Manish Vasoya

Aparna Beena Unni

Jean-Baptiste Leblond

Veronique Lazarus

Laurent Ponson

Finite size and geometrical non-linear effects during crack pinning by heterogeneities: An analytical and experimental study

Keywords: Brittle fracture, crack pinning, perturbation approach, thin film peeling

Crack pinning by heterogeneities is a central toughening mechanism in the failure of brittle materials. So far, most analytical explorations of the crack front deformation arising from spatial variations of fracture properties have been restricted to weak toughness contrasts using first order approximation and to defects of small dimensions with respect to the sample size. In this work, we investigate the non-linear effects arising from larger toughness contrasts by extending the approximation to the second order, while taking into account the finite sample thickness. Our calculations predict the evolution of a planar crack lying on the mid-plane of a plate as a function of the material parameters and the loading conditions, especially in the case of a single infinitely elongated obstacle. Peeling experiments are presented which validate the approach and evidence that the second order term broadens its range of validity in terms of toughness contrast values. The work highlights the non-linear response of the crack front to strong defects and the central role played by the thickness of the specimen on the pinning process.

Introduction

Predicting the role played by small scale heterogeneities on the macroscopic response of solids is an important challenge in mechanics. In a large range of free-boundary and free-discontinuity problems like wetting of liquids on solid substrates, magnetization of ferromagnetic materials or phase transformations, the relation between microscopic properties and macroscopic behavior is governed by some interface which can be highly sensitive to localized defects. In the context of brittle fracture, this reflects on the strong dependence of the toughness to microstructural details, like the spatial distribution of defects and their strength [START_REF] Gao | A first-order perturbation analysis of crack trapping by arrays of obstacles[END_REF][START_REF] Bower | A 3-dimensional analysis of crack trapping and bridging by tough particles[END_REF][START_REF] Xia | Toughening and asymmetry in peeling of heterogeneous adhesives[END_REF]Patinet et al., 2013b;[START_REF] Démery | From microstructural features to effective toughness in disordered brittle solids[END_REF]. The development of predictive tools connecting microstructural parameters of materials to their failure properties is central to engineer materials achieving increased resistance and lifetime. However, to address this challenge, predictive models must consider realistic situations and overcome the limitations of current theories that consider either weak variations of material properties or infinitely small defects compared to the specimen dimensions. The extension of these theories to strong heterogeneities and finite size specimens is the central point of this work.

Microstructural effects on brittle failure phenomena have been largely explored through the fracture mechanics analysis of a continuum elastic medium characterized by some heterogeneous field of local toughness. This approach has met a fair amount of success in capturing experimental observations [START_REF] Ponson | Crack propagation in brittle heterogeneous solids: material disorder and crack dynamics[END_REF][START_REF] Dalmas | Crack front pinning by design in planar heterogeneous interfaces[END_REF][START_REF] Santucci | Fracture roughness scaling: a case study on planar cracks[END_REF][START_REF] Xia | Toughening and asymmetry in peeling of heterogeneous adhesives[END_REF]Patinet et al., 2013a). In this description, small scale variations of the failure properties locally perturb crack propagation and, in fine, affect the whole failure behavior of the material. The central point in this approach is the prediction of the crack front geometry from the characteristics of local toughness field (spatial distribution of defects, toughness contrast...) and the distribution of the local stress intensity factor along a perturbed crack front [START_REF] Lazarus | Perturbation approaches of a planar crack in linear elastic fracture mechanics: a review[END_REF].

Thirty years ago, [START_REF] Rice | First-order variation in elastic fields due to variation in location of a planar crack front[END_REF] derived a first-order formula for the variations of the stress intensity factor induced by some small, but otherwise arbitrary coplanar perturbation of the front of a semi-infinite tensile crack in an infinite body. This expression has been used extensively to predict planar crack growth evolution through random arrays of defects [START_REF] Schmittbuhl | Interfacial crack pinning: Effect of nonlocal interactions[END_REF][START_REF] Ponson | Effect of peeling angle on the adhesion of heterogeneous thin films[END_REF][START_REF] Bonamy | Crackling dynamics in material failure as the signature of a self-organized dynamic phase transition[END_REF], and decipher the puzzling geometrical properties of planar cracks observed in experiments [START_REF] Delaplace | High resolution description of a crack front in a heterogeneous plexiglas block[END_REF][START_REF] Santucci | Fracture roughness scaling: a case study on planar cracks[END_REF][START_REF] Bonamy | Failure of heterogeneous materials: a dynamic phase transition?[END_REF].

However, Rice's first-order formula relies on the assumption that crack front perturbations are of small wavelength compared to the specimen dimensions, which is questionable in some experiments [START_REF] Schmittbuhl | Slow crack propagation and slip correlations[END_REF]. This was the motivation for [START_REF] Legrand | Coplanar perturbation of a crack lying on the mid-plane of a plate[END_REF]'s recent extension 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 case of coplanar perturbation of an emerging tensile crack lying on the mid-plane of a plate of finite thickness, thus accounting for the effect of the finite dimensions of the specimen. Patinet et al. (2013a) showed that the new formula did significantly improve the agreement between experimental and computed shapes of crack fronts deformed by the presence of obstacles.

Both [START_REF] Rice | First-order variation in elastic fields due to variation in location of a planar crack front[END_REF]'s and [START_REF] Legrand | Coplanar perturbation of a crack lying on the mid-plane of a plate[END_REF]'s formulae are however accurate only to first order in the perturbation of the front, which limits their application to weak variations of toughness. To explore the non-linear response of cracks pinned by defects of larger contrasts, [START_REF] Leblond | Second-order coplanar perturbation of a semi-infinite crack in an infinite body[END_REF] extended [START_REF] Rice | First-order variation in elastic fields due to variation in location of a planar crack front[END_REF]'s first-order formula for a semi-infinite crack in an infinite body to second order, under the assumption of independence of the unperturbed stress intensity factor imposed by the loading with respect to the average crack front location. Then, [START_REF] Vasoya | A geometrically nonlinear analysis of coplanar crack propagation in some heterogeneous medium[END_REF] released this hypothesis, thus extending the range of application of [START_REF] Leblond | Second-order coplanar perturbation of a semi-infinite crack in an infinite body[END_REF]'s formula to general loading conditions. Recently, [START_REF] Willis | Crack front perturbations revisited[END_REF] and [START_REF] Willis | Second order in-plane dynamic perturbation of a crack propagating under shear loading[END_REF] investigated the dynamic perturbation of a crack up to second order. Their results, taken in the elastostatic limit for mode I loading conditions, were found to be consistent with those previous works.

Still, since both [START_REF] Leblond | Second-order coplanar perturbation of a semi-infinite crack in an infinite body[END_REF][START_REF] Vasoya | A geometrically nonlinear analysis of coplanar crack propagation in some heterogeneous medium[END_REF]'s works considered only infinite bodies, the combined effect of the finite size of the specimens and the geometrical nonlinearities induced by strong defects remains unexplored, and it is the aim of this work to address this question. The results of our calculations apply to several experimental situations involving strong heterogeneities, and for which the crack front perturbation wavelength compares with the thickness of the specimen [START_REF] Santucci | Fracture roughness scaling: a case study on planar cracks[END_REF][START_REF] Chopin | Crack front dynamics across a single heterogeneity[END_REF]. This study also provides predictive tools relevant for the design of heterogeneous materials or thin films where crack pinning by strong engineered obstacles is used to control and modify their failure properties, e.g. to produce asymmetry in the peeling strength of adhesives [START_REF] Xia | Toughening and asymmetry in peeling of heterogeneous adhesives[END_REF].

The paper is organized as follows:

• Section 2 recalls, as a prerequisite, some established results for planar cracks with slightly perturbed fronts. These results pertain to cracks located (i) in some arbitrary body [START_REF] Rice | Weight function theory for three-dimensional elastic crack analysis[END_REF], and (ii) on the mid-plane of a plate [START_REF] Legrand | Coplanar perturbation of a crack lying on the mid-plane of a plate[END_REF]. • Section 3 presents an extension of [START_REF] Legrand | Coplanar perturbation of a crack lying on the mid-plane of a plate[END_REF]'s first-order results for a cracked plate to the second order, using Rice (1989)'s general results. • Section 4 applies these results to the case of a semi-infinite crack propagating quasistatically along the mid-plane of an infinite plate having some heterogeneous distribution of toughness. Assuming the stress intensity factor to be equal to the toughness at every point of the crack front, we determine the resulting shape of this front up to second order in the toughness fluctuations. • As an application, Section 5 considers the case of a crack penetrating into a single obstacle of infinite length in the direction of propagation. The equilibrium shape of the front is calculated up to second order in the contrast of toughness between the matrix and the obstacle. • In Section 6, we take inspiration from the experimental setup designed in [START_REF] Xia | Toughening and asymmetry in peeling of heterogeneous adhesives[END_REF], and explore the process of front pinning in specimens of thickness comparable to the obstacle size. The geometry of the experimental pinned front is compared with the results of our calculations for various toughness contrasts of the obstacles, providing a critical test of the theory and its range of applicability. • Section 7 finally provides a summary of, and some comments on, the results obtained.

2 Summary of some previous works 2. 1 Rice (1989)'s general formulae for perturbation of a plane crack Consider (Figure 1) a planar crack with arbitrary contour embedded in some arbitrary isotropic elastic body symmetric about the crack plane, loaded in pure mode I through some symmetric system of prescribed forces and/or displacements. Assume that the crack front is perturbed, under constant loading, by a small distance δa(s) within the crack plane in the direction perpendicular to the front, where s denotes a curvilinear abscissa along it.

a(s)

Fig. 1. A planar mode I crack with a slightly perturbed front in an arbitrary body.

Under such conditions, Rice (1989)'s first formula provides the infinitesimal variation δK of the local SIF in the following form:

δK(s 1 ) = [δK(s 1 )] δa(s)≡δa(s 1 ),∀s + P V CF Z(s 1 , s)K 0 (s)[δa(s) -δa(s 1 )]ds.
(1)

In this expression,

• [δK(s 1 )] δa(s)≡δa(s 1 ),∀s denotes the value of δK(s 1 ) for a uniform advance of the front equal to δa(s 1 ); • the symbol PV denotes the Cauchy principal value of the integral, which is taken over the crack front CF ; • K 0 (s) denotes the local unperturbed SIF at the point s of the front; • Z(s 1 , s) denotes the fundamental kernel (FK) of the cracked geometry considered, tied to Bueckner's mode I crack-face weight function.

The FK depends on the geometry, but has no dependence upon the loading other than on which portions of the body and its boundary have forces versus displacements prescribed. It is not known explicitly except in some special cases, but it is known to obey the following general properties:

       Symmetry : Z(s 1 , s 2 ) = Z(s 2 , s 1 ).
Asymptotic behavior for nearby points :

Z(s 1 , s 2 ) ∼ 1 2π(s 1 -s 2 ) 2 for s 1 → s 2 .
(2) Note that the second property shows that the integral in Eq. (1) does make sense as a Cauchy principal value (the function δa being assumed to be differentiable).

In addition, Rice (1989)'s second formula provides the infinitesimal variation of the FK in the following form:

δZ(s 1 , s 2 ) = δ * Z(s 1 , s 2 ) + P V CF Z(s 1 , s)Z(s, s 2 )[δa(s) -δ * a(s)]ds (3)
where δ * Z(s 1 , s 2 ) is the variation of the FK due to any other crack perturbation δ * a(s) such that δ * a(s 1 ) = δa(s 1 ) and δ * a(s 2 ) = δa(s 2 ). Note that there are in fact two principal values here, one at s = s 1 and the other at s = s 2 .

