Mohamed Ferjani

Daniel Averbuch

Andreï Constantinescu
email: andrei.constantinescu@lms.polytechnique.fr

Semianalytical solution for the stress distribution in notched tubes

Keywords: Notch, closed-form solution, stress concentration factor, tube

The aim of this paper is to present an alternative solution to the finite element method for the determination of the elastic stress field for an axi-symmetric notched tube under tension and bending loadings. The proposed solution is an extension of the work of Filippi and Lazzarin (2004) to tubular specimens. It provides the stress field distribution along the notch bisector. A good agreement was found between the solution proposed in the present paper and the results from a finite element analysis. A sensitivity analysis is performed to establish the domain of validity of the solution with respect to the dimensions of the notched tube

Fatigue design is a complex engineering field where materials, geometry and loading define the lifetime of a structure. The fatigue design methodology is a two step process based on a mechanical analysis devoted to the computation of the stresses and a fatigue analysis devoted to lifetime prediction. As fatigue cracks initiate favorably near notches, holes or generally in regions of stress concentration, it is important to estimate stresses precisely in these parts in order to obtain accurate lifetime predictions.

Additional difficulties to estimate the fatigue life of notched components come from different factors which are essentially related to the stress gradient [START_REF] Schijve | Stress gradients around notches[END_REF][START_REF] Pluvinage | Fatigue and fracture emanating from notch; the use of the notch stress intensity factor[END_REF][START_REF] Hofmann | A multiscale discussion of fatigue and shakedown for notched structures[END_REF][START_REF] Hofmann | A discussion at the mesoscopic scale of the stress-gradient effects in high cycle fatigue based on the dang van criterion[END_REF] or the scale effect [START_REF] Frost | Non-propagating cracks in vee-notched specimens subject to fatigue loading[END_REF][START_REF] Du Quesnay | An analysis of notch-size effects at the fatigue limit[END_REF][START_REF] Morel | A non-local theory applied to high cycle multiaxial fatigue[END_REF]. These factors influence the fatigue process leading to unsatisfactory fatigue predictions when classical fatigue criteria such as the Crossland criterion or the Findley criterion are employed.

Fortunately the application of these fatigue criteria combined with different methods such as the critical distance [START_REF] Tanaka | Engineering formulae for fatigue strength reduction due to crack-like notches[END_REF][START_REF] Lazzarin | Fatigue crack initiation and propagation phases near notch in metals with low notch sensitivity[END_REF][START_REF] Taylor | Geometrical effects in fatigue: A unified theoretical model[END_REF] or modified to incorporate the influence of the stress gradient [START_REF] Papadopoulos | Invariant formulation of a gradient dependent multiaxial high-cycle fatigue criterion[END_REF] shows good agreements to estimate fatigue limits. However, all these techniques need the knowledge of the stress distribution near the notch tip either to compute the stress gradient or to compute the stress components at a different distance from notch tip.

To achive the first step of the fatigue design methodology, engineers can use different numerical or analytical methods and tools. Numerical methods like finite boundary elements method provide solutions for a large class of geometries, but are mesh dependent in areas of stress concentration and time consuming. Moreover, sensitivity analysis with respect to different parameters will further increase the computational burden.

Classical closed-form solutions for infinite bodies containing holes have been presented by Savin [START_REF] Savin | Stress Concentration Around Holes[END_REF] based on the complex potentials proposed previously by Muskeshishvilli [START_REF] Mushkelishvili | Some Basics Problems of the Mathematical Theory of Elasticity[END_REF]. This solution employs the theory of complex elasticity which has been a powerful mathematical tool for many different problems such as contact problem [START_REF] Jäger | Half-planes without coupling under contact loading[END_REF], slope stability [START_REF] Milyayev | The action of a concentrated force inside an elastic half-plane with a step[END_REF] or crack problem [START_REF] Bui | Solutions exactes de fissure d'interface sous contact frottant avec un milieu indéformable[END_REF].

Several papers [START_REF] Williams | On the stress distribution at the base of a stationary crack[END_REF][START_REF] Creager | Elastic field equations for blunt cracks with reference to stress corrosion cracking[END_REF][START_REF] Glinka | Universal feature of elastic notch-tip stress field[END_REF]4] have proposed expressions for the stress field ahead of a notch depending on two parameters: the notch root radius ρ and the axial stress concentration factor K t . Indeed, they showed that in the vicinity of the notch tip, the stress field distribution is more influenced by the notch root radius than by the global geometry of the notch and is very similar for a variety of notches when stresses are normalized by the stresses at the notch tip.

