Laurène Legrand

Sylvain Patinet

Jean-Baptiste Leblond

Joël Frelat

Veronique Lazarus

Damien Vandembroucq

Véronique Lazarus

Coplanar perturbation of a crack lying on the mid-plane of a plate

Keywords: Crack, coplanar perturbation, plate, finite thickness, Love-Kirchhoff theory, finite elements

Introduction

In the past 20 years, a number of works have been devoted to experimental studies of the deformation of the front of plane cracks propagating quasistatically in heterogeneous materials; see [START_REF] Schmittbuhl | Direct observation of a self-affine crack propagation[END_REF], [START_REF] Delaplace | High resolution description of a crack front in a heterogeneous plexiglas block[END_REF], [START_REF] Maloy | Dynamical event during slow crack propagation[END_REF], [START_REF] Schmittbuhl | Slow crack propagation and slip correlations[END_REF], [START_REF] Maloy | Local waiting time fluctuations along a randomly pinned crack front[END_REF], [START_REF] Dalmas | Crack front pinning by design in planar heterogeneous interfaces[END_REF], [START_REF] Chopin | Statique et dynamique d'un front de fissure en milieu hétérogène[END_REF], [START_REF] Santucci | Fracture roughness scaling: A case study on planar cracks[END_REF], and the recent reviews of [START_REF] Bonamy | Intermittency and roughening in the failure of brittle heterogeneous materials[END_REF] and [START_REF] Bonamy | Failure of heterogeneous materials: a dynamic phase transition?[END_REF]. In a typical experiment, two plexiglas plates were bonded together through application of a large pressure at high temperature, an inhomogeneous distribution of toughness being generated on the interface through some suitable treatment. 1 The plates were then gradually separated by increasing the gap between them on one side, thus inducing propagation of a crack from this side along the interface. The transparency of the plates allowed for direct optical observation of the shape of the crack front as a function of the distance of propagation.

Theoretical interpretations of experiments of this type have been proposed by many authors; see [START_REF] Schmittbuhl | Interfacial crack pinning: effect of nonlocal interactions[END_REF], [START_REF] Tanguy | From individual to collective pinning: effect of long-range elastic interactions[END_REF], [START_REF] Schmittbuhl | Interfacial crack front wandering: influence of quenched noise correlations[END_REF], [START_REF] Krishnamurthy | Dynamic exponent in extremal models of pinning[END_REF], [START_REF] Roux | Effective toughness of heterogeneous brittle materials[END_REF], [START_REF] Schmittbuhl | Slow crack propagation and slip correlations[END_REF], [START_REF] Charles | Material-independent crack arrest statistics[END_REF], [START_REF] Katzav | Roughness of tensile crack fronts in heterogenous materials[END_REF], [START_REF] Bonamy | Cracking dynamics in material failure as the signature of a self-organized dynamic phase transition[END_REF], [START_REF] Laurson | Avalanches and clusters in planar crack front propagation[END_REF], [START_REF] Ponson | Crack propagation in brittle heterogeneous solids: material disorder and crack dynamics[END_REF], and the reviews of [START_REF] Alava | Statistical models of fracture[END_REF] and [START_REF]Perturbation approaches of a planar crack in linear fracture mechanics: a review[END_REF]. These interpretations were invariably based on a formula of [START_REF] Rice | First-order variation in elastic fields due to variation in location of a planar crack front[END_REF] providing the first-order expression of the variation of the local mode I stress intensity factor resulting from some small, but otherwise arbitrary in-plane perturbation of the crack front. This formula was established for a semi-infinite crack in some infinite body. Clearly, to be applicable to a crack lying on the interface between bonded plates, it requires that the characteristic distance of variation of the perturbation of the front be small compared to all other characteristic dimensions of the problem, and first of all the thickness of the plates.

Unfortunately, this condition is often violated in actual experiments. In [START_REF] Dalmas | Crack front pinning by design in planar heterogeneous interfaces[END_REF]'s experiments for instance, regularly distributed tough spots of size comparable to the plate thickness were generated, resulting in notable fluctuations of the position of the crack front over this lengthscale. Even in more customary experiments which involved randomly distributed toughness inhomogeneities of much smaller size, Fourier components of the fluctuations of position of the front of wavelength comparable to the plate thickness were present; see for instance Figure 5 of [START_REF] Schmittbuhl | Slow crack propagation and slip correlations[END_REF]'s paper.

Applicability of [START_REF] Rice | First-order variation in elastic fields due to variation in location of a planar crack front[END_REF]'s formula to the experiments discussed is therefore questionable. More accurate interpretations of these experiments would result from use of a formula specifically adapted to the geometric configuration employed. Establishing it seems all the more important since extensions of [START_REF] Rice | First-order variation in elastic fields due to variation in location of a planar crack front[END_REF]'s formula to different situations -a tunnel-crack loaded in mode I [START_REF] Leblond | The tensile tunnel-crack with a slightly wavy front[END_REF] or in some arbitrary combination of modes (Lazarus and [START_REF] Leblond | In-plane perturbation of the tunnel-crack under shear loading. II: Determination of the fundamental kernel[END_REF]; a system of two coplanar, parallel, identical mode I tunnel-cracks [START_REF] Pindra | In-plane perturbation of a system of two coplanar slit-cracks -I: Case of arbitrarily spaced crack fronts[END_REF]; a system of two coplanar, parallel semi-infinite mode I cracks [START_REF] Legrand | In-plane perturbation of a system of two coplanar slit-cracks -II: Case of close inner crack fronts or distant outer ones[END_REF] -have revealed a notable influence of the cracked geometry considered upon the "influence function" depicting the effect of a small crack advance at some point of the front upon the variation of the stress intensity factor at some other point. The effect is particularly important upon the asymptotic behavior of this function at large distances [START_REF] Legrand | In-plane perturbation of a system of two coplanar slit-cracks -II: Case of close inner crack fronts or distant outer ones[END_REF].

The aim of this paper is therefore to extend [START_REF] Rice | First-order variation in elastic fields due to variation in location of a planar crack front[END_REF]'s formula to coplanar perturbation of a crack lying on the interface between bonded plates, or equivalently on the mid-plane of a plate of double thickness.2 This will be done in two steps. First, [START_REF] Rice | First-order variation in elastic fields due to variation in location of a planar crack front[END_REF]'s treatment, appropriate for plates of very large thickness, will be adapted to the other extreme case of plates of very small thickness. This will be done by replacing the standard 3D theory of elasticity he used by the Love-Kirchhoff plate theory, which is known to be the correct asymptotic theory in the limit of an infinitesimal plate thickness [START_REF] Sanchez-Hubert | Introduction aux méthodes asymptotiques et à l'homogénéisation[END_REF]. Second, the theoretical expressions applicable to extreme values of the thickness will be supplemented with finite element computations providing results for intermediate values. These results will be used to define a simple, approximate but accurate "interpolation formula" for the variation of the local stress intensity factor, applicable in all cases.

The paper is organized as follows:

• Section 2 presents the problem to be solved.

• Section 3 recalls [START_REF] Rice | First-order variation in elastic fields due to variation in location of a planar crack front[END_REF]'s solution, applicable to plates of very large thickness.

• As a prerequisite for the next section, Section 4 expounds some generalities on cracks lying on the mid-plane of plates. • Rice (1985)'s treatment is then adapted in Section 5 to the case of plates of very small thickness. • Finally Section 6 presents the numerical study for plates of intermediate thickness and the interpolation formula deduced from it.

Statement of the problem -Notations

Figures 1 and 2 offer different 2D views of the 3D problem considered. A plate occupies the domain 0 ≤ x < +∞, -h ≤ y ≤ h, -∞ < z < +∞ in the 3D space. This plate contains a plane crack occupying the region 0 ≤ x ≤ a(z), -∞ < z < +∞ in the plane y = 0. The crack front is almost straight so that the distance a(z) separating it from the Oz axis may be written in the form

a(z) ≡ a + δa(z) , |δa(z)| a.
(1)

At each point P of the crack front, we define some local axes P x 1 , P x 2 , P x 3 with P x 1 perpendicular to the front within the Oxz plane, P x 2 parallel to the Oy axis and P x 3 tangent to the front. The plate is made of some isotropic elastic material with Young's modulus E and Poisson's ratio ν.

The loading consists of uniform, opposite displacements v 0 , -v 0 in the y direction imposed on the upper and lower halves of the left boundary.

The crack is thus loaded in a state of pure mode I at all points of its front. When this front is straight, the mode I stress intensity factor is uniform along it and denoted K. When it is curved, this stress intensity factor is non-uniform and denoted K(P ) or K(z), depending on the context. If the deviation from straightness is slight, K(z) may be written in the form

K(z) ≡ K + δK(z) , |δK(z)| K. ( 2
)
The problem is to derive an expression of the local fluctuation of the stress intensity factor at the point z of the crack front, δK(z), accurate to first order in the values of the perturbation of this front, δa(z ), -∞ < z < +∞.

Use will be made of Fourier transforms of functions in the direction of the crack front.

The definition adopted for the Fourier transform φ(k) of an arbitrary function φ(z) will be

φ(k) = +∞ -∞ φ(z)e ikz dz ⇔ φ(z) = 1 2π +∞ -∞ φ(k)e -ikz dk. ( 3
)
3 The Rice solution for an infinite 3D body [START_REF] Rice | First-order variation in elastic fields due to variation in location of a planar crack front[END_REF] derived the expression of δK(z) for a semi-infinite crack in an infinite body. Such a geometric schematization is acceptable in the present case only if the characteristic distance of variation λ of the perturbation of the crack front3 satisfies the conditions

λ h , λ a, ( 4
)
the first of which implies that the plate must be "very thick". Provided that these conditions are met, δK(z) is given by the formula

δK(z) K = 1 2π P V +∞ -∞ δa(z ) -δa(z) (z -z) 2 dz (subject to conditions (4)) (5)
where the symbol P V denotes a Cauchy principal value.

The equivalent formula in Fourier's space is readily shown to be

δK(k) K = - |k| 2 δa(k) (subject to conditions (4)). ( 6
)
4 Generalities on cracks lying on the mid-plane of very thin plates

We now consider the limit of a "very thin" plate, that is one satisfying the conditions h λ , h a.

(7)

There are two ways to consider this limit, according to whether one observes the plate as it is or adopting a magnified view of the vicinity of the crack front. Within the framework of matched asymptotic expansions, the first possibility corresponds to considering the "exterior problem", and the second one the "interior problem". For the sake of simplicity and clarity, we shall avoid a fully rigorous, but heavily formal study of these two problems, and be content with a more qualitative and intuitive presentation.

