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% <<PhysicaA.tex>> 
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% Author to be contacted: J.P. Vilotte
% Address:
%   D\'epartement de Sismolgie, 
%   Unit\'e Mixte de Recherche UMR 7580
%   Institut de Physique du Globe de Paris
%   4, Place Jussieu
%   75252 Paris Cedex 05, France
%
% Tel. +33 1 44 27 38 88
% Fax. +33 1 44 27 38 94
% email: vilotte@ipgp.jussieu.fr
%
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\documentclass{elsart}

\usepackage{epsfig}

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

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\begin{document}
\begin{frontmatter}

\title{Interfacial crack front wandering: influence of quenched
noise correlations.}

\author{Jean Schmittbuhl$^{(1)}$, Jean-Pierre Vilotte$^{(2)}$}
\address{
(1) D\'epartement de G\'eolgie,\\
    Unit\'e Mixte de Recherche UMR 8538,\\
    \'Ecole Normale Sup\'erieure,\\
    24 rue Lohmond, 75232 Paris cedex 05, France.\\
\vskip5pt
(2) D\'epartement de Sismologie,\\
    Unit\'e Mixte de Recherche CNRS UMR 7580\\
    Institut de Physique du Globe de Paris,\\
    4 Place Jussieu, 75252 Paris cedex 05, France.}
\date{\today}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
We discuss the influence of quenched noise correlations on the 
wandering exponent of the crack-front line during slow in-plane 
crack propagation. For a uniform quenched noise, the wandering 
exponent in the stastically stationary regime is $\zeta = 0.35 \pm 
0.02$, while the dynamic exponent, in the transient regime, is $z = 
0.75 \pm 0.02$. For short-range Gaussian correlations, the wandering
exponent is shown to be $\zeta = 0.50 \pm 0.03$ in the small 
fluctuations limit and $\zeta = 0.35 \pm 0.02$ in the large fluctuations
limit which can be shown to be consistent with the uniform case. In the 
case of long-range correlations, characterized by a self-affine exponent
$\zeta_t$, the wandering exponent is shown to scale as $\zeta = \zeta_t 
+ 1$ for a wide range of $\zeta_t$ between $-$1.0 and $+$1.0.
These results are discussed with reference to recent experiments of 
slow in-plane crack propagation between two annealed Plexiglas blocks.
\end{abstract}
\end{frontmatter}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}

Understanding the propagation of rupture front through heterogeneous elastic
solids emerges as a key issue for a large number of geophysical and mechanical
problems. Even when the rupture front is interfacial, \ie{} restricted to move 
in a plane, the dynamics involves complicated interactions between non local 
elastic coupling, inertia and wave effects, as well as quenched heterogeneities.
In-plane crack front propagation along an heterogeneous interface is observed as
non uniform with segments of the front that are trapped temporarily by local 
asperities that may be related to local material heterogeneities or residual
stresses. The depinning from these asperities involves local instabilities.
Even though the exact mechanism of crack/asperity
interactions has not yet been clarified, it requires explicit three dimensional
modeling due to the fact that long range bulk elastic interactions has to be
accounted for the determination of the stress intensity factor variations along
the crack front. When the elastic coupling is small, \ie{} strong pinning, the 
dynamics is controlled by individual instabilities. When the elastic coupling 
is strong, \ie{} weak pinning, the dynamics is controlled by the whole front. 
Moreover when local asperity depinning occurs, dynamical coupling can develop
from waves that result in stress overshoots elsewhere on the front.

In recent years, there has been theoretical developments within the context of
crack theory. In the framework of first-order perturbation theory a
relationship between the stress intensity factor for tensile crack (Mode I) and 
small deviations from straightness of the edge of a semi-infinite plane crack,
extending at constant speed in an infinite elastic body, was first proposed 
\cite{GaoRic86,GaoRic89} using weight functions \cite{Bue70,Ric85} in
the approximation of scalar elasticity.  This work has been extended to 
dynamic crack growth both for scalar and vectorial elasticity 
\cite{RicZioKim94,PerRic94,WilMov95,RamFis97} as well as for non planar crack
growth under mixed loading \cite{Gao92,Xuetal94,Rametal97,WilMov97}. 

In an early study \cite{SchVilRou95a}, it was shown, using a first-order linear perturbation theory for scalar elasticity, that the competition between the 
elastic energy and the pinning due to the disorder of the interface gives rise,
during slow in-plane crack propagation, to a non trivial wandering exponent 
of the crack front. The stationary state was found 
to be characterized by an in-plane wandering exponent $\zeta$ of the crack front,that is $\zeta \approx 1/3$, for uncorrelated quenched noise and by 
spatio-temporal correlations, see also \cite{ThoPac96} for a different result. 
Recently, this study was extended to take into account the effects of different
elastic interactions \cite{TanGouRou98}, from local to long range interactions,
as well as possible dynamic effects such as elastic waves \cite{RamFis97}.
The extending in-plane tensile crack problem can be regarded as a specific 
example of driven elastic manifolds in random media 
\cite{HalZan95,LesNatStepTan96,Fish97}. 