Thus the stress distribution obtained by these solutions, when considered away from the notch, is not dependent on the stress concentration factor, which is incoherent with the assumptions of the analysis [START_REF] Xu | Practical stress expressions for stress concentration regions[END_REF]. Therefore these solutions are not accurate enough [START_REF] Kujawski | Estimation of stress intensity factors for small cracks at notches[END_REF] for a distance ahead of the notch tip larger than 3ρ in the case of a sharp notch (K t > 5) and ρ in the case of blunt notch (K t < 5 ).

[ [START_REF] Lazzarin | A unified approach to the evaluation of linear elastic stress field in the neighborhood of cracks and notches[END_REF][START_REF] Filippi | Developments of some explicit formulas useful to describe elastic stress fields ahead of notches in plate[END_REF][START_REF] Atzori | Stress distribution in notched structural components under pure bending and combined traction and bending[END_REF] proposed elaborate approximate closed-form expressions for a unique solution applicable both for blunt and sharp notches. Their solutions are valid for notched bars and plates under tension and bending loading.

The present work proposes a further extension of these solutions in the form of an approximate stress field solution devoted to tubes presenting an internal or an external notch under tension and bending loadings. The application of this solution to a threaded connection is presented in [START_REF] Ferjani | Intégration des phénomènes dynamiques dans l'analyse en fatigue des garnitures de forage[END_REF][START_REF] Ferjani | A computational approach for the fatigue design of threaded connections Submitted to[END_REF] The paper is organized as follows: the next section presents a short review of the solution proposed by [START_REF] Filippi | Distributions of the elastic principal stress due to notches in finite size plates and rounded bars uniaxially loaded[END_REF] dedicated to notched bars. The third section is dedicated to the extension of the preceding solution to tubular geometries with an external or an internal notch and the determination of the different parameters introduced in the solution. The final section discusses the comparisons of the semi-analytical solution with the finite element results for a variety of notch geometries and defines the validity of the solution with regards to a dimensionless parameter ξ representing a ratio between notch depth and the tube thickness.

Stress problem for notched tubes

In this section, we are interested in the formulation of an approximate solution of the elastic stress field Σ in the vicinity of a notched axi-symmetrically tube under tension and bending loading.

The geometry of the tubes is depicted in Fig. 1. We shall consider two systems of coordinates, the former is a global system of coordinates {e R , e Θ , e Z } with the z-axis along the axis of the tube and the latter is a local system of coordinates e x , e y , e z in the neighborhood of the notch. The tube is defined by an internal and an external radius, R i and R e respectively. The notch has its tip at a distance R o from the axis of the tube and its depth p is equal to

p =      R e -R o for external notch R o -R i for internal notch . ( 1
)
The notch has a rounded V-shaped geometry with an opening angle 2α and a root radius ρ.

Along the bisector of the notch, the stress field Σ has the following matrix form for tension and bending loading in the global system of coordinates {e R , e Θ , e Z }, due to the symmetry of the problem, and to some simplifications regarding the order of magnitude of the stress components with respect to the axial component

Σ ZZ Σ =        Σ RR 0 0 0 Σ ΘΘ 0 0 0 Σ ZZ        . (2)
On the same bisector, we shall denote σ the associated stress field expressed in the local system of coordinates e x , e y , e z . In our case, its components are defined by

σ =        σ x 0 0 0 σ y 0 0 0 σ z        =        Σ RR 0 0 0 Σ ZZ 0 0 0 Σ ΘΘ        . (3)
One can note that the shear component σ xy is equal to zero everywhere along the notch ligament, i.e 0 > r -r o > R e -R o , due to the symmetry of the problem.

Recently, a set of closed-form equations valid for V-shaped notched axi-symmetric bars, i.e R i = 0, subjected to mode I loadings (tension and bending) has been proposed in [START_REF] Filippi | Distributions of the elastic principal stress due to notches in finite size plates and rounded bars uniaxially loaded[END_REF].

The local elastic stress field σ has been obtained with the use of the Kolosoff-Mushkelishvili's complex potential function [START_REF] Mushkelishvili | Some Basics Problems of the Mathematical Theory of Elasticity[END_REF] and the conformal mapping introduced by Neuber [22]. This solution has been obtained under plane stress or plane strain condition with respect to the local coordinate system, which is verified for sufficiently large R.

The solution in [START_REF] Filippi | Distributions of the elastic principal stress due to notches in finite size plates and rounded bars uniaxially loaded[END_REF] for the stress field σ along the notch bisector, i.e. y = 0 or θ = 0, of a cylindrical bar expressed in the local system of coordinates e x , e y , e z (see Fig. 1) is

σ y = σ max 4 (q -1) + qω 1 g (r) 4 (q -1) f (r) λ 1 -1 + qω 1 r r o µ 1 -1 , (4)
σ x = σ max η 1 [4 (q -1) + qω 1 ] 4 (q -1) r r o λ 1 -1 + qδ 1 r r o µ 1 -1 (5)
where σ max is the maximum stress in the y-direction and functions f (r) and g (r) will be defined later. The other parameters q, r o , λ 1 , µ 1 , ω 1 , δ 1 and η 1 involved in (4) and ( 5) are defined in Appendix 1.