Exterior problem

Consider the vicinity of some arbitrary point P of the crack front, with the local axes P x 1 , P x 2 , P x 3 defined in Figure 2. Figure 3 shows the "exterior" view of this region. In a formal asymptotic expansion of the equations of the problem, this view would correspond to using the variables x 1 , x 2 /h, x 3 , that is, considering a dilation of coordinates in the sole direction x 2 .

P

x 1

x 2

x 3 2h Fig. 3. The exterior problem

In the limit h → 0, the mechanical situation is equivalent to a system of two symmetric plates occupying the region x 1 ≤ 0 and clamped into some rigid frame at x 1 = 0. These plates must be described using Love-Kirchhoff's plate theory, since it is known to be the correct asymptotic one in the limit h → 0 (Sanchez-Hubert and Sanchez-Palencia (1992), p. 158 sqq.). Within this theory, the displacement u 2 (x 1 , x 3 ) ≡ v(x 1 , x 3 ) of the upper plate in the x 2 ≡ y direction is of the following form, in the plane x 3 = 0 and behind and near the point P :

v(x 1 , 0) = 1 2 ∂ 2 v ∂x 2 1 (0, 0)x 2 1 + O(x 3 1 ) (x 1 ≤ 0).
Using rather Love-Kirchhoff's deformation tensor (the opposite of the curvature tensor of the plate)

4 C ≡ -grad(grad v), ( 8
)
this expression may be rewritten as

v(x 1 , 0) = - C 11 (P ) 2 x 2 1 + O(x 3 1 ) (x 1 ≤ 0). (9)

Interior problem

Now Figure 4 shows the "interior" view of the vicinity of the point P . In a formal asymptotic expansion, this view would correspond to using the variables x 1 /h, x 2 /h, x 3 , that is, considering a dilation of coordinates in the directions x 1 and x 2 but not x 3 . (Dilating coordinate x 3 would mean letting λ go to zero together with h).

P x 1

x 2

x 3 h h Fig. 4. The interior problem

In the limit h → 0, the 3D elasticity problem becomes one of plane strain in the plane P x 1 x 2 , since the characteristic distance of variation λ of the position of the crack front becomes infinitely larger than the typical dimension h of the region studied. In this problem, the component C 11 (P ) of Love-Kirchhoff's deformation tensor in the exterior problem acts as a boundary condition imposed on the far left of the upper half of the geometry, which is thus subjected to some prescribed bending moment (see Figure 4). The linearity of the problem implies that the stress intensity factor K(P ) at point P must depend linearly on this boundary condition, so that there must be a constant γ such that

K(P ) = - C 11 (P ) 2γ . ( 10
)

Calculation of the constant γ

Although the calculation of the constant γ is not strictly indispensable in the present context, we shall present it for completeness.

The standard way of evaluating γ would be to explicitly solve the interior problem. It so happens, however, that this is unnecessary in the present case. Indeed, as will now be seen, γ may be calculated analytically and explicitly by combining elementary energetic considerations and Irwin's formula connecting the local energy-release-rate G(P ) to the local stress intensity factor K(P ).

Let us consider a rectangular region centered at the point P of the crack front, with sides parallel to the P x 1 and P x 3 axes (Figure 5). Conditions (7) make it possible to choose, for the dimensions L 1 and L 3 of this rectangle along these axes, values much larger than h, while still much smaller than λ and a. The conditions L 1 , L 3 h allow for the use of Love-Kirchhoff's theory to evaluate the total energy of the region (the result will be asymptotically exact in the limit h → 0), while the conditions L 1 , L 3 λ, a make it possible to consider the strain and stress states as uniform within its left, deformed part.

P a(z) x 1 x 2 x 3 L 1 L 3 L λ Fig. 5

. Definition of a domain for the calculation of the local energy-release rate

Let M denote the tensor of bending and torsion moments, related to the deformation tensor C through Love-Kirchhoff's constitutive law

M = D [(1 -ν)C + ν(tr C)1] (11)
where

D ≡ Eh 3 12(1 -ν 2 ) ( 12
)
is the bending stiffness. In the left part of the upper plate, the values of the components of the deformation tensor are C 11 = C 11 (P ), C 13 = C 33 = 0 (because of the clamping conditions v = 0, ∂v/∂x 1 = 0 at x 1 = 0). It then follows from equation ( 11) that M 11 = DC 11 (P ), so that the value of the surface density of elastic energy of the upper plate is

1 2 M : C = 1 2 M 11 C 11 = D 2 [C 11 (P )] 2 .
Accounting for the presence of two plates, one concludes that the value of the total elastic energy of the rectangular region is

W = D [C 11 (P )] 2 LL 3 ( 13
)
where L is the length of the crack within the rectangle, that is, the distance between its left side and the crack front (see Figure 5).

Now the opposite of the potential energy of prescribed forces, φ, is just the work done in the deformed configuration by the bending moments imposed on the left side of the plates. Therefore, by the work theorem,

φ = 2W. ( 14
)
It follows from equations ( 13) and ( 14) that

- ∂ ∂L (W -φ) = ∂W ∂L = D [C 11 (P )] 2 L 3 . ( 15
)
But the left-hand side of equation ( 15) is equal to

L 3 /2 -L 3 /2 G(x 3 )dx 3 = G(P )L 3
, by the very definition of the local energy-release-rate. It follows that

G(P ) = D [C 11 (P )] 2 . ( 16
)
Using now Irwin's formula, one gets from there the value of the local stress intensity factor:

K(P ) = E 1 -ν 2 G(P ) = - Eh 3/2 2 √ 3 (1 -ν 2 ) C 11 (P ), (17)
where equation ( 12) has been used and account has been taken of the obvious positiveness of K(P ) and negativeness of C 11 (P ). This means that K(P ) and C 11 (P ) are related by a formula of type (10), with

γ ≡ √ 3 1 -ν 2 Eh 3/2 . ( 18
)

Synthesis of results

Combining equations ( 9) and ( 10), one sees that the asymptotic expression of the displacement of the upper plate in the y direction, behind and near the point P of the crack front, in the exterior problem, reads

v(x 1 , 0) = γK(P ) x 2 1 + O(x 3 1 ) (x 1 ≤ 0) (19)
with γ given by equation ( 18). This expression is the equivalent, for cracks lying on the mid-plane of very thin plates, of the standard asymptotic formula for the crack opening displacement near the crack front in classical 3D fracture mechanics.5 It will play a central role in the next section.

The solution for a very thin plate

We shall now adapt [START_REF] Rice | First-order variation in elastic fields due to variation in location of a planar crack front[END_REF]'s treatment of the in-plane perturbation of a semi-infinite crack in some infinite body to the in-plane perturbation of a crack lying on the midplane of a very thin plate (that is, satisfying conditions ( 7)).

In fact [START_REF] Rice | First-order variation in elastic fields due to variation in location of a planar crack front[END_REF]'s paper contained two independent derivations of the same final formula: one based on the theory of Bueckner's 3D weight functions, and the other on a direct calculation of the Neuber-Papkovich potentials of the 3D elasticity problem. The first approach would be inapplicable here, because of lack of development of an analog of Bueckner's theory for thin plates. On the other hand the second approach can be adapted to a thin plate, as will be seen; in fact the adaptation will even turn out to be simpler than the original, in that it will directly deal with the displacement field instead of the Neuber-Papkovich potentials. It will suffice to consider the exterior problem defined in Section 4, making use of equation ( 19); the study of the interior problem will not be necessary.

Original problem for an arbitrary perturbation of the crack front

The "exterior equations" of the problem defined in Section 2, with a perturbation of the crack front of arbitrary amplitude (that is, without hypothesis (1)), within Love-Kirchhoff's plate theory, read on the upper plate:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ 2 ∂x 2 + ∂ 2 ∂z 2 2 v(x, z) = 0 for 0 ≤ x ≤ a(z), -∞ < z < +∞ v(0, z) = v 0 ∂ 2 v ∂x 2 (0, z) = 0 ⎫ ⎪ ⎬ ⎪ ⎭ for -∞ < z < +∞ v(a(z), z) = 0 ∂v ∂n (a(z), z) = 0 ⎫ ⎪ ⎬ ⎪ ⎭ for -∞ < z < +∞ (20)
where the symbol ∂/∂n denotes the normal derivative to the front. Equation ( 20) 1 is the equilibrium equation, equations (20) 2,3 are the boundary conditions on the left side, and equations (20) 4,5 are the clamping conditions on the crack front.

First-order problem for an infinitesimal perturbation of the crack front

Consider now a perturbation of small amplitude (obeying hypothesis (1)), and denote δv(x, z) the variation, under constant loading, of the normal displacement of the upper plate arising from the variation of position δa(z) of the crack front around the mean value a. The equilibrium equation and boundary conditions of the left side satisfied by δv(x, z)

are readily obtained through differentiation of equations ( 20) 1,2,3 at constant v 0 :

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ 2 ∂x 2 + ∂ 2 ∂z 2 2 δv(x, z) = 0 for 0 ≤ x ≤ a, -∞ < z < +∞ δv(0, z) = 0 ∂ 2 δv ∂x 2 (0, z) = 0 ⎫ ⎪ ⎬ ⎪ ⎭ for -∞ < z < +∞. ( 21
)
On the other hand the conditions verified by δv(x, z) on the crack front cannot be obtained from equations (20) 4,5 in such a simple way. To derive them, it is convenient to temporarily write the perturbation of the front in the form

δa(z) ≡ φ(z)
where is a small parameter and φ(z) a given function. We then introduce the power expansion of the displacement in the original problem, up to the third order, in the plane perpendicular to the front at the point (a + φ(z), z), with explicit indications of dependence of all quantities upon :

v ( ; x, z + (a -x)φ (z)) = γK( ; z) [a + φ(z) -x] 2 + χ( ; z) [a + φ(z) -x] 3 +O [a + φ(z) -x] 4 (0 ≤ x ≤ a + φ(z)). ( 22
)
(In the left-hand side, the correction of the z coordinate arises from the fact that the normal vector to the perturbed front is not exactly e x but e xφ (z)e z , see Figure 2; also, in the right-hand side, the distance from the point (x, z + (ax)φ (z)) to the point (a + φ(z), z) of the crack front is considered to be a + φ(z)x, with an error of second order in ). Differentiating this equation with respect to at = 0, one gets ∂v ∂ (0; x, z) + ∂v ∂z

(0; x, z)(a -x)φ (z) = 2γK(0; z)(a -x)φ(z) + γ ∂K ∂ (0; z)(a -x) 2 +3χ(0; z)(a -x) 2 φ(z) + O ((a -x) 3 )
or more simply, since for = 0 the problem becomes 2D in the Oxy plane and v, K and

χ independent of z: ∂v ∂ (0; x, z) = 2γK(0)(a -x)φ(z) + γ ∂K ∂ (0; z)(a -x) 2 +3χ(0)(a -x) 2 φ(z) + O ((a -x) 3 ) .
Multiplying both sides by and identifying ∂v/∂ to δv, φ to δa and ∂K/∂ to δK, one gets from there

δv(x, z) = 2γK(a -x)δa(z) + γδK(z)(a -x) 2 +3χ(a -x) 2 δa(z) + O (a -x) 3 (0 ≤ x ≤ a) ( 23
)
where henceforth superfluous indications that quantities are taken at = 0 have been discarded. It follows that

⎧ ⎪ ⎨ ⎪ ⎩ δv(a, z) = 0 ∂δv ∂x (a, z) = -2γKδa(z) for -∞ < z < +∞. ( 24
)
These are the boundary conditions on the crack front looked for.

