CAN WE HEAR THE ECHOS OF CRACKS?

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CAN WE HEAR THE ECHOS OF CRACKS?

Philippe Destuynder, Caroline Fabre

To cite this version:

Philippe Destuynder, Caroline Fabre. CAN WE HEAR THE ECHOS OF CRACKS?. 2015. �hal-

01166898v2�

CAN WE HEAR THE ECHOS OF CRACKS?

PHILIPPEDESTUYNDER

Département d’ingénierie mathématique, laboratoire M2N Conservatoire National des Arts et Métiers

292, rue saint Martin, 75003 Paris France

CAROLINEFABRE

Laboratoire de mathématiques d’Orsay UMR 8628 Université paris-sud

Orsay 91405 France

ABSTRACT. The detection of cracks in mechanical engineering is mainly based on ultra- sonic testing and Foucault currents. But even if they are efficient tools, this technology requires an important handling and is limited to the detection of cracks which are close to the source. Recently, several searchers have discussed the possibility of using waves as Lamb waves, for thin plates and shells, but also Love waves for bimaterials. In both cases the structure works as a wave guide and enables a long range propagation which is a promising possibility for detecting a crack quite far from the source. In this paper, we discuss the observability property of a crack inside an open set using Love waves. But there is a condition which connects the space representation of the excitation and the geometry of the structure containing the crack.

Introduction

The aim of this paper is to obtain a criterion of detection of a crack using local waves for bimaterial modelled by a transmission problem. The crack is located inside the material along the interface of transmission and the local waves that we consider are Love waves which are located in the low velocity side of the set occupied by the material. The detection of cracks in mechanical engineering is mainly based on ultrasonic testing and Foucault currents. But even if they are efficient tools, this technology requires an important handling and is limited to the detection of cracks which are close to the source. Recently, several searchers have discussed the possibility of using waves as Lamb waves, for thin plates and shells, but also Love waves for bimaterials (see [11], [13], [19] and [22]). In both cases, the structure works as a wave guide and enables a long range propagation which is a promising possibility for detecting a crack quite far from the source. In this paper, we discuss the observability property of a crack inside an open set using Love waves. In order to be able to detect cracks using Love waves, we state in the paper a condition which connects the space representation of the excitation and the geometry of the structure containing the crack.

Our plan is the following :

In section 1, we present the mathematical problem, its notations and the mathematical tools used in the paper such as the time-Fourier transform of solutions of the involved partial differential equations. We state some classical existence results.

Date: June 26, 2015 8:32.

2010Mathematics Subject Classification. Primary: 35C07, 65M15; Secondary: 35M12, 65T60.

Key words and phrases.Non destructive testing, Wave equation, Fracture mechanics, Inverse problems.

1

In section 2, we focus our study on the singularity of the time-Fourier transform of solutions. Let us notice that such results have been deeply studied by P. Grisvard in many cases but not for bimaterial and for stationary models (see the explanations below) and this is why we entirely detail it.

In section 3, we present different methods in order to compute the coefficients involved by the singularities due to cracks. The results presented here are based on works developed in [8].

In section 4, we introduce a criterion for detection of cracks which is based on an observ- ability result. We state here our main theorem in non destructive testing which concerns an efficient method of detection of small cracks using Love waves and that can be explicitly computed. Let us notice that our main theorem requires an assumption (52) which is still an open problem. However, we think that it is generically true (and still work on it), and our feeling is illustrated by the next section.

Section 5 is concerned with numerical simulations of our results. We compute Love waves and we give illustrations of the previous work. Computations are performed with Matlab.

1. Notations and preliminaries results. Let us consider a rectangle -sayR- as shown on figure1. Let us assume that there is a crack parallel to the axis bearing the coordinatex₁. Its two extremities are at pointsAandBwith the abscissaaandband ordinateh. Both (up and down) sides of the closed segmentABare the crack lips and they are denoted byΓ_f. The line which bears the crack is denotedΓ_i.The wave equation that we consider is set on the open setΩ =

o

R\Γf. We writeΩ+= Ω∩(x2> h)andΩ₋= Ω∩(x2< h).The material can be different from both sides ofΓiand thus the wave velocities are different. They are denoted respectively byc+inΩ+ and byc−inΩ−.For any setQ ⊂R^d(d= 1,2), we writeQ⁺=Q∩Ω+,Q⁻=Q∩Ω−and1Qdenotes the characteristic function ofQ.

The boundary ofΩisΓ = Γ1∪Γf whereΓ1 denotes the boundary of the rectangle R. OnΓ1, the free edge condition is assumed. The unit normal outwardsΩand along its boundary isν. Letc=c+1Ω₊+c₋1Ω−be the wave velocity which is piecewise constant for a bimaterial with0< c₋< c+.

The wave model that we consider is the following one, whereuis the transverse dis- placement.



















Findusuch that:

∂²u

∂t² −div(c²∇u) =f in Ω,

∂u

∂ν = 0 on Γ, u(x,0) = 0 and ∂u

∂t(x,0) = 0in Ω.