The stress field defined by (4) and ( 5) is an approximate solution, in the sense that the free surface boundary condition is satisfied only at the notch tip and on the rectilinear flanks of the V-notch, where the boundary conditions were set according to Williams [START_REF] Williams | Stress singularities resulting from various boundary conditions in angular corners of plates in tension[END_REF].

Originally [START_REF] Filippi | Distributions of the elastic principal stress due to notches in finite size plates and rounded bars uniaxially loaded[END_REF] the function f (r) in ( 4) was linear, expressed as: r r o . This solution gives good results close to the neighborhood of the notch tip but becomes less accurate far away from it. Thereby, in order to obtain a better solution all along the notch bisector, the function f (r) was modified in the form

f (r) = 1 + atan [m (r -r o )] mr o (6)
where the parameter m will be defined later. This solution has shown good agreements with respect to FEM results [START_REF] Filippi | Distributions of the elastic principal stress due to notches in finite size plates and rounded bars uniaxially loaded[END_REF] . It is worth noting that (6) matches the linear law r r o in the vicinity of the notch tip.

The function g (r) expressed in ( 4) is related to the loading mode, tension or bending, and is equal to

g (r) =      1 - r -r o κ
for bending 1 for tension [START_REF] Ferjani | A computational approach for the fatigue design of threaded connections Submitted to[END_REF] where κ is the distance from the neutral axis to the notch tip, and is equal to R o .

At this stage, the solution is complete up to the determination of the parameter m (see [START_REF] Filippi | Distributions of the elastic principal stress due to notches in finite size plates and rounded bars uniaxially loaded[END_REF]) and to the determination of σ max :

In order to identify the best values of these two parameters, for a given geometry, two additional relations are then needed. These two values will then be identified by equating the theoretical axial stress concentration factor K ZZ , with the stress concentration K t computed with the finite element method, and by writing the global equilibrium condition of the tube. As it will be shown later in section 5., the value of the parameter m varies only weakly with the notch root radius. This result, equally reported in [START_REF] Filippi | Distributions of the elastic principal stress due to notches in finite size plates and rounded bars uniaxially loaded[END_REF], permits to perform the sensitivity studies without performing additional finite element analysis, and justifying the pratical interest in the present approach.

The results can be summerized by the following expressions:

Σ RR = E 1 -ν 2 du R dR + ν u R R + Σ ZZ ν 1 -ν , (8)
Σ ΘΘ = E 1 -ν 2 u R R + ν du R dR + Σ ZZ ν 1 -ν (9)
where E and ν are the Young's modulus and the Poisson's ratio, respectively, whereas u R is the radial displacement. The first of the previous equations provides the radial displacement by integration with an initial condition u R = 0 at R = 0 which is valid for bars only. Then the distribution of the hoop stress, with respect to R, is obtained by substituting the solution u R into the second equation.

Local formulation of the solution for notched tubes

In order to extend this solution to tubular geometries subjected to tension and bending loadings, the stress component σ x must satisfy the free surface condition, i.e. σ x = 0 at R = R i and R = R e for an internal and external notched tube, respectively.

As a consequence the function h (r) is added to [START_REF] Kujawski | Estimation of stress intensity factors for small cracks at notches[END_REF] such that the boundary condition is satisfied. Thus, the stress component σ x becomes

σ x = σ max η 1 [4 (q -1) + qω 1 ] 4 (q -1) r r o λ 1 -1 + qδ 1 r r o µ 1 -1 -h (r) Γ (10)
where the function h (r) satisfies the two following conditions

h (r o ) = 0 and h (r o + l) = 1. (11)
Moreover, the coefficient Γ in [START_REF] Frost | Non-propagating cracks in vee-notched specimens subject to fatigue loading[END_REF] and the ligament length l in [START_REF] Glinka | Universal feature of elastic notch-tip stress field[END_REF] are given by

Γ = 4 (q -1) 1 + l r o λ 1 -1 + qδ 1 1 + l r o µ 1 -1 , (12)
l =      R o -R i ext R e -R o int . ( 13
)
Here 'ext' and 'int' simply denote the formula to use for an external or an internal notch, respectively.