Solution of the first-order problem in Fourier's space

To solve equations (21, 24), take their Fourier transform in the direction z and get

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ 2 ∂x 2 -k 2 2 δv(x, k) = 0 for 0 ≤ x ≤ a, -∞ < k < +∞ δv(0, k) = 0 ∂ 2 δv ∂x 2 (0, k) = 0 ⎫ ⎪ ⎬ ⎪ ⎭ for -∞ < k < +∞ δv(a, k) = 0 ∂ δv ∂x (a, k) = -2γK δa(k) ⎫ ⎪ ⎬ ⎪ ⎭ for -∞ < k < +∞. ( 25
)
The solution of equation ( 25 where A, B, C and D are constants to be determined. 6 Now equations (25) 2,3 immediately imply that

A = D = 0, (27)
and the values of B and C then follow from equations (25) 4,5 :

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ B = 2γKa δa(k) cosh(ka) cosh(ka) sinh(ka) -ka C = -2γK δa(k)
sinh(ka) cosh(ka) sinh(ka)ka .

(28)

Then, by equation (26), δv(x, k) = 2γK δa(k) a cosh(ka) sinh(kx)x sinh(ka) cosh(kx) cosh(ka) sinh(ka)ka .

(29)

Fourier transform of the variation of the stress intensity factor

The next task is to relate the Fourier transform of the variation of the stress intensity factor, δK(k), to that of the perturbation of the front, δa(k). To do so, differentiate equation ( 23) twice with respect to x at x = a:

∂ 2 δv ∂x 2 (a, z) = 2γδK(z) + 6χδa(z) ⇒ δK(z) = 1 2γ ∂ 2 δv ∂x 2 (a, z) - 3χ γ δa(z),
then take the Fourier transform of this equation:

δK(k) = 1 2γ ∂ 2 δv ∂x 2 (a, k) - 3χ γ δa(k). ( 30
)
Now, by equation ( 29),

∂ 2 δv ∂x 2 (a, k) = -4γkK δa(k) sinh 2 (ka) cosh(ka) sinh(ka) -ka . ( 31
)
Also, to calculate the constant χ, note that by equation ( 22) applied to the unperturbed configuration of the crack front ( = 0),

γK ≡ 1 2 d 2 v dx 2 (a) ; χ ≡ - 1 6 d 3 v dx 3 (a) (32)
where v(x) is the unperturbed normal displacement of the upper plate. Now it is easy to calculate this displacement from the equations of the unperturbed (2D) problem ((20) with a(z) ≡ Cst. ≡ a):

v(x) = v 0 1 - 3x 2a + x 3 2a 3 , ( 33
)
so that γK = 3v 0 2a 2 ; χ = -

v 0 2a 3 = - γK 3a . ( 34
)
(The value of K in equation ( 34) 1 , with γ given by equation ( 18), coincides with that provided by [START_REF] Tada | The Stress Analysis of Cracks Handbook[END_REF], p. 419, for the problem in question). Combining equations ( 30), ( 31) and ( 34), one gets

δK(k) K = 1 2γK ∂ 2 δv ∂x 2 (a, k) - 3χ γK δa(k) = - 2k sinh 2 (ka) cosh(ka) sinh(ka) -ka δa(k) + δa(k) a = 1 - 2ka sinh 2 (ka) cosh(ka) sinh(ka) -ka δa(k) a or equivalently δK(k) K = F (ka) δa(k) a (subject to conditions (7)) ( 35
)
where F (p) is a function of the "reduced wavenumber"

p ≡ ka (36) defined by

F (p) ≡ 2p cosh(2p) -sinh(2p) 2p -sinh(2p) . ( 37
)
Several features of the function F (p) are noteworthy:

• First, expanding cosh(2p) and sinh(2p) in powers of p, one easily shows that

F (0) = -2. ( 38
)
• Second, the very definition (37) of F (p) implies that it is indefinitely differentiable except perhaps at the origin. But near the origin, it is obviously the ratio of two power series of p, and this ratio has a finite limit for p → 0, as just noted. Hence it is also indefinitely differentiable at the origin. In conclusion, F (p) is indefinitely differentiable on the whole real line. • Finally it is easy to show that

F (p) = -2|p| + 1 + O p 2 e -2|p|
for p → ±∞.

(39)

There is a connection here with the work of [START_REF] Vandembroucq | Conformal mapping on rough boundaries. II. Applications to biharmonic problems[END_REF] on biharmonic functions satisfying certain conditions on the boundary of a slightly perturbed domain.

Although the results of these authors, obtained by a conformal mapping technique, do not directly apply to the present problem because of some differences in the boundary conditions, a link may be established in the limiting case where the depth a of the crack goes to infinity; the asymptotic behavior of the function F (p) described by equation ( 39) fully agrees with [START_REF] Vandembroucq | Conformal mapping on rough boundaries. II. Applications to biharmonic problems[END_REF]'s findings in this case.

There is also a connection with a very recent, yet unpublished work of [START_REF] Ponson | Peeling of patterned adhesive tapes: a theoretical analysis[END_REF] devoted to the problem of coplanar perturbation of a crack resulting from a "peeling test", in which some thin flexible membrane bonded onto some rigid substrate is gradually pulled from it. This problem was more complex than that envisaged here in that it required use of the Föppl-von Karman nonlinear plate theory in order to account for the effect of the in-plane tension of the deformed membrane; on the other hand the geometry was simpler in that the crack was semi-infinite. The solutions obtained may be compared by replacing the lower deformable plate by a rigid one and letting the depth a of the crack go to infinity in the present work, and dropping nonlinearities in [START_REF] Ponson | Peeling of patterned adhesive tapes: a theoretical analysis[END_REF]'s work. The two solutions do coincide with these adaptations and simplifications.

Variation of the stress intensity factor

Although equations (35, 37) for the Fourier transform δK(k) may be used directly, an explicit formula for the variation of the stress intensity factor in the physical space, δK(z), rather than in Fourier's space, may also be useful.

To derive such a formula, take the inverse Fourier transform of equation ( 35):

δK(z) K = 1 2π +∞ -∞ δK(k) K e -ikz dk = 1 2π +∞ -∞ F (ka) δa(k) a e -ikz dk.
Now use the Fourier decomposition of δa(k) in this expression:

δK(z) K = 1 2π +∞ -∞ +∞ -∞ F (ka) δa(z ) a
e ikz e -ikz dz dk.

It would seem tempting at this stage to change the order of integration; the inverse Fourier transform of the function F (p) would then appear as a result of the first integration on k. This is however not feasible because the function F (p), being nonzero at infinity as is apparent in equation ( 39), does not admit a Fourier transform.

A more elaborate reasoning is thus necessary. First, rewrite the preceding expression in the form

δK(z) K = I 1 + I 2 , ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ I 1 ≡ 1 2π +∞ -∞ +∞ -∞ F (ka) δa(z) a e ik(z -z) dz dk I 2 ≡ 1 2π +∞ -∞ +∞ -∞ F (ka) δa(z ) -δa(z) a e ik(z -z) dz dk. ( 40
)
The integrals I 1 and I 2 may now be calculated as follows:

• With regard to I 1 , performing the integration on z , one gets

I 1 = 1 2π +∞ -∞ F (ka) δa(z) a 2πδ(k) e -ikz dk = F (0) δa(z) a = -2 δa(z) a ( 41
)
where equation ( 38) has been used. • With regard to I 2 , note first that the function

k → -i +∞ -∞ δa(z ) -δa(z)
zz e ik(z -z) dz vanishes at infinity, being a Fourier transform, and that its derivative is the function

k → +∞ -∞ [δa(z ) -δa(z)] e ik(z -z) dz .
Therefore integration by parts of the expression of I 2 yields

I 2 = i 2π +∞ -∞ +∞ -∞ F (ka) δa(z ) -δa(z) z -z e ik(z -z) dz dk = i 2π +∞ -∞ F (ka) P V +∞ -∞ δa(z ) -δa(z) z -z e ik(z -z) dz dk
where the introduction of the P V is allowed by the fact that the integral is convergent in the usual sense. Now note that the function -z) dz again vanishes at infinity, being a Fourier transform, and that its derivative is the function

k → -i P V +∞ -∞ δa(z ) -δa(z) (z -z) 2 e ik(z
k → P V +∞ -∞ δa(z ) -δa(z) z -z e ik(z -z) dz .
Therefore a second integration by parts of the expression of I 2 yields -z) dz adk. The order of integration may now be changed since the function F (p), being zero at infinity as implied by equation ( 39), does admit a Fourier transform unlike F (p). One thus gets -z) adk dz or equivalently

I 2 = - 1 2π +∞ -∞ F (ka) P V +∞ -∞ δa(z ) -δa(z) (z -z) 2 e ik(z
I 2 = -P V +∞ -∞ δa(z ) -δa(z) (z -z) 2 1 2π +∞ -∞ F (ka)e ik(z
I 2 = P V +∞ -∞ f z -z a δa(z ) -δa(z) (z -z) 2 dz , ( 42
)
where f (u) is a function of the "reduced distance"

u ≡ z -z a (43) defined by 7 f (u) ≡ - 1 2π +∞ -∞ F (p)e -ipu dp = - 1 π +∞ 0 F (p) cos(pu)dp; (44)
note that f (u) is just in fact the opposite of the inverse Fourier transform of F (p).