(1)

The right hand sidef is the excitation and can be considered as a control variable. For sake of simplicity it is assumed that :

f(x, t) =z(t)q(x), (2)

wherezis for instance a wavelet function which enables one to generate an excitation on given frequencies andqis a smooth function which is the so-called space control. In fact, it has to be chosen in order to optimize the detection of the cracks, this is why we mention it as a control variable.

FIGURE1. The open set used for the wave model

LetKbe a compact set with{A, B} ∩K =∅.This setKis fixed in all the paper. For sake of simplicity, we assume thatK=^cDA∩^cDB∩Ω¯ whereDAandDBare fixed open disk centered atAandBwith same strictly positive radius. In the following it is assumed that the support ofqis a subset ofK.The existence and uniqueness of a solution are very classical results (see for instance [21]). The Helmoltz equation associated to this wave equation is obtained by taking the time Fourier transform. Setting (the initial conditions are homogeneous):

ˆ

u(x, ω) = Z ∞

−∞

e^−iωtu(x, t)dt z(ω) =ˆ Z ∞

−∞

e^−iωtz(t)dt, (3) one obtains:







−ω²uˆ−div(c²∇ˆu) = ˆz(ω)q in Ω,

∂ˆu

∂ν = 0 onΓ.

(4) Concerning existence and uniqueness of a solution one can derive the results from those known for (1) and from the Fourier transform. But, it is possible to prove it directly using Fredholm alternative : it enables one to precise for which values ofω the solutionˆuis a unique element of the spaceH¹(Ω). First of all let us introduce the eigenvalue problem which is useful in this study.

Let us set:















find(w, λ)∈H¹(Ω)×Rsuch that:

−div(c²∇w) =λwinΩ,

∂w

∂ν = 0onΓ, Z

Ω

w²(x)dx= 1.

(5)

The spectral theory (see for instance [12]) can be applied and enables one to state that there is a countable family of solutions(wn, λn)∈H¹(Ω)×R⁺, λ0= 0< λ1≤λ2≤. . .≤ λn≤λn+1≤. . . ,such that the family{wn}n∈N,

(respectively 1

p|Ω|⊕ { w_n

√λn

}n∈N^∗), is an Hilbert basis of the spaceL²(Ω)(respectively of H¹(Ω)). Furthermore, the sequence λn tends to infinity and the multiplicity of each eigenvalue is finite. The computation of these eigenmodes can be performed analytically when there is no crack, even and mainly, for a bimaterial as shown on figure1. In this case, one obtains two families of eigenvectors. One contains the so-called Love stationary waves which are mainly localized in the open setΩ₋ifc₋< c+with an exponential decay inside Ω+from the boundaryΓi. They are computed explicitly at subsection5.2. The second one contains global waves and their energy is sprayed in the whole domainΩ. Just for giving an idea on these two families of waves, we have plotted an element belonging to each of them on figure2.

FIGURE2. The two families of eigenvectors in case of a bimaterial (left is a Love wave and right a global one). The softest media is up side and the hardest is at the bottom.

The well-posedness of system (4) is resumed in the following theorem.

Theorem 1.1. Let us introduce the set:

Λ ={λn}n∈N.

Ifω²∈/ Λ, the system of equations (4) has a unique solution which is given by:

ˆ

u(x, ω) = ˆz(ω)X

n∈N

Z

Ω

q(x)wn(x)dx λn−ω² w_n(x).

Proof This is the classical Fredholm alternative as given in [12]-[21]. 2 We now turn to section 2 where we describe the singularity due to the crack of the solution of (4).

2. Singularity due to the crack. We give a more precise description of the solutionuˆin the neighborhood of the crack tip. In the case of a unique material, such characterizations are known since Kondratiev’s work. A nice presentation is given in the book by Grisvard [14] in the following case : P. Grisvard analyses the case of a unique material (thusc+ = c−) in a polyhedral domain with general boundary conditions (Robin’s type). One of the main tools is the use of polar coordinate and, in each angular sector, the use of the Hilbertian basis of eigenvector of second order differential equation in the polar coordinate satisfying the required boundary conditions. Let us notice that the case where the sector angle is2π (which is the case of a crack) with homogeneous Neumann conditions at the two extremities is surprisingly not fully studied in the reference [14]. However, proving theorem below is inspired of P. Grisvard methods with a main change due to the non unique velocity which requires a suitable and not Hilbertian basis (for the polar coordinate).

Before stating our result, let us introduce some notations near the crack. The local polar coordinates are denoted respectively by(r_A, θ_A)for Aand(r_B, θ_B)forB as shown on figure3. The global cartesian coordinates arex= (x1, x2).