Indeed this problem bare some strong similarities with other interface problems in condensed matter such as the dynamics of the solid-liquid-vapor contact 
line depinning for partially wetting liquid on a disordered substrate 
\cite{JoaGen84,Gen85}. In the latter, the wandering exponent can be derived 
for weak disorder from scaling arguments \cite{Lar70,PomVan85}, further 
developed in \cite{Gen85,RobJoa87}. From these analysis, the wandering 
exponent for the lateral fluctuations of the contact line should be $\zeta 
\approx 1/2$ on length scales less than the Larkin length and $\zeta \approx 
1/3$ on length scales between the Larkin and the capillary length. A functional 
renormalization calculation has been performed \cite{ErtKar94} for a field 
theoretical description of the dynamics of the contact line, subject to a 
uniform pulling force, using the formalism developed in 
\cite{MarSigRos73,NarFis93}. The predicted wandering 
exponent was also $\zeta = 1/3$. Using the replica method, together with a 
Gaussian variational approximation, with replica symmetry breaking 
\cite{MezPar91}, it has been possible \cite{HazMez98} to confirm the existence 
of the two scaling exponents already suggested \cite{Gen85,RobJoa87} and to 
provide a quantitative computation of the correlation function of the contact 
line.

The aim of this paper is to go beyond our previous analysis of the dynamics of
slow in-plane tensile crack along a disordered interface and to shed some light
on the influence of possible correlations in the pinning forces, \ie{} quenched
heterogeneities, on the dynamics. This will be discussed with reference to
recent experimental estimation of the scaling exponent. The paper is organized 
as follow: we introduce in section (2) the experimental data and in section (3),
the model; in section (4), we summarize the main results for
uncorrelated uniform quenched disorder; in sections (5) and (6), we 
present numerical predictions for the scaling of the correlation 
function of the crack front for short and long range correlated 
disorder along the interface; finally in section (7) we point out some 
possible extensions of our study based on the comparison with experimental 
data.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Experiment}

There has been very few experimental studies of the propagation of a single
crack front along an interface (see e.g. Ref.\cite{MowArg95} for an in-plane
experiment in a slightly different context). Recently, Schmittbuhl and
M\r{a}l{\o}y~\cite{SchMal97} provided an experimental study of the dynamics of
a planar crack front along the annealed interface between two Plexiglas blocks.
This interface was roughened, before the annealing, by sand blasting with grains
of size $\approx 50$ $\mu$m. The size of the induced geometrical flaws on the
interface can be measured experimentally and shown to scale with the grain
size. The blocks are annealed together in a press at $205^{\circ}$ C and then
cooled down slowly. Roughness fluctuations of the interface induce, at the end
of the annealing procedure, toughness variations along the interface. However, 
the details of the induced toughness disorder, such as the correlation length,
is difficult to measure and have not yet been fully characterized even though it 
plays as we shall see an important role. A step by step loading with an imposed 
crack opening allows a very slow extension of the in-plane tensile
crack. Typical velocities are within the range of $10^{-7}$ ms$^{-1}$ to 
$10^{-5}$ ms$^{-1}$ and the wandering exponent of the crack front is shown to
be independent of the mean crack front velocity. All the experiments were made 
at room temperature. Direct observation of the crack-front geometry is possible
due to the Plexiglas transparency. A microscope mounted on a translation stage 
equipped with a high resolution photographic camera (1536$\times$1024 pixels) 
provides an accurate description of the crack front geometry and its evolution 
during the crack extension. 

The correlation function of the position of the crack front is measured and 
shown to increase as a power law, which defines the wandering exponent 
$\zeta$. The experimental prediction of the wandering exponent is found to be 
$\zeta \approx 0.55 \pm 0.05$. This experiment has been recently re-explored by
Delaplace et al. \cite{SchMal97} with a much higher resolution owing to the
assembly of up to 12 pictures and a careful optical check of the set-up. The
measured wandering exponent was found to be $\zeta = 0.63 \pm 0.03$. Both
results agree within the error bars and are above the numerical prediction
of $\zeta \approx 1/3$ \cite{SchVilRou95a}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Model}

We consider here a half-plane tensile crack propagating through an unbounded
elastic solid. The crack is restricted to move in the plane $x_3 = 0$, along
the $x_1$ direction. The crack front position, at time $t$, is described by a
curve $x_1=\Phi(x_2,t)$ that is assumed to be single valued excluding 
configurations with overhangs. To first-order, $\Phi(x_2,t) = v_{\mathrm{cr}}t 
+ \phi(x_2,t)$ where $v_{\mathrm{cr}}$ is a uniform crack extension velocity and
$\phi(x_2,t)$ are small amplitude distortions around that configuration with 
zero mean. 

The energy per unit area of the interface that must be provided to fracture the 
solid is denoted $\Gamma\left(x_1=\Phi(x_2,t),x_2\right)$. Interface 
heterogeneities are considered here as fluctuations in the local energy
density: $\Gamma(x_1,x_2) = \Gamma_0 \left[1 + \gamma(x_1,x_2)\right]$ 
where $\Gamma_0$ is the mean value of the interface toughness and
$\gamma(x_1,x_2)$ random quenched fluctuations of zero mean.

