\degr{} and $\theta$\degr{}-plies on the threshold for the
initiation of transverse cracks in each ply.  The finite element model
and the CC procedure are described in Section~\ref{sec:Modeling}.
Section~\ref{sec:Results} focuses on evaluating different damage
mechanisms such as crack initiation in (i) 90\degr{} ply and (ii)
$\theta$-ply, as well as (iii) debonding (delamination) between 90\degr{}
ply and $\theta$-ply. A qualitative comparison to the damage mechanisms
observed experimentally is finally provided. 


\section{Experiments}\label{sec:Experiments}

The laminates under investigation were manufactured from Hexcel ply
prepreg with AS4 carbon fibers and 8552 matrix. The panels were
fabricated using a press machine, following the pressure and
temperature cycles recommended by the prepreg manufacturer. One ply
thickness corresponds to $t_{\mathrm{ply}}=150~\upmu\mathrm{m}$. The aim of
this section is to highlight the influence of various parameters on the
strain threshold for transverse crack initiation in the plies. To
achieve this, different layups and stacking sequences were
investigated:
\begin{itemize}
\item $[30_2/90_2/{-}30_2/0]_{\mathrm{s}}$ is the base line (see
Figure~\ref{fig:cracks_P4E1}). It is expected that the 90\degr{}
plies will crack first, and that any transverse cracking in the
30\degr{} plies will occur only at a loading level approaching
failure. 
\item The $[70_2/90_2/{-}70_2/0]_{\mathrm{s}}$ layup is designed
to investigate whether a crack initiated in a misaligned ply can
reinitiate cracking in an adjacent ply. It is important to note that,
in this case, the doubled-thickness ${-}70$\degr{} plies are embedded
within the laminate, while the other doubled-thickness 70\degr{}
plies are located on the outer surface. As shown in \cite{Camanho2006},
the position of the ply within the laminate (whether embedded or on the
outer surface) influences the in-situ strength.
\item The $[70_2/0/90_2/0/{-}70_2]_{\mathrm{s}}$ layup is used
to investigate the confinement effect of a highly oriented ply
(0\degr{} in this case) on crack reinitiation. In this case, the
outer plies are doubled-thickness 70\degr{} plies, while the ply
located in the laminate's symmetry plane is a quadrupled-thickness
${-}70$\degr{} ply.
\item The $[70_4/90_4/{-}70_4/0]_{\mathrm{s}}$ layup is used to
study the effect of ply thickness on crack reinitiation.
\end{itemize}


\begin{figure}
\includegraphics{fig01} 
\caption{Location of the first cracks in the
$[30_2/90_2/{-}30_2/0]_{\mathrm{s}}$ laminate under two levels
of external loading. A representative cross-sectional image has been
included to indicate the ply locations (not visible due to the white
paint layer). The arrows highlight the crack locations in the
90\degr{} plies.\label{fig:cracks_P4E1}}
\end{figure}


The specimen surfaces, where cracks are observed, are polished using
silicon carbide paper with grit sizes ranging from P80 to P4000. A
thin layer of white paint is then applied to these surfaces using a
Posc$^{\text{\textregistered}}$ marker pen (Uni Mitsubishi
Pencil). This thin layer of white paint facilitates crack detection, as
cracks appear black against the white background. During the test, the
gauge length is monitored using a high-resolution camera ($4504 \times 4504$
pixels, with a 50~mm lens). Particular attention has been paid to the
lighting to ensure that the light exposure is as even as possible and
remains constant throughout the test. The aim is to detect the crack
initiation in each ply and its location. The evolution in the number of
cracks during testing falls outside the scope of this study.


The results obtained in the base line are shown in
Figure~\ref{fig:cracks_P4E1}.  The first crack is initiated in the
$90_2$ plies. The number of cracks in these plies increases until final
failure occurs, driven by delamination (between the 90\degr{} plies
and the ${\pm}30$\degr{} plies) and fiber failure in the 0\degr{}
plies.


If the ${\pm} 30$ plies are replaced by ${\pm} 70$ plies,
($[70_2/90_2/{-}70_2/0]_{\mathrm{s}}$) the scenario changes
significantly. The initial failures occur simultaneously in all
off-axis plies, namely the outer 70\degr{} plies, the 90\degr{}
plies, and the ${-}70$\degr{} plies (see Figure~\ref{fig:cracks_P2E2}).
It is important to note that cracks in adjacent plies are
interconnected, suggesting that a crack initiated in one ply may lead
to the initiation of a crack in an adjacent ply. 

\begin{figure}
\includegraphics{fig02} 
\caption{Location of the first cracks in the
$[70_2/90_2/{-}70_2/0]_{\mathrm{s}}$ laminate. A representative
cross-sectional image has been included to indicate the ply locations
(not visible due to the white paint layer). The arrows highlight the
crack locations in the plies.\label{fig:cracks_P2E2}}
\end{figure}


When the 0\degr{} plies are placed between the 90\degr{} plies
and the ${\pm}70$\degr{} plies (as in the
$[70_2/0/90_2/0/{-}70_2]_{\mathrm{s}}$ laminate), transverse
cracks initiate in the ${\pm}70$\degr{} plies
(Figure~\ref{fig:P3_E1}a). The locations of cracks in the outer
doubled-thickness 70\degr{} plies differ significantly from those in
the quadrupled-thickness ${-}70$\degr{} ply. The first cracks in the
90\degr{} plies initiate at a slightly higher applied load (see
Figure~\ref{fig:P3_E1}b). The crack locations in the plies appear
random, with no observable connection between cracks in different
plies.