FIGURE3. The neighborhood of the crack tipsAandB

LetηA (respectivelyηB) be a smooth truncation function depending on the distance from pointA(respectivelyB), equal to1in a close neighborhoodDA₀(respectivelyDB₀) of A(respectivelyB) and null outside of a larger one denoted in the following byDA₁

(respectivelyDB₁). We assume that their supports are subsets of the complementary ofK.

We introduce the two local singular functionsSAandSBdefined by











S_A(r_A, θ_A) =

√rA

c² sin(θ_A 2 )η_A(x), SB(rB, θB) =

√r_B c² sin(θB

2 )ηB(x).

(6)

We denote byV^Rthe space ([ ]_Γi\Γf denotes the jump of a function across the bound- aryΓi\Γf)

V^R={v∈H¹(Ω), v_|Ω±∈H²(Ω_±), [c²∂v

∂ν]_Γi\Γf = 0, ∂v

∂ν = 0on ∂Ω}. (7) We write forv∈V^R

||v||2,Ω₊∪Ω− =q

||v||²_2,Ω

++||v||²_2,Ω−,

where||v||p,X denotes theH^p(X)−norm of the functionv.Let us finally underline that the norm in the spaceV^Ris defined by:

v∈V^R → ||v||_R=q

||v||²_1,Ω+||v||²_2,Ω

++||v||²_2,Ω

−, which takes into account the continuity ofv∈VRacrossΓi−Γf.

Let us summarize the main features of the singularity analysis in the following theorem.

Theorem 2.1. Let q ∈ L²(Ω) with a compact support inK. There exist two complex numbersK_A,K_B (depending onω) and a functionuˆ^R ∈V^Rsuch that the solutionuˆof (4) satisfies onΩ

ˆ

u(x, ω) = KA(ω) c²

√rAsin(θA

2 )ηA(x) +KB(ω) c²

√rBsin(θB

2 )ηB(x) + ˆu^R(x, ω), The numberK_A(respectivelyK_B) is called the stress intensity factor at pointA(respec- tivelyB).

Furthermore, there exists a constantc0 >0independent on the functionq, but depen- dent onω, such that:

|KA|+|KB|+||ˆu^R||2,Ω+∪Ω−≤c₀||q||0,Ω. (8) 2 Remark 1. The conclusion of the theorem remains valid if one replaces the equation

−ω²uˆ−div(c²∇ˆu) = ˆz(ω)qby−div(c²∇u) =ˆ f wheref ∈L²(Ω)is null in a neigh- borhood of the interface between Ω₊ andΩ₋ and satisfies the compatibility condition R

Ωf = 0 2

Proof

Even if the result is known for homogeneous materials and elliptic model (see [14]), we present the proof which takes into account the modifications necessary for multimaterials and stationary solutions (termω²u).ˆ

The functionqis in the spaceL²(Ω)but its support doesn’t meet a close neighborhood of the crack tips . The only righthandside of the model is therefore the termω²uˆwhich belongs to the spaceH¹(Ω). The proof is split into five steps for sake of clarity. In what follows, we focus at pointB, similar results are valid of course at pointA.

Step1: Localization. From classical Fredholm theorem [20], and for anyω² ∈/ Λ (see theorem2.1), there exists a unique solutionuˆ∈H¹(Ω)to the following model:

−ω²uˆ−div(c²∇u) =ˆ q inΩ, ∂uˆ

∂ν = 0onΓ.

Let us denote byQ_B = [R_B0 <|x−B|< R_B1]an open crown surrounded by two cir- clesC_B0 andC_B1 of radiusR_B0 andR_B1 (R_B0 < R_B1) and both centered at the crack tipB. LetC_B be any circle centered atB inside the open setQ_B. One can easily prove (using a symmetrization aroundΓi) thatu1ˆ _Q+

B

∈H²(Q⁺_B)andu1ˆ _Q− B

∈H²(Q⁻_B).Hence, the restriction ofuˆ toCB,C_B⁺orC_B⁻ satisfiesu|ˆC_B ∈ H^1/2(CB),u|ˆ_C+

B

∈ H^3/2(C_B+) andu|ˆ_C−

B ∈ H^3/2(C_B−)(trace theorem). Letρ= ρ(x) ∈ C₀^∞(R²)withρ(x) = 1for

|x−B| ≤ R andρ(x) = 0for |x−B| > RB₁.We writeuˆ = ρˆu+ (1−ρ)ˆu.The function(1−ρ)ˆuis null on |x−B| ≤ R therefore ηB(1−ρ)ˆu1Ω₊ ∈ H²(Ω+)and ηB(1−ρ)ˆu1Ω− ∈H²(Ω−).In what follows, we studyρˆuwhich isuˆon|x−B| ≤R.

Step 2: A Schauder basis forL²(]−π, π[)andH¹(]−π, π[).

Let us consider the family of functions onC(which is the circle of radiusRBminus the point onΓf) :

p^s_n(θB) = A_n

c²(θ)sin((2n+ 1)

2 θB) and p^c_n(θB) =Bncos(nθB)

wherec²(θ)is the velocity at the angleθand where coefficientsAn andBn are chosen such that

Z π

−π

p^s_n(θ)²dθ= Z π

−π

p^c_n(θ)²dθ= 1.