The analysis is restricted to a scalar approximation of elasticity 
\cite{RicZioKim94}. The displacement field is then a scalar variable 
$u(x_1,x_2,x_3,t)$ and the associated tensile stress across planes parallel to 
the crack is $\sigma = \mu \partial_{x_3} u$, with $\mu$ the elastic modulus. 
The problem is to find $u$ satisfying the three-dimensional scalar 
wave equation: $c^2 \nabla^2 u \; = \; \partial^2 u/\partial t^2$
with the sound speed $c^2 = \mu/\rho$ where $\rho$ is the density. 
The displacement field has a discontinuity across the crack surface 
$\Delta u(x_1,x_2,t) = u(x_1,x_2,0^+,t) - u(x_1,x_2,0^-,t)$, $\forall x_1 < 
\Phi(x_2,t)$. The stress must satisfy the boundary condition $\sigma = 0$ on 
the crack surface,\ie{} $\partial_{x_3} u(x_1,x_2,x_3=0^{\pm},t) = 0$ and meets 
the remote loading condition, \ie{} the static external load. When the crack 
front arc has a continuously turning tangent, the structure of the singular 
field along a 3D crack edge is shown~\cite{RicZioKim94} to be the same as 
for solutions of the 2D plane strain, or anti-plane strain, crack extension
problem at uniform (possibly zero) speed $V_{\mathrm{cr}}$, where $u = 
u(x_1-v_{\mathrm{cr}},x_2)$, and which has been extensively studied by 
Kostrov~\cite{Kos66,Kos75} and Eshelby~\cite{Esh69}. Thus the stress field
$\sigma$ at distance $r = \Phi(x_2,t)-x_1$ ahead of the crack tip, on the plane
$x_3=0$, has the square-root singularity $\sigma \propto (2\pi r)^{-1/2} 
K^{\sigma}(v)$ and the displacement discontinuity at distances $r$ behind the
crack tip satisfies $\Delta u \propto (r/2\pi)^{-1/2} K^{u}(v)/2\mu$, 
where $K^{\sigma}(v) = K_{\mathrm{stat}} (1-v/c)^{1/2}$ and $K^{u}(v) =
K_{\mathrm{stat}} (1+v/c)^{-1/2}$ are respectively the stress and strain
dynamic intensity factors, with $v$ the local propagation speed of the crack 
front, $v=\partial_t \Phi(x_2,t)$ and $v < c$. 

The fracture energy $G$ that is provided to the crack front by the flux of 
stored elastic energy per unit of new crack area can be written in terms of the 
stress and strain intensity factors $G(v) = K^{\sigma}(v)K^{u}(v)/2\mu$ and has
the general form \cite{RamFis97}: $G(v) = f\left[v_{\perp}(x_2,t)\right]
G_{\mathrm{R}}$ where $v_{\perp} = \partial \Phi(x_2,t)/\partial t [1 + 
(\partial \Phi(x_2,t)/\partial x_2)^2]^{-1/2}$ is the local velocity normal to 
the crack front. The local rest value of the energy release rate 
$G_{\mathrm{R}}$ is then independent of $v$ and depends of the prior growth 
history. When the crack font line is straight, and only outgoing waves are 
involved without effect of reflections from the opposite end of the crack, 
$G_{\mathrm{R}}$ has a simple form $G_{\mathrm{R}} = G^{\infty} = 
{K^{\infty}}^2/2\mu$ where $K^{\infty}$ is the static remote load. The function 
$f(v_{\perp})$ controls the fraction of the energy release rate available, it 
satisfies $f(0^+) = 1$ and $f(v_{\perp}) = 0$ when $v_{\perp} \rightarrow c$.