Eq. ( 3) is interesting, of course, only if special motions δ * a(s) of the crack front exist for which the variation δ * Z(s 1 , s 2 ) may be calculated independently. It is always possible to find a combination of a translatory motion, a rotation and a homothetical transformation of the crack front satisfying the constraints put on δ * a(s). In the special case of a semiinfinite crack in an infinite body subjected to prescribed forces only, a combination of a translatory motion and a rotation suffices and leaves the overall geometry unchanged so that δ * Z(s 1 , s 2 ) is zero. In finite bodies, this is no longer the case in general but there are some special cases where certain perturbations δ * a(s) generate a zero δ * Z. Under such conditions, Eq. ( 3) reads simply

δZ(s 1 , s 2 ) = P V CF Z(s 1 , s)Z(s, s 2 )[δa(s) -δ * a(s)]ds. ( 4
)
2.2 [START_REF] Legrand | Coplanar perturbation of a crack lying on the mid-plane of a plate[END_REF]'s solution for a crack lying on the mid-plane of a plate

Let us now consider (Figure 2a) a semi-infinite plate of thickness 2h. Choose the coordinate axes in such a way that this plate occupies the domain 0 ≤ x < +∞, -h ≤ y ≤ h, -∞ < z < +∞ in 3D space. Assume that there is an emerging crack on the mid-plane of the plate, occupying the region 0 ≤ x ≤ A, y = 0, -∞ < z < +∞, where A, the distance from the plate boundary to the crack front, is supposed to be much larger than h. Finally assume that the upper and lower parts of the boundary are subjected to opposite prescribed displacements perpendicular to the crack plane, generating a state of pure mode I at every point of the crack front.

Assume now that the crack front is perturbed by a small distance δa(z) within the crack plane (Ozx). Under such conditions, [START_REF] Legrand | Coplanar perturbation of a crack lying on the mid-plane of a plate[END_REF] expressed the infinitesimal variation of the local SIF in the following form:

δK(z 1 ) K 0 (A) = -2 δa(z 1 ) A + P V +∞ -∞ f ( A h ; z-z 1 h ) (z -z 1 ) 2 [δa(z) -δa(z 1 )] dz (5)
where K 0 (A) denotes the unperturbed SIF, depending on the location A of the unperturbed front, and the function f ( A h ; z-z 1 h )/(z 1z) 2 represents the FK for the semi-infinite cracked plate considered, subjected to displacements prescribed on its boundary. A fully explicit formula was provided by [START_REF] Legrand | Coplanar perturbation of a crack lying on the mid-plane of a plate[END_REF] for this specific FK.

An interesting limit-case is that of a semi-infinite crack in an infinite plate (A → +∞). The conditions of prescribed displacements on the boundary then become equivalent, away from it, to conditions of remote prescribed bending moments. In such a limit-case, Eq. ( 5) becomes δK(z 1 )

K 0 = P V +∞ -∞ f ( z-z 1 h ) (z -z 1 ) 2 [δa(z) -δa(z 1 )] dz (6)
where the function f ( z-z 1 h )/(zz 1 ) 2 ≡ f (+∞; z-z 1 h )/(zz 1 ) 2 represents the FK for an infinite cracked plate subjected to prescribed forces only. Attention will be focused on this sole special case in the sequel.

It will be necessary to use formulae in Fourier's space. The definition adopted here for the Fourier transform φ(k) of an arbitrary function φ(z) is

φ(z) ≡ +∞ -∞ φ(k)e ikz dk ⇔ φ(k) ≡ 1 2π +∞ -∞ φ(z)e -ikz dz. (7)
The Fourier transform of δK(z 1 ) is readily found by expressing the function δa in Eq. ( 6) in terms of its Fourier transform δa: one thus obtains

δK(z 1 ) K 0 = +∞ -∞ δa(k) P V +∞ -∞ f ( z-z 1 h ) (z -z 1 ) 2 e ikz -e ikz 1 dz dk.
Writing now e ikze ikz 1 as e ikz 1 (e ik(z-z 1 ) -1), one gets from there, with a few transformations,

δK(k) K 0 = -|k|X(kh) δa(k) (8)
where

X(p) ≡ 1 |p| P V +∞ -∞ f (u) u 2 1 -e ipu du ⇔ f (u) ≡ 1 2π +∞ -∞ |p|X(p) e -ipu dp. (9)
Eq. ( 9) 1 here is obtained directly, and Eq. ( 9) 2 is then easily established by calculating [|p|X(p)] .

In the work of [START_REF] Legrand | Coplanar perturbation of a crack lying on the mid-plane of a plate[END_REF], Eq. ( 6) was not obtained directly but through Fourier inversion of Eq. ( 8). This equation was itself established by solving the elasticity problem of a plate the mid-plane of which contained a crack with a sinusoidal front, both analytically in the case of an infinitesimal plate thickness and by the finite element method in the case of a finite one. The procedure yielded the following explicit expression of the function X(p):

X(p) 1 2 1 + 3 1 + 1 3 |p| 4/3 . ( 10
)
This expression is (slightly) approximate in general because of the numerical method used to determine it, but exact in the two limits p → 0 and p → ±∞.

It is finally instructive to consider a sinusoidal perturbation of the front of wavelength λ. For such a perturbation the only values of the wavenumber k to be considered are k = ±2π/λ. Then:

• if h λ, |kh| 1
so that, by Eq. ( 10), X(kh) → 1 2 and Eq. ( 8) becomes identical 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 a semi-infinite crack in an infinite body;

• if h λ, |kh| 1, so that X(kh) → 2 and Eq. ( 8) becomes identical to the formula established analytically by [START_REF] Legrand | Coplanar perturbation of a crack lying on the mid-plane of a plate[END_REF] for a semi-infinite crack lying on the mid-plane of a plate of infinitesimal thickness.

The value of δK(z) is thus 4 times larger, for a given crack front perturbation δa(z), in a very thin plate than in an infinite body.

3 Coplanar perturbation of a semi-infinite crack lying on the mid-plane of an infinite plate

Notations

For now on, we consider (Figure 2b) the case of a semi-infinite crack located on the midplane of an infinite plate of thickness 2h, loaded in pure mode I through prescribed forces only.

Assume that the crack front perturbation δa(z) is of the form

δa(z) = φ(z) (11)
where is a small parameter and φ(z) a given, fixed, smooth function. With this notation the equation of the perturbed front reads

x(z) = a + φ(z) (12)
where a denotes the distance from some arbitrary reference axis Oz to the unperturbed straight front. The position of the perturbed front is thus characterized by the parameters a and , and the position of a current point along it by the parameter z.

First-order expansion of the fundamental kernel

The FK for the crack configuration depicted by Eq.( 12) is denoted Z( ; z 1 , z 2 ), and our interest here lies in the first-order expression of Z( ; z 1 , z 2 ),

Z( ; z 1 , z 2 ) ≡ Z 0 (z 1 , z 2 ) + Z 1 (z 1 , z 2 ) + O( 2 ). ( 13
)
Since both the special loading considered by [START_REF] Legrand | Coplanar perturbation of a crack lying on the mid-plane of a plate[END_REF] when deriving Eqs. ( 6), ( 9) and ( 10), and the more general one considered here, involve prescribed forces only (no prescribed displacements), the FK Z 0 (z 1 , z 2 ) for the unperturbed configuration of the crack is the same as in Eq. ( 6),

Z 0 (z 1 , z 2 ) ≡ f ( z 1 -z 2 h ) (z 1 -z 2 ) 2 (14)
where the function f is given by Eqs. ( 9) and (10).

In the specific case considered, the ancillary perturbation δ * a(z) ≡ φ * (z) of Rice's second formula (3) may be taken as a suitable combination of a translatory motion and a rotation:

φ * (z) ≡ φ(z 1 ) + φ(z 2 ) -φ(z 1 ) z 2 -z 1 (z -z 1 ) = φ(z 2 ) + φ(z 1 ) -φ(z 2 ) z 1 -z 2 (z -z 2 ). ( 15
)
The lack of a homothetical transformation of the crack front here (which basically arises from its initial straightness) implies that it undergoes a mere rigid-body motion. Under such a motion the thickness of the plate, and therefore the overall geometry, remain unchanged1 so that δ * Z is zero. Eq. ( 4) therefore applies and gives

Z 1 (z 1 , z 2 ) = P V +∞ -∞ f ( z 1 -z h )f ( z-z 2 h ) (z 1 -z) 2 (z -z 2 ) 2 [φ(z) -φ * (z)] dz (16)
where use has been made of Eq. ( 14).

Second-order expansion of the stress intensity factor in the physical space

The mode I SIF for a given, fixed loading imposed upon the cracked geometry is denoted K(a, ; z). Our interest here lies in the second-order expression of this SIF with respect to :

K(a, ; z) ≡ K 0 (a) + K 1 (a; z) + 2 K 2 (a; z) + O( 3 ). ( 17
)
It is assumed in this equation that the loading has a translatory invariance in the direction z of the crack front, so that the unperturbed SIF K 0 (a) depends on the location a of the (straight) front but not on the position of the point of observation along it.

At order 1, the expression of K 1 (a; z) is obtained through direct application of Rice (1989)'s first formula (1) for the variation of the SIF to the straight configuration of the front:

K 1 (a; z 1 ) = dK 0 da (a) φ(z 1 ) + K 0 (a) P V +∞ -∞ f ( z-z 1 h ) (z -z 1 ) 2 [φ(z) -φ(z 1 )] dz. ( 18
)
This equation is identical to Eq. ( 6) of [START_REF] Legrand | Coplanar perturbation of a crack lying on the mid-plane of a plate[END_REF] except for the additional first term in the right-hand side. This term was zero for the special loading envisaged by [START_REF] Legrand | Coplanar perturbation of a crack lying on the mid-plane of a plate[END_REF] because the unperturbed SIF was independent of the location of the (straight) crack front within the crack plane, but this does not remain true for the more general loading considered here.

In the same way, at order 2, K 2 (a; z) may be obtained by again applying Rice (1989)'s first formula (1), considering some pre-perturbed configuration of the front upon which is superimposed a secondary, infinitesimal proportional perturbation. When doing so, one must use formulae for the SIF and the FK on the pre-perturbed configuration accurate to first order in the primary perturbation; the first of these formulae is provided by Eq. ( 18) and the second by Rice (1989)'s second formula (3), which takes the form (16) in the present case. The output is an expression of ∂K(a, ; z)/∂ accurate to first order in , which immediately yields the second-order expression of K(a, ; z) upon integration. This was the procedure followed by [START_REF] Leblond | Second-order coplanar perturbation of a semi-infinite crack in an infinite body[END_REF] and [START_REF] Vasoya | A geometrically nonlinear analysis of coplanar crack propagation in some heterogeneous medium[END_REF] in the case of an infinite body. The reasoning for a plate is analogous and therefore will not be repeated. The final result reads:

K 2 (a; z 1 ) = 1 2 d 2 K 0 da 2 (a) [φ(z 1 )] 2 + 1 2 dK 0 da (a) P V +∞ -∞ f ( z-z 1 h ) (z -z 1 ) 2 [φ 2 (z) -φ 2 (z 1 )] dz + K 0 (a) 2 P V +∞ -∞ +∞ -∞ f ( z-z 1 h )f ( z -z h ) (z -z 1 ) 2 (z -z) 2 [φ(z) -φ(z 1 )][φ(z ) -φ(z)] dzdz +P V +∞ -∞ +∞ -∞ f ( z -z 1 h )f ( z -z h ) (z -z 1 ) 2 (z -z) 2 φ(z ) -φ(z 1 ) - φ(z) -φ(z 1 ) z -z 1 (z -z 1 ) ×[φ(z) -φ(z 1 )] dzdz . (19)
3.4 Second-order expansions of the stress intensity factor and the energy-release-rate in Fourier's space

With regard to the SIF, at order 1, the expression of the Fourier transform of K 1 (a; z 1 ) is easily established by expressing the function φ in Eq. ( 18) in terms of its Fourier transform φ and using Eq. ( 9) 1 ; one thus gets

K 1 (a; k) = dK 0 da (a) -K 0 (a)|k|X(kh) φ(k). (20)
This expression extends Eq. ( 8), due to [START_REF] Legrand | Coplanar perturbation of a crack lying on the mid-plane of a plate[END_REF], by including the term dK 0 da (a) φ(k), which was absent in this equation since for the loading considered by these authors, the unperturbed SIF K 0 was independent of the location of the crack front within the crack plane.