\begin{figure}
\includegraphics{fig03}
\caption{Location of the cracks in the
$[70_2/0/90_2/0/{-}70_2]_{\mathrm{s}}$ laminate. (a) Initial
cracks in the ${\pm}70$\degr{} plies and (b) first crack in the
90\degr{} ply, occurring at a higher load. A representative
cross-sectional image indicates ply locations (not visible due to the
white paint layer). The arrows highlight crack locations within the
plies.\label{fig:P3_E1}}
\end{figure}


To conclude this experimental section, it is important to note that, on
the one hand, when the angular difference between two adjacent plies is
large, a matrix crack in the more misaligned ply does not initiate a
crack in the adjacent ply, as in the baseline case
$[30_2/90_2/{-}30_2/0]_{\mathrm{s}}$. On the other hand, when
the mismatch angle between two adjacent plies is small, a crack in one
ply can initiate interconnected cracks in the adjacent plies, as
observed in the
$[70_{\mathrm{n}}/90_{\mathrm{n}}/{-}70_{\mathrm{n}}/0]_{\mathrm{s}}$
laminate with $n = 2$ or $4$, a phenomenon not predicted by classical
design approaches. A single ply with a large mismatch angle (in this
case, the 0\degr{} ply between the ${\pm}70$\degr{} and 90\degr{}
plies) could prevent this phenomenon. Another notable result concerns
the effect of ply thickness: in the
$[70_2/0/90_2/0/{-}70_2]_{\mathrm{s}}$ laminate, the first
matrix cracks appear in the outer doubled-thickness 70\degr{} plies
and the quadrupled-thickness ${-}70$\degr{} ply, rather than in the
doubled-thickness 90\degr{} ply, which would be expected to fail
first according to a simple failure criterion. The various observed
mechanisms will be examined in the following sections using the CC.


\section{Fracture modeling and simulation}\label{sec:Modeling}

\subsection{The coupled criterion}\label{ssec:CC}

The CC states that crack initiation occurs if the two following conditions are met:
\begin{itemize}
\item The incremental energy release rate ($\Ginc$), \ie~the ratio
between the elastic strain energy release (${-}\Delta
\Wel(S)=\Wel(0)-\Wel(S)$) due to the initiation of a crack and its
surface ($S$) must be larger than the material critical energy release
rate $\Gc$: $\Ginc \geq  \Gc$
\item The stress over the whole crack path prior to initiation must be
sufficiently large. In homogeneous isotropic materials, it reverts to
comparing the opening stress to the tensile strength. However, in
anisotropic materials such as composite laminates, or for interfaces
subjected to mixed mode loadings, a criterion combining the opening
($\sigmann$) and shear ($\sigmant$) stress components and the
corresponding strengths can be used. For a lamina under tensile
loading, the transverse crack initiation criterion can be described
using a Hashin-based failure criterion:
$f(\underline{\underline{\sigma}}, Y_{\mathrm{t}}, S_{\mathrm{c}}) = \sqrt{
( \sigma_{\mathrm{nn}}/Y_{\mathrm{t}} )^2 + (
\sigma_{\mathrm{nt}}/S_{\mathrm{c}} )^2 } \geq 1$ where
$\sigma_{\mathrm{nn}}$ and $\sigma_{\mathrm{nt}}$ represent the transverse
and shear components of the stress tensor in the ply coordinate system.
The parameters $Y_{\mathrm{t}}$ and $S_{\mathrm{c}}$ denote the transverse
tensile strength and the shear strength, respectively.
\end{itemize}

Under linear elasticity and small deformation assumptions, the stress
components are proportional to the applied strain (denoted
$\varepsilon_0$) and the energy is proportional to the square applied
strain.  These conditions write:
{\begin{eqnarray}
\left\lbrace
\begin{array}{l} 
\displaystyle
\sigmaeq = \sqrt{( \sigma_{\mathrm{nn}} )^2+ \left(\frac{\sigma_{\mathrm{nt}}}{\alpha} \right)^2} = \xi(S) \varepsilon_0 \geq \sigmac
\vspace*{4pt}\\
\displaystyle
\Ginc = A(S) \varepsilon_0^2 \geqslant \Gc
\end{array}
\right.
\label{eq:Stress_Energy}
\end{eqnarray}}\unskip
where $\alpha = S_c/Y_t$, $\sigmac=Y_t$ and $\xi$ and $A$ are functions
depending on the geometry and material properties. Note that in the
following, we study configurations in which the stress field in the ply
is either uniform (Sections~\ref{ssec:90deg} and
\ref{ssec:90deg_and_theta}) or present a gradient
(Section~\ref{ssec:90deg_and_theta_reinitiation}) The strain that must
be prescribed in order to fulfill either the stress or the energy
conditions can be calculated from Equation~(\ref{eq:Stress_Energy}):
{\begin{eqnarray}
\left\lbrace
\begin{array}{l} 
\displaystyle
\varepsilon_0^{\mathrm{stress}}(S) = \frac{\sigmac }{\xi(S)} 
\vspace*{4pt}\\
\displaystyle
\varepsilon_0^{\mathrm{energy}}(S) = \sqrt{\frac{\Gc }{A(S)}}
\end{array}
\right.
\label{eq:Epsilon_Stress_Energy}
\end{eqnarray}}\unskip

