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\DOI{10.5802/crmeca.302}
\datereceived{2025-02-06}
\daterevised{2025-05-17}
\dateaccepted{2025-05-19}
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\dateposted{2025-06-16}
\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{research-article}

\title{3D finite fracture mechanics under mode I loading: the flat
elliptical crack}

\alttitle{M\'{e}canique de la rupture finie en 3D sous chargement en
mode I : la fissure elliptique plane}

\author{\firstname{Pietro} \lastname{Cornetti}\CDRorcid{0000-0001-9063-9913}\IsCorresp}
\address{Department of Structural, Geotechnical and Building
Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129
Torino, Italy}
\email[P. Cornetti]{pietro.cornetti@polito.it}

\author{\firstname{Vladislav} \lastname{Manti\v{c}}\CDRorcid{0000-0002-7569-7442}}
\address{Grupo de Elasticidad y Resistencia de Materiales, Escuela
T\'{e}cnica Superior de Ingenier\'{i}a, Universidad de Sevilla, Camino
de los Descubrimientos s/n, Sevilla, 41092, Spain}

\author{\firstname{Zohar} \lastname{Yosibash}\CDRorcid{0000-0002-0826-2553}}
\address{Computational Mechanics and Experimental Biomechanics Lab,
School of Mechanical Engineering, Iby and Aladar Fleischman Faculty of
Engineering, Tel Aviv University, Ramat Aviv 69978, Israel}

\shortrunauthors

\begin{abstract} 
The determination of the remote stress causing crack propagation in an
infinite 3D domain with an embedded flat elliptical crack is here
revisited in the framework of the Coupled Criterion of Finite Fracture
Mechanics. We started reviewing Linear Elastic Fracture Mechanics
approaches, which differ by accounting for different infinitesimal
crack growths. Then, we provide the solution based on Finite Fracture
Mechanics: if the elliptical flaw is sufficiently small, the crack
grows along iso-stress lines. For larger sizes, other crack growths may
take place. Thus, the present investigation shows that assuming an
iso-stress crack front may effectively provide the exact Finite
Fracture Mechanics solution, particularly for small defects; on the
other hand, it can be wrong for larger size, providing moreover
un-conservative predictions. However, for the geometry at hand, it
yields failure stress estimates differing from the actual one by a few
percents. Thus, the iso-stress assumption, conjectured by Leguillon [D.
Leguillon, ``An attempt to extend the 2D coupled criterion for crack
nucleation in brittle materials to the 3D case'',  \textit{Theor. Appl.
Fract. Mech.} \textbf{74} (2014), pp. 7--17]---implying strong
simplifications in the numerical implementation of the coupled
criterion in 3D problems---seems to be largely justified by the present
results. Moreover, regardless of the initial crack size, the finite
growth predicted by the model results in a new elliptical crack shape
closer to the circular one, meaning the eccentricity consistently
decreases as the crack propagates. 
\end{abstract}

\begin{altabstract} 
La d\'{e}termination de la contrainte \`{a} distance provoquant la
propagation d'une fissure dans un domaine 3D infini contenant une
fissure elliptique plane est ici revisit\'{e}e dans le cadre du
Crit\`{e}re Coupl\'{e} de la M\'{e}canique de la Rupture Finie. Nous
commen\c{c}ons par passer en revue les approches de la M\'{e}canique
Lin\'{e}aire de la Rupture, qui diff\`{e}rent selon la prise en compte
de diff\'{e}rentes croissances infinit\'{e}simales de la fissure.
Ensuite, nous pr\'{e}sentons la solution bas\'{e}e sur la M\'{e}canique
de la Rupture Finie : si le d\'{e}faut elliptique est suffisamment
petit, la fissure cro\^{i}t le long de lignes iso-contraintes. Pour des
tailles plus grandes, d'autres modes de croissance de fissure peuvent
se produire. Ainsi, cette \'{e}tude montre que supposer un front de
fissure iso-contraintes peut effectivement fournir la solution exacte
en M\'{e}canique de la Rupture Finie, en particulier pour les petits
d\'{e}fauts ; en revanche, cela peut \^{e}tre erron\'{e} pour des
d\'{e}fauts de plus grande taille, entra\^{i}nant de surcro\^{i}t des
pr\'{e}dictions non conservatrices. Toutefois, pour la
g\'{e}om\'{e}trie consid\'{e}r\'{e}e, cela donne des estimations de
contrainte de rupture ne diff\'{e}rant que de quelques pourcents de la
valeur r\'{e}elle. Ainsi, l'hypoth\`{e}se d'iso-contraintes,
avanc\'{e}e par Leguillon [D. Leguillon, ``An attempt to extend the 2D
coupled criterion for crack nucleation in brittle materials to the 3D
case'',  \textit{Theor. Appl. Fract. Mech.} \textbf{74} (2014), pp.
7-17], impliquant des simplifications importantes dans
l'impl\'{e}mentation num\'{e}rique du crit\`{e}re coupl\'{e} dans des
probl\`{e}mes 3D, semble largement justifi\'{e}e par les r\'{e}sultats
pr\'{e}sents. En outre, quelle que soit la taille initiale de la
fissure, la croissance finie pr\'{e}dite par le mod\`{e}le aboutit
\`{a} une nouvelle forme elliptique de la fissure, plus proche d'un
cercle, ce qui signifie que l'excentricit\'{e} diminue
syst\'{e}matiquement au fur et \`{a} mesure de la propagation de la
fissure. 
{\vspace*{-2pt}}
\end{altabstract}

\keywords{\kwd{Coupled criterion}
\kwd{Finite fracture mechanics}
\kwd{3D linear elastic fracture mechanics}
\kwd{Elliptical cracks}
\kwd{Quasi-brittle materials}}

\altkeywords{\kwd{Crit\`{e}re coupl\'e}
\kwd{M\'{e}canique de la rupture finie} 
\kwd{M\'{e}canique de la fracture \'{e}lastique lin\'{e}aire en 3D} 
\kwd{Fissures elliptiques} 
\kwd{Mat\'{e}riaux quasi-fragile}}

\thanks{NEWFRAC Project (Marie Sklodowska-Curie grant agreement No.
861061), Pazy Research Foundation, Spanish Ministry of Science and
Innovation, European Regional Development Fund (PID2021-123325OB-I00).}

\maketitle

\end{noXML}

{\vspace*{-2pt}}

\section{Introduction} \label{sec1}

{\vspace*{-2pt}}

The Coupled Criterion of Finite Fracture Mechanics (CCFFM), introduced
for the first time by Leguillon~\cite{1} in 2002, has proven to be an
effective, yet simple, fracture criterion for obtaining the failure
load in a variety of structural problems, spanning from size effect
(e.g.~\cite{2}) to stress concentration/intensification
(e.g.~\cite{3,4,5}) in homogeneous materials, from composite materials
(e.g.~\cite{6,7,8}) to bonded joints (e.g.~\cite{9}), from static
loadings to dynamic (e.g.~\cite{10,11}) and fatigue (e.g.~\cite{12,13}) 
loadings. With respect to Linear Elastic Fracture Mechanics
(LEFM), a major advantage is its applicability to any geometry, cracked
or plain (i.e.\ not only cracked). With respect to more sophisticated
models like the Cohesive Crack Model (CCM) or the Phase Field (PF)
model for fracture, the numerical implementation of the CCFFM is
usually much easier, often allowing for an analytical or
semi-analytical solution for the problem at hand. Moreover, the CCFFM
has proven to be often in agreement with CCM and PF models: for a
comparison between CCFFM and CCM the reader is referred
to~\cite{14,15,16,17,18,19}; and for a comparison between CCFFM and PF
to~\cite{20,21,22,23,24}.