For small crack-front distortions, the energy release rate can be expanded about
a uniformly extending crack \cite{RicZioKim94,WilMov95,RamFis97}, and 
to first-order:
\begin{equation}
G(v) = G_0(v_{\mathrm{cr}}) \left[1 -
\frac{1}{c \alpha^2(v_{\mathrm{cr}})}\,\frac{\partial\phi(x_2,t)}{\partial t} + \mathcal{H}(x_2,t)\right]
\end{equation}
where $\alpha(v_{\mathrm{cr}}) = (1-v^2_{\mathrm{cr}}/c^2)^{1/2}$ and
$G_0(v_{\mathrm{cr}}) = ([1-v_{\mathrm{cr}}/c]/[1+v_{\mathrm{cr}}/c])^{1/2} 
G^{\infty}$. The non local term is given by
\begin{equation}
\mathcal{H}(x_2,t) = \frac{1}{\pi}\,\int_{-\infty}^{\infty} \int_0^{\tau_m}
\frac{c (t-t') \left[\partial_{t'}\phi(x_2',t') - \partial_{t'}\phi(x_2,t)
\right]}{(x_2-x_2')^2 \left[c^2 \alpha^2(v_{\mathrm{cr}}) (t-t')^2 -
(x_2-x_2')^2\right]^{1/2}} dx_2' dt'
\end{equation}
with $\tau_m = \max\left\{0, t-|x_2-x_2'|/\alpha(v_{\mathrm{cr}})c\right\}$.
If the equation of motion is supposed to be ruled by a local energy
conservation, \ie{} the local energy release rate criterion, we get
\begin{equation}
\frac{1}{c \alpha^2(v_{\mathrm{cr}})} \,
\frac{\partial\phi(x_2,t)}{\partial t}\; = \; \mathcal{H}(x_2,t) \, - \,
\tau(x_1=\Phi(x_2,t),x_2)
\end{equation}
where the last term is the actual random driving force on the crack front line 
\begin{eqnarray}
\tau(x_1=\Phi(x_2,t),x_2) \, &=& \, \frac{\Gamma(x_1=\Phi(x_2,t),x_2)}{G_0(v_{\mathrm{cr}})} - 1 \\
 &=& \, \frac{\Gamma_0}{G_0(v_\mathrm{cr})}\, \left[1 + 
\gamma(x_1=\Phi(x_2,t),x_2)\right] - 1 \nonumber
\end{eqnarray}
With reference to the experimental conditions, discussed in the previous 
section, we make several simplifications. First, the averaged speed of the 
crack front is supposed to be slow, \ie{} $v_{\mathrm{cr}} \ll c$, thus 
$G_0(v_{\mathrm{cr}}) \approx G^{\infty}$. Second, we assume that inertia is 
negligible, \ie{} the dynamics is purely dissipative. The actual dynamics
is therefore replaced by the adiabatic approximation, \ie{} at each instant 
$t$ of the evolution forces at every points balance exactly: 
\begin{equation}
\mathcal{H}(x_2)\; - \; \tau(x_1=\Phi(x_2),x_2) \; = \; 0
\end{equation}
where the non local kernel is given by
\begin{equation}
\mathcal{H}(x_2) \; = \; \frac{1}{\pi}\,\int_{-\infty}^{\infty} \frac{\phi(x_2')
- \phi(x_2)}{(x_2 - x_2')^2}\, dx_2' 
\end{equation}
The final Hamiltonian for the quasi-static evolution of the crack front line is 
thus given by
\begin{equation}
\begin{array}{c}
H \, = \, \displaystyle\frac{1}{2} \displaystyle\int_k |k| |\phi(k)|^2 dk \, + 
\, \displaystyle\int_0^L{V\left(\Phi(x_2),x_2)\right) dx_2} \\
V(\Phi(x_2),x_2) \, = \, \displaystyle\int_0^{\Phi(x_2)}{ \tau(x_1,x_2) dx_1} 
\end{array}
\end{equation}
where $V(\Phi(x_2),x_2)$ is a random potential.

We shall consider in the following a periodic in-plane crack front in $x_2$, 
with a spatial period $L$ and slow a propagation speed $v/c \ll 1$. This 
requires a summation of the non local kernel over all the periods and modifies 
the $1/x^2$ interaction to be $(\pi/L)^2/\sin^2(\pi x/L)$. The non local kernel 
becomes therefore, at any time $t$
\begin{equation}
\mathcal{H}(x_2) \; = \; \frac{\pi}{L^2} \int_0^L 
\frac{\phi(x_2') - \phi(x_2)}{\sin^2\left[ \frac{\pi}{L} (x_2' - x_2)\right]} 
\,dx_2'
\end{equation} 
The model is then discretized for numerical simulations. In the $x_2$
direction, the crack front line is described as a periodic array of sites. The
unit length along the $x_2$ direction is given by the distance between the
sites and therefore the system size is $L$ with $L$-nodes. In all the
computations we make use of large system size $L = 4096$.
Since we are interested in the case of very slow extension of the in-plane 
crack, extremal dynamics (ED) is very appealing because it allows to 
grasp the many metastable configurations of the problem. Extremal dynamics is 
based on the assumption that only one site of the crack front line is allowed 
to evolve during one time step, \ie{} at each step the system is loaded by 
tuning $G_0$ in order to bring the most susceptible site along the crack 
front up to instability:
\begin{equation}
G_0 \; = \; \Gamma_0 \min_{x_2}\left\{\frac{1 + \gamma(x_1,x_2)}{1 + \mathcal{H}(x_2)}\right\}
\end{equation}
This search defines also the position of the most susceptible site on the
crack front. The local dynamics is not solved exactly and the crack front
line is depinned at this site with a jump of a fixed amount $\delta x$ 
that gives the unit length in the direction of propagation. The local 
random driving force is locally updated according to the adopted random 
distribution. At that stage, the crack is unloaded and all the forces along 
the crack front line are modified according to the discretized non-local kernel.
The procedure is then repeated. The time scale is fixed by the 
frequency of the updates of single sites and not directly related to the 
real time measured in experiment. All the simulations start with a flat uniform 
crack front and run over a long sequence of such steps, more than 10$^6$, after
the initial transient.