Combining equations ( 40), ( 41) and ( 42), one finally gets

δK(z) K = -2 δa(z) a + P V +∞ -∞ f z -z a δa(z ) -δa(z) (z -z)
2 dz (subject to conditions ( 7)) (45) with f (u) given by equation (44). Formula ( 45) is analogous to that derived by [START_REF] Rice | First-order variation in elastic fields due to variation in location of a planar crack front[END_REF] for perturbation of a semiinfinite crack in an infinite body, equation ( 5), except for two differences. First, the formula involves a term proportional to δa(z) in addition to the integral term, which represents the variation of the stress intensity factor for a uniform advance of the crack front (δa(z ) ≡ δa(z), ∀z ).8 Second, the "influence function" in the integral term is more complex since it involves an extra function f ((zz )/a).

Figure 6 provides a graphical representation of the function f (u) obtained by performing the second integral in equation ( 44) numerically, using an explicit expression of F (p) deduced from equation ( 37). The slightly negative values of f (u) for large values of u in this figure have been checked to be significant and not arise from numerical errors. Also, one may note the following additional features of this function:

• First, by equation ( 44) 2 ,

f (0) = - 1 π [F (+∞) -F (0)] = 2 π (46)
where use has been made of the properties F (0) = 0 (consequence of the fact that the function F (p) is even) and F (+∞) = -2 (consequence of equation ( 39)). Note that the value of f (0) is higher than its equivalent for a semi-infinite crack in an infinite body, that is, the constant prefactor of 1 2π in [START_REF] Rice | First-order variation in elastic fields due to variation in location of a planar crack front[END_REF]'s formula (5), by a factor of 4. In other words, the influence function is 4 times larger for a thin plate than for an infinite body, when the distance between the points considered is much smaller than a (but still of course much larger than h).

• Second, it has been noted in Subsection 5.4 that the function F (p) is indefinitely differentiable on the entire real line, and the same is of course true of its second derivative.

It then follows from a classical theorem that the opposite f (u) of the inverse Fourier transform of F (p) vanishes at infinity more quickly than any power function of u. The behavior at large distances of the influence function f ((zz )/a)(zz) -2 in equation ( 45) is thus very different from that of the simple influence function (zz) -2 in equation (5). Clearly, in the cracked geometry envisaged here, the distance a from the left boundary to the mean position of the crack front acts as a "cutoff length" above which the influence function quickly vanishes. 6 Numerical study for a plate of arbitrary thickness

Generalities

We shall now envisage the general case of a plate or arbitrary thickness. Therefore, neither conditions (4) nor conditions (7) will be imposed. We shall however introduce the hypothesis that h a.

(47) Indeed this condition is always fulfilled in practice, and considerable simplifications result from there. Note immediately for instance that with such a hypothesis, λ h suffices to warrant conditions (4) and therefore applicability of [START_REF] Rice | First-order variation in elastic fields due to variation in location of a planar crack front[END_REF]'s formulae ( 5) and ( 6); whereas λ h suffices to warrant conditions ( 7) and therefore applicability of formulae ( 35) and ( 45).

We shall look for some "interpolation formula", in Fourier's space, between the expression (6), appropriate for λ h, and the expression ( 35), appropriate for λ h, in the form

δK(k) K = -(1 -θ(kh)) |k| 2 + θ(kh) F (ka) a δa(k), ( 48
)
where θ(q) denotes some even function of the "reduced wavenumber"

q ≡ kh (49)
to be determined, obeying the properties

θ(0) = 1 ; θ(±∞) = 0. ( 50
)
Formula ( 48) does possess the desired features: if λ h, |kh| = 2πh/λ 1 so that θ(kh) 0 and the expression of δK(k)/K reduces to (6); and if λ h, |kh| = 2πh/λ 1, θ(kh) 1 and it reduces to (35). Formula (48) may seem to be criticizable on the grounds that since the problem involves three characteristic lengths, a, h and λ ≡ 2π/|k|, and thus two dimensionless geometric parameters, for instance kh and ka, the function θ should depend on both of these parameters instead of just one, kh. However the dependence of θ upon ka may safely be disregarded: indeed • if λ is much smaller than a, then the latter length is much larger than the other two by hypothesis (47); it is thus safe to consider it as infinite and replace θ(kh, ka) by θ(kh, ±∞); • if λ is comparable to a or larger, then it is necessarily much larger than h by hypothesis (47); then θ(kh, ka) 1 by equation ( 50) 1 , whatever the value of ka.

Hence with hypothesis (47), one may safely consider the interpolation function θ as a function of the sole variable kh, as assumed in equation ( 48).

Principle of numerical calculations

The function θ(q) will be determined by performing finite element elastic computations of cracked plates of finite dimensions. The depth a of the crack and the length of the unbroken ligament ahead of the crack front will be chosen much larger than the thickness h of the plate. Two configurations of the front will be considered, one straight and one with slight sinusoidal in-plane undulations of wavelength λ. Because of the periodicity of the problem in the z direction, only one slice of material lying between the planes z = 0 and z = λ will be meshed, periodic boundary conditions being imposed on these planes.

The oscillations of the local stress intensity factor along the crack front, δK(z), will be determined by comparing the values of this factor in the original and perturbed configurations of the front. One test of the relevance of the first-order perturbation approach will be that these oscillations will have to be sinusoidal like those of the local crack advance, δa(z); this test will serve to fix a maximum admissible value of the amplitude of variation of δa(z). For acceptable values of this amplitude, the value of the ratio δK(k)/ δa(k) (k = ±2π/λ) will be deduced from that of the ratio of the amplitudes of variation of δK(z) and δa(z).

Various values of λ will be considered, all of them comparable to h and therefore much smaller than a. Under such conditions |ka| 1 so that F (ka)/a ∼ -2|k| by equation (39). Hence equation ( 48) reduces to

δK(k) K = - 1 2 [1 + 3θ(kh)] |k| δa(k) (subject to the condition |ka| 1), (51)
which shows that the function θ(kh) is related to the quantity

X ≡ - δK(k)/K |k| δa(k) (52)
through the equivalent relations

X = 1 2 [1 + 3θ(kh)] ⇔ θ(kh) = 2X -1 3 . ( 53
)
Equations ( 52) and ( 53) will serve to determine numerical values of the quantity X and the function θ(kh) from those of the ratio δK(k)/ δa(k). Note in particular that by equations ( 50) and ( 53), X must take the values 2 and 1 2 in the limits |kh| 1 (λ h) and |kh| 1 (λ h) respectively.

Presentation of numerical calculations

All computations are performed with the finite element code CAST3M developed by the French Commissariat à l'Energie Atomique (CEA).

Two types of computations are performed with different boundary conditions on the upper and lower halves of the left boundary of the cracked plate: opposite vertical displacements versus opposite bending moments. Boundary conditions of the first type precisely correspond to those employed in actual experiments. Boundary conditions of the second type are representative of those prevailing on the left side of a rectangular subregion of the plate of dimensions much smaller than a, though much larger than h.9 (That dimensions satisfying these two conditions may be found is a consequence of hypothesis ( 47)). Hence the two types of conditions are expected to yield identical results for sufficiently large plates; this will serve as a test that the dimensions considered are sufficient.

The two types of computations also use different dimensions, meshes and types of elements, although the plate thickness is 1 mm in all cases. In these involving prescribed displacements, the distance a between the left side and the unperturbed crack front is 100 mm or 200 mm, depending on whether λ is smaller or larger than 10 mm; the mesh consists of 449,760 (for λ < 10 mm) or 646,440 (for λ > 10 mm) bilinear 8-node parallelepipedic elements and 471,713 or 688,568 nodes. In those involving prescribed moments, which require a smaller mesh since they focus on a subregion of the plate, the distance a between the left side and the unperturbed front is 50 mm, and the mesh consists of 182,400 bilinear 6-node prismatic elements and 106,896 nodes. In both cases the elements are strictly parallelepipedic or prismatic only for the unperturbed straight configuration of the front; for the perturbed configuration the same mesh is used with a slight distortion of elements.

The value of Poisson's ratio is the same in all cases, namely ν = 0.3.

Local values of the stress intensity factor along the crack front are evaluated using [START_REF] Destuynder | Quelques remarques sur la mécanique de la rupture élastique[END_REF]'s "Gθ method", the accuracy of which is well established.

In a first step, a straight front is considered. The unperturbed stress intensity factors calculated for the two types of boundary conditions are compared to the exact analytic values provided by [START_REF] Tada | The Stress Analysis of Cracks Handbook[END_REF], pp. 417 and 419, for the problems in question.

The relative error found decreases from about 10% for a/h = 10 to less than 0.1% for the largest values of this ratio. (The explanation of the relatively large difference for the smallest value of a/h does not lie in numerical errors but in the fact that [START_REF] Tada | The Stress Analysis of Cracks Handbook[END_REF]'s solutions are asymptotic ones in the limit a/h → +∞).

In a second step, wavy fronts of sinusoidal shape with different amplitudes δa max are considered. Figures 7 and8 show, in the typical case where λ = h, the distribution of the local energy-release-rate, normalized by half the sum of its maximum and minimum values, along the crack front, for values of δa max equal to 1% and 10% of λ. In each case the blue curve represents numerical results and the red one the sinusoidal approximation of these results having the same minima and maxima. The oscillation of the energy-releaserate can be observed to be sinusoidal for the smaller amplitude, but not so for the larger one. This shows that the linear perturbation approach is applicable only to the smaller one, which is therefore adopted in subsequent calculations.

Results

Figure 9 shows numerical values of the quantity X defined by equation ( 52), as a function of kh. (Only positive values of kh are considered here since the function X(kh) is obviously even). The two types of boundary conditions can be observed to yield identical results, which shows that the dimensions of the plates considered are large enough. Also, the quantity X goes to the values 2 (thin plate solution) and 1 2 (3D solution) in the limits kh → 0 and kh → +∞ respectively, as expected. Figure 9 shows that a decrease of kh from values much larger than unity to values much smaller than it, or equivalently an increase of λ from values much smaller than h to values much larger than it, induces an increase of X. This clearly shows that in contrast to the distance between the crack front and the boundary of the plate, the plate thickness does not play the role of a "cutoff length" above which the ratio δK(k)/ δa(k) should quickly decrease. (There is in fact nothing surprising in this conclusion: in Love-Kirchhoff's thin plate theory, the deformation of a plate is never confined by the thickness but by the boundary conditions).