Most of the CCFFM applications address two-dimensional problems, where
cracks/V-notch tips are straight lines through-the-thickness. However,
recently, attention has also been focussed to 3D problems, starting
from Leguillon's pioneering work~\cite{25}. The application of the
CCFFM to 3D problems is challenging because the finite crack advance,
differently from the 2D case, can be of any shape~\cite{26}. To
overcome this difficulty, researchers often assumed a finite crack
growth occurring along an iso-stress line, see e.g.~\cite{27,28,29}.
One of our main purposes is investigating this assumption for a model
problem allowing an analytical derivation in the framework of the
CCFFM: a flat elliptical flaw in an infinite linear elastic medium
subjected to a remote tensile stress orthogonal to the crack plane.

The linear elastic stress-strain solution for an elliptical flaw under
remote tensile stress dates back to Green and Sneddon~\cite{30},
following (and including) the one for a penny-shaped crack provided by
Sneddon~\cite{31}. Later, Irwin~\cite{32} provided the Stress Intensity
Factor (SIF) values along the elliptical crack front. More recent
contributions related to the stress field in the vicinity of the crack
front can be found in~\cite{33,34}. For what concerns crack
propagation, Lazarus~\cite{35}, among different flaw shapes, considered
the elliptical one and analysed the crack growth for a brittle fracture
(assuming propagation where the SIF reaches the fracture toughness and
regularising the crack front) and fatigue (using Paris' equation). More
numerical/practical contributions along with experimental data
validation (under cycling loading) about crack propagation from flat
elliptical cracks can be found, e.g., in~\cite{36,37,38}. Finally, we
refer to a recent numerical investigation where the CCFFM criterion has
been exploited to deal with free edge delamination in angle-ply
laminates, assuming a semi-elliptical crack shape, under
static~\cite{39}, fatigue~\cite{40} and thermal~\cite{41} loadings.
These papers provide details on the numerical implementation of the
Coupled Criterion for geometries similar to the one addressed herein.

The paper is organised as follows. In Section~\ref{sec2} we focus on
LEFM, deriving the general expression of the mode I failure stress in
the presence of a flaw of any shape under the assumption of an
infinitesimal iso-stress crack growth. Then, we specify the failure
stress for the flat elliptical crack, providing its closed form
estimate by means of SIFs. Thereafter, we show the same result can be
achieved by evaluating the Strain Energy Release Rate (SERR). Following
this latter procedure, we also consider the infinitesimal elliptical
crack growth along the minor axis alone, showing it provides failure
stress estimates lower than the iso-stress one.

In Section~\ref{sec3} we derive the failure stress provided by the
CCFFM for the flat elliptical crack. We assume the finite crack
propagation to be characterised by an elliptical crack front of any
shape/size. Thus, the new crack front is characterised by two
parameters (the increments of the semi-axes); CCFFM implies solving a
minimisation problem upon the variation of these two parameters. It
will be shown that, based on the crack shape and size, two different
scenarios (i.e.\ iso-stress and minor-axis crack propagations) can occur
and the corresponding fracture stress is finally provided. The results
are commented and in Section~\ref{sec4} some conclusions are drawn.

\section{LEFM approach}\label{sec2}

We first provide the failure stress according to LEFM for a flat crack
with an arbitrary shape subject to a mode I loading, then specify it
for the elliptical crack. Two procedures, based on SIF or SERR, are
outlined and exploited.

\subsection{Planar crack of arbitrary shape: iso-stress crack
growth}\label{sec21}

Let us consider a planar crack of arbitrary shape in an infinite body,
made by a homogeneous isotropic linear elastic brittle material. Let
the plane of the crack be ($x,y$). The remote loading is a uniform
tensile stress ${\sigma}$ in the $z$ direction as in Figure~\ref{fig1}.
Under this assumption the crack is in pure mode I condition (or---the
same---in opening mode, since, because of isotropy, the displacement
field is symmetrical to the crack plane).

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig1}A planar crack of arbitrary shape in an infinite
3D domain under uniform tensile stress normal to the crack plane (a).
Crack geometry (b).}
\end{figure}

Except for the penny-shaped crack, the SIF varies along the contour
${C}$ of the crack, i.e.\ $K_I = K_I({s})$, ${s}$ being the curvilinear
abscissa along the contour ${C}$ (Figure~\ref{fig1}b). Denoting by
$K_{I,\min}$ and the by $K_{I,\max}$ the minimum and maximum value of
the SIF respectively, we can write:
{\begin{equation}\label{eq1}
K_{{I},\min }\leq K_{{I}}(s)\leq K_{{I},\max}.
\end{equation}}\unskip
Assume $K_I = K_I({s})$ is available in an analytic or numerical form,
and the fracture toughness ${K}_{\mathrm{Ic}}$ of the material is
known. One is interested in ${\sigma}_{\mathrm{f}}$, the remote stress causing
crack growth, i.e.\ failure, according to LEFM. Because of mode I, the
crack expands in its plane ($x,y$). However, if one wishes to consider
the Griffith's SERR, there are an infinite possible shapes of
infinitesimal crack growth, unlike in 2D domains where only a collinear
crack growth is possible along an infinitesimal length $\mathrm{d}{a}$
($a$ being the crack length).

A reasonable starting point is assuming (yet, an assumption) an
infinitesimal crack growth defined by an iso-stress line. Since the
asymptotic stress field in the direction normal to the crack contour
($r$ is the coordinate along the normal $\hat{n}$ starting from the
crack contour ${C}$, see Figure~\ref{fig1}b) is:
{\begin{equation}\label{eq2}
\sigma _{z}\cong \frac{K_{{I}}}{\sqrt{2\uppi r}}
\end{equation}}\unskip
The same stress level (e.g.\ ${\sigma}_{0}$) is achieved at different
distances from the contour ${C}$, larger where $K_I$ is larger, smaller
where $K_I$ is smaller (see Figure~\ref{fig2}a). Where $K_I$ is maximum,
the stress ${\sigma}_{0}$ is achieved at a distance
$(\Delta a)_{\max}$:
{\begin{equation}\label{eq3}
\left(\Delta a\right)_{\max }\cong \frac{{K}_{{I},\max }^{2}}{2\uppi {\sigma }_{0}^{2}}
\end{equation}}\unskip
while in a generic point we have:
{\begin{equation}\label{eq4}
\left(\Delta a\right)\cong \frac{{K}_{{I}}^{2}}{2\uppi {\sigma }_{0}^{2}}
\end{equation}}\unskip

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig2} Asymptotic stress field ahead the crack front (a)
and iso-stress crack growth (b).}
\end{figure}

Dividing Equation~(\ref{eq3}) by (\ref{eq4}), one obtains:
{\begin{equation}\label{eq5}
\frac{\Delta a}{\left(\Delta a\right)_{\max }}\cong 
\left(\frac{K_{{I}}}{K_{{I},\max }}\right)^{2}=\frac{G}{G_{\max }}
\end{equation}}\unskip
where the last expression is a consequence of Irwin's relationship
$G={K}_{I}^{2}/E'$, ${G}$ {being the SERR and} $E'${ the Young modulus
of the material under plane strain conditions. As ${\Delta}{{a}}$}
{tends to zero, Equation~(\ref{eq5}) defines the shape of the
infinitesimal iso-stress crack growth (see Figure~\ref{fig2}b).}