One gets

A_n=A₀= r2

π

c₊c₋ q

c²₊+c²₋

and B_n=B₀= 1

√π

Let us notice that these coefficients do not depend onnand we thus write them A₀ andB₀. If the functionsp^c_n areC^∞ functions at the point θ = 0, this is not the case of the functionsp^s_n which are discontinuous at the point θ = 0 (since the functionc is discontinuous).

For a given functionfdefined on]−π, π[,we introduce the symmetrical and antisym- metrical parts off:

fa(θB) =f(θB)−f(−θB)

2 , fs(θB) = f(θB) +f(θB)

2 .

The following Lemma will be useful for our study with the local polar coordinates near the pointB(for instance) :

Lemma 2.2. Letg∈L²(]−π, π[).There exists unique coefficientsanandbnsuch that g=X

n≥0

a_np^s_n+X

n≥0

b_np^c_n. We have











an =A0

Z π

−π

c²(θ)g(θ)p^s_n(θ)dθ, bn=B0

Z π

−π

[g(θ)−X

n≥0

an(p^s_n)s(θ)]p^c_n(θ)dθ.

(9)

Furthermore,

X

n≥0

[a²_n+b²_n]<+∞.

Ifg∈H¹(]−π, π[), then its norm is equivalent to

||g||_1,]−π,π[∼ sX

n≥0

(1 +n²)a²_n+X

n≥0

(1 +n²)b²_n.

2 Proof of Lemma2.2

From classical results in Fourier analysis on]0, π[, one can write ga(θB) =X

n≥0

an

2 ( 1 c²₊ + 1

c²₋) sin((n+1

2)θB), (10)

with

an= 4c²₊c²₋ π(c²₊+c²₋)

Z π 0

ga(θB) sin((n+1

2)θB)dθB.

The convergence of the previous series occurs in the spaceL²(]−π, π[)ifg∈L²(]−π, π[) and in the spaceH¹(]−π, π[)ifg∈H¹(]−π, π[).

Settingg_a =g−g_sand the value ofA²₀= 2c²₊c²₋

π(c²₊+c²₋), we get a_n= 2c²₊c²₋

π(c²₊+c²₋) Z π

−π

g(θ_B) sin((n+1

2)θ_B)dθ_B

= 2c²₊c²₋ π(c²₊+c²₋)

Z π

−π

c²(θ_B)

A g(θ_B)p^s_n(θ_B)dθ_B

=A²₀ Z π

−π

c²(θB) A0

g(θB)p^s_n(θB)dθB

=A0

Z π

−π

c²(θB)g(θB)p^s_n(θB)dθB,

(11)

which is the value given in Lemma2.2.

The functionp^s_ncan be split into its symmetrical and antisymmetrical part and we obtain (p^s_n)_a(θ_B) =1

2( 1 c²₊ + 1

c²₋) sin((n+1 2)θ_B), and

(p^s_n)s(θB) = 1 2( 1

c²₊ − 1

c²₋) sin((n+1 2)θB), therefore we proved that

g_a=X

n≥0

a_n(p^s_n)_a=X

n≥0

a_np^s_n−X

n≥0

a_n(p^s_n)_s. Let

h(θ_B) =X

n≥0

a_n(p^s_n)_s=X

n≥0

a_n 2 ( 1

c²₊ − 1

c²₋) sin((n+1 2)θ_B), be the symmetrical part of the seriesX

n≥0

anp^s_n.

The functiongs−his symmetrical and furthermore belongs to the spaceH¹(]−π, π[) ifg∈H¹(]−π, π[). From Fourier theory, one can write:

gs(θB)−h(θB) =X

n≥0

bncos(nθB), with

bn= 2 π

Z π 0

[gs(θB)−h(θB)] cos(nθB)dθB.

(12)

The convergence occurs in the spaceL²(]−π, π[)and also in the spaceH¹(]−π, π[)if g∈H¹(]−π, π[).

We have :

bn= 1 π

Z π

−π

[gs(θB)−h(θB)] cos(nθB)dθB

= 1 π

Z π

−π

[g(θB)−h(θB)] cos(nθB)dθB

=B0

Z π

−π

[g(θB)−h(θB)]p^c_n(θB)dθB.

(13)

Finally it has been proved thatgcan be written in a unique way as follows:

g(θ_B) =g_a(θ_B) +g_s(θ_B) =X

n≥0

a_np^s_n(θ_B) +X

n≥0

b_np^c_n(θ_B). (14) The assertionP

n≥0[a²_n+b²_n]<+∞is obvious from Fourier’s theory.