It is also shown in Figure 9 that the numerical results may be accurately represented by the following very simple approximate formula for the function θ(q), which is therefore adopted hereafter:

θ(q) 1 1 + 1 3 |q| 4/3 .
(54)

Final formulae in Fourier's space and the physical space

We shall now provide final formulae for the variation of the stress intensity factor, distinguishing between the cases a < +∞ and a = +∞ (which often represents an acceptable approximation in actual experiments). The formulae will be parametrized by the dimensionless quantity ξ ≡ a h ( 1 by condition (47)).

In Fourier's space, formula (48) for δK(k) may be written in the form

δK(k) K = G a h ; kh δa(k) h (subject to condition (47)) (56)
where

G(ξ; q) ≡ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ -[1 -θ(q)] |q| 2 + θ(q) F (ξq) ξ if ξ < +∞ - 1 2 [1 + 3θ(q)] |q| if ξ = +∞. (57)
In equation ( 57), the expression of the function θ(q) is given by equation ( 54), and that of the function F (p) by equation (37) (or (39) when ξ is infinite).

The following special features of the function G(ξ; q) are of interest:

G(ξ; 0) = - 2 ξ ; G(ξ; q) ∼ - |q| 2 for q → ±∞. ( 58
)
Of course, the values of G(ξ; .) at the origin and near infinity are connected to those of δK(k) for crack front perturbations of very large wavelength in a thin plate (equations ( 35) and ( 38)), and of very small wavelength in an infinite body (equation ( 6)), respectively.

In the physical space, the formula for δK(z) may be obtained through Fourier inversion of that for δK(k). The reasoning is completely analogous to that presented in Subsection 5.5 for the case of a very thin plate (the only difference being that the reference length used here is h instead of a), and therefore will not be repeated. The resulting formula reads δK

(z) K = -2 δa(z) a + P V +∞ -∞ g a h ; z -z h δa(z ) -δa(z) (z -z) 2 dz (59)
where g(ξ; v) is a function of the dimensionless parameter ξ and the "reduced distance"

v ≡ z -z h (60) defined by g(ξ; v) ≡ - 1 2π +∞ -∞ ∂ 2 G ∂q 2 (ξ; q)e -iqv dq = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ - 1 π +∞ 0 ∂ 2 G ∂q 2 (ξ; q) cos(qv)dq if ξ < +∞ 2 π - 1 π +∞ 0 ∂ 2 G ∂q 2 (+∞; q) cos(qv)dq if ξ = +∞.
(61) The extra term 2 π in the final expression of g(+∞; v) arises from the fact that the function ∂G/∂q(+∞; q) being discontinuous at the point q = 0, with ∂G/∂q(+∞; 0 ± ) = ∓2 (see equations (50) 1 and (57) 2 ), its derivative ∂ 2 G/∂q 2 (+∞; q) in the integral from -∞ to +∞ must be understood in the sense of generalized functions, with an additional term -4δ(q).

Figure 10 shows the function g(ξ; v) obtained numerically from equations (61) 2 for various values of ξ, including infinity.10 Again, like in Figure 6, the slightly negative values of g(ξ; v) for moderate ξ and large v have been checked to be significant. The numerical results may also be supplemented by analytical ones in some special cases. First, at the origin, by equations (61),

g(ξ; 0) = ⎧ ⎪ ⎨ ⎪ ⎩ -1 π ∂G ∂q (ξ; +∞) -∂G ∂q (ξ; 0 + ) if ξ < +∞ 2 π -1 π ∂G ∂q (+∞; +∞) -∂G ∂q (+∞; 0 + ) if ξ = +∞ ⎫ ⎪ ⎬ ⎪ ⎭ = 1 2π in all cases (62)
where use has been made of the properties ∂G/∂q(ξ; +∞) = -1 2 in all cases, whereas ∂G/∂q(ξ; 0 + ) = 0 or -2 depending on whether ξ < +∞ or ξ = +∞. Equation ( 62) means that the influence function g( a h ; z-z h )(zz) -2 in equation ( 59) behaves like 1 2π(z -z) 2 at short distances, like in an infinite body, of course (see equation ( 5)). Also,

lim v→±∞ g(ξ; v) = ⎧ ⎪ ⎨ ⎪ ⎩ 0 if ξ < +∞ 2 π if ξ = +∞ (63)
since the integrals in equations ( 61) 2 go to zero when v goes to ±∞. All of these features agree with what can be observed in Figure 10. It is easy to understand the behavior of the function g(ξ; v). When the distance |zz | increases from values much smaller than h to values much larger than it, but still much smaller than a (that is, when the normalized distance |v| increases from values much smaller than unity to values much larger than it, but still much smaller than ξ), this function increases from the value for an infinite body, 1 2π , to that for a thin plate, 2 π . However, since the distance a has been observed to play the role of a cutoff length, when |zz | becomes comparable to or much larger than it (that is, when |v| becomes comparable to or much larger than ξ), the function decreases back to zero. Of course, the larger the value of a or ξ, the slower the decrease; for a = +∞ or ξ = +∞, there is no decrease at all. The role of cutoff length played by the distance a is probably best illustrated by Figure 11, which shows g(ξ; v) as a function of the variable v/ξ ≡ (zz )/a. (The use of this new variable obviously forbids to consider the case ξ = +∞ ⇔ a = +∞). This figure clearly shows that the decrease of g(ξ; v) occurs above some critical distance comparable to a.

Conclusion

The aim of this paper was to provide a first-order formula for the local variation of the mode I stress intensity factor resulting from some small but otherwise arbitrary coplanar perturbation of the front of a planar crack, in the specific case where this crack results from gradual separation of two bonded plates from one side. This configuration is frequently used in experiments, and it was our hope to provide an alternative to [START_REF] Rice | First-order variation in elastic fields due to variation in location of a planar crack front[END_REF]'s formula for the theoretical interpretation of such experiments. Indeed, although this formula has systematically been used for this purpose up to now, it was derived for a semi-infinite crack in some infinite body, and thus applies to the configuration considered only when the characteristic distance of variation of the perturbation of the front is much smaller than the plate thickness.

In a first step, [START_REF] Rice | First-order variation in elastic fields due to variation in location of a planar crack front[END_REF]'s treatment, appropriate for infinitely thick plates, was adapted to the other extreme case of plates of infinitesimal thickness. This was done by replacing the standard 3D theory of elasticity he used by Love-Kirchhoff's theory of plates. The validity of this procedure is ensured by the fact that the latter theory is known to be the correct asymptotic one for very thin plates. An interesting outcome of the analysis is that the distance between the crack front and the boundary of the plate plays the role of a "cutoff length" for elastic interactions along the crack front; that is, the influence of the crack advance at some point of the front upon the stress intensity factor at some other point quickly vanishes when the separation between these points becomes larger than this distance. The plate thickness, on the other hand, plays no similar role.

In a second step, the analytical formulae applicable to extreme, infinite or infinitesimal, values of the plate thickness, were completed by finite element computations for a number of intermediate values. In addition to confirming the validity of the analytical formulae for the two limit cases, these computations allowed us to define an approximate but accurate formula for the local variation of the stress intensity factor applicable to plates of arbitrary thickness.

Fig. 2 .

 2 Fig. 1. The general problem -View in the Oxy plane

  ) 1 must be looked for in the form of an exponential e μx . Then μ satisfies the characteristic equation (μ 2k 2 ) 2 = 0, which admits two double solutions, μ = k and μ = -k. It follows that δv(x, k) must be a linear combination of the functions e kx , xe kx , e -kx , xe -kx , or equivalently of the form δv(x, k) = (Ax + B) sinh(kx) + (Cx + D) cosh(kx) (26)

Fig. 6 .

 6 Fig. 6. The function f (u) for a very thin plate

Fig. 7 .Fig. 8 .

 78 Fig. 7. Distribution of the normalized energy-release-rate along the crack front for δa max /λ = 0.01.

Fig. 9 .

 9 Fig. 9. The quantity X for a plate of arbitrary thickness

Fig. 10 .

 10 Fig. 10. The function g(ξ; v) for a plate of arbitrary thickness

Fig. 11 .

 11 Fig. 11. Plot of g(ξ; v) versus v/ξ ≡ (z -z )/a

The general consensus is that the sole role of the interface is to enforce coplanar propagation of the crack between the plates; its lower elastic stiffness is unimportant and may be ignored.

That is, in the case of a sinusoidal perturbation, simply the wavelength 2π/|k|.

The more customary notation K for Love-Kirchhoff's deformation tensor is avoided in order to prevent any possible confusion with the stress intensity factor K.

It may be noted incidentally that equation (19) is just a straightforward 3D extension of that applicable to 2D problems, which underlies the derivation of some solutions of thin elastic cracked bodies using beam theory; see[START_REF] Tada | The Stress Analysis of Cracks Handbook[END_REF] for specific examples.

The constant D should not be confused with the bending stiffness defined by equation (12), which no longer plays a role.

Use is been made here of the fact that the function F (p) being even, see equation (37), its second derivative is also even.

In fact a term of this kind does appear in[START_REF] Rice | First-order variation in elastic fields due to variation in location of a planar crack front[END_REF]'s formula but must logically be discarded when this formula is applied to a cracked plate, its contribution becoming negligible then.

It is implicitly assumed here that the crack advance does not induce any variation of the bending moments exerted on the two halves of the left side; this is true in all calculations since λ a.

The curve corresponding to ξ = 10 is shown for illustrative purposes but involves some error since the necessary hypothesis (47) is barely satisfied for such a value.

Acknowledgement. The authors wish to express their sincere thanks to Dr. Laurent Ponson for several stimulating discussions.


Julien Chopin
email: julien.chopin@espci.fr

Aditya Bhaskar

Atharv Jog

Laurent Ponson
email: laurent.ponson@upmc.fr

Depinning dynamics of crack fronts

We investigate experimentally and theoretically the dynamics of a crack front during the microinstabilities taking place in heterogeneous materials between two successive equilibrium positions. We focus specifically on the spatio-temporal evolution of the front, as it relaxes to a straight configuration, after depinning from a single obstacle of controlled strength and size. We show that this depinning dynamics is not controlled by inertia, but instead, by the rate dependency of the dissipative mechanisms taking place within the fracture process zone. This implies that the crack speed fluctuations around its average value vm can be predicted from an overdamped equation of motion (v -vm)/v0 = (G-Gc(vm))/Gc(vm) involving the characteristic material speed v0 = Gc(vm)/G c (vm) that emerges from the variation of fracture energy with crack speed. Our findings pave the way to a quantitative description of the critical depinning dynamics of cracks in disordered solids and open up new perspectives for the prediction of the effective failure properties of heterogeneous materials.