Starting with a straight crack front line, the system organizes after a
transient regime into a statistically highly correlated stationary state. 
Experimentally the correlation function of the position of the crack front 
line in the $x_2$ direction $\langle\left(\Phi(x_2)-\Phi(x_2')\right)^2\rangle 
= \langle \left(\phi(x_2)-\phi(x_2')\right)^2\rangle$ can be computed. One 
manifestation of these long range correlations is the roughening of the crack 
front line that is characterized by the scaling of the correlation 
function that increases as a power law, and defines locally the wandering 
exponent $\zeta$  
\begin{equation}
\langle\left(\phi(x_2)-\phi(x_2')\right)^2\rangle \propto |x_2-x_2'|^{2\zeta}
\end{equation}
The power-law behavior of the correlation function have been analyzed using 
several methods: variable bandwidth, return probability, Fourier spectrum
\cite{SchVilRou95b} and the recently proposed wavelet analysis \cite{SimHan98}. 
The results are shown here using the latter technique. Averaging out the 
dependency on the translation parameter, the wavelet transform of the crack-front
line is shown to obey:
\begin{equation}
\mathcal{W}[\phi](\lambda a) = \langle\mathcal{W}[\phi](\lambda a,\lambda
b)\rangle_b \propto \lambda^{\frac{1}{2}+\zeta} \, \mathcal{W}[\phi](a)
\end{equation}
where $\mathcal{W}[\phi](a,b)$ is the wavelet transform of the crack-front line
perturbation and $(a,b)$ the dilatation and translation parameters of the
analyzing wavelet.

\begin{figure}[ht]
\begin{center}
\epsfig{file=Figures/wav_as.epsi,angle=-90,width=0.68\linewidth}\\[2mm]
\epsfig{file=Figures/wav_uniform.epsi,angle=-90,width=0.68\linewidth}
\end{center}
\caption{\label{CL_unif1}Scaling of the crack-front line for uniform 
quenched disorder. (a) Family-Viscek scaling in the
transient using average wavelet analysis. Symbols correspond to a 
different time steps (using a geometry serie). The power law fit corresponds 
to $\zeta = 0.35$ and the collapse is obtained for $z = 0.75$. (b): 
scaling of the correlation function in the statistically stationary 
state using average wavelet transform. The straight line shows a power-law 
fit that corresponds to $\zeta = 0.35$.}
\end{figure}
\begin{figure}[ht]
\centerline{\epsfig{file=Figures/furu.epsi,angle=-90,width=0.8\linewidth}}
\caption{\label{CL_unif2} Furuberg analysis of the spatio-temporal correlations in 
the stationary state for uniform quenched disorder. The power law fit
corresponds to $b = 2.0$ and the best collapse is obtained for $\beta = 1.33$.}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Uniform quenched disorder}

We shall suppose that the fluctuations, $\Gamma_0 \gamma$, of the local
toughness, \ie{} the pinning energy per unit area, are quenched and are 
described by a flat distribution. This leads to a random force problem that was 
analyzed in details \cite{SchVilRou95a,TanGouRou98}. We just recall here the 
main results of these studies.

The power-law behavior of the correlation function of the position of the crack
front in the statistically stationary state, using the wavelet analysis, are 
shown on Fig.~\ref{CL_unif1}b. The wandering exponent $\zeta$ is found to be 
$\zeta \approx 0.35 \pm 0.02$ that is consistent with previous studies 
\cite{SchVilRou95a,TanGouRou98}.

The roughness over a local length scale $l$, $W(l,t) = \langle\left[\Phi(x_2,t) 
- <\Phi>(t)\right]^2 \rangle^{1/2}_{x_2}$ evolves with time during a 
transient regime. We checked that, in the case of large fluctuation regimes, 
such an evolution follows a Family-Viscek scaling. The wavelet
transform of the crack-front profile scales therefore as :
\begin{equation}
\mathcal{W}[\phi(\cdot,t)](a) \, = \, a^{\zeta + \frac{1}{2}}
g\left(\frac{a}{t^{1/z}}\right)
\end{equation}
where $\zeta$ and $z$ are the wandering and the dynamic exponents. The scaling
function behaves as $g(u) = 1 $ if $u\ll1$, and $g(u) \propto
u^{-\zeta - 1/2}$ if $u\gg1$. The dynamic exponent controls the time evolution 
of the roughness during the transient regime. The local roughness saturates at 
time $l^z$ as $W(l,t\gg l^z) \propto l^{\zeta}$ independently of the system size
and therefore is statistically self-affine up to a correlation length $\xi$ 
that grows with time as $\xi \propto t^{1/z}$. We find, Fig.~\ref{CL_unif1}a, 
a dynamic exponent $z \approx 0.75$ in close agreement with our previous study 
\cite{SchVilRou95a} and the functional renormalization group analysis 
\cite{ErtKar94,RamFis97b}.