Sinceg ∈H¹(]−π, π[)⇔(ga, gs)∈H¹(]−π, π[)⇔ (h, gs)∈H¹(]−π, π[),the H¹(]−π, π[)−norm ofgis equivalent to||ga||_H1(]−π,π[)+||gs−h||_H1(]−π,π[)which ends

the proof of lemma2.2. 2

Remark 2. The family{p^s_n, p^c_n}is a basis inL²(]−π, π[)and even inH¹(]−π, π[). Let us point out that it is not an Hilbert basis as far as the functions are not two by two orthogonal (even if one can normalize them). The functionsp^s_nbelong to the spaceH¹(]−π, π[), but not toH²(]−π, π[)because of the discontinuity of the first order derivatives atθB = 0 when c+ 6= c−, which is the interesting case. But c²p^s_n does belong to H²(]−π, π[).

The functionsp^c_nare polynomials in cartesian coordinates and thus they areC^∞([−π, π[).

Moreover they are symmetrical onQ_B. Both functionsp^s_nandp^c_nsatisfy the homogeneous Neumann boundary conditions atθ_B=±πwhich correspond to the crack . 2 Remark 3. It is also possible to consider an Hilbert basis ofL²(]−π, π[)by computing the eigenfunctions solution of:























find(λ, w)∈R⁺×v∈H¹(]−π, π[)such that:

∀v∈H¹(]−π, π[), Z π

−π

c²dw dθ

dv dθ =λ

Z π

−π

wv, Z π

−π

w²(θ)dθ= 1.

(15)

The analytical computation leads to the functionsp^c_n, (n≥0)on the one hand, and to the functions

p^sc_n(θ) =D_n

c² sin(2n+ 1 2 θ)

on the other hand. The coefficient Dn is defined in order to satisfy the normalization condition:

Dn = 2c²₊c²₋ π(c²₊+c²₋).

The analysis is exactly the same as before if one choose to use this Hilbert basis of the

spaceL²(]−π, π[). 2

Let us introduce a family of Hilbert spaces denoted byD^s,and defined by:

D^s={g∈L²(]−π, π[), g=X

n≥0

anp^s_n+bnp^c_n, X

n≥0

(1 +n^2s)(a²_n+b²_n)<∞} (16)

The spaceD^sis endowed with its natural norm

||v||²_Ds=X

n≥0

(1 +n²)^s(a²_n+b²_n)

One has the following lemma which enables one to characterize few spacesD^s. Lemma 2.3. One has:

D⁰=L²(]−π, π[), D¹=H¹(]−π, π[),

D^3/2⊂H¹(]−π, π[)∩H^3/2(]−π,0[∪]0, π[), D²={v∈H¹(]−π, π[)∩H²(]−π,0[∪]0, π[), dv

dθB

(±π) = 0, [c² dv dθB

]Γ_i−Γf(0) = 0}.

2 Proof

The two first equalities are a consequence of what has been done before. Therefore, we focus ons= 2ands= 3/2. Letg∈H¹(]−π, π[). From Lemma2.2, one can write in H¹(]−π, π[):

g=X

n≥0

[a_np^s_n+b_np^c_n].

Let us now consider the second order derivative of the series (term by term), on]0, π[for instance:

a0

d²p^s₀ dθ_B² +X

n≥1

[an

d²p^s_n dθ²_B +bn

d²p^c_n dθ_B² ]

=−1

4a₀sin(θ_B 2 )−X

n≥1

[(n+1

2)²a_nsin((n+1

2)θ_B) +n²b_ncos(nθ_B)].

The square of theL²(]0, π[)-norm of the series is upper bounded inL²(]0, π[)by (c0is a constant independent ong):

c₀[a²₀+b²₀+X

n≥1

n⁴(a²_n+b²_n)].

Therefore the convergence of this series would ensure that d²g

dθ²_B ∈ L²(]0, π[)and would prove the result. In fact ifg∈D²this is satisfied; thus:

D²⊂ {v∈H¹(]−π, π[)∩[H²(]−π,0[∪]0, π[), dv dθB

(±π) = 0, [c² dv dθB

](0) = 0}.

Conversely, if:

g∈ {v∈H¹(]−π, π[)∩[H²(]−π,0[∪]0, π[), dv dθB

(±π) = 0, [c² dv dθB

](0) = 0}, from the definition of the coefficients an andbn given earlier, the series X

n≥0

n²anp^s_n+ n²bnp^c_nconverges for instance inL²(]0, π[)and finallyg∈D². The interpolation between D¹andD²(see [17]) leads to the result fors= 3/2and Lemma2.3is proved. 2

We now turn to the study of the explicit local solutionu.ˆ

Step 3 Analytical solution of the local modelLet us look for analytical solutions of the local model and we still focus at pointB. One can set on the circleCBsurrounding the neighborhoodQBofB:

ˆ

u(r, θB) =X

n≥0

an(r)p^s_n(θB) +bn(r)p^c_n(θB) with

∀s∈[0,3/2], sX

n≥0

(1 +n^2s)(a_n(r)²+b_n(r)²)' ||ˆu(r)||_s,]−π,π[.