The simplest function satisfying [START_REF] Glinka | Universal feature of elastic notch-tip stress field[END_REF] is the linear function, thus

h (r) = r -r o l . ( 14
)
For the component σ y , its expression is given by

σ y = σ max 4 (q -1) + qω 1 g (r) 4 (q -1) f (r) λ 1 -1 + qω 1 r r o µ 1 -1 (15)
with the modified g function

g (r) =                     1 - r -r o κ ext 1 + r -r o κ int bending 1 tension . ( 16
)
Therefore the axial Σ ZZ and the radial Σ RR components of the stress field Σ along the ligament are equal to

Σ ZZ = σ max 4 (q -1) + qω 1 g (r) 4 (q -1) f (r) λ 1 -1 + qω 1 r r o µ 1 -1 , (17)
Σ RR = σ max η 1 [4 (q -1) + qω 1 ] 4 (q -1) r r o λ 1 -1 + qδ 1 r r o µ 1 -1 -h (r) Γ] . (18)
Once again, the hoop stress can be determined with the help of the constitutive equations 9. One can however remark that the initial condition used for bars : u R = 0 for R = 0, has to be modified. For this reason, we assume that, far away from the notch tip, the mechanical fields are the same as for a tube without the notch. In other words, we assume that Σ ΘΘ = 0 at R = R i for the case of an external notch and Σ ΘΘ = 0 at R = R e for the case of an internal notch. Solving for u R in ( 8) and ( 9) and using the boundary condition, one obtains:

u R | R=R i = - νR i E Σ ZZ | R=R i external (19) u R | R=Re = - νR e E Σ ZZ | R=Re internal (20)
Finally the hoop stress Σ ΘΘ is determined by introducing u R and Σ ZZ into (9).

Global formulation of the solution for notched tubes

At this stage, the solution is complete up to the determination of the parameter m (see [START_REF] Ferjani | Intégration des phénomènes dynamiques dans l'analyse en fatigue des garnitures de forage[END_REF]) which belongs to the open range [0, +∞[ and depends on the notch geometry and the relative dimensions of the tubular components. This parameter is obtained numerically by equaling the theoretical axial stress concentration factor K ZZ with the stress concentration K t computed with the finite element method

K t = K ZZ = σ max σ nom ( 21
)
where σ nom is the applied gross nominal stress and σ max is the maximal axial stress as depicted in Fig. 2. It is well known, that the precision of the numerical computed stress concentration is influenced by the mesh density. A short discussion of this point will be presented in the next section and is illustred in Fig. 4.

Invoking the global equilibrium condition for tension and bending loadings, one can obtain an equation which can be solved numerically as a function of the parameter m.

The global equilibrium condition for tension loading takes the following form

T =      Ro Ri Σ ZZ (R) 2πRdR ext Re Ro Σ ZZ (R) 2πRdR int , (22)
R =      R o -r + r o ext R o + r -r o int ( 23
)
where T is the tension load linked to the gross nominal stress σ nom by

T = π R 2 e -R 2 i σ nom . (24)
Substituting ( 17) into [START_REF] Neuber | Theory of notch stresses: Principles for exact stress calculation[END_REF] and rearranging the obtained equation, one obtains the following relation for K t ZZ for tension loading

K t ZZ = (R 2 e -R 2 i ) [4 (q -1) + qω 1 ] 2 (A m + B) (25)
with

A m = 4 (q -1) l 0 1 + atan (mx) mr o λ 1 -1 (R o ± x) dx, (26)
B = qω 1 r o µ 1 R o 1 + l r o µ 1 -1 ±   r o µ 1 + 1 1 + l r o µ 1 +1 - l r o 1 + l r o µ 1 -1   ( 27
)
where the signs + andin ( 26) and ( 27) stand for an internal and external notch, respectively.

For bending loading (see Fig. 3), the balance of angular moment is employed after having specified which is the peak stress σ max to use in [START_REF] Filippi | Distributions of the elastic principal stress due to notches in finite size plates and rounded bars uniaxially loaded[END_REF] and which is the distance κ to introduce in [START_REF] Ferjani | A computational approach for the fatigue design of threaded connections Submitted to[END_REF]. To carry out the calculation, we assume that along a vertical strip dX, as depicted in Fig. 3, the axial stress component Σ ZZ is given by ( 17) and the peak stress σ max is only a function of X such that

σ max (X) = σ ′ max 1 - X R o 2 (28)
where σ ′ max is the maximum axial stress at X = 0.

Regarding the strip dX, the parameter κ is now defined as the distance between the notch tip at X and the plan Y = 0 and, as for the stress peak, we assume that κ is equal to

κ = R o 1 - X R o 2 . ( 29
)
Moreover, in the case of an internal notch, the stress component Σ ZZ is not defined in the region R e ≥ X ≥ R o , therefore we assume that, in this region, Σ ZZ is equal to the axial stress of the same geometrical tube without notch, thus

Σ * ZZ = Y κ σ nom . (30)
Therefore, under the preceeding assumptions, the balance of angular moment becomes