Expressing the Griffith infinitesimal energy balance according to LEFM
($G_{\mathrm{c}}$ being the material fracture energy) by an integral along the
curve ${C}$, one obtains:
{\begin{equation}\label{eq6}
\oint _{C}G(s)\Delta a\,\mathrm{d}s=\oint _{C}G_{\mathrm{c}}\Delta a\,\mathrm{d}s
\end{equation}}\unskip
Dividing both sides of Equation~(\ref{eq6}) by $(\Delta a)_{\max}$
and substituting  Equation~(\ref{eq5}) into Equation~(\ref{eq6}), one
obtains the condition for crack growth:
{\begin{equation}\label{eq7}
G_{\mathrm{iso}}=\frac{\displaystyle\oint _{C}G^{2}(s)\,\mathrm{d}s}
{\displaystyle\oint _{C}G(s)\,\mathrm{d}s}=G_{\mathrm{c}}
\end{equation}}\unskip
The ratio between the integrals is somehow an equivalent-2D SERR, since
crack growth occurs whenever this value reaches the material fracture
energy $G_{\mathrm{c}}$, like in 2D problems. We named it $G_{\mathrm{iso}}$ since it is
the equivalent-2D SERR under the assumption of iso-stress crack growth.
Note that, from a mathematical point of view, $G_{\mathrm{iso}}$ is the
contra-harmonic mean (sum of the squared values divided by sum of
values, see Appendix~\ref{appA}) of the SERR values evaluated along the crack
contour. Among different averages (i.e.\ harmonic, geometric,
arithmetic, quadratic, etc.) the contra-harmonic mean is the highest
one, thus affected by large values and close to the maximum value of
the variable. By Irwin's relationship we can get also the equivalent-2D
SIF ${K}_{I,\mathrm{iso}}$, which provides the failure stress when it
equals the material fracture toughness ${K}_{\mathrm{Ic}}$:
{\begin{equation}\label{eq8}
K_{{I},\mathrm{iso}}=\sqrt{\frac{\displaystyle\oint _{C}{K}_{{I}}^{4}(s)\,\mathrm{d}s}
{\displaystyle\oint _{C}{K}_{{I}}^{2}(s)\,\mathrm{d}s}}=K_{\mathrm{Ic}}
\end{equation}}\unskip


\subsection{Elliptical crack: iso-stress crack growth by SIF values}\label{sec22}

As a particular case, we consider the elliptical flat crack shown in
Figure~\ref{fig3}a. The failure stress according to LEFM assuming an
iso-stress crack growth is here obtained by the results in
Section~\ref{sec21} (Equation~(\ref{eq8})). The ellipse in
Figure~\ref{fig3}b is defined by semi-axes $a$ and $b$
($a \geq b$) defined as:
{\begin{equation}\label{eq9}
\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1
\end{equation}}\unskip
or in parametric form ($0 \leq \varphi &lt; 2\uppi$):
{\begin{equation}\label{eq10}
\mathbf{P}(\varphi)=\begin{cases} x={a}\cos\varphi \\
y={b}\sin\varphi \end{cases} 
\end{equation}}\unskip

\begin{figure}
\includegraphics{fig03}
\caption{\label{fig3}An elliptical (planar) crack in an infinite medium
under uniform tensile stress normal to the crack plane (a). Crack
geometry (b).}
\end{figure}

The aspect of the ellipse is univocally defined either by the aspect
ratio $\gamma = b/a\ (0 &lt; \gamma &lt; 1)$ or by the eccentricity
$k=\sqrt{1-(b/a)^{2}}\ (0 &lt; k &lt; 1)$. 

The SIF along the crack front is~\cite{32}:
{\begin{equation}\label{eq11}
K_{{I}}=\frac{\sigma \sqrt{\uppi b}}{E(k^{2})}
\sqrt[4]{1-k^{2}\cos ^{2}\varphi }
\end{equation}}\unskip
where ${E}({k}_{2})$ is the complete elliptic integral of the second
kind (see Appendix~\ref{appB}). From Equation~(\ref{eq11}) the maximum
value of the SIF $K_{I,\max}$ is at point $B$ (i.e.\ on the minor axis,
$\varphi = \uppi/2$) while the minimum value of the SIF $K_{I,\min}$ is
at point $A$ (i.e.\ on the major axis, ${\varphi} = 0$):
{\begin{eqnarray}\label{eq12}
K_{{I},\max }&amp;=&amp;(K_I)_{\mathrm{B}}=\frac{\sigma \sqrt{\uppi b}}{E(k^{2})} \Seqnsplit
\label{eq13}
K_{{I},\min }&amp;=&amp;(K_I)_{\mathrm{A}}=\frac{\sigma b}{E(k^{2})}
\sqrt{\frac{\uppi }{a}}=K_{{I},\max }\sqrt{\frac{b}{a}} 
\end{eqnarray}}\unskip

For what concerns $K_{I,\max}$, Equation~(\ref{eq12}) encompasses the
limit cases of a penny-shaped crack ($k = 0$ or $\gamma =1$):
{\begin{equation}\label{eq14}
K_{{I}}=\frac{2}{\uppi }\sigma \sqrt{\uppi b}
\end{equation}}\unskip
constant all around the crack front, and of a through-the-thickness
crack of length $2b$ (corresponding to $k = 1$ or $\gamma =0$):
{\begin{equation}\label{eq15}
K_{{I}}=\sigma \sqrt{\uppi b}
\end{equation}}\unskip
To obtain the failure stress, we have to compute the 2D-equivalent SIF
by means of Equation~(\ref{eq8}). Using the parametric expression of
the ellipse (Equation~(\ref{eq10})), the equivalent SIF
(Equation~(\ref{eq8})) reads:
{\begin{equation}\label{eq16}
K_{{I},\mathrm{iso}}=\sqrt{\frac{{\int }_{0}^{\uppi /2}{K}_{{I}}^{4}(\varphi)
\left\|\frac{\mathrm{d}\mathbf{P}}{\mathrm{d}\varphi }\right\|
\mathrm{d}\varphi }{{\int }_{0}^{\uppi /2}{K}_{{I}}^{2}(\varphi)
\left\|\frac{\mathrm{d}\mathbf{P}}{\mathrm{d}\varphi }\right\|
\mathrm{d}\varphi }}=\frac{\sigma \sqrt{\uppi b}}{E(k^{2})}
\sqrt{\frac{{\int }_{0}^{\uppi /2}\left(1-k^{2}\cos ^{2}\varphi \right)^{3/2}
\mathrm{d}\varphi }{{\int }_{0}^{\uppi /2}\left(1-k^{2}\cos ^{2}
\varphi \right)\mathrm{d}\varphi }}
\end{equation}}\unskip
where the double symmetry of the ellipse has been exploited to limit
the integration interval to [$0, \uppi/2$]. By analytical
manipulations, the integrals in Equation~(\ref{eq16}) can be cast in
terms of complete elliptic integrals of first  $({K}({k}^{2}))$ and
second $(E(k^2))$ kind (see Appendix~\ref{appB}) as:
{\begin{equation}\label{eq17}
K_{{I},\mathrm{iso}}=\frac{2\sigma \sqrt{b}}{E(k^{2})}
\sqrt{\frac{2(2-k^{2})E(k^{2})-(1-k^{2})K(k^{2})}{3(2-k^{2})}}
\end{equation}}\unskip
Thus, according to LEFM and iso-stress crack growth, failure is
achieved whenever the above quantity reaches the material fracture
toughness. Hence, the corresponding failure stress
$({\sigma}_{\mathrm{f}})_{\mathrm{LEFM}\hyphen\mathrm{iso}}$ is:
{\begin{equation}\label{eq18}
(\sigma _{\mathrm{f}})_{\mathrm{LEFM}\hyphen\mathrm{iso}}=\frac{\sqrt{3}}{2}
\frac{K_{\mathrm{Ic}}}{\sqrt{b}}E(k^{2})\sqrt{\frac{2-k^{2}}{2(2-k^{2})E(k^{2})-(1-k^{2})K(k^{2})}}
\end{equation}}\unskip
Note that, as expected for any LEFM approach, the failure stress tends
to infinity as the crack vanishes (i.e.\ 
${\sigma}_{\mathrm{f}}\rightarrow\infty$ as $b\rightarrow 0$).

\subsection{Elliptical crack: iso-stress crack growth by SERR evaluation}\label{sec2.3}

We show in this sub-section that same results in Section~\ref{sec22}
are obtained also by proper evaluation of the SERR. This procedure will
be exploited in the following to remove the assumption of iso-stress
crack growth.