Remark 4. This convergence property is used as follows in the following (see step 3). Let us consider a seriesαnsatisfying:

X

n≥3

α²_nn³<∞. (17)

Setting

k(r_B) =X

n≥3

α_nr_Bⁿ², |r| ≤1,

and let us consider the second order derivative of this series with respect torB. In fact, the integration for the computing theH²norms are performed on the two dimensional open sets: Q_B∩Ω₊andQ_B ∩Ω₋. The surface element isr_Bdr_Bθ_B. Therefore, in order to prove that the second order derivative ofkis locally (in a neighborhood ofB) in the space L², one has to check the integration with respect tor_B, of the following term:

δ= Z 1

0

X

n≥3

α_n²(n⁴

16)r_Bⁿ⁻⁴rBdrB≤X

n≥3

α²_n(n⁴ 16)

Z 1 0

rⁿ⁻³_B dr≤X

n≥3

α²_n n⁴ 16(n−2). And one can ensure from (17) thatδ∈L²(]0,1[)and therefore:

k∈H²(]0,1[).

2 Let us now introduce the functions

H_n^s(rB, θB) = (rB

R_B)ⁿ⁺¹²p^s_n(θB)andH_n^c(rB, θB) = (rB

R_B)ⁿp^c_n(θB). (18) Since(an, bn) ∈C([R/2, R])², coefficientsAn = an(RB)andBn = bn(RB)are well defined. We have



















−div(c²∇H_n^s) =−div(c²∇H_n^c) = 0 in Ω,

∂H_n^s

∂ν (r,±π) = ∂H_n^c

∂ν (r,±π) = 0, X

n≥0

AnH_n^s(RB, θB) +BnH_n^c(RB, θB) = ˆu(RB, θB).

Let us set onQB:

H(rB, θB) =X

n≥0

AnH_n^s(rB, θB) +BnH_n^c(rB, θB). (19)

The differenced = ˆu−H ∈ H¹(QB)is the unique solution in the spaceH¹(QB)of (q= 0onQB):







−div(c²∇d) =ω²u in Qˆ B,

∀rB ∈]0, RB[: ∂d

∂ν(rB,±π) = 0; ∀θB ∈]−π, π[: d(RB, θB) = 0.

(20)

Clearly, ifω= 0, one would haved= 0. We assumeω6= 0. Lemma2.2leads to : ω²uˆ=X

n≥0

[e^s_n(r_B)p^s_n(θ_B) +e^c_n(r_B)p^c_n(θ_B)], where















e^s_n(r_B) =A₀ω² Z π

−π

c²u(rˆ _B, θ_B)p^s_n(θ_B)dθ_B,

e^c_n(r_B) =B₀ Z π

−π

[ω²u(rˆ _B, θ_B)−X

j

e^s_j(r_B)p^s_j(θ_B)]p^c_n(θ_B)dθ_B. We look fordas follows :

d(r_B, θ_B) =d^s(r_B, θ_B) +d^c(r_B, θ_B), where we have set

d^s(r_B, θ_B) =X

n≥0

ξ_n^s(r_B)p^s_n(θ_B), and d^c(r_B, θ_B) =X

n≥0

ξ_n^c(r_B)p^c_n(θ_B).

(21)

Let us computeξ^s_n andξ_n^c inH¹(]0, R_B[), focusing onξ_n^s (analogous computations are valid forξ_n^c).

Since the functionsp^s_n are orthogonal inL²(]0, π[),(and the same is true for the func- tionsp^c_n) it is easy to prove thatξ^s_nis solution of (we setµn=n+1

2)











−rB

d drB

(rB

dξ^s_n drB

) +µ²_nξ_n^s =r_B²e^s_n, 0< rB< RB, ξ_n^s(RB) = 0.

(22)

Let us notice that there is no boundary condition atrB = 0,since it is replaced in this case by the fact thatξ^s_n andξ^c_n must be inH¹(]0, RB[)). By integrating this differential equation, we obtain

ξ_n^s(r_B) =r^µ_Bⁿ Z RB

rB

1 s^2µn+1

Z s 0

η^µn+1e^s_n(η)dηds. (23) Let us now estimated^s(recalling thatωuˆ∈H¹(QB)) with the following lemma Lemma 2.4. Letv(r, θ_B) =X

n≥0

a_n(r)p^s_n(θ_B)be inL²(D_R).Then

1. ∂v

∂r ∈L²(QB) ⇔ Z R

0

rX

n≥0

(dan

dr )²dr <∞,

2. ∂v

∂θ ∈L²(QB) ⇔ Z R

0

rX

n≥0

(1 +n²)an(r)²dr <∞,

3. ∂²v

∂r² ∈L²(Q⁺_B) ⇔ Z R

0

rX

n≥0

(d²an

dr² )²dr <∞,

4. ∂²v

∂r∂θ ∈L²(Q⁺_B) ⇔ Z R

0

rX

n≥0

(1 +n²)(dan

dr )²dr <∞,

5. ∂²v

∂θ² ∈L²(Q⁺_B) ⇔ Z R

0

rX

n≥0

(1 +n⁴)an(r)²dr <∞.