M 4 =                        R i 0 √ R 2 o -X 2 √ R 2 i -X 2 Σ ZZ Y dY dX + Ro R i √ R 2 o -X 2 0 Σ ZZ Y dY dX for an external notch Ro 0 √ R 2 e -X 2 √ R 2 o -X 2 Σ ZZ Y dY dX + Re Ro √ R 2 e -X 2 0 Σ * ZZ Y dY dX for an internal , (31)
Y =      R 2 o -X 2 + r o -r for an external R 2 o -X 2 -r o + r for an internal and for X ≤ R o (32)
where M denotes the bending load that related to the maximal gross nominal stress σ nom by

M = I R e σ nom , (33)
I = π 4 R 4 e -R 4 i . (34)
Substituting ( 17) into [START_REF] Williams | On the stress distribution at the base of a stationary crack[END_REF], one obtains for bending loading the expression for the axial stress concentration factor

K f ZZ K f ZZ = R o I 4R e -G 4 (q -1) + qω 1 C m (35)
with

C m =          R i 0 √ R 2 o -X 2 √ R 2 i -X 2 • + Ro R i √ R 2 o -X 2 0 • G m (Y, X) Y 2 dY dX ext Ro 0 √ R 2 e -X 2 √ R 2 o -X 2 G m (Y, X) Y 2 dY dX int , (36)
G =      0 ext Re Ro √ R 2 e -X 2 0 Σ * ZZ Y dY dX int , (37)
G m (Y, X) = 4 (q -1)

   1 ± atan m Y - √ R 2 o -X 2 mro    λ 1 -1 + qω 1 1 ± Y - √ R 2 o -X 2 ro µ 1 -1 (38)
where the signs + andin (38) stand respectively for an internal and external notch.

Results and limitations of the solution for notched tubes

In this section, we shall compare the predictions of the solution with FEM results along the notch ligament. If not mentioned, all the results presented below are done for a notch geometry with an opening angle 2α of 60 o and a notch root radius ρ of 1mm.

For accuracy purpose, a study of the mesh size has been performed for tension and bending loading using the ABAQUS commercial software [1]. For the tension case, 2D axi-symmetric model has been used with linear four noded elements CAX4R and for the bending case, 3D axi-symmetric model has been used with linear eight-node elements C3D8R. Both models have been set up to obtain a regular and smooth mesh in the neighborhood of the notch tip.

The geometry of the mesh as well as the results are depicted in Fig. 4. This figure presents the value of the axial concentration factor K t with respect to the mesh size ratio ǫ ρ . To reach a comfortable accuracy, the mesh size ratio has been chosen equal to 0.01 for tension loading and equal to 0.02 for bending loading. This choice is also motivated by the limitation of the number of degrees of freedom created; about 3000 and 10000 elements respectively for tension and bending loading.

Figs. 5 and 6 illustrate the distribution of the stress components Σ ZZ , Σ RR and Σ ΘΘ along the notch ligament for the case of an external and internal notch under tension and bending loadings. A good agreement is found between the FEM results (markers) and the closed-form solution (line) for the axial stress component Σ ZZ .

Concerning the radial stress component Σ RR , the semi-analytical solution is more accurate in the neighborhood of the notch tip and becomes less accurate for r > ρ; this error is principally due to the choice of the linear function h (r) in ( 18). This mismatch influences slightly the distribution of the hoop stress component Σ ΘΘ except for the case of an internal notched tube under bending (see Fig. 6).

A way to correct this mismatch is to choose another function h (r). Instead of the initial linear function introduced in ( 14), we propose the following hyperbolic function

h (r) = tanh r-ro ρ tanh l ρ . ( 39
)
However, let us remark that the form has been chosen without any a priori information or optimization, and therefore we expect that better functions can be found for this role.

In this case, an improvement is attained on the accuracy of the hoop stress Σ ΘΘ , as for example depicted in Fig. 7. This comparison with FEM results has been obtained for the same notch geometry as before in tension loading. The improvement is less impressive in the case of bending loading.

Figs. 8 and9 illustrate, for a variety of geometrical parameters α and ρ, the distribution of the principal stress component Σ ZZ along the notch ligament for the case of an external and internal notch under tension. Again good agreements are found between the FEM results (markers) and the closed-form solution (line).

Let us now explore the limitation of the solution with respect to the dimensions of the tube and the notch. This discussion is important in the case of a tubular geometry when the ligament length l is of the same order of magnitude as the radius of the notch ρ. Indeed, the solution derived in the last sections has been obtained under some assumptions with respect to the dimensions of the tube [START_REF] Lazzarin | A unified approach to the evaluation of linear elastic stress field in the neighborhood of cracks and notches[END_REF], therefore the closed-form solution expressed here will not necessarily be suitable for any type of geometry. This section is therefore devoted to define the range of application of the solution.