Computation of the SERR $G$ can be performed without the need of SIF
(i.e.\ without exploiting Irwin's relationship). It is the way followed
originally by Griffith in his 1921 seminal work. Accordingly, under
load control, one has:
{\begin{equation}\label{eq19}
G=\lim _{\Delta A\rightarrow 0}\frac{\Delta \Phi }{\Delta A}
\end{equation}}\unskip
To compute $G_{\mathrm{iso}}$ (i.e.\ the SERR assuming an iso-stress
crack growth) by Equation~(\ref{eq19}) one need to know (i) the
iso-stress contour lines and (ii) the change in strain energy
$\Delta \Phi$ due to the (finite) variation $\Delta A$ of
the crack surface. These ingredients can be derived by 
Green and Sneddon~\cite{30} solution. Accordingly, because of the remote stress
the planar elliptical crack takes the shape of an ellipsoid whose
semi-axes are ${a}, {b}$ and ${w}_{\max}$ (the maximum crack opening
displacement), the last one given by:
{\begin{equation}\label{eq20}
w_{\max }=\frac{2\sigma b}{E(k^{2})E'}
\end{equation}}\unskip
By looking at the load as a uniform remote tensile stress field plus a
uniform compressive stress applied on the crack faces, and by means of
Clapeyron's theorem, the strain energy ${\Phi}$ increment due to the
presence of the crack can be computed as:
{\begin{equation}\label{eq21}
\Phi =\frac{\sigma V}{2}
\end{equation}}\unskip
where ${V}$ is the volume of the (deformed) crack (i.e.\ the ellipsoid), whose value is:
{\begin{equation}\label{eq22}
V=\textstyle\frac{4}{3}\uppi \,a\,b\,w_{\max }
\end{equation}}\unskip
Hence, by combining Equations~(\ref{eq20}) to (\ref{eq22}):
{\begin{equation}\label{eq23}
\Phi =\frac{4\uppi }{3E(k^{2})}\frac{\sigma ^{2}ab^{2}}{E'}
\end{equation}}\unskip
The iso-stress lines can also be derived from Green and
Sneddon~\cite{30}. From their solution, the ${\sigma}_{z}$ stress field
on the crack plane (outside the crack faces) is amenable of the
following analytical expression:
{\begin{equation}\label{eq24}
\frac{\sigma _{{z}}}{\sigma }=1+\frac{1}{E(k^{2})}\left[\frac{a}
{\sqrt{\upxi }}\sqrt{\frac{b^{2}+\upxi }{a^{2}+\upxi }}-E
\left(\arcsin \frac{a}{\sqrt{a^{2}+\upxi }}|\,k^{2}\right)\right]
\end{equation}}\unskip
where $E(\varphi|k^2)$ is the incomplete elliptic integral of the
second kind (see Appendix~\ref{appB}) and ${\upxi}$ is an ellipsoidal
coordinate. On the crack plane ($z= 0$), $\upxi = \mathrm{constant}$
($\upxi \geq 0$) corresponds to a family of ellipses with equation:
{\begin{equation}\label{eq25}
\frac{x^{2}}{a^{2}+\upxi }+\frac{y^{2}}{b^{2}+\upxi}=1
\end{equation}}\unskip
Hence, we get a relevant information: \textit{the iso-stress curves are
a particular family of ellipses}. More precisely,
Equations~(\ref{eq24})  and (\ref{eq25}) show that, as ${\upxi}$
increases from 0 to ${\infty}$, the stress value decreases from
${\infty}$ to ${\sigma}$ and the corresponding isostress lines are
ellipses of increasing size and decreasing eccentricity. As an example,
some of them are plotted in Figure~\ref{fig4}a for an elliptical crack
with aspect ratio $\gamma = 0.5$. Naming by $\Delta a$ and
$\Delta b$ the increment of the semi-axes of the generic iso-stress
line with respect to the ones of the original elliptical flaw, from
Equation~(\ref{eq25}) the relationship between them may be obtained
(see details in Appendix~\ref{appC}):
{\begin{equation}\label{eq26}
\Delta a=\sqrt{a^{2}+2b\,\Delta b+\left(\Delta b\right)^{2}}-a
\end{equation}}\unskip
Let us assume the crack grows by a finite amount up to a given
iso-stress line, which in turns is defined by a given $\upxi$ value.
From Equation~(\ref{eq23}), the energy variation $\Delta\Phi$ is:
{\begin{equation}\label{eq27}
\Delta \Phi =\frac{4\uppi \sigma ^{2}}{3E'}
\left\{\frac{\left(b^{2}+\upxi \right)\sqrt{a^{2}+\upxi }}
{E\left[(a^{2}-b^{2})/(a^{2}+\upxi )\right]}-
\frac{ab^{2}}{E(1-b^{2}/a^{2})}\right\}
\end{equation}}\unskip
while the newly created crack surface is (difference between elliptical
areas):
{\begin{equation}\label{eq28}
\Delta A=\uppi \,\left[\sqrt{a^{2}+\upxi }\sqrt{b^{2}+\upxi }-ab\right]\,
\end{equation}}\unskip
The following step is inserting Equations~(\ref{eq27})  and
(\ref{eq28}) into Equation~(\ref{eq19}). Then, the limit for $\Delta
A \rightarrow 0$ (i.e.\ $\upxi \rightarrow 0$) has to be evaluated. The
limit takes the undetermined form 0/0, but the application of De
l'H\^{o}pital rule along with property (\ref{eqb4}) allows the
computation of the limit. Finally equating the SERR $G_{\mathrm{iso}}$
to the fracture energy $G_{\mathrm{c}}$ along with Irwin relationship yields
failure stress $(\sigma_{\mathrm{f}})_{\mathrm{LEFM-iso}}$, which coincides with
Equation~(\ref{eq18}). Thus, despite the different lines of thought
(local vs.\ global), we checked that the  SIF- and SERR-based procedures
yield the same outcome, Equation~(\ref{eq18}).

\begin{figure}
\includegraphics{fig04}
\caption{\label{fig4}Possible crack growths: (elliptical)
\textit{iso-stress crack growth} (a); elliptical, with increment along
minor axis alone (b), named \textit{minor-axis crack growth};
non-elliptical crack growth, not considered in the current
investigation (c).}
\end{figure}


\subsection{Elliptical crack: minor axis crack growth}\label{sec24}

On the basis of the analysis in the previous section, we are able to
evaluate the failure stress for any elliptical crack growth (i.e.\ not
only the iso-stress crack growth). The criterion that mostly agrees
with LEFM is the $G$-max, i.e.\ the crack growth actually occurring is the
one providing the maximum SERR and consequently the minimum failure
stress. Since the SIF is higher at the edge of the minor axis, among
all elliptical crack extensions, the one resulting in the largest SERR
is the one at which only the minor axis is increasing.

Let us denote by $\Delta b$ the increment of the minor axis (see
Figure~\ref{fig4}b). The energy available for a finite elliptical crack
growth along the minor axis by $\Delta b$ is:
{\begin{equation}\label{eq29}
\Delta \Phi =\frac{4\uppi \sigma ^{2}}{3E'}
\left\{\frac{a\left(b+\Delta b\right)^{2}}{E\left[1-
\left(b+\Delta b\right)^{2}/a^{2}\right]}-
\frac{ab^{2}}{E\left(1-b^{2}/a^{2}\right)}\right\}
\end{equation}}\unskip
The energy needed to create the new surface is:
{\begin{equation}\label{eq30}
\Delta \Phi =G_{\mathrm{c}}\uppi \,\left[a\left(b+\Delta b\right)-ab\right]
\end{equation}}\unskip
The corresponding failure stress $(\sigma_{\mathrm{f}})_{\mathrm{LEFM}\hyphen\mathrm{minor}}$
is obtained by equating Equations~(\ref{eq29})--(\ref{eq30}) and taking
the limit for $\Delta b\rightarrow 0$, or, by introducing the ratio
$\upvarepsilon_{b}$  between the final value of the minor axis and its
initial value, letting $\upvarepsilon_b\rightarrow 1$:
{\begin{eqnarray}\label{eq31}
\upvarepsilon _{b}&amp;=&amp;\frac{b+\Delta b}{b} \Seqnsplit
\label{eq32}
(\sigma _{\mathrm{f}})_{\mathrm{LEFM}\hyphen\mathrm{minor}}
&amp;=&amp;\lim _{\upvarepsilon _{b}\rightarrow 1}\sqrt{\frac{3}{4}
\frac{{K}_{\mathrm{Ic}}^{2}}{b}\frac{\upvarepsilon _{b}-1}
{\frac{{\upvarepsilon }_{b}^{2}}{E\left[1-{\upvarepsilon }_{b}^{2}(1-k^{2})\right]}-
\frac{1}{E(k^{2})}}}\nonumber\\
&amp;=&amp;\frac{\sqrt{3}}{2}\frac{K_{\mathrm{Ic}}}{\sqrt{b}}E(k^{2})
\frac{k}{\sqrt{(1+k^{2})E(k^{2})-(1-k^{2})K(k^{2})}} 
\end{eqnarray}}\unskip
where again De l'H\^{o}pital rule along with property (\ref{eqb4}) has
been used.