For a given crack surface $S$, the strain that must be prescribed in
order to fulfill Equation (\ref{eq:Stress_Energy}) is the maximum
between $\varepsilon_0^{\mathrm{stress}}(S)$ and
$\varepsilon_0^{\mathrm{energy}}(S)$. The applied strain at initiation
$\varepsilon_{\mathrm{c}}$ is thus determined for the crack surface
$S_{\mathrm{c}}$ that minimizes the applied strain: 
{\begin{eqnarray}
\left\lbrace
\begin{array}{l} 
\displaystyle
\varepsilon_{\mathrm{c}} = \min_{S} (\max(\varepsilon_0^{\mathrm{stress}}(S),\varepsilon_0^{\mathrm{energy}}(S)) )\\
\displaystyle
S_{\mathrm{c}} = \mathrm{arg}\,\min_{S} (\max(\varepsilon_0^{\mathrm{stress}}(S),\varepsilon_0^{\mathrm{energy}}(S)) )
\end{array}
\right.
\label{eq:Eps_initiation}
\end{eqnarray}}\unskip



\subsection{Finite element model of laminates}\label{ssec:FEmodel}

The configuration under investigation is
$[0_{\mathrm{n}}/{-}\theta_{\mathrm{n}}/\theta_{\mathrm{n}}/90_{\mathrm{n}}]_{\mathrm{s}}$,
except in Section \ref{ssec:90deg_and_theta_reinitiation} where
$[{-}\theta_{\mathrm{n}}/\theta_{\mathrm{n}}/0_{\mathrm{n}}/90_{\mathrm{n}}]_{\mathrm{s}}$
stacking is studied to highlight the influence of the presence of a
0\degr{} ply in between 90\degr{} and $\theta$\degr{} plies. Even if
the stacking sequences studied numerically slightly differ from the
tested ones, they allow assessing the different damage mechanisms
observed experimentally. The FE models presented in the following are
set-up using Abaqus{\texttrademark} Standard. Due to the loading and
geometry symmetry, only half the laminate thickness is modeled. The
thickness of $n$ plies is denoted $t$, so that $t=n t_{\mathrm{ply}}$. It
means that when studying $\theta$-ply crack initiation, one crack is
present in the $n$~$\theta$-plies (thickness $t$) in the FE model,
which actually represents the simultaneous cracking of the two sets of
$n$\ $\theta$-plies by symmetry. In addition, when studying 90\degr{}
ply crack initiation, one crack is present in the $n$ 90\degr{}
plies (thickness $t$) in the FE model, which actually represents
cracking in a set of $2n$ 90\degr{} consecutive plies by symmetry
since the 90\degr{} ply lies on the symmetry plane. Crack opening for
all the studied mechanisms (transverse cracking in a 90\degr{} ply, in
$\theta$-ply or debonding) are simulated by doubling the nodes along
the corresponding surface. The ply angle refers to the angle formed
between the loading direction and the fiber orientation within the ply.
Since crack paths are oriented by the microstructure, they are likely
to follow the fiber direction therefore cracks in $\theta$-ply form a
$\theta$ angle with respect to the loading direction. The material
orientation of each ply is assigned and the same ply properties are
defined in the corresponding ply local axis system. The transversely
isotropic ply properties are $E_{\mathrm{L}}=127$~GPa,
$E_{\mathrm{T}}=9.2$~GPa, $\nu_{\mathrm{LT}}=0.302$, $\nu_{\mathrm{TT}}=0.4$,
$G_{\mathrm{LT}}=4.8$~GPa \cite{Marlett2010,Garcia2019}, where the index
$L$ denotes the longitudinal fiber direction and $T$ the direction
perpendicular to the fibers in the ply plane. The~ply transverse
tensile and shear strengths are respectively $Y_t=63.9$~MPa and
$S_c=91$~MPa, the ply critical energy release rate is $248~\mathrm{J}/\mathrm{m}^2$
\cite{Garcia2019}, which corresponds to transverse crack propagation
under opening mode. Boundary conditions are set-up as imposed
displacement on the specimen faces along the tensile loading direction
($z$ in the Figure~\ref{fig::geometry}). Due to the symmetry of the
laminate, only half of the layup is modeled, with symmetric boundary
conditions applied along the plane $(O\vec{x}, \vec{z})$. The meshes
are made of 10-nodes quadratic tetrahedrons, the mesh size being
refined in the vicinity of the crack or debonding locations, resulting
in meshes with around 250,000 nodes. The mesh size is chosen so that a
finer mesh provides a similar initiation\break loading. 


\begin{figure}
\includegraphics{fig04} 
\caption{Geometry and dimensions of (a) the tested specimens and of two
possible mechanisms occurring after transverse crack initiation in a 
90\degr{} ply: (b) cracking in $\theta$-ply or (c) debonding between
90\degr{} and $\theta$-ply. The loading direction is along
$\vec{z}$.\label{fig::geometry}}
\end{figure}