As mentioned earlier, the parameter m will take a value between 0 and +∞ for a given set of notch geometry (ρ, α and p) and tube dimension (R i and R e ) and therefore a given stress concentration factor K t . On this range, the axial stress concentration factor K ZZ defined by [START_REF] Pluvinage | Fatigue and fracture emanating from notch; the use of the notch stress intensity factor[END_REF] for tension loading and (35) for bending loading is continuously decreasing, leading to a maximum value for m = 0 and a minimum value for m = +∞.

For this two limits, the function f (r) introduced in (4) takes the following expressions

lim m→0 f (r) = r r o , ( 40
)
lim m→+∞ f (r) = 1. (41)
Introducing this two limiting functions into [START_REF] Pluvinage | Fatigue and fracture emanating from notch; the use of the notch stress intensity factor[END_REF], one can derive analytically the two bounds of the axial stress concentration factor K t ZZ in the case of tension loading

K t,max ZZ = (R 2 e -R 2 i ) [4 (q -1) + qω 1 ] 2 (A o + B) , (42)
K t,min ZZ = (R 2 e -R 2 i ) [4 (q -1) + qω 1 ] 2 (A ∞ + B) (43)
where the upper fix max and min are respectively related to m = 0 and m = ∞.

The coefficient B is given by ( 27) and the coefficients A o and A ∞ are given by

A o = 4 (q -1)        r o (R o ± r o ) 1 + l r o λ 1 -1 λ 1 ± r 2 o 1 + l r o λ 1 +1 -1 λ 1 + 1        , (44)
A ∞ = 4 (q -1)    (R o ± r o ) l ± r 2 o 2   1 + l r o 2 -1      . (45)
For the case of bending loading, the bounds for the axial stress concentration factor K f ZZ are obtained numerically

K f,max ZZ = I 4R e -G 4 (q -1) + qω 1 C o , (46)
K f,min ZZ = I 4R e -G 4 (q -1) + qω 1 C ∞ (47)
where the coefficients G, C ∞ and C o are given respectively by ( 36) and (37).

In order to define the range of application of the solution, we define a parameter ξ characterizing the geometry of the tube which is equal to

ξ = R i R e × p e (48)
where p is the depth of the notch defined by ( 1) and e is the thickness of the tube

e = R e -R i . (49)
The parameter ξ is equal to zero for a notched bar (i.e. R i = 0) and tends to ∞ for thin tubes with the tickness going to zero. We can remark that this parameter ξ is not dependent on the geometry of the notch as the notch root radius ρ and the half opening angle α and therefore the following results shall only be applied to the notch geometry tested (ρ = 1mm and α = 30 o ).

Thereby to define the range of application of the closed-form solution, we have to find the value of the parameter ξ at which the maximum axial stress concentration factor K max ZZ becomes smaller than the real axial stress concentration (computed with FEM).

Figs. 10 and 11 illustrate the range of validity of the closed-form solution with respect to ξ. These results show the evolution of the axial stress concentration factor K t computed with FEM solution (marked dashed lines) and the maximum axial stress concentration factor K max ZZ (solid lines) obtained by (42) for tension loading and (46) for bending loading. These results are obtained by varying the parameter ξ for tubes presenting an internal and external notch submitted to tension and bending loading and for three different notch depths p (1mm, 2mm and 5mm). To vary the parameter ξ, all dimensions have been kept constant excepted the internal radius R i for an external notch and the external radius R e for an internal notch.

The figures display the intersections between K t and K max ZZ represented with a bigger marker, and one remark that all the intersection lie on a small interval [0.21, 0.27]. Therefore we conclude that the closed-form solution is only valid for a notched tubes with a parameter ξ smaller than a critical value ξ * . The fact that ξ higher than the critical value ξ * express the fact that the notch depth becomes large with respect to the tube thickness. This can be interpreted by saying that the solution in [START_REF] Filippi | Distributions of the elastic principal stress due to notches in finite size plates and rounded bars uniaxially loaded[END_REF] provided in equations ( 4) and [START_REF] Kujawski | Estimation of stress intensity factors for small cracks at notches[END_REF] can not be applied for this configuration, as the free internal surface of the tube is too close to the notch tip.

Moreover, as mentioned before this critical value ξ * should depend on the notch geometry and we have found that for ρ = 1mm and α = 30 o , the critical value is equal to 0.086. An alternative to the numerical procedure to determine the parameter m has been developed based on an empirically function depending of the notch root radius for notched bar geometry. Investigation has been performed to determine the influence of the notch root radius ρ, the half-opening angle α and the parameter ξ on the parameter m for notched tube geometry. The results are presented in Figs. 12 and 13 for which the computations have been done with ξ equal to 0.1. We remark in Fig. 12.a that the notch root radius ρ does not influences the parameter m both for an external and an internal notched tube, but has a greater influence on the opening angle α and the parameter ξ.