Woods, nacre, bones or rationally designed artificial materials, are all heterogeneous solids, with mechanical properties far exceeding those of their constitutive components. Understanding the role of microscale heterogeneities on the macroscale fracture behavior of solids still remains a query. This becomes especially relevant now, as rapid developments in microfabrication techniques allow the tailoring of microstructures at ever smaller scales, yielding new types of composites, known as meta-materials, with unprecedented mechanical properties [1][2][3][4][5][6]. Recently, significant progresses were made for weakly heterogeneous brittle solids where models describing a crack front as a deformed interface pinned by tough obstacles have been successfully applied [7][8][9][10][11]. The homogenized fracture properties can be computed exactly within the so-called weak pinning limit [12], where the elastic energy release rate G balances the fracture energy G c at any time and any position along the front. This approach holds for weak variations of toughness along the propagation direction. The crack evolution is then smooth and can be properly approximated by a continuous succession of equilibrium front configurations [13,14]. This approach was successfully used to design weakly heterogeneous systems with improved and new macroscopic failure properties [15][16][17][18].

However, most natural and engineered materials have a microstructure composed of discontinuous heterogeneities which cannot be described within the weak pinning regime. The strong pinning regime that predominates for large toughness gradients challenges standard homogenization approaches. Crack propagation is not quasi-static but proceeds by intermittent and local micro-instabilities. Further, for a disordered distribution of obstacles, crack growth takes place close to the so- Rectangular obstacles of larger toughness. (c) c(z, t) and δf (z, t) = c(z, t) -c(0, t) are the crack front profile and deformation, respectively. δfs = δf (z, t < 0), δf0 = δf (d, 0) and w(t) are the pinned (stationary) crack front deformation, the characteristic height and half-width, respectively. (d) Sequence of crack front deformation after depinning and theoretical prediction.

called depinning critical transition [19][20][21], so that the crack front dynamics is dominated by avalanches spanning over a large range of length and time scales [22][23][24][25][26].

The precise understanding of the front evolution during these rapid events is a prerequisite to predict and further, control the fracture energy of heterogeneous solids. Beyond fracture, the behavior of driven disordered mechanical systems with long-range interactions is still an open question whose tremendous difficulty resides in the subtle interplay between fast, localized, depinning events and larger macroscopic avalanches forming a complex energetic landscape composed of many metastable states [27][28][29].

In this letter, we address experimentally and theoreti-cally the basic problem of the interaction of a crack front with a tough obstacle in the strong pinning regime. In our experiment, a planar crack is driven at a constant speed over a tough region of finite length along the propagation direction, triggering a depinning instability between two well-defined metastable states. The size and strength of the obstacles are fully controlled and adjusted using our patterning technique. The sample allows insitu visualization of the crack front dynamics which is resolved spatially and temporally.

After normalizing all the length scales by the obstacle width d, we show that the relaxation dynamics follows a universal law which only depends on v 0 , the crack speed at depinning for an obstacle of unit strength. v 0 is also found to vary linearly with the crack speed v m imposed prior depinning by the loading rate. Next, we develop a theoretical model based on linear elastic fracture mechanics to quantitatively capture the observed behavior. Here, inertial effects can be neglected as the crack speed remains several orders of magnitude lower than the wave speed. Instead, we take into account the rate dependency of the fracture energy to quantitatively capture the effect of crack speed on the dissipative mechanisms taking place within the process zone. Thus, unlike perfectly brittle solids, crack may propagate at finite speed in dissipative materials as the elastic energy release rate may be constantly balanced by a rate dependent fracture energy. Linearizing the equation of motion around v m , we obtain an analytical solution for the depinning of a crack from a single obstacle that is shown to capture quantitatively all our experimental observations. The implications of our results on the energy dissipated during fast fracture events and the fracture behavior of materials with randomly distributed obstacles are discussed in the final part of our paper.

We start by describing our experimental setup. A 5 mm plate made of Polymethylmethacrylate (PMMA, Young modulus E p = 1.8 GPa) with a heterogeneous coating is detached from a thick elastomer block using the beam cantilever geometry shown in Fig. 1(a). A vertical upward point like force is exerted at the extremity of the PMMA plate by means of a string connected to a mechanical testing machine allowing to impose the deflection speed. The elastomer is a crosslinked PolyDiMethyl-Siloxane (PDMS Sylgard184, Dow Corning) with a much lower Young modulus E s = 1.5 MPa than PMMA and a Poisson's ratio ν s 0.5. It is prepared by mixing an oligomer together with a silicon oil and degased for 2 hours under mild vacuum. It is then cured in an oven at 75 o C for at least 2 h. The resulting crosslinked PDMS block of size W × L = 50 × 80 mm 2 with thickness h s = 20 mm is then demoulded. The crack is driven at an average speed v m in the range 5 -100 µm/s that is set by the deflection rate imposed by the testing machine.

Taking inspiration from the experiments of Xia et al. [10,15], we control the local fracture properties of the interface by printing obstacles on a commercial transparency, taking advantage of the high toughness G c1 of the printed regions on PDMS compared to the neat one noted G c0 . Unlike G c1 which does not show significant variations with the crack speed v m , G c0 is found to increase as v γ m where γ = 0.37 ± 0.05 [30]. The microscopical origin of the exponent is still highly debated [31,32]. As a consequence, the contrast C = (G c1 -G c0 )/G c0 can be varied by exploring different crack speeds. As shown in Fig. 1(b), rectangles of width 2 d and length 6 d are aligned along the propagation direction where d is varied between 0.1 and 0.5 mm. A spacing of 6 d between two successive obstacles is chosen to allow a complete relaxation of the front before it reaches the next obstacle. The transparency is then bonded onto the PMMA plate by means of a double-sided adhesive tape, the heterogeneous side faced up. Finally, a thin liquid film of PDMS is laid on the substrate before bringing the coated PMMA plate in contact allowing an intimate bonding between materials after curing at 40 o C for 48h.

The transparency of the materials used in our setup is exploited to visualize the front geometry and its evolution as it interacts with the obstacle. Images of 3900 × 2600 pixels are taken normal to the mean fracture plane by a CCD camera through a semi-transparent mirror oriented at 45 o . An LED panel is placed horizontally beneath the sample to increase the contrast between the bonded and unbonded regions of the interface. A home-made algorithm extracts then the crack position c(z, t) for each image taken at time t where the depinning onset defines t = 0 (see Fig. 1(c)). The front deformation is defined as δf (z, t) = c(z, t) -c(0, t). An acquisition rate of 10 Hz allows resolving in detail the front evolution during the depinning regime.

In a typical experiment, the front propagates initially in a homogeneous interface as a straight line. For the range of velocity explored, no stick-slip instability occurs [33]. When crossing the obstacle, the profile gradually deforms until reaching a stationary shape composed of a pinned region of amplitude δf 0 (C, d) and logarithmic tails δf [8,9,11,34,35]. When reaching the end of the obstacle, the crack front is suddenly out-of-equilibrium as the deformed profile is not stable in a homogeneous interface. We observe a fast motion of the pinned region and a slower motion of the remote part resulting in a relaxation towards a straight configuration (Fig. 1(d)). This behavior is reminiscent of avalanches which are sudden fracture events observed between metastable configurations in fully disordered materials driven close to the so-called depinning transition [23,25,27]. We will see later that both phenomena are actually closely related.

s (z) 2δf 0 (C, d) ln(|z|/d) for |z| d. For weak obstacles, δf 0 (C, d) varies linearly with C but non-linearities appear when C is finite, yield- ing δf 0 (C, d) = d C N L /π, where C N L = C (1 -C/2 + C 2 /6)
We first focus on the initial dynamics of the insta- Next, we address the relaxation dynamics at longer times beyond the onset of instability. We first measure the amplitude δf (d, t) of the front deformation, and its evolution during depinning (see Fig. 1(c)). As shown in the inset of Fig. 3(a), we observe that δf (d, t) relaxes towards zero at a rate strongly depending on v m and C. However, we found a good collapse of the relaxation profiles by normalizing δf (d, t) and t by C N L × d and d/v m , respectively. These rescalings are found to be also relevant for the evolution of the half-width w(t) of the perturbation, where w(t) is defined from the relation δf (w(t), t) = δf s (d) (see Fig. 1(c)). Here, w(t) quantifies the lateral spreading of the perturbation through time. As shown in Fig. 3(b), we also find a good collapse of the data normalizing w(t) by d. Further, after a short transient, the width is found to grow linearly with time, following w(t) = 5.7v m t.

To explain quantitatively the observed dynamics, we develop a model within the framework of Linear Elastic Fracture Mechanics (LEFM) including a physically based dissipation mechanism to account for the viscoelastic dissipation in the process zone (PZ). Imposing that the energy release rate is balanced by the dissipated work within the PZ, the equation controlling crack evolution reads

G[c(z, t)] = G c [c(z, t), v(z, t)] . (1)
Here, G c not only depends on the crack configuration c(z, t) resulting from the interaction of the front with the obstacle but also on the local speed v(z, t) = ∂c(z, t)/∂t owing to the rate dependency of the dissipation. A firstorder perturbation of Eq. ( 1) around the mean front position v m t yields δG[δc] = ∂Gc(vm) ∂v δv where δc(z, t) = c(z, t) -v m t and δv(z, t) = v(z, t) -v m . The l.h.s. term corresponds to a non-local elastic restoring force [36] while the r.h.s term represents a local friction term increasing linearly with v. Terms such as ∂Gc ∂c are not relevant since depinning occurs in a homogeneous region of the interface. The fracture toughness is taken in the form of

G c = G 0 c (v/v c ) γ
, where G 0 c , γ and v c are material parameters characterizing the dissipation mechanisms taking place in the process zone. Upon linearization of G c around the macroscopic driving velocity v m in a slow propagation regime v m v c , we obtain

1 G 0 c δG c δv ≡ 1 v 0 = γ v m . (2)
Here we take γ = 1/3 in agreement with the value measured for the neat regions of the interface between the PDMS substrate and the cantilever [30]. Using the expression of δG derived for an interfacial crack between an incompressible substrate and a much stiffer material [37], we obtain the equation of motion

1 v 0 ∂δc(z, t) ∂t = 1 π P V +∞ -∞ dz δc(z , t) -δc(z, t) (z -z) 2 . (3)
It is noteworthy that Eq. 3 is formally equivalent to the linear order to the equation of motion of a contact line of Newtonian fluids partially wetting a solid surface [38,39]. The steady-state pinned profile with logarithmic tails given in [34] is taken as initial condition. The equation of propagation can be solved exactly [30], yielding

π C δv v 0 = arctan z + d v 0 t -arctan z -d v 0 t . (4)
From Eq. ( 4), we obtain the depinning velocity v dep = Cv 0 ≈ 3Cv m which is in good agreement with the experimental data of Fig. 2(c) (solid line). Note that the existence of a characteristic depinning speed v 0 emerging from the kinetic law G c (v m ) had already been noticed by Kolvin et al. [40] in the context of microbranching induced crack pinning. The speed profile can then be readily integrated to provide the general form of the front profile. Fig. 4 shows a spatio-temporal map of δf (z, t) where the entire relaxation to a straight configuration can be observed. To avoid cumbersome equations, we will just give analytical expressions of δf (z, t) in some limits which are useful to interpret our experimental data [

Eq. 5 is in good agreement with the experimental data of Fig. 3(b) (dashed line) provided that the amplitude of the perturbation is normalized by C N L which is justified owing to the rather large contrasts values explored in our experiments. In the limit |z| d, we obtain δf (z, t) = Cd/π ln 1 + (z/(tv 0 )) 2 which generalizes the model of Marsh & Cazabat [41] for the depinning of a contact line, obtained in the long time and vanishing mean velocity limits [30]. Thus, we obtain

w(t) √ 3v 0 t = ( √ 3/γ)v m t. (6)
As shown in Fig. 4, the linear spreading of the perturbation provides a good approximation even at relatively short time.