The active sites, in the statistically stationary state ($t \gg L^z$), were shown 
to be spatially and temporally correlated \cite{SchVilRou95a,TanGouRou98}. In 
order to analyze these correlations, it is of interest to study the probability 
distribution $p(|x_2-x_2'|,\Delta t)$ of having a distance $|x_2-x_2'|$ between 
the sites active at time $t$ and $t+\Delta t$ \cite{FurFed98}. From the 
numerical simulations, it is possible to estimate the distribution as 
\begin{equation}
p(|x_2-x_2'|,\Delta t) \, = \, \Delta t^{1/\beta} g\left(\,\frac{|x_2-x_2'|}{\Delta
t^{1/\beta}}\,\right)
\end{equation}
with a dynamic exponent $\beta$ that describe the spreading of the activity
over a zone of size $\xi \propto \Delta t^{1/\beta}$. The scaling function $g$
behaves as $g(u) \propto u^a$ for $u \ll 1$ and $g(u) \propto u^{-b}$ for $u
\gg 1$. The numerical results, Fig.~\ref{CL_unif2}, show a data collapse for $a = 0$, 
$b\approx1.9$ 
and $\beta \approx 1.33$, that is in close agreement with the result of 
\cite{TanGouRou98}. It is possible in this case to link the exponent $\beta$ to 
the wandering exponent using a simple scaling argument. Since the average time 
for the spreading of the activity is $\tau(\xi) \propto \xi^{\beta}$ and the 
number of depinned sites to cover this area after a time $\Delta t$ is $\propto 
\xi \delta x \propto \xi^{1+\zeta}$, we get $\beta = 1 + \zeta$. If we 
consider two consecutive events, the probability distribution decays as a power 
law $p(|x_2-x_2'|,1) \propto |x_2-x_2'|^{-b}$ and is controlled by the 
singularity of the non-local elastic kernel. On the other hand, the probability 
of having a recurrent event at the same location on the crack-front line 
$p(1,\Delta \rightarrow \infty)$ is high and does not evolve with time, since 
$a=0$. This is consistent with the observed clustering in space.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Quenched disorder with short-range correlations}

We shall suppose now that the fluctuations, $\Gamma_0 \gamma$, of the local 
toughness,results from a large number of small scale interactions and obey 
Gaussian distributions with zero mean. These fluctuations are assumed to be
spatially correlated on short range and describe by a Gaussian correlation
function with a correlation length $\Delta$: $\langle\tau(x_1,x_2) 
\tau(x_1',x_2')\rangle \, = \, \nu \langle \gamma(x_1,x_2) - \gamma(x_1',x_2')
\rangle \, = \, \nu \, \Upsilon(x_1-x_1',x_2-x_2')$ where $\nu = \Gamma_0/G_0$
and $\Upsilon$ is a function that decays rapidly for large values of its 
argument. In most physical situations, the distribution of disorder can be 
assumed to be isotropic in the $x_1-x_2$ plane. However, the correlation in 
the $x_2$ direction has only small effects on very short distances, \ie{} the 
coarse-graining length, and is not relevant experimentally. The pair 
correlation function is thus assumed to be asymmetric \cite{HazMez98}
\begin{equation}
\Upsilon(x_1-x_1',x_2-x_2') \, = \, \frac{W}{\Delta^2} \delta(x_2-x_2')
C\left(\,\left|\frac{x_1-x_1'}{\Delta}\right|\,\right)
\end{equation}
where the correlation function satisfies $C(0) = 1$, $C''(0) = -1$, with $C(r) 
\approx 0$ for $|r| \gg 1$, and is assumed for simplicity to be $C(r) = 
\e^{-r^2/2}$. The random potential term $V(\Phi(x_2),x_2)$, as the sum of 
independent Gaussian variables, is Gaussian distributed \cite{HazMez98} and, 
for the specific choice of the correlation function $C$, given by 
\begin{eqnarray}
\langle V(\Phi,x_2)V(\Phi',x_2')\rangle \, &=& \, - 
W \delta(x_2-x_2') f\left(\left[\frac{\Phi-\Phi'}{\Delta}\right]^2\right)\\
\mbox{where} \qquad f(u^2) \, &=& \, |u| \int_0^{|u|} \e^{-v^2/2} dv - \left(1 - \e^{-u^2/2}\right)
\end{eqnarray}

\subsection{Roughness in the regime of small fluctuations}

We first consider the regime where the fluctuations of the crack front line is
small compared with the correlation length $\Delta$, \ie{} $|\phi(x_2) -
\phi(x_2')| \ll \Delta$.  In this case, the potential term can be linearized and
to first order 
\begin{equation}
V(\Phi(x_2),x_2) \simeq V(<\Phi>,x_2) - \nu \tau(<\Phi>,x_2) \phi(x_2)
\end{equation}
with a force correlation function $\nu \langle \gamma(<\Phi>,x_2) 
\gamma(<\Phi>,x_2') \rangle = \nu \frac{W}{\Delta^2} \delta(x_2,x_2')$. The 
Hamiltonian can be rewritten as \cite{HazMez98}:
\begin{equation}
H \, = \, \frac{1}{2} \int_k \left|k| |\phi(k) - \frac{\gamma(k)}{\nu |k|}\right|^2 dk - 
\frac{1}{\nu^2 }\int_k \frac{|\gamma(k)|^2}{|k|} dk
\end{equation}
and therefore at large separations $|x_2-x_2'| \gg \Delta$, 
\begin{eqnarray}
\langle\left(\phi(x_2) - \phi(x_2')\right)^2\rangle &=& \frac{W}{\nu^2\Delta^2}
\int_k \frac{2-2\cos(k(x_2-x_2'))}{k^2} dk \nonumber\\
&=& \frac{W}{\nu^2\Delta^2} |x_2-x_2'|
\end{eqnarray}
The predicted exponent in this regime, \ie{} the Larkin regime, is thus given 
by $\zeta = 1/2$ for $|x_2-x_2'| \ll \xi$, where $\xi$ is the 
Larkin length, $\xi = \frac{\nu^2\Delta^2}{W}$. 