In the same way, at order 2, the expression of the Fourier transform of K 2 (a; z 1 ) can be established by expressing the function φ in Eq. ( 19) in terms of its Fourier transform φ; one thus gets

K 2 (a; z 1 ) = +∞ -∞ +∞ -∞ φ(k) φ(k ) 1 2 d 2 K 0 da 2 (a)e i(k+k )z 1 + 1 2 dK 0 da (a)P V +∞ -∞ f ( z-z 1 h ) (z -z 1 ) 2 e i(k+k )z -e i(k+k )z 1 dz + K 0 (a) 2 P V +∞ -∞ +∞ -∞ f ( z-z 1 h )f ( z -z h ) (z -z 1 ) 2 (z -z) 2 e ikz -e ikz 1 e ik z -e ik z dzdz +P V +∞ -∞ +∞ -∞ f ( z -z 1 h )f ( z -z h ) (z -z 1 ) 2 (z -z) 2 e ikz -e ikz 1 - e ikz -e ikz 1 z -z 1 (z -z 1 )
× e ik ze ik z 1 dzdz dkdk (21) Now, taking e i(k+k )z 1 as a common factor in the integrands and using Eq. ( 9) 1 , one gets

K 2 (a; z 1 ) = +∞ -∞ +∞ -∞ φ(k) φ(k ) 1 2 d 2 K 0 da 2 (a) - 1 2 dK 0 da (a)|k + k |X((k + k )h) + K 0 (a) 2 I 1 (h; k, k ; z 1 ) + I 2 (h; k, k ; z 1 ) e i(k+k )z 1 dkdk (22)
where I 1 (h; k, k ; z 1 ) and I 2 (h; k, k ; z 1 ) are double integrals, the expression and calculation of which is presented in Appendix A, with the following results:

             I 1 (h; k, k ; z 1 ) ≡ I 1 (h; k, k ) = |k |X(k h) |k + k |X((k + k )h) -|k |X(k h) I 2 (h; k, k ; z 1 ) ≡ I 2 (h; k, k ) = 1 2 k 2 X 2 (kh) -(k + k ) 2 X 2 ((k + k )h) -k 2 X 2 (k h) +2|k + k ||k |X((k + k )h)X(k h) .
(23) Equation ( 22) becomes, upon use of these formulae and "symmetrization" of the integrand with respect to k and k ,

K 2 (a; z 1 ) = K 0 (a) +∞ -∞ +∞ -∞ R(a; h; k, k ) φ(k) φ(k )e i(k+k )z 1 dkdk (24)
where

R(a; h; k, k ) ≡ 1 2 d 2 K 0 K 0 da 2 (a) - 1 2 dK 0 K 0 da (a)|k + k |X((k + k )h) + |k + k | 2 X((k + k )h) |k|X (kh) + |k |X (k h) - 1 4 (k + k ) 2 X 2 ((k + k )h) + k 2 X 2 (kh) + k 2 X 2 (k h) . (25)
Using the change of variable k 1 = k + k , the preceding expression of K 2 (a; z 1 ) may be rewritten in the form

K 2 (a; z 1 ) = K 0 (a) +∞ -∞ +∞ -∞ R(a; h; k, k 1 -k) φ(k) φ(k 1 -k)dk e ik 1 z 1 dk 1
which implies, upon comparison with the definition (7) of the Fourier transform, that

K 2 (a; k 1 ) = K 0 (a) +∞ -∞ R(a; h; k, k 1 -k) φ(k) φ(k 1 -k)dk. ( 26
)
Note the remarkable property that K 2 (a; k 1 ), just like K 1 (a; k 1 ), depends upon the FK of the cracked geometry considered through the sole function X defined by Eq. ( 9) (since the expression (25) of R(a; h; k, k ) involves only X).

A comparison with the results of [START_REF] Vasoya | A geometrically nonlinear analysis of coplanar crack propagation in some heterogeneous medium[END_REF] for an infinite body is in order here, by letting h go to infinity. The function R becomes in this limit, since X(kh) → 1 2 (see Eq. ( 10)):

lim h→+∞ R(a; h; k, k ) = 1 2 d 2 K 0 K 0 da 2 (a) - 1 4 dK 0 K 0 da (a)|k + k | + 1 16 2|k + k | (|k| + |k |) -(k + k ) 2 -k 2 -k 2 (27)
which is another form of the function R found by [START_REF] Vasoya | A geometrically nonlinear analysis of coplanar crack propagation in some heterogeneous medium[END_REF] (and noted P in their paper). This establishes the consistency of the results with those found in the case of an infinite body. Interestingly, Eq. ( 27) is also consistent with the calculations up to second order of [START_REF] Willis | Crack front perturbations revisited[END_REF] who considered the more general situation of the dynamic in-plane perturbation of a mode I crack. Indeed, specializing his results to some sinusoidal front perturbation φ(z) = cos(kz) in the elastostatic limit, his calculations lead to K 2 = -1 8 K 0 (k ) 2 sin 2 (kz) in agreement with our results for loading conditions invariant in the propagation direction, i.e. such that dK 0 da = 0.

The expansion of the Fourier transform G(a, ; k) of the elastic energy-release-rate G(a, ; z) follows from Irwin's formula and the expressions ( 20) and ( 26) of K 1 (a; k) and K 2 (a; k):

G(a, ; k) ≡ G 0 (a)δ(k) + G 1 (a; k) + 2 G 2 (a; k) + O( 3 ) (28)
where δ denotes Dirac's function, G 0 (a) the unperturbed elastic energy-release-rate, and

         G 1 (a; k) = dG 0 da (a) -2G 0 (a)|k|X(kh) φ(k) G 2 (a; k 1 ) = G 0 (a) +∞ -∞ S(a; h; k, k 1 -k) φ(k) φ(k 1 -k) dk (29)
where

S(a; h; k, k ) ≡ 1 2 d 2 G 0 G 0 da 2 (a) - 1 2 dG 0 G 0 da (a) |k + k |X((k + k )h) + |k|X(kh) + |k |X(k h) - 1 2 |k + k |X((k + k )h) -|k|X(kh) -|k |X(k h) 2 +2|k||k |X(kh)X(k h). ( 30
)
4 The equilibrium shape of the front of a crack propagating in a heterogeneous plate

We shall now apply the preceding results to the study of coplanar propagation of a crack along the mid-plane of a plate, governed by Griffith's criterion with a heterogeneous fracture toughness G c (x, z) given by

G c (x, z) ≡ G c [1 + g c (x, z)] , (31)
where G c is a "mean toughness", a small parameter and g c (x, z) a given dimensionless function describing the toughness fluctuations. For a given loading, provided that G is equal to G c at every point of the crack front, the distribution of toughness determines the shape of this front in the form

x = a + φ 1 (a; z) + 2 φ 2 (a; z) + O( 3 ) (32)
where a, φ 1 (a; z) and φ 2 (a; z) are a parameter and some functions to be determined.

For the crack front shape depicted by Eq. ( 32), corresponding to the perturbation function δa ≡ φ 1 + 2 φ 2 , the expression (28) of G takes the form

G(a, ; k) = G 0 (a) δ(k) + G 1 a; { φ 1 } (k) + 2 G 1 a; { φ 2 } (k) + G 2 a; { φ 1 } (k) + O( 3 ) (33)
where G 1 a; { φ} and G 2 a; { φ} are the linear and quadratic functionals of the function φ defined by Eqs. (29) 1 and (29) 2 respectively. Also, the value of the local toughness along the crack front is

G c x = a + φ 1 (a; z) + 2 φ 2 (a; z) + O( 3 ), z = G c 1 + g c (a, z) + 2 ∂g c ∂x (a, z)φ 1 (a; z) + O( 3 ).
The Fourier transform of this expression at the point k 1 is

G c δ(k 1 ) + g c (a, k 1 ) + 2 +∞ -∞ ∂ g c ∂x (a, k) φ 1 (a; k 1 -k)dk + O( 3 ).
Equating the right-hand side of equation ( 33) (at k = k 1 ) to this expression, one gets the following conditions:

• At order 0:

G 0 (a) = G c . ( 34
)
This condition determines the mean location a of the crack front. • At order 1:

G 1 a; { φ 1 } (k) = G c g c (a, k),
which implies, by the expression (29) 1 of the functional G 1 and equation ( 34), that

φ 1 (a; k) = - g c (a, k) 2|k|X(kh) -dG 0 G 0 da (a) . ( 35
)
• At order 2:

G 1 a; { φ 2 } (k 1 ) = -G 2 a; { φ 1 } (k 1 ) + G c +∞ -∞ ∂ g c ∂x (a, k) φ 1 (a; k 1 -k)dk,
which implies, by equations ( 29), ( 34) and ( 35), that

φ 2 (a; k 1 ) = 1 2|k 1 |X(k 1 h) -dG 0 G 0 da (a) +∞ -∞ S(a; h; k, k 1 -k) × g c (a, k) 2|k|X(kh) -dG 0 G 0 da (a) g c (a, k 1 -k) 2|k 1 -k|X((k 1 -k)h) -dG 0 G 0 da (a) dk + +∞ -∞ ∂ g c ∂x (a, k) g c (a, k 1 -k) 2|k 1 -k|X((k 1 -k)h) -dG 0 G 0 da (a)
dk .

(36)

As discussed in [START_REF] Vasoya | A geometrically nonlinear analysis of coplanar crack propagation in some heterogeneous medium[END_REF], the convergence of the integrals involved in the expressions ( 35) and (36) of φ 1 (a; k) and φ 2 (a; k) and their inverse Fourier transforms requires

dG 0 da (a) < 0. ( 37
)
This condition indeed ensures that the denominators of the various fractions never vanish since X(p) > 0. This hypothesis is natural since it warrants stability of crack propagation in the case of a homogeneous toughness (straight crack front). It was satisfied in all experiments mentioned in the Introduction, and will also be so in those discussed in the sequel.

5 The shape of a crack front encountering a single obstacle

Toughness map

As an application, we shall determine the equilibrium shape of the front of a crack lying on the mid-plane of an infinite plate and penetrating into a single obstacle of infinite length in the direction of propagation, up to second order in the contrast of toughness.