2 The proof of this lemma is straightforwards using the orthogonality of the functionsp^s_n

onH^s(]0, π[)(s= 0,1,2)and remark4. 2

The remaining proof concerning the estimate onξ^s_nis rather technical and can be omitted in a first reading. Accepting it, the reader can jump to equation (30).

Let us now obtain the estimates ofξ^s_nusing thatX

n≥0

e^s_np^s_n∈H¹(QB)and thus satisfies assertions 1 and 2 of Lemma2.4.

We write A_n(s) = [ Z s

0

ηe^s_n(η)²dη]^1/2 and, for sake of clarity during the proof, R instead ofRB.

First estimate : We have forn≥0:

|ξ^s_n(r)| ≤r^µn Z R

r

1 s^2µn+1[

Z s 0

η^2µn+1dη]^1/2An(s)ds

≤r^µn Z R

r

1 s^2µn+1

s^µn+1

p2(µn+ 1)A_n(s)ds

≤ r^µn

p2(µ_n+ 1)An(r)[ 1 µ_n−1( 1

r^µn−1 − 1

R^µn−1)] ifn≥1

≤c₀rAn(r) µ_n√

µ_n ifn≥1 and thus forn≥1 :

|ξ^s_n(r)|²≤c0

r²An(r)²

µ³_n . (24)

The functiond^sthus satisfies Z R

0

X

n≥1

(1 +n⁵)r|ξ_n^s(r)|²dr≤c₀X

n≥0

(1 +n²)A_n(R)²<∞.

With Lemma2.4,d^s−ξ₀^sp^s₀satisfies assertions 2 and 5 whenωuˆ∈L²(Q_B).

Let us now turn to the study of the others assertions of Lemma2.4: they involve the computations of the derivatives with respect torof the functionξ_n^s.

Second estimate : We have:

dξ_n^s

dr(r) = µn

r ξ_n^s(r)− 1 r^µn+1

Z r 0

η^µn+1e^s_n(η)dη, hence:

|dξ_n^s

dr (r)| ≤µn

r |ξ_n^s(r)|+ 1

r^µn+1An(r) r^µn+1 p2(µ_n+ 1)

≤µn

r |ξ_n^s(r)|+ An(r) p2(µ_n+ 1)

≤c₀A_n(r)

õ with (24).

We thus get (recall thatµ_n=n+ 1/2):

Z R 0

X

n≥0

(1 +n²)r|dξ^s_n

dr (r)|²dr≤c0

X

n≥0

(1 +n)An(R)²<∞ which proves assertions 1 and 4 of Lemma2.4.

Let us now turn to assertion 3 which involvesd²ξ_n^s dr² .Since d²ξ_n^s

dr² = µ²_n

r²ξ_n^s−e^s_n+1 r

dξ_n^s dr,

we study separately each term of the right hand side, starting with µ²_n r²ξ_n^s. Integrating by parts, we get forµ_n≥1(which requiresn≥1):

ξ^s_n(r) =r^µn Z R

r

1

s^2µn+1 [η^µn+2

µ_n+ 2e^s_n(η)]^s₀− 1 µ_n+ 2

Z s 0

η^µn+2de^s_n dr(η)dη

ds

= r^µn µn+ 2

Z R r

1

s^µn−1e^s_n(s)ds− r^µn µn+ 2

Z R r

1 s^2µn+1

Z s 0

η^µn+2de^s_n dr (η)dη

ds

def= ξ_n¹(r) +ξ_n²(r).

Forn≥2,ξ¹_nsatisfies ξ¹_n(r) = r^µn

µn+ 2 [s^2−µn 2−µn

]^R_r − Z R

r

de^s_n

dr (s)s^2−µn 2−µn

ds

= r^µn

µ²_n−4[e^s_n(r)

r^µn−2 − e^s_n(R)

R^µn−2]− r^µn µ²_n−4

Z R r

1 s^µn−2

de^s_n dr(s)ds, thus (still forn≥2)

|ξ_n¹(r)| ≤ 1

µ²_n−4[r²|e^s_n(r)|+ (r

R)^µnR²|e^s_n(R)|+r^µn Z R

r

1 s^µn−2|de^s_n

dr (s)|ds]