\subsection{Transverse crack initiation}\label{ssec:Transverse}

The main difference between 2D and 3D simulation of transverse crack
initiation in laminates arises from the fact that the stress field is
not perfectly homogeneous in the whole ply but only in the ply center,
the stress increasing closer to the ply free edge. Nevertheless, it was
shown by Garcia et~al.~\cite{Garcia2016} that the 2D or the 3D
applications of the CC for transverse crack initiation leads to similar
initiation strain, especially for plies thinner than the Irwin's length
$\ell_{\mathrm{mat}}=E_{\mathrm{T}}\Gc/\sigmac^2$. The differences slightly
increase with increasing ply thickness, they however remain smaller
than 10\%. Therefore, in this case it is possible to simplify the 3D
application of the CC for transverse crack initiation by neglecting the
stress variation near the 90\degr{} ply free edge and considering a
constant homogeneous stress within the ply. Under this assumption,
Equation (\ref{eq:Eps_initiation}) reduces to:
{\begin{eqnarray}
\varepsilon_{\mathrm{c}} = \max(\varepsilon_0^{\mathrm{stress}}(S_{90}),\varepsilon_0^{\mathrm{energy}}(S_{90})) = 
\max\left(\frac{\sigmac}{\xi_{90}(S_{90})},\sqrt{\frac{\Gc}{A_{90}(S_{90})}}\right) 
\label{eq:Stress_Energy_transversecrack}
\end{eqnarray}}\unskip
where $S_{90}=wt_{90}$ is the surface of the crack crossing the whole
ply. The functions $\xi_{90}(S_{90})$ and $A_{90}(S_{90})$
(corresponding to the functions $\xi$ and $A$ involved in Equation
(\ref{eq:Epsilon_Stress_Energy}) in the case of transverse cracking)
are computed from two FE calculations, one before crack opening that
enables the calculation of the stress condition (and thus
$\xi_{90}(S_{90})$) and the elastic strain energy before initiation,
then crack surface nodes are unbuttoned to calculate the elastic strain
energy release due to crack initiation and to deduce $A_{90}(S_{90})$.
Transverse crack initiation loading is finally calculated from Equation
(\ref{eq:Stress_Energy_transversecrack}).


\subsection{Debonding}\label{ssec:debonding}

A possible damage mechanism occurring after transverse crack initiation
is debonding between the 90\degr{} and $\theta$-ply. This occurs because
of the presence of a transverse crack in the 90\degr{} ply induces a
stress singularity at the interface between both plies. It can be noted
that under in-plane tensile loading, the interface is parallel to the
loading direction thus debonding is not likely to occur without the
presence of a crack in a surrounding ply. For the sake of simplicity,
we assume that symmetric debonding occurs ahead of the transverse crack
and along the whole ply width. This assumption allows us to describe
the inter-ply debonding by only one parameter, namely the debonding
length $2d$, the corresponding debonding surface being calculated as
$S_{\mathrm{d}}=2dw$, where $w$ is the sample width. Thus, the CC for
debonding length calculation can be applied following the reasoning
described in Section~\ref{ssec:CC}. We make the hypothesis that the
same material strength and fracture toughness as for transverse crack
can be used in the case of debonding. This is justified by the fact
that both mechanisms involve matrix cracking and fibre-matrix debonding
at the microscopic scale. The function $\xi_{\mathrm{d}}$ and
$A_{\mathrm{d}}$ (corresponding to the functions $\xi$ and $A$ involved
in Equation (\ref{eq:Epsilon_Stress_Energy}) in the case of inter-ply
debonding) are computed based on FE calculations for several debonding
lengths. Since debonding occurs after transverse crack initiation, two
possible cases may arise: 
\begin{itemize}
\item If both the stress and the energy criteria are fulfilled for the
loading corresponding to 90\degr{} ply transverse crack initiation, thus
debonding is likely to occur instantaneously after transverse crack
initiation.
\item Otherwise, no debonding is observed when the transverse crack
initiates so that the loading must be increased in order to initiate a
debonding ahead of the transverse crack, the imposed debonding
initiation strain can be calculated from Equation
(\ref{eq:Eps_initiation}).
\end{itemize}


\subsection{Re-initiation in $\theta$-ply}\label{ssec:REinitiation}

Another possible damage mechanism occurring after transverse crack
initiation is cracking in the $\theta$-ply (Figure
\ref{fig:cracks_P2E2}). In next section we study different scenarii,
namely crack initiation in $\theta$-ply alone, simultaneous crack
initiation in 90\degr{} and $\theta$-ply, or crack initiation in 90\degr{}
followed by crack re-initiation mechanism in $\theta$ ply. For crack
initiation in $\theta$-ply alone, the exact same approach as for 
90\degr{} ply crack initiation presented in
Section~\ref{ssec:Transverse} is adopted. For simultaneous crack
initiation in 90\degr{} and $\theta$-ply, a similar formulation as in
Section~\ref{ssec:Transverse} is adopted, except that the elastic
strain energy release is calculated for a total crack surface which is
the sum of both 90\degr{} and $\theta$ ply cracks.