The toughness of the matrix will be denoted G M c , and the toughness and width of the obstacle, G O c and 2d, respectively (see Fig. 3). We shall be particularly interested in the limit-case where dG 0 da (a) → 0 -, corresponding to the situation where the typical distance of variation of the unperturbed elastic energy-release-rate is much larger than d.

The toughness distribution may be represented by formula (31) with

G c ≡ G M c ; ≡ G O c -G M c G M c ; g c (x, z) ≡ g c (z) ≡      1 if |z| < d 0 if |z| > d. ( 38
) G c G c M G O c O z d d

Expressions of the crack front shape up to the second order

The Fourier transform of the function g c (x, z) is given by

g c (x, k) ≡ g c (k) = 1 2π d -d e -ikz dz = sin(kd) πk . ( 39
)
At order 1, one gets from equations ( 35) and (39):

φ 1 (a; k) = - sin(kd) πk 2|k|X(kh) -dG 0 G 0 da (a) so that φ 1 (a; z) = - 1 π +∞ -∞ sin(kd) k 2|k|X(kh) -dG 0 G 0 da (a) e ikz dk = - 2 π +∞ 0 sin(kd) k 2kX(kh) -dG 0 G 0 da (a)
cos(kz) dk.

In the limit dG 0 da (a) → 0 -, the integral defining φ 1 (a; z) here diverges. However we are interested only in the deviation of the crack front from straightness. This deviation can be characterized at order 1 by the quantity

φ 1 (a; z) ≡ φ 1 (a; z) -φ 1 (a; 0) = 2 π +∞ 0 sin(kd) k 2kX(kh) -dG 0 G 0 da (a) [1 -cos(kz)] dk . ( 40
)
This quantity has a well-defined limit φ 1 (z) for dG 0 da (a) → 0 -given by

φ 1 (z) ≡ 1 π +∞ 0 sin(kd) k 2 X(kh) [1 -cos(kz)] dk . ( 41
)
At order 2, equations ( 36) and ( 39) yield, since the function

g c (x, k) is independent of x: φ 2 (a; z) = 1 π 2 +∞ -∞ +∞ -∞ S(a; h; k, k ) sin(kd) k 2|k|X(kh) -dG 0 G 0 da (a) sin(k d) k 2|k |X(k h) -dG 0 G 0 da (a) × e i(k+k )z 2|k + k |X((k + k )h) -dG 0 G 0 da (a) dkdk = 2 π 2 k+k ≥0 S(a; h; k, k ) sin(kd) k 2|k|X(kh) -dG 0 G 0 da (a) sin(k d) k 2|k |X(k h) -dG 0 G 0 da (a) × cos[(k + k )z] 2(k + k )X((k + k )h) -dG 0 G 0 da (a)
dkdk where we have grouped the terms (k, k ) and (-k, -k ) in the double integral and accounted for the fact that S(a; h; -k, -k ) = S(a; h; k, k ), see equation ( 30).

Again, we are interested only in the deviation of the crack front from straightness, characterized at order 2 by the quantity

φ 2 (a; z) ≡ φ 2 (a; z) -φ 2 (a; 0) = 2 π 2 k+k ≥0 S(a; h; k, k ) sin(kd) k 2|k|X(kh) -dG 0 G 0 da (a) × sin(k d) k 2|k |X(k h) -dG 0 G 0 da (a) cos[(k + k )z] -1 2(k + k )X((k + k )h) -dG 0 G 0 da (a)
dkdk .

(

) 42
This quantity has a well-defined limit φ 2 (z) for dG 0 da (a) → 0 -given by

φ 2 (z) ≡ 1 4π 2 k+k ≥0 S 0 (h; k, k ) sin(kd) k|k|X(kh) sin(k d) k |k |X(k h) cos[(k + k )z] -1 (k + k )X((k + k )h) dkdk (43)
where

S 0 (h; k, k ) ≡ lim dG 0 /da→0 -S(a; h; k, k ) = - 1 2 |k + k |X((k + k )h) -|k|X(kh) -|k |X(k h) 2 + 2|k||k |X(kh)X(k h).
(44) The integral in equation ( 43) is convergent because the function S 0 (h; k, k ) verifies the properties S 0 (h; k, 0) = S 0 (h; 0, k ) = 0.

Since the function S 0 is obviously invariant upon interchange of k and k , the integral in Eq. ( 43) can be simplified by reducing the integration domain {(k, k ), k + k ≥ 0}: this domain consists of two sub-domains, where k ≥ k and k ≥ k respectively, which yield equal contributions; hence the integral is equal to twice the integral over the first subdomain. This sub-domain consists of two sub-sub-domains, {(k, k ), k ≥ 0, 0 ≤ k ≤ k} and {(k, k ), k ≥ 0, -k ≤ k ≤ 0}; re-noting k as -k in the integral over the second subsub-domain, one finally gets

φ 2 (z) = 1 2π 2 +∞ 0 k 0 S 0 (h; k, k ) cos((k + k )z) -1 (k + k )X((k + k )h) +S 0 (h; k, -k ) cos((k -k )z) -1 (k -k )X((k -k )h) sin(kd) k 2 X(kh) sin(k d) k 2 X(k h) dk dk.
One may now write k ≡ λk, 0 ≤ λ ≤ 1 and use the variables of integration (k, λ) instead of (k, k ); the preceding equation then becomes

φ 2 (z) = 1 2π 2 +∞ 0 1 0 S 0 (h; k, λk) cos((1 + λ)kz) -1 (1 + λ)X((1 + λ)kh) +S 0 (h; k, -λk) cos((1 -λ)kz) -1 (1 -λ)X((1 -λ)kh) sin(kd) X(kh) sin(λkd) λ 2 X(λkh) dλ dk k 4 .
Introducing the notation

S 0 (h; k, λk) k 2 X(kh)X(λkh)X((1 + λ)kh) ≡ T 0 (kh, λ) (45)
in the preceding integral and changing the order of integration, one gets the final expression of φ 2 (z):

φ 2 (z) = 1 2π 2 1 0 J [h; λ; d, λd, (1 + λ)z] -J(h; λ; d, λd, 0) 1 + λ + J [h; -λ; d, λd, (1 -λ)z] -J(h; -λ; d, λd, 0) 1 -λ dλ λ 2 (46)
where

J(h; λ; α, β, γ) ≡ +∞ 0 T 0 (kh, λ) sin(αk) sin(βk) cos(γk) dk k 2 . ( 47
)

Limiting cases of infinite and infinitesimal thickness

In the limit of an infinite body (h → +∞), for which X(kh) → 1 2 , the integrals in Eqs. ( 41), ( 46) and ( 47) may be calculated analytically, and the final results for φ 1 and φ2 read [START_REF] Chopin | Crack front dynamics across a single heterogeneity[END_REF][START_REF] Vasoya | A geometrically nonlinear analysis of coplanar crack propagation in some heterogeneous medium[END_REF]:

                 lim h→+∞ φ 1 (z) = d π (1 + u) ln (|1 + u|) + (1 -u) ln (|1 -u|) lim h→+∞ φ 2 (z) =          - d 2π (1 + u) ln(1 + u) + (1 -u) ln(1 -u) if |u| ≤ 1 - d 2π (|u| -1) ln |u| + 1 |u| -1 + 2 ln 2 if |u| ≥ 1 , u ≡ z d .
(48) Note that for u = ±1, the values of the functions here are defined only as limits using the property lim x→0 + x ln x = 0.

On the other hand, for an infinitely thin plate (h → 0), X is multiplied by a factor of 4 with respect to the case h → +∞, so that by Eqs. ( 41), ( 44), ( 45), ( 46) and ( 47), φ 1 and φ 2 are simply divided by a factor of 4; that is,

                 lim h→0 φ 1 (z) = d 4π (1 + u) ln (|1 + u|) + (1 -u) ln (|1 -u|) lim h→0 φ 2 (z) =          - d 8π (1 + u) ln(1 + u) + (1 -u) ln(1 -u) if |u| ≤ 1 - d 8π (|u| -1) ln |u| + 1 |u| -1 + 2 ln 2 if |u| ≥ 1.
(49)

Numerical solutions for arbitrary values of the thickness

For arbitrary values of h, the integrals appearing in Eqs. ( 41), ( 46) and ( 47) may be calculated numerically. Instead of plotting both φ 1 (z) and φ 2 (z), we choose to plot the sole perturbation of the front defined by: 2

δa(z) = φ 1 (z) + 2 φ 2 (z) (50)
In the following figures, aiming at validating these numerical calculations, its normalized value is displayed for ε = 1. obtained numerically (full lines) and analytically (dotted lines). The agreement is excellent in both cases, which shows that the numerical calculation of the integrals is correct and accurate. Figure 5 now shows the same quantity as a function of the normalized plate thickness h/d, for the same normalized toughness contrast of unity. The dependence of δa(z)/d upon h/d is quite clear here: when the plate becomes thinner, the deformation of the crack front decreases, that is, this front becomes "stiffer". Note that although the value = 1 here is admittedly high, the perturbation of the front remains reasonably small. It is worth noting that the effect of the finite thickness of the plate on the crack front geometry is more pronounced in the thick plate regime, h d, than in the thin plate regime, h d. Indeed the solution for an infinitely thick plate provides an acceptable approximation (within 10% in the domain |z/d| ≤ 6) only for values of h/d exceeding 100, whereas that for an infinitely thin plate remains acceptable (with the same error on the same domain) up to values of h/d of the order of 0.4. This observation suggests that the finite thickness of the specimen may have had a significant impact on the experimental front geometries reported in the works of [START_REF] Delaplace | High resolution description of a crack front in a heterogeneous plexiglas block[END_REF] and [START_REF] Dalmas | Crack propagation through phaseseparated glasses: effect of the characteristic size of disorder[END_REF].

Another interesting feature is that the range of validity of the first order approximation is independent of the specimen thickness, and extends (within 10% for all values of z) up to contrasts 0.2. Indeed, this range of validity may be estimated from the ratio of the first-order contribution to the second-order one, and this ratio is the same for infinitely thin and thick plates, see Eqs. ( 48) and ( 49). One of the aims of the following section is to study experimentally this range of validity.

Experimental investigation of crack pinning by strong designed obstacles

We shall now explore experimentally the process of crack pinning by strong heterogeneities. We shall focus on the infinitely thin plate limit h/d → 0, and investigate the relevance of the second-order formula derived in the previous section to describe the deformation of cracks due to a single obstacle. For this, we investigate experimentally the peeling of a thin film (F) from a patterned substrate (S) (see fig. 6).

Experimental setup

Pa#erned( substrate( PDMS( thin(film( Peeling( direc7on( a normalized toughness contrast of unity.

ntal setup system interaction of a crack front with strong heterogeneities on experimental film peeling experiments are carried out (Fig. 7). In the following, we show e specific loading conditions, the deformation of peeling fronts under the cles of large adhesion energy can be described by the theoretical model sly in the context of crack fronts pinning by tough heterogeneities, when ess limit h → 0 is considered (Fig. 3). The specimen preparation follows the procedure detailed in [START_REF] Xia | Toughening and asymmetry in peeling of heterogeneous adhesives[END_REF]. To introduce an obstacle of controlled geometry and toughness and permit a quantitative comparison with the predictions of Section 5, an obstacle of width 2d aligned with the peeling direction is printed on a transparency using a standard commercial printer. The neat side of the transparency is subsequently glued onto a glass plate to form the substrate. The thin film is then spin-coated directly over the printed face from a liquid solution of PDMS obtained by mixing an elastomer and a curing agent of PDMS (Sylgard 184, Dow Corning Co.) at a weight ratio of 10:1 by spinning the machine at a rate of 300 rpm during 20 s. This procedure produces homogeneous thin films with thickness h = 400 µm. This phase is followed by curing of the thin film and its substrate at a temperature of 60 • C for two hours. Both curing temperature and time control the cross linking of PDMS and lead to a Young's modulus E and a Poisson's ratio ν of 1.2 MPa and 0.5 (a value characteristic of incompressible elastomers), respectively. The heat treated samples are then kept at room temperature for 48 hours prior to the test. This two-stage curing treatment is used to minimize the residual stress in cured PDMS.