≤ 1

µ²_n−4[[r²|e^s_n(r)|+ (r

R)^µnR²|e^s_n(R)|+ r² p2(µ_n−2)(

Z R 0

s|de^s_n

dr (s)|²ds)^1/2] Sincee^s_n∈H¹(]R/2, R[),the sequence(e^s_n(R))n∈l^∞and forn≥2,

µ²_n

r²|ξ_n¹(r)| ≤c₀[|e^s_n(r)|+r^µn−2|e^s_n(R)|+ 1

õn

Z R 0

s|de^s_n

dr(s)|²ds1/2

]. (25) With simlar computations, one can prove that

|ξ²_n(r)| ≤ 1 µ_n√

µ_n(µ_n−2)r²( Z R

0

s|de^s_n

dr (s)|²ds)^1/2 thus forn≥2,

µ²_n

r²|ξ_n²(r)| ≤ 1

õn

r²( Z R

0

s|de^s_n

dr (s)|²ds)^1/2. (26) From (25) and (26), we deduce that

X

n≥2

Z R 0

rµ²_n

r²|ξ_n^s(r)|²dr <∞. (27) Let us now turn to the term involving 1

r dξ_n^s

dr.Writing 1

r dξ_n^s

dr =µn

r²ξ^s_n− 1 r^µn+2

Z r 0

η^µn+1e^s_n(η)dη, (28) and integrating by parts, it is easy to check that

| 1 r^µn+2

Z r 0

η^µn+1e^s_n(η)dη| ≤c0[ 1 µn

|e^s_n(r)|+ 1 µn

õn

( Z R

0

s|de^s_n

dr (s)|²ds)^1/2]. (29) With (27), (28) and (29), we deduce that

X

n≥2

Z R 0

r(d²ξ_n^s

dr² )²dr≤c0[X

n≥2

Z R 0

r(|e^s_n(r)|²+|de^s_n

dr (r)|²)dr+X

n≥2

r^2µn−4||e^s_n||²_C[R/2,R]] where the terms||e^s_n||²_C[R/2,R]are uniformly bounded inntherefore

d^s−ξ₀^sp^s₀−ξ₁^sp^s₁∈H²(Q⁺_B)∩H²(Q⁻_B).

Sinceµ1 = 3/2, we getξ^s₁(r) = r^3/2 Z R

r

1 s⁴

Z s 0

η^5/2e^s₁(η)dηdsand one can easily verifies that:

ξ₁^s(r)p^s₁(θB) =ξ^s₁(r)A

c²sin(3θB

2 )∈H²(Q⁺_B)∩H²(Q⁻_B).

Hence (with the notations used in theorem1.1)

d^s−ξ₀^sp^s₀=d^s−a₀S_B ∈H²(Q_B∩[Ω₊∪Ω₋]).

Finally, it has been proved that onQBand withµn=n+1 2: d^s(r_B, θ_B) =X

n≥0

[A_n(r_B RB

)^µn+ξ_n^s(r_B)]p^s_n(θ_B). (30) In (30), the first series (with the coefficientsAn) takes into account the non homoge- neous Dirichlet boundary condition on the circleCand the second one which satisfies an homogeneous Dirichlet boundary condition onC, takes into account the right handside of the equation (30) satisfied byξ_n^s. The expression ofuˆonQBis the sum of two series (d^sand d^c). It is worth noting that the functions(r_B

RB

)ⁿp^c_nwhich appear in the expression ofd^c, are polynomials with respect to the cartesian coordinates(x1, x2)and therefore areC^∞. Fur- thermore the convergence of the corresponding series is uniform insideQB, (rB < RB).

But this is no more true for the terms(r_B RB

)ⁿ⁺¹²p^s_nwhich appear in the expression ofd^s, because of the multiplicative term√

r_B. In fact the first term which appears in the expres- sion ofd^s-say

rrB

RB

p^s₀(θB)is not smooth. It is named the singularity term (in the sense that it isn’t in the spaceVRsee (7)) and denoted in the following by:

SB(rB, θB) = 1 c²

rrB

RB

sin(θB

2 ).

It is in the spaceH¹(QB)but not inH²(Q⁺_B)orH²(Q⁻_B).The second series which appears in the expression ofd^scontains the termξ^s_nwhich is explicit at (23). It has been noticed that the series converges to an element of the spaceH²(QB).

Step 4 : Thea prioriestimate (8).

Let us recall thatKis a compact set with{A, B} ∩K =∅.LetηA andηB be fixed such as in step 2 with compact supports inK.¯ LetL²_K = {q ∈ L²(Ω), supp(q) ⊂ K, R

Ωq(x)dx= 0}.Forq∈L²_K, there exists a unique solutionwof







−div(c²∇w) =q in Ω

∂w

∂ν = 0 on Γ

and we proved in step 3 that there exists a unique(k_A, k_B, v) ∈ R²×V^R such that w=KASAηA+KBSBηB+v^R.

Applying the closed graph Theorem [16], il it easy to prove that the linear mapping q∈L²_K→(KA, KB, v^R)∈R²×V^R

is continuous and estimate (8) is proved and so is Theorem2.1. 2 Let us now turn to the explicit computation of constantsK_AandK_B.

3. Computation of the stress intensity factors. There are several possibilities for com- puting the coefficientKAandKB. Each of them has its own advantages and drawbacks depending of the goal one aims at. Let us sketch three of those methods recalling thatq (see (3)) has a support far from the two crack tipsAandB.