Conclusions

We proposed a semi-analytical stress solution suitable for notched axi-symmetrically tube submitted to tension and bending loading. This closed-form solution is a generalization of the stress solution [START_REF] Filippi | Distributions of the elastic principal stress due to notches in finite size plates and rounded bars uniaxially loaded[END_REF] devoted to notched axi-symmetric bar. The proposed semi-analytical solution is based on an approximate elastic stress solution where the local balance condition is always satisfied by the use of a biharmonic stress function but boundary conditions are not satisfied except at specific points.

The solution has been adapted to tubular geometries by adding a supplementary term in the expression of the radial stress components Σ RR . The free stress boundary condition at the inner or the outer radius could therefore be respected for externally or internally notched tubes.

The proposed semi-analytical solution has shown good agreements with respect to finite element solution in the case of an external notched tube. The match is less accurate with an internal notch on the radial and the circumferential stress components. This error has been reduced by changing the linear term in the expression of the radial stress component with a hyperbolic function as example.

Finally a study of the validity of the semi-analytical solution regarding to the dimension of the notched tube has been performed. This study has highlighted the fact that the solution is not capable to model too high stress concentration factors. A numerical study has shown that it is possible to define a parameter ξ which characterizes the geometry of the notched tube and the solution is applicable for only notched tube presenting a ξ parameter smaller than ξ * which can be considered as independent on the notch depth but only on the notch opening angle and the notch root radius.

Moreover we have shown that the m parameter introduced in ( 6) is highly dependent on the half opening angle α and the ξ parameter and slightly dependent on the notch root radius ρ.

The application of this solution for the design of threaded components is presented in a companion paper [START_REF] Ferjani | A computational approach for the fatigue design of threaded connections Submitted to[END_REF].

2α [ o ] q λ 1 µ 1 χ b 1 χ c 1 χ d 1 δ 1 η 1 ω 1 0 2 0.5 -0.5 1 4 0 -2 1

σ

  stress field associated to the local system of coordinates Σ stress field associated to the global system of coordinates ρ root radius of the notch α half opening angle of the notch R e , R i and R o external, internal and notch tip radii of the tube p depth of the notch l ligament length of the notch e thickness of the tube K t ZZ axial stress concentration factor for tension loading K f ZZ axial stress concentration factor for bending loading ξ dimensionless parameter characterizing the geometry of the notched tube 1 Introduction

Figure 1 .Figure 2 .Figure 3 .

 123 Figure 1. Geometry of tubular components with an internal (right) and external (left) notch and geometry of the notch (center).

Figure 4 .

 4 Figure 4. Mesh size sensitivity for an externally notched tube with 2α = 60 o and ρ = 1mm.

Figure 5 .

 5 Figure 5. Stress field distribution along the ligament of an external notch.

Figure 6 .

 6 Figure 6. Stress field distribution along the ligament of an internal notch.

Figure 7 .

 7 Figure 7. Distribution of the radial (a) and the hoop stress (b) components under tensile loading in the case of an internal notch for differents choices of h: linear see equation (14) and the new nonlinear h see equation (39).

Figure 8 .Figure 9 .Figure 10 .Figure 11 .Figure 12 .Figure 13 .

 8910111213 Figure 8. Distribution of the axial stress component for different notch root radius under tension loading and with α = 40 o for (a) an externally notched tube (b) an internally notched tube.

Figure 14 .

 14 Figure 14. Representation of the auxiliary system of coordinates {u, v}. The curve u = u o represents the geometry of the notch.

  Value parameters of the closed-form solution for various opening angle 2α.

					2
	30	1.83 0.502 -0.456 1.07 3.79 0.063 -1.82 1.04 1.88
	60	1.67 0.512 -0.406 1.31 3.28 0.096 -1.6	1.17 1.55
	90	1.5	0.544 -0.345 1.84 2.51 0.105 -1.33 1.47 1.08
	120 Table 1	1.33 0.616 -0.268 3	1.51 0.087 -1	2.25 0.57

Appendix 1

Recently, a set of closed-form equations valid for V-shaped notched axi-symmetric bar under Modes I and II loading has been proposed in [START_REF] Filippi | Distributions of the elastic principal stress due to notches in finite size plates and rounded bars uniaxially loaded[END_REF]. The stress field has been obtained by the use of the Kolosoff-Mushkelishvili's complex potential function [START_REF] Mushkelishvili | Some Basics Problems of the Mathematical Theory of Elasticity[END_REF] and the following conformal mapping introduced by Neuber [START_REF] Neuber | Theory of notch stresses: Principles for exact stress calculation[END_REF] z = x + iy = (u + iv) q = ξ q (50) where z is the complex representation of the cartesian coordinates {x, y} and {u, v} is the auxiliary system of curvilinear coordinates (see Fig. 14).