A standard peel test configuration is employed to measure the peel force required to peel off the PDMS thin film. As shown in Fig. 6, the free end of the PDMS thin film is peeled off at a constant speed, v p = 1.65 mm.s -1 , and at a peel angle θ p = 45 • , from the inkpatterned rigid substrate. For ease of experimental implementation, the film is always peeled vertically, and the rigid substrate is rotated about its hinge to obtain any desired peel angle. In order to maintain a constant peel angle θ p during the test, the substrate is moved in the horizontal direction at a speed v x = v y sin θ p /(1cos θ p ). In this way, a stationary regime of propagation is reached.

Compatibility with the theoretical framework

The defect width 2d is chosen to be small enough so that d 2πr b where the bending length r b = Eh3 /[24(1ν 2 )G c ] represents the thin film's radius of curvature near the peeling front, but large enough so that d h. The first condition ensures that the nonlinear effects arising from the large displacements of the film can be neglected and the LEFM approach developed in the previous sections applies [START_REF] Xia | Adhesion of heterogeneous thin films: II. Adhesive heterogeneity[END_REF]. The second condition ensures that the geometry of the detachment front can be described by the previous calculations in the limit h/d → 0. In practice, d = 1 mm while r b = 1.4 mm, so that h/d = 0.4 and d/(2πr b ) 0.1. Finally, the choice of a relatively large peeling angle, θ p = 45 • , is motivated by the occurrence of large stretching deformations for small peeling angles that dominate over the bending mode of deformation of the film considered in our calculation. This transition from a peeling mode governed by stretching to a mode governed by bending occurs for a peeling angle

θ c p 4 G c /Eh 15 • (Ponson et al., in preparation).
It is now necessary to explain why, in the limit of very thin plates h/d → 0, the LEFM analysis of a crack lying at the interface between two elastic plates (see Fig. 2(b)) applies to the peeling of an elastic plate from a rigid substrate. It so occurs that for the problem considered in Fig. 2(b), the replacement of the bottom elastic plate by a rigid one does not affect the calculation, as long as the limit h/d → 0 is considered. Indeed in this limit, as a result of the Love-Kirchhoff theory of thin plates, all components of the displacement are identically zero on the unbroken part of the interface between the two plates, irrespectively of the elasticity of the bottom plate; hence the boundary conditions imposed onto the top plate by the bottom one are the same as if the latter were rigid 3 . Of course, for a given front geometry, the perturbation δG of the elastic energy-release-rate is twice larger when the upper plate is bonded onto another elastic plate rather than on some rigid substrate, since the same amount of elastic energy is released from both plates; but since the same argument also holds for the elastic energy-release-rate G 0 for a straight crack front, Eqs. (49) derived above using only the value of the ratio δG/G 0 apply indifferently to both cases.

Characterization of the adhesion energies

To introduce defects pinning the detachment front, we take advantage of the strong adhesion of PDMS on ink, characterized by the high interfacial fracture energy G O c , as compared to that on neat transparency, characterized by the lower energy G M c . To vary the strength of these defects, we tune their grayscale c gray (c gray = 0 without any obstacle and c gray = 1 for a black obstacle).

The first task is to measure the interfacial fracture energies of PDMS with the transparency (G M c ), and with the printed obstacle (G O c ) as a function of c gray . To do so, we first deduce the mean interfacial fracture energy G c from Rivlin (1944)'s equation

G c = F p 2b (1 -cos θ p ) (51)
where F p is the steady-state peel force measured and b (= 24 mm in the tests) the halfwidth of the adhesive4 (see Fig. 6). An alternative, more refined equation due to [START_REF] Kendall | Thin film peeling-elastic term[END_REF] incorporating the membrane elasticity of the film is actually used in the following, without significantly changing the results obtained (see Fig. 7). The inset (a) of Fig. 7 shows the peel force F p as a function of the peel displacement, for various values of the width 2d of the obstacle. One sees that the force quickly increases to reach a stationary value. This reflects the stationary loading conditions imposed to the adhesive that is peeled with an angle maintained constant during the experiment.

To now deduce the values of the toughness contrast =

G O c (c gray ) -G M c G M c from that of G c ,
we perform peel strength measurements of samples with defects with the same gray level c gray but various widths 2d, and use the equation

G c = 1 - d b G M c + d b G O c ⇒ G c G M c = 1 + d b (52)
expressing the "mixture rule'" for the effective fracture energy of heterogeneous interfaces invariant along the propagation direction. This procedure is illustrated in Fig. 7 where the normalized effective fracture energy G c /G M c is plotted as a function of the normalized defect width d/b. The slope of the linear fit provides the fracture energy contrast = 0.66± 0.06 where the uncertainty is determined from the quality of the fit and the uncertainty on the fracture energies (see Appendix B). The adhesion energy G M c = 2.21±0.02 J.m -2 of the PDMS-transparency is measured independently through additional peel tests performed on homogeneous specimens. Note that the values of fracture energy are found to slightly vary with the velocity of the peeling front. In the experiments reported in this paper, the velocity is therefore fixed to a value of v p = 1.65 mm.s -1 .

The very same procedure is applied for various grey levels, so that the contrast can be measured as a function of c gray , as shown in the inset (b) of Fig. 7. One observes that (c gray ) evolves linearly with c gray , according to the law (c gray ) = 1.77 c gray . The range 0 ≤ ≤ 1.77 of contrasts achieved by our experimental setup allows to study crack front pinning beyond the domain of validity of the first-order theory and to investigate the relevance of the second-order correction.

Deformation of the front pinned by obstacles of varying strength

A digital camera equipped with a high-magnification lens is positioned above the substrate surface, perpendicularly to it, and used to record in situ images of the peel-front configuration. An example of the picture obtained is shown in Fig. 8(c). Using an edequate lighting device, the front shape appears as a thin bright line. The other, thicker bright line that appears on the raw image results from the reflection of the light on the curvy part of the debonded thin film and is discarded in the analysis. Image processing is then used to enhance the contrast and discrete positions of the front are obtained by manual selection on a zoomed image. In the stationary experimental regime, G 0 does not depend on the crack size a, hence it is not possible to define a mean position of the front using Eq. (34). Instead, from now on, we take as reference the straight line which intersects the front at z = 0 to define the crack front perturbation δa, in agreement with Eq. ( 50). The result is plotted on Fig. 8(a) with an amplification in the y-direction. One immediately recognizes the characteristic shape of a crack front pinned by an isolated obstacle of larger toughness, as has been observed by [START_REF] Dalmas | Crack propagation through phaseseparated glasses: effect of the characteristic size of disorder[END_REF]; Patinet et al. (2013a); [START_REF] Budzik | Adhesive fracture of heterogeneous interfaces[END_REF]; [START_REF] Chopin | Crack front dynamics across a single heterogeneity[END_REF]. The first-and second-order approximations (Eqs. ( 49)), corresponding to the limit h/d → 0 , are also plotted in Fig. 8(a) using the contrast value = 0.66 measured independently from peel force measurements. To take into account the experimental uncertainties on the contrast value, we also plot in thinner lines the lower and upper bounds of the theory that correspond to min = 0.60 and max = 0.72, respectively. It is observed that the second-order theory brings a significant correction to the first-order one, by predicting a less deformed ("stiffer") crack front, and considerably improves the agreement with the experiment.

The "tunability" of the printed obstacles may be exploited to explore other contrasts. Fig. 8(b) shows the normalized amplitude of the crack front perturbation, defined as the amplitude of this deformation within the obstacle, that is the difference of position δa(d) of the front at the edge and center of this obstacle, normalized by the defect half-width d. This quantity depends only on the contrast, and varies linearly with it according to the first-order theory. In contrast, the experiments exhibit some "saturation" of the amplitude as the contrast increases. This saturation is correctly predicted by the second-order theory. This quantitative comparison with experiments permits to define the practical limits of the first-and second-order theories. We retrieve experimentally that the first-order calculation provides an accurate prediction of the deformation of the front of a pinned crack for low contrasts, 0.2 -0.3. It also provides the domain of validity of the secondorder calculation that describes accurately the experimental results for 0.7. For larger contrasts, the second-order theory underestimates the deformation of the front; either a higher-order development or a numerical approach (see [START_REF] Bower | A 3-dimensional analysis of crack trapping and bridging by tough particles[END_REF]; [START_REF] Lazarus | Brittle fracture and fatigue propagation paths of 3D plane cracks under uniform tensile loading[END_REF]) would be needed to accurately describe the geometry of the crack front in such situations.

Effect of the sample width on the deformation of the front

In the previous set of experiments, we limited our analysis to the region |z/d| ≤ 5. Beyond this domain, we observe deviations from the theoretical predictions that suggest that the effect of the finite width neglected in the calculations may begin to play a significant role.

To investigate this question, we measure the crack front deformation for a fixed contrast = 0.66 and a fixed defect half-width d = 1 mm, but various specimen widths 2b. Figure 9(a) shows the results obtained. 5 The comparison with the first-and second-order theories, strictly valid in the limit b/d → +∞, is also displayed in Fig. 9(a). It shows that the specimen width has a negligible effect on the crack front deformation in the region of interest |z| < 5 d as long as b 12 d. Provided this condition is met, the second-order expansion captures the front geometry fairly well, confirming the results of the previous section and the relevance of the second-order correction for stronger obstacles. For the smallest sample having b/c = 8, the crack front geometry is clearly influenced by the finite width of the specimen. To account for this effect, Patinet et al. (2013a) proposed to correct the crack front deformation δa(z) observed by subtracting the deformation δa hom (z) measured for a homogeneous interface, before comparing it to that predicted theoretically for an infinitely wide specimen. This procedure is tentatively applied to our experiments: First, the crack front geometry is measured in homogeneous specimens of various widths 2b, as shown by the open symbols in the inset of Fig. 9(b). A parabolic fit δa hom (z) = 0.037 z 2 /b shown by continuous lines provides an accurate description of the experimental points. This deformation is then subtracted from the original deformation and the corrected crack front geometry is compared to the theoretical predictions in Fig. 9(b). One observes that this procedure actually degrades the agreement between experimental and theoretical curves.

The apparent inappropriateness of this procedure might relate to the implicit assumption that the zeroth-order crack front shape corresponding to a homogeneous interface is influenced by the finite width of the specimen, but that the first-and second-order corrections due to the presence of the obstacle are not. There is no clear reason why this assumption should be correct, especially considering the large influence of the specimen thickness h upon the first-and second-order corrections.