A further, third, estimate of the failure stress is obtained by setting
$K_{I,\max} = K_{\mathrm{Ic}}$. This condition is sometime referred to
as Irwin criterion (as opposed to the previous ones, based on
Equation~(\ref{eq19}), referred to as Griffith criterion, see
e.g.~\cite{42}). Note that, differently from 2D problems where the two
criteria coincide, the equality $K_{I,\max} = {K}_{\mathrm{Ic}}$ does
not rely on any energy balance; however, it is interesting because it
provides conservative predictions with respect to the previous
estimates Equations~(\ref{eq18}) and~(\ref{eq32}). Let us denote
$({\sigma}_{\mathrm{f}})_{\mathrm{Irwin}}$ the new failure stress estimate. By
Equation~(\ref{eq12}) one easily obtains:
{\begin{equation}\label{eq33}
\left(\sigma _{\mathrm{f}}\right)_{\mathrm{Irwin}}=
K_{\mathrm{Ic}}\frac{E(k^{2})}{\sqrt{\uppi b}}
\end{equation}}\unskip

In Figure~\ref{fig5} the dimensionless failure stress estimates 
Equations~(\ref{eq18}),~(\ref{eq32}),~(\ref{eq33}) are plotted vs.\ the
ellipse eccentricity. Note that all predictions are close, since the
contra-harmonic mean (of $G \sim K_I^{2}$), as already observed, is
close to its maximum value. In Figure~\ref{fig5}a the estimates are
compared at constant minor axis length. As ${k}$ increases, so does the
major axis and, obviously, the failure stress decreases. LEFM based on
minor-axis crack growth provides predictions closer (or equal) to $G$-max
criterion: however, we cannot state Equation~(\ref{eq32}) coincides
with $G$-max criterion since infinitesimal crack extensions other than
the elliptical ones could take place (see e.g.\ Figure~\ref{fig4}c). It
is apparent that, according to $G$-max criterion the effective failure
stress lies in between  Equations~(\ref{eq32}) (upper bound) 
and~(\ref{eq33}) (lower bound). Reasonably, it will be closer to
minor-axis crack growth for small eccentricity and closer to Irwin
estimate for large eccentricity, where an elliptical crack growth is
unlikely: e.g., for ${k} = 1$, we know the failure stress is
${\sigma}_{\mathrm{f}}={K}_{\mathrm{Ic}}/\surd{({\uppi}{b})}$, since the
geometry becomes the 2{b}-long through-the-thickness crack, while
LEFM-minor (and LEFM-iso) provides
$\sigma_{\mathrm{f}}=K_{\mathrm{Ic}}\surd{3}/\surd{(8b)}$, see
Figure~\ref{fig5}a.

\begin{figure}
\includegraphics{fig05}
\caption{\label{fig5}LEFM failure stress estimates vs.\ ellipse
eccentricity ${k}$ for an infinitesimal iso-stress crack growth (LEFM,
iso-stress), an infinitesimal minor axis crack growth (LEFM,
minor-axis), $K_{I,\max} = K_{\mathrm{Ic}}$ (Irwin): at constant defect
minor axis ${b}$ (a) and at constant defect area ${A}$ (b); ${k} = 0$
refers to a penny-shaped crack while ${k} =1$ refers to a
through-the-thickness crack.}
\vspace*{4pt}
\end{figure}

In Figure~\ref{fig5}b the same estimates are plotted at constant defect
area  ${A} = \uppi a b$. It is evident that the effect of eccentricity
is very low up to eccentricity values close to unity, i.e.\ what really
affect failure stress is the defect area and not its shape. This result
agrees with the ones provided by David and Lazarus~\cite{43}, related to
(flat) defects of almost any shape.

\section{FFM approach}\label{sec3}

The estimates provided in the previous section are based on a LEFM
approach. As such, they share the well-known drawback that can be
applied only to cracked bodies, or, the same, they provide infinite
failure stress as the defect size vanishes. On the other hand, for
quasi-brittle materials, we expect the failure stress to approach the
material tensile strength ${\sigma}_{\mathrm{c}}$ as the defect size vanishes.
Thus, more refined models able to take into account the material
tensile strength beyond the fracture toughness (or fracture energy)
have to be used to deal with defects of any size. Herein, we resort to
the CCFFM, whose predictions will match LEFM just for relatively large
defect size, where energy alone rules crack growth and propagation.

The CCFFM assumes crack growth by a finite increment ${\Delta}{A}$
(whence the name FFM), see Figure~\ref{fig6}a. This increment as well
as the failure load are determined based on the finite energy balance
and a stress condition (whence the name CC)---the stress must exceed
the material tensile strength over the newly created crack surface
${\Delta}{A}$. While in 2D the finite crack extension is simply
determined by its length and, possibly, by its direction, in 3D
problems the crack extension can have any shape (the actual one will be
the one minimising the failure load), even in cases where the crack
plane is already known, as in the present case. The application of FFM
to 3D problems is thus a challenging task. Up to now, a common approach
in the literature has been to consider finite crack growth following
the iso-stress lines~\cite{25,27,28,29}. This assumption greatly
simplifies the problem, allowing the crack growth to be described by
just one parameter; moreover, it makes the fulfilment of the stress
requirement trivially checked. However, a check of the approximation
introduced by considering just (finite) iso-stress crack growths is not
available; our goal is to provide it for the model problem at hand (the
flat elliptical\break crack).

\begin{figure}
\vspace*{2pt}
\includegraphics{fig06}
\vspace*{2pt}
\caption{\label{fig6}An elliptical (planar) crack in an infinite medium
under uniform tensile stress normal to the crack plane: solution by FFM
(a); crack geometry and elliptical finite crack growth of any shape (b).}
\end{figure}


Thus, let us derive the CCFFM failure stress for the planar elliptical
crack (Figure~\ref{fig2}). Here we don't restrict the analysis to
finite iso-stress crack growth, since the results in the previous
section allow us to consider the crack growth of any (yet elliptical)
shape.

Hence, let us assume a finite crack growth where the new crack front
has the shape of an ellipse (see Figure~\ref{fig6}b) (whose symmetry
axes coincide with the ones of the original flaw elliptical geometry).
Its shape is univocally determined by two parameters, namely the
increment of the minor axis $\Delta b$ and the increment of the
major axis $\Delta a$. As such, this approach includes as
particular cases the iso-stress crack growth (Figure~\ref{fig4}a) and
the minor-axis crack growth (Figure~\ref{fig4}b). Since the iso-stress
lines are also elliptical, the lowest stress level within the crack
extension (darker area in Figure~\ref{fig6}) is reached either at $A'$ or
$B'$ (Figure~\ref{fig6}b). Thus, in order the stress condition to be
fulfilled, it is enough to check that ${\sigma}_{{z}}(A') \geq
\sigma_{\mathrm{c}}$ and ${\sigma}_{{z}}(B') \geq \sigma_{\mathrm{c}}$;
meanwhile FFM also requires that the energy available for crack growth
$\Delta \Phi$ is larger than the energy dissipated to create
the new fracture surface $\Delta A$. That is:
{\begin{equation}\label{eq34}
\begin{cases} \sigma _{z}({A}')\geq \sigma _{\mathrm{c}}\\
\sigma _{z}({B}')\geq \sigma _{\mathrm{c}}\\
\Delta \Phi \geq G_{\mathrm{c}}\,\Delta A
\end{cases} 
\end{equation}}\unskip
It is worth noting the analogy with the approach presented
in~\cite{44}, where the growth of an embedded elliptical crack under
fatigue loading was analysed by analytically coupling and solving the
Paris laws at points A and B, while assuming that the crack front
maintains an elliptical shape throughout its propagation.