The main difference in the CC implementation compared to the two
previous scenarii arises for crack re-initiation in $\theta$-ply. This
is mainly due to the fact that the presence of a transverse crack in
the 90\degr{} ply induces a stress singularity at the crack tip, \ie~at
the interface between 90\degr{} and $\theta$-ply, which results in a
heterogeneous stress field in $\theta$-ply. Therefore, the CC can be
applied to assess crack re-initiation in $\theta$-ply after the
nucleation of a first transverse crack in the 90\degr{} ply. The stress
heterogeneity in the $\theta$-ply prevents us to adopt a simplified
application of the CC and 3D crack initiation has to be studied. The
critical step in the 3D CC application concerns the definition of the
possible initiation crack shapes. We follow the approach proposed in
\cite{Doitrand2018IJSS,Doitrand2018IJAA,Doitrand2018IJF,Doitrand2019EJM,Doitrand2020JMPS} 
to define the possible crack shapes based on the stress criterion
isocontours (Figure~\ref{fig:stress_isocontour_mesh}).  
The main interest of this method is the definition of a unique relation
between the stress level and the crack surface. Besides, for a given
crack size, the stress criterion is strictly fulfilled since the crack
shape is defined from the stress criterion isocontour. The
corresponding crack surface is calculated based on the face area of the
FE mesh elements attached to the crack. For a given configuration
including a $\theta$-ply, the functions $\xi_{\theta}$ and $A_{\theta}$
(corresponding to the functions $\xi$ and $A$ involved in Equation
(\ref{eq:Epsilon_Stress_Energy}) in the case of crack re-initiation in
$\theta$-ply after a first transverse crack initiation in the 90\degr{} 
ply) are computed by successively unbuttoning the crack nodes
corresponding to decreasing stress isovalues or equivalently increasing
crack surfaces. The re-initiation crack surface and imposed strain are
finally determined by solving Equation (\ref{eq:Eps_initiation}). In
the sequel, the numerical results corresponding to the different damage
mechanisms are presented.


\begin{figure}
\includegraphics{fig05} 
\caption{(a) Geometry including both 90\degr{} and $\theta$-ply cracks. (b)
Stress criterion isovalues in $\theta$-ply after 90\degr{} ply crack
initiation and (c) corresponding mesh including the stress criterion
isocontour based possible crack shapes, (d)--(e) crack visualization in
$\theta$-ply.\label{fig:stress_isocontour_mesh}}
\vspace*{-4pt}
\end{figure}


\section{Results}\label{sec:Results}

We recall that the configuration under investigation is
$[0_{\mathrm{n}}/{-}\theta_{\mathrm{n}}/\theta_{\mathrm{n}}
/90_{\mathrm{n}}]_{\mathrm{s}}$,
except in Section~\ref{ssec:90deg_and_theta_reinitiation} where
$[-\theta_{\mathrm{n}}/\theta_{\mathrm{n}}/0_{\mathrm{n}}
/90_{\mathrm{n}}]_{\mathrm{s}}$
stacking is studied to highlight the influence of the presence of a
0\degr{} ply in between 90\degr{} and $\theta$\degr{} plies.




\subsection{Crack initiation in 90\degr{} or~$\theta$\degr{} ply}\label{ssec:90deg}

We first study transverse crack initiation in 90\degr{} or
$\theta$\degr{} ply. The CC enables the calculation of the strain that
must be imposed in order to fulfill the stress criterion, the energy 
criterion, or both (Figure~\ref{fig:ExampleCC_Initiation90deg}a).  
Crack initiation in a thick enough ($t>0.6$~mm) 90\degr{} ply is
controlled by the stress criterion. Indeed, a thick ply stores a
sufficient amount of elastic energy that could be used to form a crack
for imposed strain smaller than the imposed strain required to fulfill
the stress criterion. Therefore, crack initiation occurs as soon as the
stress criterion is fulfilled. This is not the case for a thin ply, for
which not enough energy is available to create the crack when the
stress criterion is fulfilled. Therefore, it results in a larger crack
initiation imposed strain which corresponds to the strain required to
fulfill the energy criterion. 
Figure~\ref{fig:ExampleCC_Initiation90deg}b shows the 90\degr{}
crack initiation imposed strain for several $\theta$-ply
configurations. The larger the disorientation between the 90\degr{} ply
and the $\theta$-ply, the larger the initiation imposed strain, even if
the difference remains smaller than 10\% for the studied
configurations.


\begin{figure}
\includegraphics{fig06}
\caption{(a) Imposed strain to fulfill the stress (circles, dotted
line), the energy (squares, dashed line) or both (solid line) criteria
corresponding to 90\degr{} ply crack initiation as a function of the ply
thickness for $\theta=30$\degr{} ply orientation. (b) Imposed strain at 
90\degr{} ply crack initiation as a function of the ply thickness for
several $\theta$-ply orientations of the adjacent ply. (c) Imposed
strain at $\theta$-ply crack initiation as a function of the ply
thickness for several $\theta$-ply
orientations.\label{fig:ExampleCC_Initiation90deg}}
\end{figure}


First crack initiation in a $\theta$-ply is also studied and the
variation of the initiation strain as a function the ply thickness is
shown in Figure~\ref{fig:ExampleCC_Initiation90deg}c.
Whatever the ply orientation, the transition between energy-driven and
stress-driven crack initiation is observed depending on the ply
thickness. For a given ply thickness, the larger the ply
misorientation, the larger the imposed strain at initiation. It can be
concluded that if the $\theta$-ply is not located at the specimen
surface, 90\degr{} ply crack initiation is likely to be the first
observed damage mechanism. In case the $\theta$-ply is located at the
specimen surface, crack initiation in $\theta$-ply may be more
favorable than crack initiation in inner 90\degr{} depending on the ply
thickness, as shown in Section \ref{sec:Experiments}. 

For plies thicker than 0.6~mm, the stress criterion is fulfilled in
the 90\degr{} ply at the crack initiation imposed strain, however it is
not fulfilled in $\theta$-ply because of the ply disorientation with
respect to the loading direction. However, for plies thinner than
0.6~mm, crack initiation is driven by the energy criterion which
means that the imposed strain may be large enough so that the stress
criterion is fulfilled not only for the 90\degr{} ply but also for the
$\theta$-ply. Thus, another cracking mechanism could be envisioned,
namely simultaneous crack initiation in both 90\degr{} ply and
$\theta$-ply. 