A discussion of the results of Patinet et al. (2013a) is finally possible. These authors performed similar measurements of the crack front deformation when pinned by a single obstacle, and concluded that the predictions of the first-order theory were quantitatively correct over a much larger range of values of the contrast than found here. Since this range of validity was shown to be independent of the specimen thickness, we might attribute this discrepancy to the finite width of the specimen: Patinet et al. (2013a) used specimen widths in the range 9 b/d 13, for which the effect of the finite width of the specimen was significant in our setup, and used the procedure of correction depicted above. Now in the double cantilever beam they used, the sign of the curvature of the crack front for a homogeneous interface is opposite to that in our peeling test. As a result, the subtraction of δa hom (z) from δa(z) resulted in an increase of the crack front deformation instead of a decrease like in our case (see Figs. 9(a) and (b)). If, as suggested by the above results, the procedure of correction was in reality inappropriate, it might have led to an overestimation of the crack front deformation bringing it closer to the predictions of the first-order theory, and thus giving the impression of correctness of this theory beyond its real range of applicability.

While the effect of the specimen thickness on the pinning of crack fronts can now be quantitatively taken into account, the role played by the specimen width is less obvious and deserves further investigations. To circumvent this difficulty in our experiments, we used thin films of sufficiently large width b 12d with respect to the obstacle width. Note however that this range of validity might vary with the test geometry employed.

Conclusion

The primary objective of this work was to provide a quantitative description of the process of crack pinning by heterogeneities (i) taking into account the finite size of the specimens and (ii) going beyond the linear first order approximation.

The problem was addressed theoretically by calculating the second-order expansion of the stress intensity factor resulting from some small, but otherwise arbitrary coplanar perturbation of the front of a semi-infinite crack lying on the mid-plane of a plate of finite thickness and subjected to general mode I loading conditions. The results obtained were used to predict the equilibrium geometry of the crack for a given slightly heterogeneous distribution of fracture energy within its plane. Specializing to the pinning by a single infinitely elongated obstacle, we found a formula for the shape of the crack front accurate to second order in the toughness contrast. This formula was fully explicit in all cases, and entirely analytic (i.e. did not require the numerical evaluation of any integrals) in the two extreme limits of infinitely thick and infinitely thin plates.

The remainder of the paper focused on experiments performed in the latter limit, using peeling tests with fully controlled fields of local fracture energy. The experimental crack front deformations were observed to be in good agreement with the theoretical formulas and the second order term to (i) greatly improve the agreement and (ii) broaden their range of applicability in terms of admissible values of the toughness contrast. This work highlights the following generic features of crack pinning in samples of finite size:

• For a given distribution of obstacles, the thinner the sample, the smaller the deformation of the front. In the limit of a very thin specimen, this deformation is exactly four times smaller than in the limit of an infinitely thick one. • The second-order correction relevant for large toughness contrasts shows that the crack front stiffens under the action of strong obstacles; in other words, the first-order approximation overestimates the deformation of the crack pinned by such obstacles. This approximation is sufficient for normalized contrasts up to 0.2, whereas the secondorder one correctly predicts the crack front deformation up to 0.7, irrespective of the specimen thickness.

• Although the effect of specimen thickness can now be quantitatively taken into account, the finite width of the specimen might play a significant role and deserve further investigations. In practice, we found that in peeling tests, films at least 12 times wider than the defect size are necessary in order to apply the theory developed in this paper under the hypothesis of infinite width. • Generally speaking, the non-linear correction arising from large contrasts of toughness is expected [START_REF] Delaplace | High resolution description of a crack front in a heterogeneous plexiglas block[END_REF][START_REF] Xia | Toughening and asymmetry in peeling of heterogeneous adhesives[END_REF] to play a significant role in various experimental situations and should be taken into account to predict the failure behavior of engineered heterogeneous materials with new and improved properties. We have shown that the second order approximation is useful to increase the accuracy. However numerical methods become unavoidable to consider strong heterogeneities. An iterated perturbation approach is then a promising method [START_REF] Vasoya | Crack Front Fingering During Planar Crack Propagation in Highly Heterogeneous Toughness Field[END_REF].

A Appendix : Calculation of integrals I 1 and I 2

A.1 Integral I 1

This integral is given by

I 1 (h; k, k ; z 1 ) = P V +∞ -∞ +∞ -∞ f ( z-z 1 h )f ( z -z h ) (z -z 1 ) 2 (z -z) 2
× e ik(z-z 1 ) -1 e ik (z -z) -1 e ik (z-z 1 ) dzdz .

Quite remarkably, one may express it in terms of the sole function X defined by Eq. (9). To do so, the first step consists in performing the changes of variables zz 1 = ζ, zz = ξ; one thus gets I 1 (h; k, k ; z 1 ) ≡ I 1 (h; k, k Writing then e i(k+k )ζe ik ζ as e i(k+k )ζ -1 + 1e ik ζ , performing the changes of variables ζ/h = u and ξ/h = v, and using the definition (9) of X, one gets Eq. ( 23) 1 of the text.

) = P V +∞ -∞ +∞ -∞ f ( ζ h )f ( ξ h ) ζ 2 ξ 2 e ikζ -

A.2 Integral I 2

This integral is given by

I 2 (h; k, k ; z 1 ) = P V +∞ -∞ +∞ -∞ f ( z -z 1 h )f ( z -z h ) (z -z 1 ) 2 (z -z) 2 ×
e ik(z -z 1 ) -1 -e ik(z-z 1 ) -1 zz 1 (zz 1 ) e ik (z-z 1 ) -1 dzdz .

Again, one may express it in terms of the sole function X. Indeed, one may first perform the changes of variables zz 1 = ζ, zz = ξ; one gets

I 2 (h; k, k ; z 1 ) ≡ I 2 (h; k, k ) = P V +∞ -∞ +∞ -∞ f ( ζ h )f ( ξ h ) ζ 2 ξ 2 × e ikζ -1 - e ik(ζ+ξ) -1 ζ + ξ ζ e ik (ζ+ξ) -1 dζdξ.
Now, "symmetrizing" the integrand with respect to the variables of integration ζ and ξ, one gets

I 2 (h; k, k ) = 1 2 P V +∞ -∞ +∞ -∞ f ( ζ h )f ( ξ h ) ζ 2 ξ 2
e ikζ + e ikξ -1e ik(ζ+ξ) e ik (ζ+ξ) -1 dζdξ

= 1 2 P V +∞ -∞ +∞ -∞ f ( ζ h )f ( ξ h ) ζ 2 ξ 2
e ikζ -1 e ikξ -1 1e ik (ζ+ξ) dζdξ

= 1 2 P V +∞ -∞ f ( ζ h ) ζ 2 e ikζ -1 dζ . P V +∞ -∞ f ( ξ h ) ξ 2 e ikξ -1 dξ - 1 2 P V +∞ -∞ f ( ζ h ) ζ 2 e i(k+k )ζ -e ik ζ dζ . P V +∞ -∞ f ( ξ h )
ξ 2 e i(k+k )ξe ik ξ dξ.

Reasoning then like for integral I 1 , one gets Eq. ( 23) 2 of the text.

B Appendix: Experimental uncertainties on the toughness contrast

We detail here the method employed to calculate the uncertainty on the experimental values of the toughness contrast . This contrast is obtained from a linear fit of the variations of the normalized fracture energy G c /G M c with the normalized obstacle width d/b following Eq. ( 52). Therefore, we calculate first the error on the values of the fracture energies at some fixed obstacle width. The effective toughness G c is obtained from the value of the peeling force F p averaged over a propagation distance of 20 mm as illustrated in Fig. 7(a). The standard deviation of the force fluctuations over this distance normalized by the average peeling force provides, through Eq. ( 51), the relative error

  Fig. 2. A slightly perturbed emerging crack lying on the mid-plane of (a) a semi-infinite plate; (b) an infinite plate.

Fig. 3 .

 3 Fig. 3. Distribution of fracture toughness on a plane containing a single infinitely elongated obstacle.

Figure 4

 4 Figure 4 first compares, in the two limiting cases h → +∞ and h → 0, the values of δa(z) d

Fig. 4 .

 4 Fig. 4. Equilibrium shape of a crack front penetrating into a single obstacle of normalized toughness contrast = 1, in plates of infinite and infinitesimal thicknesses.

Fig. 5 .

 5 Fig. 5. Equilibrium shape of a crack front penetrating into a single obstacle of normalized toughness contrast = 1, in plates of various thicknesses.

Fig. 7 .

 7 Fig. 7. Schematic diagram of the peeling test.

Fig. 6 .

 6 Fig. 6. Experimental setup: (a) Schematic representation of the peeling test. The substrate is moved horizontally at the velocity v x in order to maintain a constant peeling angle θ p . (b) Snapshot of the experiment.

Fig. 7 .

 7 Fig. 7. Normalized effective adhesion energy of the interface as a function of the normalized width d/b of the obstacle for c gray = 0.4. The slope of the linear fit gives the toughness contrast = 0.66 ± 0.06. Inset (a): Peel force versus displacement curves for the same grayscale but different obstacle half-widths. Inset (b): Contrast of interfacial fracture energy versus grayscale of the obstacle.

Fig. 8 .

 8 Fig. 8. (a) Equilibrium shape of a crack front pinned by a single obstacle of contrast = 0.66 ± 0.06, and comparison with the first-and second-order theoretical predictions (Eqs. (49)). (b) Variations of the amplitude δa(d)/d of the crack front deformation as a function of the contrast, and comparison with the predictions of the first-and second-order theories. (c) Typical snapshot of the peeling front as it crosses an obstacle.

Fig. 9 .

 9 Fig. 9. Effect of the specimen width on the front perturbation: (a) Geometry of the pinned front for = 0.66 for various specimen widths 2b. (b) Geometry of the pinned front after correction by the unpinned front geometry.

  1 e ik ξ -1 e ik ζ dζdξ = P V +∞ -∞ f ( ζ h ) ζ 2 e i(k+k )ζe ik ζ dζ . P V +∞ -∞ f ( ξ h ) ξ 2 e ik ξ -1 dξ.

Note that the hypothesis of infiniteness of the plate plays a central role here; if it were semiinfinite, like at the beginning of

Section 2.2, the rotation of the crack front would not leave the geometry unchanged since this front would not remain parallel to the plate boundary.

This definition differs from that used in Section 4, φ 1 (z) + 2 φ 2 (z), through the quantity φ 1 (0) + 2 φ 2 (0). The new definition means that the perturbation of the crack front is observed using a new reference straight line passing through the point z = 0 of the perturbed front. It will also be used in Section 6.4 below.

This does not remain true for elastic plates of finite thickness, because as a result of the theory of 3D elasticity, the horizontal components of the displacement are no longer zero on the unbroken part of the interface.

This width is kept constant during the first set of experiments discussed here, but will be varied in a second set discussed in Section 6.5 below.

Because of the symmetry (z → -z) of the peeling test geometry, we only show one half of the front and represent the mean value of the deformations δa(z) measured in the regions z < 0 and z > 0.

Acknowledgements

The support of the European Union throughout a Marie Curie integration grant "Tough-Bridge" (LP), of the city of Paris throughout the Emergence program (LP) and of the ANR Program SYSCOMM (ANR-09-SYSC-006) (VL), is gratefully acknowledged. We also wish to thank Damien Vandembroucq and Sylvain Patinet for stimulating discussions.

σ Gc 2.0 % on G c . The value of the reference fracture energy G M c that corresponds to a neat interface is measured from a large set of independent experiments, leading to an improved accuracy σ G M c 1.0 %. From these two values, one obtains the relative error

2.2 % on the normalized values of fracture energy G c /G M c , that correspond to absolute errors in the range 0.02 ≤ δ ≤ 0.03.