Beyond the ratio $\upvarepsilon_{b}$ between the final value of the
minor axis and its initial value---Equation~(\ref{eq31})---we now also
introduce the ratio $\upvarepsilon_{a}$ {between the final value of the
major axis and its initial value:
{\begin{equation}\label{eq35}
\upvarepsilon _{a}=\frac{a+\Delta a}{a}
\end{equation}}\unskip


The first inequality in Equation~(\ref{eq34}) is achieved by means of
Equation~(\ref{eq24}) along with the following ${\upxi}$ value (coming
from Equation~(\ref{eq25})):
{\begin{equation}\label{eq36}
\sqrt{a^{2}+\upxi }=a+\Delta a\Rightarrow 
\upxi =({\upvarepsilon }_{a}^{2}-1)a^{2}
\end{equation}}\unskip
yielding (by Equation~(\ref{eq35}) as well):
{\begin{equation}\label{eq37}
\frac{\sigma }{\sigma _{\mathrm{c}}}\geq E(k^{2})\left[\frac{1}
{\upvarepsilon _{a}}\sqrt{\frac{{\upvarepsilon }_{a}^{2}-k^{2}}
{{\upvarepsilon }_{a}^{2}-1}}+E(k^{2})-E\left(\arcsin 
\frac{1}{\upvarepsilon _{a}}\left| \,k^{2}\right.\right)
\right]^{-1}=f_{Sa}(\upvarepsilon _{a},k^{2})
\end{equation}}\unskip
The second inequality in Equation~(\ref{eq34}) is achieved by means of
Equation~(\ref{eq24}) along with the following ${\upxi}$ value (coming
again from Equation~(\ref{eq25})):
{\begin{equation}\label{eq38}
\sqrt{b^{2}+\upxi }=b+\Delta b\Rightarrow \upxi =\left({\upvarepsilon }_{b}^{2}-1\right)b^{2}
\end{equation}}\unskip
yielding (by Equation~(\ref{eq31}) as well):
{\begin{equation}\label{eq39}
\frac{\sigma }{\sigma _{\mathrm{c}}}\geq E(k^{2})\left[\frac{{\upvarepsilon }_{b}^{2}}
{\sqrt{{\upvarepsilon }_{b}^{2}-1}\sqrt{\left(1-k^{2}\right)
{\upvarepsilon }_{b}^{2}+k^{2}}}+E(k^{2})-E\left(\arcsin 
\frac{1}{\sqrt{\left(1-k^{2}\right){\upvarepsilon }_{b}^{2}+k^{2}}}
\left| k^{2}\!\right.\right)\right]^{-1}\!
=f_{{Sb}}\left(\upvarepsilon _{b},k^{2}\right)
\end{equation}}\unskip

\begin{figure}
\includegraphics{fig07}
\caption{\label{fig7}Graphical representation of the minimum
search---Equation~(\ref{eq44})---for ellipse eccentricity ${k}$
(flaw shape) equal to 0.8. The dots correspond to the position of the
minimum. Small flaw size (${\beta} = 0.8$), corresponding to
iso-stress crack growth scenario (a); large flaw size (${\beta} =10$), 
corresponding to minor-axis crack growth scenario (b); limit
(${\beta} \cong 1.45$) case (c).}
\end{figure}

The third inequality in Equation~(\ref{eq34}) follows from
Equation~(\ref{eq23}) and the area of the crack increment  $\Delta
A$:
{\begin{eqnarray}\label{eq40}
\Delta \Phi &amp;=&amp;\frac{4\uppi }{3}\frac{\sigma ^{2}}{E'}
\left\{\frac{\left(a+\Delta a\right)\left(b+\Delta b\right)^{2}}
{E\left[1-\left(\frac{b+\Delta b}{a+\Delta a}\right)^{2}\right]}-
\frac{ab^{2}}{E\left[1-\left(\frac{b}{a}\right)^{2}\right]}\right\} \Seqnsplit
\label{eq41}
\Delta A&amp;=&amp;\uppi \left[\left(a+\Delta a\right)\,\left(b+\Delta b\right)-a\,b\right]
\end{eqnarray}}\unskip

Let us introduce the dimensionless flaw size ${\beta}$ as:
{\begin{equation}\label{eq42}
\beta =\frac{b}{\left(K_{\mathrm{Ic}}/\sigma _{\mathrm{c}}\right)^{2}}=\frac{b}{l_{\mathrm{ch}}}
\end{equation}}\unskip
where ${l}_{\mathrm{ch}}$ is (Irwin's) material characteristic length.
Equation~(\ref{eq42}) highlights that the crack size is a relative
concept: what really matters is the ratio of the size to the
characteristic length. Note also that ${\beta}$ is the square of the
inverse of the so-called brittleness number introduced by
Carpinteri~\cite{45}, ${s} = {K}_{\mathrm{Ic}} / ({\sigma}_{\mathrm{c}}
{\surd{b}})$. By Equations~(\ref{eq40}) and (\ref{eq41}), the third
inequality in Equation~(\ref{eq34}) in dimensionless form finally
reads:
{\begin{equation}\label{eq43}
\frac{\sigma }{\sigma _{\mathrm{c}}}\geq \frac{\sqrt{3}}{2\sqrt{\beta }}
\sqrt{\frac{\upvarepsilon _{a}\upvarepsilon _{b}-1}{\frac{\upvarepsilon _{a}
{\upvarepsilon }_{b}^{2}}{E\left[1-\frac{(1-k^{2}){\upvarepsilon }_{b}^{2}}
{{\upvarepsilon }_{a}^{2}}\right]}-\frac{1}{E(k^{2})}}}=
\frac{1}{\sqrt{\beta }}f_{E}(\upvarepsilon _{a},\upvarepsilon _{b},k^{2})
\end{equation}}\unskip
Hence, according to FFM, for a given defect shape (${k}$) and size
(${\beta}$), the relative failure stress ${\sigma}_{\mathrm{f}}/{\sigma}_{\mathrm{c}}$
is univocally determined. More precisely, ${\sigma}_{\mathrm{f}}$ is the minimum
value, for any crack increment ${\upvarepsilon}_{{a}} &gt; 1$ and
$\upvarepsilon_{{b}}&gt; 1$, satisfying the three inequalities in
Equation~(\ref{eq34}). That is:
{\begin{equation}\label{eq44}
\sigma _{\mathrm{f}}=\sigma _{\mathrm{c}}\times \min _{
\upvarepsilon _{a}&gt; 1\atop
\upvarepsilon _{b}&gt; 1}\max \left\{f_{Sa}\left(\upvarepsilon _{a},k^{2}
\right),\,f_{{Sb}}
\left(\upvarepsilon _{b},k^{2}\right),\frac{1}{\sqrt{\beta }}f_{E}
\left(\upvarepsilon _{a},\upvarepsilon _{b},k^{2}\right)\right\}
\end{equation}}\unskip
Let us fix ${k}$, e.g.\ $k = 0.8$. Depending on ${\beta}$, we may have
just two scenarios, as evident in Figure~\ref{fig7}. For small
${\beta}$ (Figure~\ref{fig7}a), i.e.\ for relatively small flaw size,
the minimum load is achieved for iso-stress crack growth
(Figure~\ref{fig4}a). The finite crack growth is significant (i.e.\ 
${\upvarepsilon}_{{a}}$ and ${\upvarepsilon}_{{b}}$ significantly
larger than unity) and the relationship between the increment of the
two semi-axes is given by Equation~(\ref{eq26}) that, in terms of
${\upvarepsilon}_{{a}}$ and ${\upvarepsilon}_{{b}}$, can be more
conveniently expressed as (see Appendix~\ref{appC} for details):
{\begin{equation}\label{eq45}
\upvarepsilon _{a}=\sqrt{k^{2}+\left(1-k^{2}\right){\upvarepsilon }_{b}^{2}}
\end{equation}}\unskip
Note that in this case the three inequalities in Equation~(\ref{eq34})
are strictly fulfilled. This former scenario is the iso-stress one.