\subsection{Simultaneous crack initiation in 90\degr{} ply and
$\theta$-ply}\label{ssec:90deg_and_theta}

The CC is applied in order to determine if simultaneous 90\degr{} ply and
$\theta$-ply crack initiation may occur instead of 90\degr{} ply crack
initiation only for thin plies.
Figure~\ref{fig:CC_Epsc_vs_t_simultaneous90_and_theta_30_and70} shows
the corresponding imposed strain to fulfill the stress criterion, the
energy criterion or both for configurations with 30\degr{} or 70\degr{}
$\theta$-ply.

\begin{figure}
\includegraphics{fig07}
\caption{Imposed strain to fulfill the stress (circles), the energy
(squares) or both criteria corresponding to either 90\degr{} ply crack
initiation (dashed line) or simultaneous 90\degr{} and $\theta$-ply crack
initiation (solid line) as a function of the ply thickness for (a)
$\theta=30$\degr{} and (b)~$\theta=70$\degr{}.
\label{fig:CC_Epsc_vs_t_simultaneous90_and_theta_30_and70}}
\end{figure}

The imposed strain required for crack initiation in thick plies is
larger for simultaneous crack initiation than for 90\degr{} ply crack
initiation only, thus resulting in a larger initiation imposed strain.
Whatever the ply thickness and orientation, the imposed strain to
fulfill the energy criterion is smaller for 90\degr{} ply crack initiation
only than for simultaneous initiation. Therefore, simultaneous crack
initiation in both plies is not likely to occur, the mechanism of crack
re-initiation in $\theta$-ply after first crack initiation in the 
90\degr{} ply is therefore selected in the sequel.

\subsection{Debonding ahead of a 90\degr{}-ply crack}\label{ssec:debonding90}

Debonding at the interface between the 90\degr{} ply and the $\theta$-ply
is not likely to occur before any intra-ply cracking because the stress
criterion is not fulfilled since the interface between both plies is
parallel to the loading direction. However, the presence of a
transverse crack in the 90\degr{} ply induces a stress singularity at the
ply interface and thus enables a possible debonding between plies. The
CC is applied in order to evaluate if debonding is likely to occur just
after 90\degr{}, as depicted in Figure~\ref{fig::geometry}c. ply crack
initiation or if a larger imposed strain is required. Two
configurations arise leading either to a possible debonding or not.  
If the imposed strain is large enough so that there exists at least a
debonding surface for which both criteria are fulfilled, a possible
debonding may be observed as a consequence of transverse crack
initiation and it occurs at the same imposed strain. Otherwise, if both
criteria are not fulfilled for any debonding surface, debonding ahead
of the transverse crack is not expected without imposed strain
increase. The CC enables the calculation of the imposed strain required
to initiate a debonding, its variation as a function of the ply
thickness is shown in Figure~\ref{fig:Epsc_vs_t_debonding} together
with transverse crack initiation strain for $\theta=50$\degr{} and
$\theta=70$\degr{} ply orientations.

\begin{figure}
\includegraphics{fig08}
\caption{Imposed strain at debonding ahead of a transverse crack for (a)
$\theta=70$\degr{} and $\theta=50$\degr{} ply
orientations.\label{fig:Epsc_vs_t_debonding}}
\end{figure}

For the studied configurations, debonding is not likely to occur just
after transverse crack initiation except for the largest ply
thicknesses $\theta$-ply orientation close enough to 90\degr{}.
Otherwise, the imposed strain must be increased in order to promote
debonding ahead of the transverse crack.

\subsection{$\theta$-ply crack re-initiation}\label{ssec:90deg_and_theta_reinitiation}

We first study the case for which the 90\degr{} ply and the $\theta$-ply
are separated by a 0\degr{} ply. The first damage mechanism is transverse
crack initiation in the 90\degr{} ply. Similarly to previous analysis,
simultaneous crack initiation in 90\degr{} ply and $\theta$-ply is not
likely to occur because it is energetically less favorable. Contrary to
the configuration for which the 90\degr{} ply and the $\theta$-ply are
adjacent, crack initiation in the 90\degr{} ply does not induce any
stress concentration in $\theta$-ply. Therefore, crack re-initiation in
the $\theta$\degr{} ply may occur if both the stress and the energy
criteria are fulfilled, \ie~for a larger imposed strain than the one
corresponding to 90\degr{} ply crack initiation 
(Figure~\ref{fig:Fig_epsc_vs_t_Theta_separated_0_theta80_and_50}).  If the
90\degr{} ply and the $\theta$-ply are separated by a 0\degr{} ply, crack
re-initiation is thus not likely to be observed~immediately after the
first crack initiation whatever the ply thickness or $\theta$-ply
orientation. 


\begin{figure}
\includegraphics{fig09}
\caption{Imposed strain at crack re-initiation in $\theta$-ply after
the first crack initiation in the 90\degr{} ply as a function of the ply
thickness for (a) $\theta=80$\degr{}. (b) $\theta=50$\degr{} in case the
$\theta$-ply and the 90\degr{} ply are separated by a 0\degr{}
ply.\label{fig:Fig_epsc_vs_t_Theta_separated_0_theta80_and_50}}
\end{figure}