One can now estimate the uncertainty on the contrast value since the linear regression illustrated on Fig. 7 does not only provide the value of , but also the absolute error δ on this quantity through the formula [START_REF] Taylor | An introduction to error analysis: The study of uncertainties in physical measurements[END_REF]

where the summation is made on the N = 5 width values d i considered in our experiments.

For the specific gray level c gray = 0.4 considered in Fig. 7, this gives δ = 0.06, leading to the range of contrast values = 0.66 ± 0.06 used in the first-and second-order theoretical predictions for comparison with experiments (see Fig. 8(a)). Note that the calculation of the error δ through a more elaborated formula than Eq. (B.1), taking into account variations of the error δ i from one value of d i to another, gives similar results. Finally, this procedure is applied for various gray levels in order to obtain the uncertainties on the values of (c gray ) as represented in Fig. 7(b) for 0 ≤ c gray ≤ 1. This uncertainty is used in Fig. 8(b) when comparing experimental and theoretically predicted values of the front deformation amplitude.


Enoch Opeyemi Oluwole

THE RIEMANN HYPOTHESIS AND THE EULER'S QUADRATIC EQUATION

Keywords: Riemann zeta function, Euler's equation, meromorphic function, nontrivial zeros, and zeros of the analytic continuation formula

We present a general formula for obtaining the zeros of the analytic continuation formula of the Riemann zeta function from Euler's quadratic equation and show that these zeros are all real. Furthermore, substituting this formula into the analytic continuation formula, the root of the Analytic Continuation Formula (ACF) of the Riemann zeta function is determined.

Introduction

We presented in [START_REF] Enoch | From the zeros of the Riemann zeta function to its analytical continuation formula[END_REF] that the Euler's quadratic equation is connected to the ACF of the Riemann Zeta Function by showing that the equation;

41 2 + + x x (1)
could be structured to give the ACF of the Riemann Zeta Function by doing the following:

• instead of the coefficients of 2 x and x, being 1, we wrote k;

• for the constant integral, 41, we replaced it with ( ); t G

• multiplied the resulting equation with the trivial zeros of the Riemann zeta function;

Then the resulting equation was obtained as;

( () ) ( ) . 2 2 n z t G kz kz + + - (2)
By discretizing (2), we obtained;

( ) ( ) ( ) ( () ) ∑ ∑ ∞ ≥ ∞ ≥ + - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = ζ = γ 1 1 2 . 2 1 2 n n E t G kz kz n z n z z (3)
By the method of partial summation [START_REF] Atle | An elementary proof of the prime-number theorem[END_REF], we [START_REF] Enoch | From the zeros of the Riemann zeta function to its analytical continuation formula[END_REF] showed that

∑ ∑ ∑ ≤ ≤ ≤ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + n dq n d d n q n z n n z n 2 1 2 2 1 2 (4)
where

n d 2 = and . 2 1 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = n z q
We eventually obtained (3) as

( ) ( ) ( ) ( () ) ∑ ∑ ∑ ∞ ≥ ≤ ≤ + - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = ζ = γ 1 2 . 2 1 2 n n d d n q E t G kz kz n z n z z ( 5
)
Such that equation (5) was written as;

( ) ( ) ( ) ( ) ( ) ∑ ≤ - ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ π ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - + Γ π - - = γ n d z z n z t G kz z z z z . 2 1 2 1 2 2 2 (6)
By using the expression;

∑ ∞ = - = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - 0 . 1 1 1 1 n nz z p p [1, 2]
We arrived at

The Riemann Hypothesis and the Euler's Quadratic Equation 3( ) ( ) ( )

1 2 2 1 1 1 - ≤ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ π ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - γ ∑ n d z z n z t G kz p z ( ) ( ) ∑ ∞ = -Γ π - = 0 2 . 1 2 1 2 n nz z p z z z (7)
Then we obtained equation [START_REF] Enoch | The Eigenvalues (energy levels) of the Riemann zeta function[END_REF], by multiplying equation ( 7) over prime numbers;

( ) ( ) ( ) ∏ ∑ - ≤ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ π ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - γ p n d z z n z t G kz p z 1 2 2 1 1 1 ( ) ( ) ∏∑ ∞ ≥ -Γ π - = p n nz z p z z z 1 2 . 1 2 1 2 (8)

Obtaining the Zero of the ACF of the Riemann Zeta

Function from ( ) t G

We went on to set the LHS of (8) as follows:

( ) ( ) ( ) . 1 2 1 1 1 2 - ∞ ≤ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ π ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - γ = ε ∏ ∑ p n d z z e z t G kz n p z (9)
Then obtained a general formula for ( ) [START_REF] Enoch | From the zeros of the Riemann zeta function to its analytical continuation formula[END_REF] And on using the definition of ( )

t G ( ) ( ) ( ) ( ) 1 2 1 1 1 1 2 - - ε ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - π γ - = ∏ ∑ ∞ ≥ z kz n p z z t G p n e z z
z γ
as in [START_REF] Enoch | A Validation of the Real Zeros of the Riemann Zeta Function via the Continuation Formula of the Zeta Function[END_REF], we got

( ) ( ) , 1 ϑ - σ - = t z kz t G ( 11
)
where

( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - Γ π = σ ∑ ∞ = - 0 2 1 1 1 1 2 2 n nz z z p p z z (12) ( ). 1 - ϑ = σ Such that; ( ) ∑ ∞ = - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - Γ π = ϑ 0 2 1 1 1 2 2 n nz z z p p z z (13)
And

( ) ( ) ( ) ∏ ∑ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ - - π + - + π + - = τ ∞ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - π - p z n z z n p z n z n e z z . 1 1 1 2 1 2 2 1 1 2 1 1 2 2 1 2 1 2 2 (14)
From equation ( 11), we obtained a general equation for the zeros of the Analytic Continuation formula as;

( ) ( ) ( ) ( ) . 1 1 1 1 2 ϑ - τ - ϑ - = ϑ - τ σ - = + = z kz z kz kt t G (15)
Such that;

( ) ( ) ( )( ) ( ) 4 ; 1 1 1 1 1 2 1 2 1 = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - ϑ - τ - ϑ - = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - ϑ - τ σ - ± = k k z z k z z t (16) 4 = k
and t is the zero of the ACF of the Riemann Zeta Function.

The Realness of the Zeros of

The Analytic Continuation Formula, ( )

t ε Theorem 1. Let ( ) t G
be defined as it is in equation [START_REF] Enoch | From the zeros of the Riemann zeta function to its analytical continuation formula[END_REF] then it can be shown that the LHS RHS = of [START_REF] Tao | The Euler-Maclaurin formula[END_REF] and that the zeros of the analytic continuation formula, ( ),

t ε
of the Riemann Zeta Function will always be real as

. ∞ → n Proof. Let ( ) ( ) ( ) ( ). 1 2 1 1 1 1 2 2 - - ε - π γ - = + ∏ ∑ ∞ ≥ - z kz n P z z kt p n e z z (16) ( ) ( ) ( ) ( ) ; 1 1 2 1 1 2 1 1 1 2 ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ - - - ε - π γ - ± = ∑ ∏ ∞ ≥ - - - k z z n k P z z t n e P z z such . 4 = k (17)
If we take the limit of (17) as

∞ → n and , 2 1 it z + =
we obtain;

( ) ( ) ( ) ( ) 2 1 1 1 2 1 1 2 1 1 lim lim ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ - - - ε - π γ - ± = ∑ ∏ ∞ ≥ - - - ∞ → ∞ → k z z n k P z z t n e P z z n n (18) ( ) . 1 lim 1 lim 0 lim 2 1 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - - - ± = ∞ → ∞ → ∞ → k z z t n n n (19) If we choose it z ± = 2 1 and 4 = k in (19), we arrival at: [ ] . , lim lim 2 1 2 R ∈ ± = ± = ∞ → ∞ → t t t t n n (20)

The Analytic Continuation Formula of the Riemann Zeta Function

Theorem 2. Let the Analytical continuation formula of the Riemann Zeta function be given as:

( ) ( ) ∫ ∞ - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ψ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + - = ε 1 4 3 2 ; log 2 cos 4 1 2 1 dx x t x x t t ( ) ∑ ∞ = π - = ψ 1 . 2 n x n e x ( 21
)
If t is the zero of ( )

t ε then it can be shown that ( ) 0 = ε t anytime ( ) ( ) ( ) ( ) . 4 1 1 2 4 1 1 2 1 1 2 1 ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ - - - ε π - γ - ± = ∑ ∏ ∞ ≥ - - z z n P z z t n e z P z Proof. We show that ( ) ( ) ( ) ( ) ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ - ε - π γ - - = ε ∑ ∏ ∞ ≥ - - - z n P z z t n e P z z 1 1 2 2 4 1 1 2 1 ( ) ∫ ∑ ∞ ∞ = - π - ϕ × 1 1 4 3 , log cos 2 n x n dx x x e (22)
where

( ) ( ) ( ) 2 1 1 1 2 4 1 2 4 1 1 2 1 2 ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ - ε - π γ - = = ϕ ∑ ∏ ∞ ≥ - - - z n P z z t n e P z z (23) ( ) . 0 = ε t Implies that; ( ) ( ) ( ) ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ - ε - π γ - = ∑ ∏ ∞ ≥ - - - z n P z z n e P z z 1 1 2 2 4 1 1 2 1 ( ) ∫ ∑ ∞ ∞ = - π - ϕ × 1 1 4 3 . log cos 2 n x n dx x x e ( 24
)
It follows that on further simplification of the RHS of (24), we obtain [START_REF] Patterson | An introduction to the theory of the Riemann zeta function[END_REF];

( ) ( ) . 2 ; 2 1 log cos 2 1 2 1 t x x x z z = ϕ + = ϕ - - (25)
This will mean [START_REF] Patterson | An introduction to the theory of the Riemann zeta function[END_REF];

. 2

1 2 1 ϕ + = ⇒ ϕ = - i z i z (26) That is . 2 2 1 2 2 1 t i z t i z + = ⇒ = - (27)
Such that it follows that;

( ) ( ) ( ). 2 1 2 1 log cos 2 2 t i t i i i x x x x x - ϕ - ϕ + = + = ϕ (28)
Because, if we have [START_REF] Patterson | An introduction to the theory of the Riemann zeta function[END_REF];

( )

x r log cos such that ; 2 1 ir z = - then ( ) ( ). 2 1 log cos 2 1 2 1 z z x x x r - -+ =
One can see that the intergrade in equation ( 24) is the same as;

( ) ∫ ∑ ∞ ∞ = - π - ϕ 1 1 4 3 log cos 2 n x n dx x x e ( ) ∫ ∑ ∞ ∞ = ϕ - ϕ - π - + = 1 1 4 3 2 2 1 n i i x n dx x x x e (29) ∫ ∑ ∫ ∑ ∞ ∞ = ∞ ∞ = ϕ - - π - ϕ + - π - + = 1 1 1 1 4 3 4 3 . 2 1 2 1 2 2 n n i x n i x n dx x e dx x e (30)
The equation was solved and the resulting solution is given as:

⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ϕ + π + + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ϕ - π + = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - ϕ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ϕ + - 4 3 2 4 3 2 1 4 3 1 4 3 2 1 i i i n i n . 4 3 2 2 2 1 2 ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ϕ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π + × ∑ ∞ = π - n e n n (31)
By using some transformations in complex analysis such that

i n e π = 2 1 in
Enoch Opeyemi Oluwole 8 (31), we obtain;

( )