To summarize, our study of the depinning of a brittle crack from a single obstacle reveals a characteristic velocity v 0 that sets the relaxation time λ/v 0 of the Following are the implications of these findings. First, it sheds light on the nature of the dissipation accompanying avalanches in failure of heterogeneous solids. During an avalanche, the depinning region of the front reaches the speed v 0 that may be much larger than the average crack speed v m . Owing to the increase of the fracture energy with crack speed and the continuity of the elastic energy at the onset of depinning, the dissipation rate during an avalanche is close to the toughness of the impurities, leading to an additional dissipation that reduces to C 2 G c0 d 2 per heterogeneity for the case of a periodic array of obstacles. For disordered distributions, in the strong pinning regime where the front motion consists of a succession of avalanches, we then expect the energy dissipated by unit fractured surface to be significantly larger than the matrix toughness, and closer to the obstacle fracture energy, even for relatively low obstacle density. The proposed crack evolution equation that is amenable to the exploration of more complex toughness landscape embedding multiple obstacles predict the total energy dissipated, including the contribution due to depinning instabilities, and so can serve as a tool for the design of patterned interfaces with improved mechanical performance.

Secondly, our findings allow to address a long standing question about the failure of disordered solids and its relationship with critical phenomena. For randomly dis-tributed obstacles, cracks exhibit a jerky dynamics characterized by universal scaling laws that were shown to be reminiscent of the so-called depinning transition of an elastic interface driven in a random medium [22,25,42]. However, the control parameter that sets the distance of the system to the critical point was not identified yet, in particular under displacement controlled conditions where the front velocity v m is imposed. From the description of the crack dynamics during unstable events brought by this study, this can now be achieved through the comparison of the driving velocity v m with the characteristic speed v 0 of the avalanches, leading to the control parameter δ = v m /v 0 . As expected for dynamical phase transition, this parameter controls the crack front behavior, like the correlation time of the speed fluctuations that was recently shown to diverge as 1/δ [28,29]. Interestingly, v 0 may not be independent of v m . For many material systems like the one considered in this study, G c increases as a power law of v m so that v 0 = v m /γ (see Eq. ( 2)). As a result, the control parameter may often take a fixed value δ = γ, explaining why the crack response is independent of its average speed over several orders of magnitude [24,43]. Overall, the introduction of the parameter δ = v m /v 0 that controls the critical behavior of fracturing material opens new perspective for the quantitative description of fracture in terms of depinning transition.

Supplemental Material for "Depinning dynamics of crack fronts"

Julien Chopin

RATE DEPENDENCE OF THE FRACTURE ENERGY AND FRACTURE ENERGY CONTRAST

The rate dependence of the fracture energy G c0 of the neat (ink free) transparency/pdms interface is shown in Fig. 1(b). It has been obtained using the following procedure: the cantilever is pulled up at different rates in the range ∆ = 1 -10 µm.s -1 . The force-displacement curve F (∆) is then analyzed using the compliance method that provides the elastic energy release rate G = F 2 /(2b)dλ/dc as a function of time. Here, b is the specimen width along the crack front and λ = ∆/F the measured value of the compliance. The crack position and its evolution is determined using a camera, taking advantage of the transparency of the materials. The crack speed determined from the crack position is then represented in Fig. 1(b) as a function of the rate of dissipated energy that balances the elastic energy release rate. The data are modeled using the kinetic law G c0 = (v m /v c0 ) n0 , leading to n 0 = 1/γ = 0.37 ± 0.05. The same procedure could not be performed with a homogeneous printed interface because the ink made the sample opaque preventing a visual detection of the crack front. However, the fracture energy of the inked interface can be derived from the fracture energy contrast. Assuming a similar kinetic law

G c1 = A 1 (v m /v c1 ) n1 for the inked interface, the contrast C = (G c1 -G c0 ) /G c0 now reads C + 1 = G c1 G c0 = v n0 0 v n1 1 v n1-n0 m . (1)
Therefore, the contrast is found to be speed dependent with an exponent ∆n = n 1 -n 0 . Driving the crack front over a single defect at various speed, we can extract the contrast as explained in the main text. Figure 1(a) shows C + 1 as a function of the crack front mean speed v m . From a fit of the data, we find C + 1 ∼ v ∆n m with ∆n = -0.33 ± 0.05. Using the measured value n 0 = 0.37, we obtain n 1 = ∆n + n 0 = 0.04 ± 0.05. We thus conclude that the adhesion of the pdms on an inked interface has a negligible rate dependence compared to the neat interface.

THEORY General solution of the crack evolution equation describing the depinning dynamics

Linear elastic fracture mechanics is used to model the depinning of an interfacial crack front from a single defect. The mechanical energy release rate, G, balances to the fracture energy, G c , at any time along the evolving crack front, leading to

G c (c(z, t), v(z, t)) = G(c(z, t)), (2)
where c(z, t) and v(z, t) = ∂c(z, t)/∂t are the absolute position and velocity fields of the crack front, respectively. In a homogeneous interface, G c (c(z, t), v(z, t)) ≡ G c (v(z, t)), since the fracture energy is identical along the crack front. Developing the fracture energy, G c (v(z, t)), around the average speed, v m , we get

G c (v(z, t)) = G c (v m ) + ∂G c ∂v v=vm × (v(z, t) -v m ). (3)
Introducing the characteristic velocity v 0 defined by v 0 = G c (v m )/ ∂Gc ∂v v=vm and defining δc(z, t) = c(z, t) -v m t, Eq. ( 3) is rewritten as

G c = G c (v m ) + G c (v m ) v 0 ∂δc(z, t) ∂t . (4)
Additionally by expanding the energy release rate, G, using a first-order perturbation valid for an interfacial crack between an incompressible elastic material and a rigid substrate, we have,

G = G ext + G ext π P V +∞ -∞ δc(z , t) -δc(z, t) (z -z) 2 dz . (5)
Using the Griffith criterion, G c (v m ) = G ext , and Eqs. ( 2), ( 4) and ( 5), we get the following governing integro-differential equation for the evolving crack front

1 v 0 ∂δc(z, t) ∂t = 1 π P V +∞ -∞ δc(z , t) -δc(z, t) (z -z) 2 dz . (6)
Using the technique of variable separation, δc(z, t) = α(t)β(z), we rewrite Eq. ( 6) as

1 v 0 1 α(t) dα(t) dt = 1 π 1 β(z) P V +∞ -∞ β(z ) -β(z) (z -z) 2 dz = k ∈ C, (7)
⇒ α(t) = e kv0t . (8)
The multiplicative constant in the solution for α(t) has been dropped as it can be absorbed in the solution for β(z).

Owing to the relaxation process, we require the front deformation δf (z, t) = δc(z, t) -δc(0, t) defined from the point of the crack line located in z = 0 to vanish at very long time. This restricts the range of k to negative real numbers. Further, assuming the solution β(z) to be equal to its Taylor expansion and bounded, we have, by integration-by-parts applied to the finite-part integral,

1 π P V +∞ -∞ β (z ) z -z dz = kβ(z). (9)
We perform a Hilbert transform on Eq. ( 9) and then use the anti-involution property of the Hilbert transform. We also differentiate Eq. ( 9) and then use the commutative property of the Hilbert operator and the derivative operator, resulting in two equations,

β (z) = k.H{β(z)} = -H{β (z)} k .
Using the Hilbert transform on the equation and the anti-involution property again we get

β (z) = -k 2 β(z), ⇒ β(z) = K 1 e ikz + K 2 e -ikz , (10)
⇒ δc k (z, t) = K 1 e ikz+kv0t + K 2 e -ikz+kv0t , k ∈ R -. (11)
Since any linear combination of the solutions of Eq. ( 6) is also a solution to the equation, we obtain the most general solution by taking a linear combination of solutions for all allowed values of k, which passes to an integral in the limit,

δc(z, t) = 0 -∞ ψ(k)(K 1 e ikz+kv0t + K 2 e -ikz+kv0t )dk, ⇒ δc(z, t) = +∞ -∞ φ(k)e ikz-|k|v0t dk. (12)
φ(k) can be specified using the initial crack shape, δc(z, 0) = c(z, 0). For a square-integrable function, c(z, 0), and due to Plancherel's theorem, one obtains

φ(k) = 1 2π +∞ -∞ c(z, 0)e -ikz dz. (13)
Since φ(k) can be found for any given c(z, 0), it implies that this linear combination of the variable-separable solutions is in fact the most general solution.