We now study the configuration for which the 90\degr{} ply and the
$\theta$-ply are adjacent. A first crack initiation in the 90\degr{} ply
induces a stress singularity at the crack tip located at the interface
between both plies and thus a stress gradient in the $\theta$-ply
(Figure~\ref{fig:stress_isocontour_mesh}) which may promote
$\theta$-ply crack re-initiation. We use the term of re-initiation
because we concluded previously that $\theta$-ply crack initiation
cannot occur simutaneously or before 90\degr{} ply crack initiation.
Therefore, this mechanism is a consequence of a first crack initiation
in the 90\degr{} ply which induces the stress singularity necessary to
further enable a possible crack re-initiation. The stress and the
energy criterion can be implemented in order to determine $\theta$-ply
re-initiation imposed strain as well as crack surface
(Figure~\ref{fig:Example_CC_ReInitiation}).  If there exists a crack
surface for which both criteria are fulfilled at the imposed strain
corresponding to a first crack initiation in the 90\degr{} ply
(Figure~\ref{fig:Example_CC_ReInitiation}a), crack
initiation immediately following 90\degr{} ply crack initiation is likely
to occur. From an experimental point of view, it is difficult to detect
both mechanisms separately since they are expected to occur
sequentially in a short time interval. However, it is possible to
observe whether a crack nucleated or not in $\theta$-ply just after 
90\degr{} crack initiation. 


\begin{figure}
\includegraphics{fig10}
\caption{Imposed strain that must be prescribed in order to fulfill
stress (dashed line) and energy (solid line) criteria and corresponding
minimum imposed strain and crack surface for which both criteria are
fulfilled (circle) for $t=1$~mm ply thickness and (a) $\theta=80$\degr{}
or (b)~$\theta=50$\degr{}.\label{fig:Example_CC_ReInitiation}}
\end{figure}


If fulfilling both criteria requires an imposed strain larger than the
imposed strain at 90\degr{} ply crack initiation
(Figure~\ref{fig:Example_CC_ReInitiation}b), crack
re-initiation in $\theta$-ply is not likely to occur immediately after
90\degr{} ply initiation but requires an increase in the imposed strain.
From an experimental point of view, it means that single crack
initiation in the 90\degr{} ply is likely to be observed before
$\theta$-ply crack re-initiation for a larger imposed strain.

In both cases, it can be noted that the incremental energy release rate
(and thus the differential energy release rate $\mathcal{G}$) is an
increasing function of the crack surface. Therefore, as soon as crack
initiation occurs, unstable crack propagation is likely to occur
($\mathcal{G} \geqslant \mathcal{G}_c$ and 
$\mathrm{d}\mathcal{G}/\mathrm{d}S \geqslant
0$). It means that without any increase in the imposed strain, the
initiated crack will instantaneously grow through the entire ply.
Therefore, the crack is expected to be observed experimentally on the
specimen surface.

Figure~\ref{fig:Epsc_vs_t_vs_theta} shows the imposed strain at
$\theta$-ply crack re-initiation as a function of the ply thickness and
orientation.  The imposed strain at crack re-initiation decreases with
increasing ply thickness~or decreasing ply disorientation (difference
between 90\degr{} and the ply orientation~$\theta$).
Figure~\ref{fig:Epsc_vs_t_vs_theta_reinititiation} shows $\theta$-ply
crack re-initiation imposed strain as a function of the ply thickness
compared to the imposed strain at 90\degr{} ply crack initiation for
several ply orientations, resulting in different possible scenarii. If
the ply orientation is close to 90\degr{}
(Figure~\ref{fig:Epsc_vs_t_vs_theta_reinititiation}a), crack
re-initiation is likely to occur whatever the ply thickness. We thus
expect to observe experimentally cracks in both plies just after
initiation. On the contrary, if the ply disorientation with respect to
90\degr{} is too large
(Figure~\ref{fig:Epsc_vs_t_vs_theta_reinititiation}c), no crack
re-initiation is expected to occur just after transverse crack
initiation, it would require a larger imposed loading. For intermediate
orientations
(Figure~\ref{fig:Epsc_vs_t_vs_theta_reinititiation}b), crack
re-initiation is expected to occur just after transverse crack
initiation only for thick enough plies ($t>0.6$~mm).


\begin{figure}
\includegraphics{fig11}
\caption{Imposed strain at $\theta$-ply crack re-initiation as a
function of (a) the ply thickness for $\theta=70$\degr{} and of the ply
angle for (b) $t=1.2$~mm or (c) for several ply
thicknesses.\label{fig:Epsc_vs_t_vs_theta}}
\end{figure}


\begin{figure}
\includegraphics{fig12}
\caption{Imposed strain at $\theta$-ply crack re-initiation as a
function of the ply thickness compared to the imposed strain at 90\degr{}
crack initiation for (a) $\theta=80$\degr{}, (b) $\theta=70$\degr{}, (c)
$\theta=50$\degr{}.\label{fig:Epsc_vs_t_vs_theta_reinititiation}}
\end{figure}