\begin{figure}[ht]
\begin{center}
\epsfig{file=Figures/carte_gauss.epsi,angle=-90,width=0.6\linewidth}\\[3mm]
\epsfig{file=Figures/wav_gauss.epsi,angle=-90,width=0.6\linewidth}
\end{center}
\caption{\label{CL_gauss}Roughness of the crack-front line in the statistically 
stationary state for small fluctuations : top, crack front line geometry at 
different time intervals; bottom, average wavelet transform of the displacement 
crack-front position. The straight line shows a power-law fit that corresponds 
to $\zeta = 0.50$. The lower cut-off corresponds to length scales $|x_2-x_2'| 
\approx \Delta$.}
\end{figure}

The crack-front line evolution and the power-law behavior of the correlation
function in the statistically stationary state, using a wavelet analysis, are 
shown on Fig.~\ref{CL_gauss}. In order to resolve this regime, the parameters 
of the numerical simulation have been set to $W=10^{-2}$, $\Delta = 3$ and 
$\delta x = 10^{-2}$. At small length scales, we determine a wandering 
exponent $\zeta = 0.50 \pm 0.02$ in good agreement with the perturbation 
theory prediction and the analytic replica method \cite{HazMez98}.

\subsection{Roughness in the regime of large fluctuations}

On larger length scales, $|x_2-x_2'| > \xi$, the fluctuations of the line are 
greater than the correlation length $\Delta$, $|\phi(x_2)-\phi(x_2')| > \Delta$,
and the potential term in the Hamiltonian can no longer be linearized. One can 
then estimate the wandering exponent by a simple scaling argument of M\'ezard 
and Parisi \cite{MezPar91,HazMez98}. The scale transformation $x_2 \rightarrow 
l x_2$, $\Phi(x_2) \rightarrow l^{\zeta} \Phi(x_2)$, $V(\Phi(x_2),x_2) 
\rightarrow$\linebreak $l^{\lambda} V(\Phi(x_2),x_2)$,imposing that the two 
terms in the Hamiltonian scale the same way and that the potential term keeps 
the same statistics after rescaling, leads $\lambda = 2\zeta -1$ and $2\lambda 
= -1 + \zeta$ and so $\zeta = 1/3$ which is less than the value obtained in the 
Larkin regime and coincides with our result for the uncorrelated quenched 
uniform disorder case. This value of the wandering exponent can also be
recovered by the Imry-Ma estimate of Huse given in \cite{Gen85}. It also consistent with the renormalization group analysis using the formalism developed in 
\cite{MarSigRos73,NarFis93,ErtKar94} and, more recently, the analytical results 
obtained using the replica method approach \cite{HazMez98}.

This regime can be numerically study for a fixed system size $L$ by assigning a
very low value to $\Delta$. Due to discreteness of the numerical model,
assigning a small value $\Delta$, compared to the grid size, is the same as 
considering a uniform random distribution for the driving force $\tau(x_1,x_2)$ 
and all the previous scaling, for uniform quenched disorder, can be recovered.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Quenched disorder with long-range correlations}

\begin{figure}[ht]
\begin{center}
\epsfig{file=Figures/profil_sa_-0.5.epsi,angle=-90,width=0.6\linewidth}\\[2mm]
\epsfig{file=Figures/profil_sa_0.epsi,angle=-90,width=0.6\linewidth}\\[2mm]
\epsfig{file=Figures/profil_sa_0.5.epsi,angle=-90,width=0.6\linewidth}
\end{center}
\caption{\label{Prof_SA} Crak-front line profile in the case of long range 
correlations. The quenched disorder correlations are controlled by a 
self-affine exponent $\zeta_t$: from top to bottom: $\zeta_t=-0.5$, 
$\zeta_t = 0$, $\zeta_t = 0.5$.}
\end{figure}

We consider here the case where the fluctuations, $\Gamma_0 \gamma$, of the
local toughness are Gaussian with zero mean and are spatially correlated with 
long range correlations, \ie{} the toughness correlations are controlled by a 
self-affine exponent $\zeta_t$. In this case the correlation function 
$\Upsilon$ has now a slow algebraic decay. The random force correlation 
function is assumed to behave as 
\begin{equation}
\Upsilon(x_1-x_1',x_2-x_2') \propto \left|(x_1-x_1')^2 + (x_2-x_2')^2\right|^{\zeta_t}
\end{equation}
and in Fourier space $ \Upsilon(\mathbf{k}) \propto k^{2-2\zeta_t}$.