Kinematic equations of a crack depinning from a single defect

We apply the result obtained in the previous section to the special case of a crack depinning from a single defect. The initial crack front is described by the function [1]

φ(k) = -C sin(kd) πk|k| ( 14
)
where d is the half-width of the obstacle with z = ±d being its sides. C is the contrast parameter defined by the relation

C = (G c1 -G c0 )/G c0
, where G c1 is the fracture toughness of the obstacle and G c0 is the fracture energy of the matrix. This initial crack profile, φ(k), is obtained by considering a single elongated obstacle and calculating the Fourier transform of the perturbed crack front up to the first order. The expression of the perturbed profile in real space reads [2]:

δc(z, 0) = c(z, 0) = C d π 1 + z d ln 1 + z d + 1 - z d ln 1 - z d . (15)
In the limit where |z/d| 1, we obtain the logarithmic tails of the perturbation δc(z, 0) ≈ 2Cd/π ln |z/d|. Second order corrections can be found in [1]. The steady state profile is used as an initial condition for analyzing the dynamical propagation of the crack. Using Eq. ( 12) and performing symmetrical integration, we obtain

δv(z, t) = ∂δc(z, t) ∂t = Cv 0 π ∞ 0 2
Upon using the limiting conditions, we get the following two cases

δv(z, t) = Cv 0 π arctan v 0 t z -d -arctan v 0 t z + d , |z| > d (19) δv(z, t) = Cv 0 π π -arctan v 0 t d + z + arctan v 0 t d -z , |z| < d. (20)
We note in particular that

v 0 = δv(|z| < d, 0 + )/C. (21)
Thus we get a new interpretation for the parameter, v 0 , as being proportional to the speed jump inside the defect, δv(|z| < d, 0 + ) at the onset of depinning. This jump in the velocity is a consequence of the continuity of the crack profile with respect to time and the sudden turn off of the toughness contrast. Before depinning, the elastic restoring force due to the crack fluctuations are balanced by the pinning force. At the depinning transition, the toughness contrast is set to zero, thus, the same elastic restoring force are now balanced by viscous dissipation. This results in a speed jump in the region |z| < d whose amplitude is adjusted so that viscous friction is equal to the pinning force. This condition of continuity yields Eq. ( 21) We further note that in the long-time limit, v 0 t/d 1, Eqs. ( 19) and (20) reduce to an expression independent of z to give

v(z, t)| v0t/d 1 = v m + 2dC πt . (22)
The crack displacement field can be evaluated by direct integration of Eqs. (19) and (20) with respect to time. To circumvent mathematical difficulties related to integration, we evaluate the displacement field measured from a frame attached to the moving origin, δf (z, t) = c(z, t) -c(0, t), by integrating, δ ḟ (z, t) = v(z, t) -v(0, t). The boundary condition used is lim t→∞ δf (z, t) = 0. Again considering two separate cases of |z| > d and |z| < d, we obtain

∞ t δ ḟ (|z| > d, t)dt = 0 -δf (|z| > d, t) = ∞ t Cv 0 π arctan v 0 t z -d -arctan v 0 t z + d -π + 2 arctan v 0 t d dt ⇒ δf (|z| > d, t) = Cv 0 π t -π + 2 arctan v 0 t d + arctan v 0 t z -d -arctan v 0 t z + d - d v 0 ln d 2 + (v 0 t) 2 + z + d 2v 0 ln (z + d) 2 + (v 0 t) 2 - z -d 2v 0 ln (z -d) 2 + (v 0 t) 2 (23)
∞ t δ ḟ (|z| < d, t)dt = 0 -δf (|z| < d, t) = ∞ t Cv 0 π 2 arctan v 0 t d -arctan v 0 t d + z -arctan v 0 t d -z dt ⇒ δf (|z| < d, t) = Cv 0 π t 2 arctan v 0 t d -arctan v 0 t d + z -arctan v 0 t d -z - d v 0 ln d 2 + (v 0 t) 2 + d + z 2v 0 ln (d + z) 2 + (v 0 t) 2 + d -z 2v 0 ln (d -z) 2 + (v 0 t) 2 . (24)
Thus, in this work we have found the temporal evolution of the steady state shape when the front leaves the pinning sites. It can be noted that at the initial time i.e. t → 0, we recover the steady state profile obtained by Chopin et al. [2] (see Eq. ( 15)).

The displacement function of the origin is also calculated using Eq. ( 20 Using the Taylor approximation, for ∆ ≈ 0, we can write,

f (x + ∆) -f (x -∆) -2f (∆) = 2∆(f (x) -f (0)) -2f (0). ( 27
)
Substituting Eq. ( 27) into the trigonometric and logarithmic terms in Eq. ( 23 Eq. 30 can be further simplified considering the long time limit, v 0 t/d 1, slow mean speed, v m /Cv 0 1, and set the arbitrary reference c(0, 0) = 0. From Eq. ( 29 Finally, we would like to discuss the linear approximation made in our model. The distribution of elastic energy release rate along the front is given by a first-order development of G[f (z)], leading to a linear variation of δG with the front perturbations δc (see Eq. ( 5)). This amounts to assume that the crack front is weakly perturbed, and more specifically, that its slope is small, f (z)

1. This is somehow contradictory with the initial condition Eq. ( 15) of the crack configuration as the slope δc (z) actually diverge in z = ±d. This may explain why the predicted values of w lie below the experimental data in Fig. 3b, suggesting that the theory underestimates the speed at which the relaxation process and the spreading of the front perturbation take place. Indeed, as shown in Ref. [1], geometrical non-linear

  FIG. 1. (a) Schematics of the experimental setup showing an interfacial crack front pinned by a heterogeneity. (b) Rectangular obstacles of larger toughness. (c) c(z, t) and δf (z, t) = c(z, t) -c(0, t) are the crack front profile and deformation, respectively. δfs = δf (z, t < 0), δf0 = δf (d, 0) and w(t) are the pinned (stationary) crack front deformation, the characteristic height and half-width, respectively. (d) Sequence of crack front deformation after depinning and theoretical prediction.

FIG. 2 .

 2 FIG. 2. Depinning velocity v dep defined as the jump in crack speed as the front escapes from the obstacle varying (a) C and (b) vm. (c) v dep increases linearly with Cvm. Error bars smaller than the symbol are not shown.

FIG. 3 .

 3 FIG. 3. (a) Relaxation of the normalized deformation amplitude with theoretical prediction (dashed line). Inset : data before normalization. (b) Spreading of the deformation width w(t) and theoretical prediction (dashed line).

FIG. 4 .

 4 FIG. 4. Analytical spatio-temporal map of the front deformation δf during depinning. The contour line for δf = δf0 obtained using the exact solution (dashed line) and the asymptotics solution for |z| d (Eq. 6, solid line) are shown.

FIG. 1 :

 1 FIG.1:(a) Evolution of the fracture energy contrast with the mean crack speed. A fit of the data provides 1 + C ∼ v ∆n m with ∆n = -0.33 ± 0.07. (b) Evolution of the fracture energy for a neat (ink-free) interface as a function of the crack mean speed obtained at different driving rates to cover a wider range. A fit of the data gives Gc0 = (vm/vc0) n 0 with vc0 = 0.45 µm.s -1 and γ = 0.37 ± 0.05.

ln z 2 + (v 0 t) 2 d 2

 22 ) we get, c(z, t)| d/|z| 1,v0t/d 1,vm/Cv0 1 = Cd π(30)Thus we have retrieved the depinning dynamics of a contact line, derived by Marsh & Cazabat[3] for thin defect (|z|/d 1), in the long time limit (v 0 t/d 1) and low mean velocity (v m /Cv 0 1). Incidentally, our derivation illuminates the various approximations implicitly used in the model developed byMarsh & Cazabat.

  30]. For z = d, we obtain

	πδf (d, t) Cd	= t v0 d + ln	arctan 2d t v0 -2 arctan d t v0 2 4 + t v0 d d 1 + t v0 2

  1,2,3 , Aditya Bhaskar 2 , Atharv Jog 2 and Laurent Ponson 2 1 Gulliver UMR 7083, CNRS, ESPCI ParisTech, PSL Research University, F-75005 Paris, France 2 Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7190, Institut Jean Le Rond d'Alembert, F-75005 Paris, France 3 Instituto de Física, Universidade Federal da Bahia, Salvador-BA 40170-115, Brazil

	(Dated: January 31, 2019)

  Since the front is driven at a velocity v m , after a long period of time, we apply the boundary condition lim

	To compute the integral, we rewrite δv(z, t) as
		δv(z, t) = lim t→∞	δv(z, t) -	t	∞	∂δv(z, t) ∂t	dt.	(17)
										t→∞	δv(z, t) = 0
	and calculate the velocity field, δv(z, t), using the Leibniz integral rule
	δv(z, t) = -	Cv 0 π	t→∞ t=t	∂ ∂t	k→∞ k=0	2 sin(kd) cos(kz) k	e -kv0t dk dt,
	⇒ δv(z, t) =	Cv 0 π	arctan	v 0 t d + z	t→∞ t=t	+ arctan	v 0 t d -z	t→∞ t=t	.
									sin(kd) cos(kz) k	e -kv0t dk.	(16)

  ), Front evolution in the case of a thin defect and slow mean speed: d/|z| 1, v0t/d 1 and vm/Cv0 1

	0	t	v(0, t)dt =	0	t	v m +	Cv 0 π	π -2 + arctan	v 0 t d	dt	(25)
	c(0, t) = c(0, 0) + v m t + Cv 0 t -	2Cv 0 π	t × arctan	v 0 t d	-	d 2v 0	ln 1 +	v 0 t d	2	.	(26)

  ) we get, We now express Eq. 28 in the moving frame of reference. Using Eqs.(26) and 28, we have,

	δf (z, t)| d/|z| 1 = 2d	d dz	C	v 0 t π	arctan	z v 0 t			-	d dz	C	v 0 t π	arctan	z v 0 t	z=0
	+2d	d dz		C	v 0 π		z 2v 0	ln(z 2 + (v 0 t) 2 )	-	d dz	C	v 0 π	z 2v 0	ln(z 2 + (v 0 t) 2 )	z=0	,
	= C	-2dz 2 π[z 2 + (v 0 t) 2 ]	+ C	2dv 0 π	2z 2 2v 0 [z 2 + (v 0 t) 2 ]	+	ln(z 2 + (v 0 t) 2 ) 2v 0	-	ln((v 0 t) 2 ) 2v 0	,
			⇒ δf (z, t)| d/|z| 1 =	Cd π	ln 1 +	z v 0 t	2	.	(28)
	c(z, t)| d/|z| 1 =	Cd π	ln 1 +	z v 0 t	2	+ c(0, 0) + v m t + Cv 0 t	(29)
						-	2Cv 0 π	t × arctan	v 0 t d	-	d 2v 0	ln 1 +	v 0 t d

2

.

effects result in a stronger elasticity (i.e. an increased stiffness) of the crack front, and so in a larger depinning speed. Even though this effect might explain a slightly faster depinning process in the experiments as compared to the predictions of our linear theory, the region where the crack front slope is large is actually very small, so that the linear theory proposed in this study captures overall well the relaxation dynamics of a crack from a tough obstacle.