The second scenario (Figure~\ref{fig7}b) is linked to large
${\beta}$, i.e.\ for relatively large flaw size: the minimum load is
achieved for minor-axis crack growth (Figure~\ref{fig4}b). The finite
crack growth is relatively small (i.e.\ ${\upvarepsilon}_{{b}}$ slightly
higher than unity) and, of course, ${\upvarepsilon}_{{a}} = 1$ since
there is no increment in the major axis of the flaw. Note that in this
latter case the second and third inequalities in Equation~(\ref{eq34})
are strictly fulfilled while the first one is over fulfilled. This
latter scenario is the minor-axis one. The discriminant between the two
scenarios is when the failure stress is the same (see
Figure~\ref{fig7}c), i.e.:
{\begin{equation}\label{eq46}
f_{E}\left(\upvarepsilon _{a}=1,\upvarepsilon _{b}\right)=f_{E}
\left(\upvarepsilon _{a}=\sqrt{k^{2}+(1-k^{2})
{\upvarepsilon }_{b}^{2}},\upvarepsilon _{b}\right)
\end{equation}}\unskip
This is an equation in the unknown ${\upvarepsilon}_{{b}}$. Let us name
$\overline{\rmvarepsilon }_{b}$ its solution. The corresponding
dimensionless threshold size ${\beta}_{\mathrm{th}}$ is obtained by equating
the stress condition (either the first or the second one in
Equation~(\ref{eq34})) with the energy one (third condition in
Equation~(\ref{eq34})). That is:
{\begin{equation}\label{eq47}
\beta _{\mathrm{th}}=\left[\frac{f_{E}\left(\upvarepsilon _{a}=1,
\overline{\upvarepsilon }_{b}\right)}{f_{{Sb}}
\left(\overline{\upvarepsilon }_{b}\right)}\right]^{2}
\end{equation}}\unskip
For ${k} = 0.8$, we found ${\beta}_{\mathrm{th}} \cong 1.45$. Then,
if ${\beta} &lt; \beta_{\mathrm{th}}$, iso-stress scenario takes
place. Accordingly, the finite crack growth (through
${\upvarepsilon}_{{b}}$) is given by the solution of the following
equation:
{\begin{equation}\label{eq48}
\sqrt{\beta }\,f_{{Sb}}\left(\upvarepsilon _{b}\right)=
f_{E}\left(\upvarepsilon _{a}=\sqrt{k^{2}+(1-k^{2})
{\upvarepsilon }_{b}^{2}},\upvarepsilon _{b}\right)
\end{equation}}\unskip
On the other hand, if ${\beta} &gt; {\beta}_{\mathrm{th}}$,
minor-axis scenario takes place. Accordingly, the finite crack growth
(through ${\upvarepsilon}_{{b}}$) is provided by solving:
{\begin{equation}\label{eq49}
\sqrt{\beta }\,f_{{Sb}}\left(\upvarepsilon _{b}\right)=
f_{E}\left(\upvarepsilon _{a}=1,\upvarepsilon _{b}\right)
\end{equation}}\unskip
Let us denote by ${\upvarepsilon}_{b\mathrm{c}}$ the solution of either
Equation~(\ref{eq48}) or (\ref{eq49}). In both cases the failure stress
is given by either the stress or energy condition. Taking the first we
have:
{\begin{equation}\label{eq50}
\sigma _{\mathrm{f}}=\sigma _{\mathrm{c}}\,f_{{Sb}}\left(\upvarepsilon _{b\mathrm{c}}\right)
\end{equation}}\unskip
For clarity, the flow chart providing the finite crack growth and the
failure stress is given in Figure~\ref{fig8}.

\begin{figure}
\includegraphics{fig08}
\caption{\label{fig8}Flowchart to determine the failure stress
${\sigma}_{\mathrm{f}}$ for a given flaw shape ($k$) and size~(${\beta}$).}
\end{figure}

In Figure~\ref{fig9} the failure stress vs.\ flaw size plots for
different ellipse eccentricity values are drawn. The threshold size
according to which there is the switch between scenarios is also
highlighted. While the iso-stress scenario prevails for small
eccentricity (being the only one for a penny-shaped crack), the
minor-axis scenario becomes predominant for large eccentricity. Note
also that FFM reverts to LEFM (in its minor-axis crack growth version)
as the flaw size increases; on the other hand, for small flaw size,
LEFM provides unrealistically high failure stresses, while FFM
predictions remain always lower than the material tensile strength.

\begin{figure}
\includegraphics{fig09}
\caption{\label{fig9}Failure stress vs.\ dimensionless flaw size
${\beta} = {b}/{l}_{\mathrm{ch}}$ for different flaw shape (i.e.\ 
ellipse eccentricity ${k}$ or aspect ratio ${\gamma} = {b}/{a}$)
according to LEFM (minor axis crack growth) and to FFM: ${\gamma}
=0.8$,  ${k} = 0.6$ (a); ${\gamma} = 0.5$, ${k} \cong 0.85$ (b);
${\gamma} = 0.2$, ${k} \cong 0.98$ (c). Threshold value of
${\beta}$ dividing iso-stress (left) and minor axis (right) finite
crack growth are also highlighted.}
\end{figure}

In Figure~\ref{fig10}a the different curves in Figure~\ref{fig9} are
compared altogether, including the ones referring to a
through-the-thickness crack of semi-length ${b}\ ({k} = 1)$ and to a
penny-shaped crack of radius ${b}\ ({k} = 0)$. Note that these two latter
cases are 2D geometries (the first one is actually a 2D problem; the
second one is a pseudo 3D problem, due to the radial symmetry); as
such, their solution is simpler and already available in the
literature~\cite{16,17}. Of course, the comparison being made at
constant ${b}$, the failure stress decreases as the eccentricity
increases. In Figure~\ref{fig10}b the same comparison is provided at a
constant flaw area (the through-the-thickness crack case is somewhat
meaningless, since ${A}$ constant and ${a}$ infinite yield ${b}$ null).
Again, as in the LEFM analysis, it is apparent that, for relatively
small eccentricity, the parameter governing failure stress is the flaw
area. In other words, for flaw aspect ratio ${b}/{a}$ between 0.5 and 1,
the failure stress due to the presence of an elliptical crack is
(almost) equal to the one due to a penny-shaped crack of the same area.

\begin{figure}
\includegraphics{fig10}
\caption{\label{fig10}FFM estimates of the failure stress for different
flaw shape  (${k}\cong 0$, 0.6, 0.85, 0.98, 1; i.e.\ ${\gamma} = 1$,
0.8, 0.5, 0.2, 0) vs.\ dimensionless flaw size ${\beta}
={b}/{l}_{\mathrm{ch}}$ (a) and vs.\ dimensionless flaw area
$({A}/{l}_{\mathrm{ch}}^{2})$ (b).}
\end{figure}

Note that, whatever is the scenario, the elliptical crack always grows
toward an elliptical shape closer to that of a circle (with respect to
the original elliptical shape), i.e.\ the eccentricity diminishes. This
is a common finding in the literature, even for original flaws of shape
other than the elliptical one~\cite{35,38,43,44}.