, 4 3 4 3 2 2 2 2 1 2 2 3 2 ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ϕ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ϕ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π + + ∑ ∞ = π - πϕ - πϕ π - n e i n e e e n n n n i n (32)
where

⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ϕ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π + = β ∑ ∞ = π -
1 2 ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ϕ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π + ϕ = α ∑ ∞ = π - n e n n (34) Such that; ( ) ( ) ( ) . 4 3 4 1 2 4 1 1 2 1 2 2 2 1 2 1 1 2 1 2 ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ϕ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π + ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ - π ε - γ - = α ∑ ∑ ∏ ∞ = π - ∞ ≥ - -
3 cos sinh ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π α - π β - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π α + π β πϕ = n n i n n n (37)
By substituting (37) into equation ( 22), we will obtain;

( ) ( ) ( ) ( ) πϕ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ - π ε - γ - - ∑ ∏ ∞ ≥ - - n z n P z z n z e P z sinh 2 4 1 1 2 1 1 2 1 . 0 2 3 cos 2 3 sin 2 3 sin 2 3 cos = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π α - π β - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π α + π β ⋅ n n i n n (38)

Conclusion

We conclude by considering the analytical continuation formula as follows:

If ( ) , 0 = ε t then in equation (38) the term ( ) ( ) ( ) ( ) πϕ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ - π ε - γ - ∑ ∏ ∞ ≥ - - n z n P z z n z e P z sinh 2 2 1 2 1 1 2 1 2 1 2 3 cos 2 3 sin 2 3 sin 2 3 cos = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π α - π β - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π α + π β ⋅ n n i n n (39) ( ) ( ) ( ) ( ) πϕ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ - π ε - γ - ∑ ∏ ∞ ≥ - - n z n P z z n z e P z sinh 2 2 1 1 1 2 1 . 1 2 3 cos 2 3 sin 2 3 sin 2 3 cos = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π α - π β - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π α + π β ⋅ n n i n n ( 40
)
Such that (39) implies; If we rationalize the RHS of (41), it will result into;

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π α - π β + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π α + π β ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π α - π β + n n n n n n i ( 42
)
And with this reality, equation (38) will always be equal to zero anytime t as defined in (16).

  can write the RHS of (29) as;

Acknowledgement

The author thanks the anonymous referees for their valuable suggestions and comments which improved the presentation of the paper.


Enoch Opeyemi Oluwole

The Fourier Laplace Approach of Proving The Riemann Hypothesis

Keywords: Riemann zeta function, Fourier transform, Laplace transform, Analytic Continuation Formula (ACF). 2010 Mathematics Subject Classification: 11H05, 11M06, 11M26

The Author presents a Fourier Laplace transformation of The Analytic Continuation Formula of The Riemann Zeta Function and showed that the zeros of the Analytic Continuation Formula will always be real. A general formula for these zeros is also obtained which when substituted into the ACF it is shown to be the root of the ACF of the Riemann Zeta Function.

Introduction

Over the years, mathematicians have studied the Analytic Continuation Formula (ACF) of the Riemann Zeta Function by representing the exponential and the trigonometric functions in the ACF as Taylors Series. The author in this work chose to represent the exponential and the trigonometric functions as Fourier Cosine transform and Laplace transform respectively since Fourier and Laplace transformations have a meeting.

The Relationship between Fourier and Laplace transform can be explained as follows:

Consider the function Then the Fourier transform of is given by Thus, the Fourier transform of the Laplace transform of as defined above.

(2.0) A closer look at the Riemann defined the analytical continuation formula as [1];

Such that

The Fourier Cosine transform of (2) gives

Examining the trigonometric term in (1), it is obvious that one can write in its Laplace transform as;

By injecting (3) and ( 4) into (1), one obtains:

This becomes:

By using the Classical estimate [15] Substituting ( 8) into [START_REF] Enoch | A general representation of the zeros of the Riemann zeta Function via Fourier series Expansion[END_REF], one arrives at If we take the limit of ( 9) as , we will have;

On integrating [START_REF] Erdos | Factorization of n! American Mathematics Monthly[END_REF]

  , we obtain; Looking closely at the bound of the Integrand in (11),b must be equal to x in the RHS. Thus, we

			124.256818554
		17 19347.80796 69.546401711 42 65043.01859	127.516683880
		18 20775.70086 72.067157674 43 67163.56233	129.578704200
		19 22925.80077 75.704690699 44 68736.92831	131.087688531
		20 23806.30539 77.144840069 45 71287.58335	133.497737203
		21 25178.67630 79.337375020 46 72638.26763	134.756509753
		22 27497.52501 82.910380854 47 76305.16423	138.116042055
		23 28721.41508 84.735492981 48 78105.83231	139.736208952
		24 30573.71456 87.425274613 49 79664.60316	141.123707404
		25 31549.23293 88.809111208 50 81925.00163	143.111845808
	have;
	Since	, equation (12) gives;

, it implies that; This results to: On further simplification This gives: We can see that (20) is the same as: Making the subject of (21), we obtain: and Thus This means that Such that: One can now determine the general expression for t, the root of from (26) such that: We can write equation (27) as Such that The author in his work [15] obtained a general formula for the zeros of the analytic continuation formula of the Riemann Zeta Function as: Where and If we equate equations (29) and (30), we will get; We can show that: On further Simplification, we obtain: Equation (35) can be elegantly rewritten as; Since Then RHS of equation (36) equals to: Thus by this equation (35) gives: The followings are the Numerical values of A for the first fifty zeros of the Riemann Analytic Continuation Formula. S/N S/N 1 800.1618191 14.134725142 26 34220.00572 92.491899271


Matteo Gaeta
email: gaeta@crmpa.unisa.it

Pierluigi Ritrovato
email: ritrovato@crmpa.unisa.it

Saverio Salerno
email: salerno@unisa.it

Implementing New Advanced Learning Scenarios Through GRID Technologies Implementing New Advanced Learning Scenarios Through GRID Technologies

Keywords:

POSITION PAPER

In this paper we present our vision concerning the use of GRID technologies for the implementation of the next generation of learning solutions and in particular for supporting distance learning. We believe that the new requirements that new Learning environment should satisfy could transform these solutions into killing applications for GRID technologies.

Our ideas move essentially from two main thoughtfulness: from one side there are the lacks characterising the existing software platforms for supporting traditional and distance learning processes and, from the other side there is the continuous achievement of GRID technologies as a solution for truly implementation of e-science infrastructure. The main drawbacks of current distance learning solutions are twofold: they only support a part of the whole learning process (the simple one), and how ICT is used.

Concerning the former issue, many of the market-available e-learning solutions are focused on supporting only a specific aspect of the learning process notably the content delivery (content centred approach). Nowadays is emerging a strongly need for supporting other fundamental aspects of the learning process such as initial skill assessment, learner progress metering through definition of specific learning goals, formal representation of the knowledge to be transferred to the learner (metadata standards, ontology, taxonomy, etc), automatic discovering and self-adaptation to individual learning styles, continuous assessment, etc. This lack is mainly due to the fact of considering the didactical communication as the information transfer from the addresser (teacher) to an addressee (student), where the propaedeutic (preparatory) mean is a written text.

This has a direct impact on the latter issues. In fact, most used ICT technologies in learning are the more traditional one such as video (TV programmes, videoconferencing sessions, web casting events) and in some cases traditional CBT enriched with multimedia contents.

It is clear to everybody that technology is not a "panacea" for all the educational ills, but today's technologies are essential tools of the teaching trade and are essential for the achievement of a breakthrough in this field. As showed by several EU and US studies the new research paths are moving towards supporting the whole learning process for individuals and organisations facilitating LeGE-WG 1 st Workshop Toward an European Learning GRID Infrastructure: educational models for grid based services the representation, acquisition and sharing of knowledge and skills in any place, at any time and any pace.

This implies the need of new advanced solution allowing the implementation of user centred collaborative environment for ubiquitous learning (CSCL -Computer Supported Collaborative Learning) through a fully exploitation of emerging networking facilities (e.g. broadband and wireless) simplifying the access to advanced software solutions (simulation environments, 3D visualisation systems, digital libraries, etc.).

In this frame, the term "ubiquitous" is not strictly referred to geographic location but it means the ability to support multiple learning contexts and automatically adapt to them. This more wide vision of ubiquitous learning means that these new environments should support, at the same time, new emerging pedagogical approaches such the one based on inductive-simulative and the experience models facilitating the construction of personalized learning paths.

Although Internet is perhaps the most transformative technology in history, reshaping business, media, entertainment, and society in an astonishing way, it is just now being tapped to transform education. Nowadays, Standard Internet technologies, due to their data oriented nature, have well supported the implementation and exploitation of traditional content based on learning solutions. A lot of efforts are currently undergoing for defining standards for learning content representation, and their interoperability. Now, as for the learning solutions a shift is needed for Internet technologies.

GRID technologies are rising as the next generation of Internet. These technologies promise to do what the internet has done with data for the applications. GRID technologies define a new powerful computing paradigm by analogy to the electric Power Grid. In this vision, a customer of the GRID will be able to use his or her private work place (Workstation, PC, UMTS phone....) to invoke any application from a remote system, use the system best suited for executing that particular application, access data securely and consistently from remote sites, exploit multiple systems to complete complex tasks in an economical manner, or use multiple systems to solve large problems that exceed the capacity of a single one.

Another interesting aspect of GRID technologies is their support for resource sharing and problem solving in dynamic, multi-institutional virtual organizations. In this vision, the sharing doesn't mean simply exchange of data or files but rather a concrete access to resources (e.g. computers, software, data, etc.). This "sharing capability" imposes the definition and implementation of welldefined resource management policies to specify what is accessible, from whom and under which conditions. A set of individuals and/or institutions defined by such sharing rules form what is called a Virtual Organization (VO). This sharing capability and support for VO implementation has been one of the main key success factors of GRID as enabling technology for e-science infrastructure. This kind of VO implemented using GRID technologies seems to provide the right answers to the needs rising from the emerging ubiquitous learning solutions.

Actually, the Grid could be used as a technology glue (seamless network combining wireless and broadband network, advanced software solutions for personalised mobile learning), providing the user with a uniform way to access these resources by means of several kinds of devices. These technologies will provide, in a natural way, a support for both distance and traditional learning activities facilitating the experimentation and evaluation of new emerging didactical models based on the learning by doing approach.

LeGE-WG 1 st Workshop Toward an European Learning GRID Infrastructure: educational models for grid based services

In order to support these new scenarios it is important that GRID move from a niche technology to a commodity one. Fortunately, it seems that the new research directions for the development of GRID technologies are moving towards that way. In deed, in the framework of the Global Grid Forum (the official organism for managing GRID standards) many efforts have been devoting to the definition of the OGSA (Open Grid Service Architecture) that essentially defines a common extensible framework for Grid services specification. The Grid Services will be accessible through web service standard mechanism. This standardized approach allows, from a one hand, to create new GRID services need for the learning domain (GRID domain specific) and, on the other, to integrate existing solutions making them accessible through the GRID, facilitating the interoperability from the existing learning solutions that is actually a real issue in this field.

Although the GRID technologies and their development seem to go towards the right direction for the implementation of new and advanced learning solutions, as for all new ideas plenty of work is still required.

In particular, this effort should be oriented to: § the definition of methodologies and techniques for making grid-aware existing applications (in order to preserve existing investments) § the definition of learning GRID services § solutions usability and their further integration with the new ambient intelligent user interface.