Several algorithms exist for the generation of such self-affine surfaces. We
used in this study the Voss algorithm \cite{Vos85} and the Fourier transform.
The latter make use of the scaling property of the Fourier spectrum. In the
Fourier space, one generate a surface $\tau(\mathbf{k})$ with a module that 
follows the scaling $|\tau(\mathbf{k})|\propto |k|^{-2-\zeta_t}$ with a 
random phase. The conditions $\tau(-\mathbf{k}) = -\tau(-\mathbf{k})$ 
insures a nil imaginary part after the inverse transform.

The case of long range correlations in the quenched disorder has not yet been
studied using the functional group analysis. Numerically, the evolution of the 
crack-front line geometry, in the statistically stationary regime, is shown on 
Fig.~\ref{Prof_SA} and the evolution of the wandering exponent of the crack
front line correlations as a function of the self-affine exponent of the
quenched disorder correlations is shown on Fig.~\ref{zf_zb}.

\begin{figure}[ht]
\centerline{\epsfig{file=Figures/zf_zb.epsi,angle=-90,width=0.6\linewidth}}
\caption{\label{zf_zb}Wandering exponent $\zeta$ of the crack-front line 
correlation function as a function of the self-affine exponent of the long 
range correlated quenched disorder $\zeta_t$.}
\end{figure}
As the exponent $\zeta_t$ increases, small scale fluctuations decreases and the
crack-front line becomes smoother and smoother. For small values of $\zeta_t$,
the crack-front line geometry is closed to the one obtained in the case of
uniform random disorder. The elastic interactions are shown to significantly
increase any long range disorder correlations in the system. For 
$\zeta_t > -1.0$, small scale correlations decrease and one can approximate the
quasi-static evolution as
\begin{equation}
|k| \phi(k) \approx - \nu \gamma(<\Phi>,k)
\end{equation}
That would lead to a scaling, to first order in small scale correlations
\begin{equation}
\langle|\phi(k)|^2\rangle \propto \frac{\Upsilon(\mathbf{k})}{|k|^2} \propto 
|k|^{-2-2(\zeta_t+1)}
\end{equation}
which is effectively observed numerically up to $\zeta_t \approx 1$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}

We have studied the slow dynamics of in-plane crack propagation including the
effects of long range elasticity and those of spatial correlations in
the toughness of the interface. Quenched noise correlations are shown to 
influence significantly the development of the crack-front line correlations 
during in-plane crack propagation at low speed. In the case of short-range 
Gaussian correlations, with correlation length $\Delta$, two limit behaviors can
be identified. In the small fluctuations limit, $|\phi(x_2)-\phi(x_2')| \ll 
\Delta$, on length scales smaller than the Larkin length, the wandering 
exponent of the crack-front line is $\zeta\approx1/2$. For large fluctuations, 
$|\phi(x_2)-\phi(x_2')| \gg \Delta$, on length scales larger than the Larkin 
length, the wandering exponent is smaller with $\zeta\approx1/3$. The latter is 
in good agreement with the wandering exponent prediction using functional 
renormalization group analysis \cite{ErtKar94} or the replica method
\cite{HazMez98}. The scaling in the large fluctuations limit is also in good 
agreement with previous numerical studies with a uniform quenched disorder. In 
the case of long-range correlations, for which no analytical results exist, the 
numerical study predict a strong amplification of the crack-front line 
correlations that scale directly as $\zeta_t + 1$, where $\zeta_t$ is the 
self-affine exponent that describes the long range correlations of the quenched 
disorder.

Recent experimental studies \cite{SchMal97} of in-plane crack propagation
between two annealed blocks of Plexiglas suggest a wandering exponent of the
crack-front line $\zeta =0.63 \pm 0.03$. This is higher than the value obtained 
for a uniform quenched disorder in the interface toughness and for the large
fluctuations regime in case of short-range Gaussian correlations. The crack
speed in the experiment is very low and the fluctuations of the crack-front
line position remain small. This seems to be closer to the small
fluctuations regime of short-range Gaussian correlations. In this regime the
predicted wandering exponent is closer to the experimental one but still
underestimate the experimental value. In case of long-range fluctuations, the 
experimental wandering
exponent may be recovered when assuming toughness long-range correlations with a
self-affine exponent of $\zeta_t\approx -0.4$. However, it is difficult to
analyze experimentally the toughness fluctuations induced by the experiment
set-up and such an analysis has not yet been done. Other explanation for this
experimental value has been proposed. The first one \cite{RamFis97b} is that 
the experiments are not quasi-static and even though the averaged crack speed 
remains slow, crack front wave instabilities may enhance the crack-front line 
correlations. An other possibility, is that at slow speed viscoelastic effects 
may operate. 
 
%\acknowledgements
%\vspace{.5cm}
{\sl The present text benefited from discussions with St\'ephane Roux,
Anne Tanguy, Arnaud Delaplace and Knuth Jorgen M\r{a}l{\o}y.}

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\end{document}