Regarding the difference between iso-stress and non-iso-stress finite
crack growth, for small sizes fracture propagates actually by
iso-stress lines. For larger size (i.e.\ ${\beta}&gt;
\beta_{\mathrm{th}}$) the iso-stress failure stress prediction
becomes larger than the minor axis one. Let us consider for instance
the case considered in Figure~\ref{fig7}b, where ${k} = 0.8$ and
${\beta} = 10$: although it is clear that the minimum load is
achieved for minor-axis growth (${\upvarepsilon}_{{a}} = 1$), it is
also apparent that the failure stress corresponding to iso-stress crack
growth is just slightly larger. Actually the difference between the two
predictions increases with size, i.e.\ ${\beta}$. For
$\beta\rightarrow \infty$, FFM reverts to LEFM and the difference
between minor-axis and iso-stress predictions can be directly
determined from Figure~\ref{fig5}a. For instance, for ${k} =0.8$,  the
relative difference (i.e.\ the error made using the iso-stress
assumption) is about 3\%, which is almost negligible from an
engineering point of view. Figure~\ref{fig5}a also shows  that the
largest error takes place when $\beta\rightarrow\infty$ and ${k} =
1$, i.e.\ a large through-the-thickness crack: in this extreme case the
difference is
$[\surd{(3/8)}-\surd{(1/{\uppi})}/\surd{(1/{\uppi})}] \cong
8.5\%$. 
Thus, we can conclude that, for the geometry at hand and for
ellipse aspect ratios not really close to zero (i.e.\ except for case
${b} \ll {a}$), the error made by using the iso-stress crack growth
assumption is reasonably small. Of course, this does not mean that this
is always the case, but the present case corroborates the conjecture
made by Leguillon~\cite{25}, i.e.\ iso-stress crack growth can be a
reasonable and effective simplifying assumption.

Finally, note that extending our analysis to a more complex stress
state, such as the one occurring to the present geometry when the
remote tensile stress is not normal to the crack plane, would broaden
the applicability of the paper. However, given the mode mixity (I, II,
III) and the expected non-planar (unknown) crack growth, this is a
major, challenging task going beyond the scope of the current
manuscript. In this sense, it would be reasonable to start with an
inclined penny shaped crack, which is a configuration investigated in
the past with simpler fracture criteria, see e.g.~\cite{46}; see
also~\cite{47} for recent interesting experimental data.

\section{Conclusions}\label{sec4}

The failure remote stress causing (unstable) crack propagation in an
infinite linear elastic 3D domain containing a flat elliptical crack
has been obtained in an analytical form by means of the CCFFM, under
the assumption that the finite crack growth can be of any elliptical
shape. It is found that finite crack growth always leads to elliptical
crack geometries with lower eccentricity, i.e.\ the crack shape tends to
that of a penny-shaped crack. Differently from other investigations
available in the literature, in this 3D application of FFM we removed
the assumption of iso-stress crack growth. Particularly for large flaws
we found failure stress values lower than the ones obtained by the
iso-stress assumption, which, thus, must be seen as potentially
dangerous since providing non-conservative predictions. Nevertheless,
the difference appears to be of a few percents and, thus, the
iso-stress assumption when applying CCFFM is regarded as more than
reasonable in engineering practice. It is noteworthy that for small
elliptical defects, the failure remote stress predicted by the present
FFM procedure can be significantly smaller than that obtained by LEFM
assuming infinitesimal crack growth. Notably, the failure remote stress
predicted by LEFM depends on the assumed shape of the infinitesimal
crack increment.


\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*{Acknowledgements}

The present manuscript is the final version of what presented in the
Sevilla workshop within NEWFRAC Project (Marie Sklodowska-Curie grant
agreement No. 861061), which is gratefully acknowledged by all the
Authors, and later, at the International Conference of Fracture held in
Atlanta, June 2023 (ICF15). In Atlanta the Authors had---as
usual---fruitful and stimulating discussions about the topic
investigated in the present manuscript with Prof.~Dominique Leguillon:
although he passed away, we wish to thank him and dedicate the present
work to his memory. ZY is grateful for the partial support of this
research by the Pazy Research Foundation. The research of VM was
partially supported by the Spanish Ministry of Science and Innovation
and the European Regional Development Fund (PID2021-123325OB-I00).
Finally, PC wishes to thank also Prof.~Veronique Lazarus for pleasant
and useful discussions had at Sorbonne Universit\'{e}, where he was
upon Prof.~Leguillon's invitation.

\CDRGrant[NEWFRAC]{861061}
\CDRGrant[ERDF]{PID2021-123325OB-I00}

\back{}

\appendix{}

\def\appendixlabel{Appendix \Alph{section}}

\setcounter{equation}{0}

\def\theequation{\thesection\arabic{equation}}
\section{}\label{appA}


The Contra-harmonic mean ($C$) was introduced by Eudoxus from
Cnidus (408--355 B.C.) as the ratio between the sum of the squares of
the values and the sum of the values themselves. The name is due to the
fact that, if we consider two values $a$ and $b$, the distance
between the Arithmetic mean ($A$) and the Harmonic mean ($H$)
is equal to the one between the Contra-harmonic mean ($C$) and
the Arithmetic mean ($A$):
{\begin{eqnarray}\label{eqa1}
A(a,b)&amp;=&amp;\frac{a+b}{2}\Seqnsplit
H(a,b)&amp;=&amp;\frac{2}{1/a+1/b}\Seqnsplit
C(a,b)&amp;=&amp;\frac{a^{2}+b^{2}}{a+b}\Seqnsplit
C(a,b)-A(a,b)&amp;=&amp;A(a,b)-H(a,b)
\end{eqnarray}}\unskip

For instance, if ${a} = 9$ and ${b} = 1$, ${A} = 5$, ${H} = 1.8$, ${C}
= 8.2$ (${C}{-}{A} = 3.2 = {A}-H$).


\section{}\label{appB}
\setcounter{equation}{0}

The complete elliptic integral of the first kind reads:
{\begin{equation}\label{eqb1}
K\left(m\right)={\int }_{0}^{\uppi /2}
\frac{1}{\sqrt{1-m\sin ^{2}\varphi }}\,\mathrm{d}\varphi
\end{equation}}\unskip
The complete elliptic integral of the second kind reads:
{\begin{equation}\label{eqb2}
E\left(m\right)={\int }_{0}^{\uppi /2}\sqrt{1-m\sin ^{2}\varphi }
\,\mathrm{d}\varphi 
\end{equation}}\unskip

The derivative of the elliptic integral of the second kind with respect
to ${m}$ is:
{\begin{equation}\label{eqb3}
\frac{\mathrm{d}E(m)}{\mathrm{d}m}=\frac{E(m)-K(m)}{2m}
\end{equation}}\unskip

The incomplete elliptic integral of the second kind is:
{\begin{equation}\label{eqb4}
E\left(\varphi \,\left| \,m\right.\right)={\int }_{0}^{\varphi }
\sqrt{1-m\sin ^{2}\upvartheta }\,\mathrm{d}\upvartheta
\end{equation}}\unskip

The relationship between the incomplete elliptic integral and its
complete counterpart is:
{\begin{equation}\label{eqb5}
E\left(\frac{\uppi }{2}\left| \,m\right.\right)=E\left(m\right)
\end{equation}}\unskip


\section{}\label{appC}

\setcounter{equation}{0}

Here we derive the relationship between the semi-axes increments in
case of iso-stress crack growth, in
dimensional---Equation~(\ref{eq26})---and
dimensionless---Equation~(\ref{eq45})---form.

From Equation~(\ref{eq36}) we have:
{\begin{equation}\label{eqc1}
\Delta a=\sqrt{a^{2}+\upxi }-a
\end{equation}}\unskip
while, from Equation~(\ref{eq38}):
{\begin{equation}\label{eqc2}
\upxi =\left(\Delta b+b\right)^{2}-b^{2}
\end{equation}}\unskip

Replacing (\ref{eqc2}) into (\ref{eqc1}), we get Equation~(\ref{eq26}).


From Equation~(\ref{eq36}) we have also:
{\begin{equation}\label{eqc3}
{\upvarepsilon }_{a}^{2}=\frac{\upxi }{a^{2}}+1
\end{equation}}\unskip
while, from Equation~(\ref{eq38}):
{\begin{equation}\label{eqc4}
\upxi =b^{2}\left({\upvarepsilon }_{b}^{2}-1\right)
\end{equation}}\unskip

Replacing (\ref{eqc4}) into (\ref{eqc3}) and recalling that, by
eccentricity definition, $(b/a)^{2}=1-k^{2}$, we get
Equation~(\ref{eq45}).}

\printbibliography
\refinput{crmeca20250112-reference.tex}

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