We finally compare the three damage mechanisms, namely transverse crack
initiation, debonding and $\theta$-ply crack re-initiation
(Figure~\ref{fig:Epsc_vs_t_vs_theta_allmechanisms}) for several ply
orientation.  For ply orientation close to 90\degr{}
(Figure~\ref{fig:Epsc_vs_t_vs_theta_allmechanisms}a) and thick enough
plies ($t>1$~mm), both debonding or crack re-initiation are likely to
occur just after transverse crack initiation. For thin plies
($t<6$~mm), the damage sequence that is likely to occur is the
following order:
\begin{enumerate}[(i)]
\item  transverse crack initiation
\item  $\theta$-ply crack re-initiation at a slightly larger imposed strain
\item  possible inter-ply debonding.
\end{enumerate}
For intermediate ply thicknesses, crack re-initiation is likely to
occur just after transverse crack initiation without any imposed strain
increase, and debonding may occur for a larger imposed strain. For ply
orientation different enough from 90\degr{}
(Figure~\ref{fig:Epsc_vs_t_vs_theta_allmechanisms}c), whatever the ply
thickness, the most likely damage sequence is transverse crack
initiation followed by debonding at a larger imposed strain. Crack
re-initiation could possibly occur at a larger imposed strain. For
intermediate ply orientation
(Figure~\ref{fig:Epsc_vs_t_vs_theta_allmechanisms}b), whatever the ply
thickness, the most likely damage sequence is transverse crack
initiation followed by crack re-initiation at a larger imposed strain.
The observed change in the predicted sequence of damage mechanisms for
different theta ply angles can be explained due to the critical strain
for re-initiation changes a lot with $\theta$ ply misorientation,
whereas the critical strain for debonding changes much less. 


\begin{figure}
\includegraphics{fig13}
\caption{Imposed strain at (i) 90\degr{} ply transverse cracking (solid
line), (ii) inter-ply debonding  (square symbols) or (iii) $\theta$-ply
crack re-initiation (disk symbols) as a function of the ply thickness
$t$ obtained for (a) $\theta=70$\degr{}, (b) $\theta=50$\degr{}, (c)
$\theta=30$\degr{}.\label{fig:Epsc_vs_t_vs_theta_allmechanisms}}
\end{figure}


\section{Discussions and conclusion}\label{sec:conclus}

Composite materials offer significant potential for optimization in
terms of the number of plies, ply orientations, and stacking sequences.
However, the potential offered by new manufacturing techniques, such as
automated fibre placement, is not being fully exploited due to the
conservative practices currently in place. These practices are used to
address the lack of predictive tools for mechanisms that may lead to
domino cracking through the thickness of the laminate. In this article,
we propose an approach based on the use of the coupled criterion,
implemented using 3D finite element calculations.

To attain this objective, numerical simulations were conducted to
investigate the damage mechanisms occurring in laminates with
$\theta$-plies adjacent to 90\degr{} plies. The results indicate that
the damage sequence depends on the orientation mismatch between the
90\degr{} ply and the $\theta$-ply. Several mechanisms can be
identified. The deformation leading to the initiation of the first
crack depends on both the ply thickness and the angle between the fiber
orientation in the ply and the applied loading direction. For
equivalent ply thicknesses, the greater the misorientation of the ply
with respect to the loading axis, the smaller the deformation leading
to the initiation of the first crack. However, this conclusion is
called into question by the thickness of the plies. For instance, a
thin ply oriented at 90\degr{} may undergo greater deformation before
failure than a thicker, less misoriented ply. This conclusion is
supported by experimental results. In the
$[70_2/0/90_2/0/{-}70_2]_{\mathrm{s}}$ laminate, the first
cracks are initiated almost simultaneously in the doubled-thickness
outer 70\degr{} ply and in the quadrupled-thickness inner
${-}70$\degr{} ply. 


This mechanism possibly enables two other mechanisms (that cannot occur
without the presence of this first crack), namely either inter-ply
debonding or crack re-initiation in adjacent $\theta$-ply. These two
mechanisms may possibly occur at the same imposed loading as the one
necessary for transverse crack initiation, if the ply is the
$\theta$-ply orientation close enough to 90\degr{}  This conclusion is
supported by experimental results. In the
$[70_2/0/90_2/{-}70_2/0]_{\mathrm{s}}$ laminate, the first
cracks are initiated simultaneously in the ${\pm}70$\degr{} and
90\degr{} plies. Moreover, all the cracks in the adjacent plies are
continuous. Based on the results obtained for the previous laminate, it
can be concluded that the crack first initiates in the outer
70\degr{} ply, leading to the failure of the adjacent plies.


If the angle offset between two adjacent plies is large, the most
likely mechanism is debonding at the interface, initiated at the tip of
the transverse crack.  As observed experimentally in the
$[70_2/0/90_2/0/{-}70_2]_{\mathrm{s}}$ layup,  a highly oriented
ply (0\degr{} in this case), acts as a barrier, preventing crack
re-initiation.


This study thus provides insight towards optimal, less empirical and
less restrictive design of laminate stacking sequences. The results
presented in this article were obtained using a thermoset matrix
composite. Future work will focus on extending this study to
thermoplastic matrix composites, which have much higher critical energy
release rates. Moreover, for the
$[30_2/90_2/{-}30_2/0]_{\mathrm{s}}$ laminate, the cracks in the
90\degr{} ply appear to be slightly inclined, which motivates the
investigation of the underlying phenomenon.   

\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.

\section*{Dedication}
The manuscript was written through contributions of all authors. All
authors have given approval to the final version of the manuscript.

\section*{Acknowledgment}
This manuscript is dedicated to our friend and colleague Dominique
Leguillon, whose remarkable contributions to the field of Fracture
Mechanics---both through his groundbreaking research and his profound
personal influence---continue to inspire us all.

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