The parameter q in (50) is linked to the opening angle 2α by the following relation

The origin of the cartesian coordinate system is located at a distance r o from the tip of the notch, where r o depends on the notch root radius ρ and the opening angle 2α according to

For Mode I loading, the stress field along the notch bisector, i.e. θ = 0, is

where the parameters introduced in (53) and ( 54) are equal to

and the constants λ 1 and µ 1 are such that they satisfy the following equations sin (λ 1 qπ) + λ 1 sin (qπ) = 0, (62)

Table 1 provides useful values of the different parameters for various opening angle 2α


Iris Brémaud
email: iris_bremaud@hotmail.com

Pierre Cabrolier

Joseph Gril -Bruno

Clair -Jean Gérard -Kazuya

Minato -Bernard Thibaut

Kazuya Minato

Joseph Gril

Bruno Clair

Jean Gérard

Bernard Thibaut

Identification of anisotropic vibrational properties of Padauk wood with interlocked grain

Keywords: Anisotropy, Damping coefficient, Extractives, Grain angle, Interlocked grain, Microfibril angle, Vibrational properties, Young's modulus

Grain deviations and high extractives content are common features of many tropical woods. We aimed at clarifying their respective impact on vibrational properties, referring to African Padauk (Pterocarpus soyauxii Taub.), a species selected for its interlocked grain, high extractives content, and uses in xylophones. Specimens were cut parallel to the trunk axis (L), and local variations in grain angle (GA), microfibril angle (MFA), specific Young's modulus (E' L /ρ, where ρ stands for specific gravity) and damping coefficient (tanδ L ), were measured. GA dependence was analyzed by a mechanical model which allowed to identify the specific Young's modulus (E' 3 /ρ) and shear modulus (G'/ρ) along the grain (3), as well as their corresponding damping coefficients (tanδ 3 , tanδ G ). This analysis was done for native and then for extracted wood. Interlocked grain resulted in 0-25° GA and in variations of a factor 2 in E' L /ρ and tanδ L . Along the grain, Padauk wood was characterized, as compared to typical hardwoods, by a somewhat lower E' 3 /ρ and elastic anisotropy (E'/G'), due to a wide microfibril angle plus a small weight effect of extracts; and a very low tanδ 3 and moderate damping anisotropy (tanδ G /tanδ 3 ). Extraction affected mechanical parameters in the order: tanδ 3 ≈ tanδ G > G'/ρ >> E' 3 /ρ. That is, extractives' effects were nearly isotropic on damping but clearly anisotropic on storage moduli.

Introduction

Viscoelastic vibrational properties of wood are mainly affected by the orientation of wood elements (grain angle GA and microfibril angle MFA) and by chemical composition. Due to the highly anisotropic nature of wood, the actual orientation of grain inside a wood piece will strongly affect its apparent mechanical properties (e.g. [START_REF] Bodig | Effect of extractives on vibrational properties of African Padauk (Pterocarpus soyauxii Taub[END_REF]. Wood grain inside a trunk can be straight, but also spiralled, interlocked or wavy. At least one kind of deviation may be found in almost any tree [START_REF] Harris | Spiral grain and wave phenomena in wood formation[END_REF]. "Grain" represents the orientation of all axial elements [START_REF] Ogata | Macroscopic and anatomical investigation of interlocked grain in Acacia mangium[END_REF] and "Grain angle" usually refers to the tangentiallongitudinal plane. Interlocked grain oscillates alternatively to the left-or right-hand along the radius. Wood pieces cut along the main axis of a trunk will thus systematically include GA, either cross-grained or not.

In straight-grained woods, microfibril angle is recognized as the primary factor affecting axial vibrational properties (specific modulus of elasticity E'/ρ and damping coefficient tanδ), and their axial-to-shear anisotropy [START_REF] Obataya | Vibrational properties of wood along the grain[END_REF][START_REF] Ono | Study on Young's modulus and internal friction of wood in relation to the evaluation of wood for musical instruments[END_REF][START_REF] Norimoto | Specific Dynamic Young's Modulus and Internal Friction of Wood in the Longitudinal direction (in Japanese)[END_REF]. Cellulose microfibrils govern elastic response, while viscoelastic damping depends on the "matrix" of lignins and hemicelluloses. Wood, however, is not only composed of polymers, but also contains secondary metabolites, variable in their amounts as in their nature, known as extractives. In some species these are found to modify damping coefficient by as much as a 2-fold order, either decreasing it [START_REF] Matsunaga | Vibrational property changes of spruce wood by impregnation with water-soluble extractives of pernambuco (Guilandina echinata Spreng.)[END_REF][START_REF] Minato | The vibrational properties of wood impregnated with extractives of some species of Leguminosae[END_REF][START_REF] Minato | Extractives of muirapiranga (Brosimun sp.) and its effects on the vibrational properties of wood[END_REF][START_REF] Yano | The changes in the acoustic properties of Western Red Cedar due to methanol extraction[END_REF][START_REF] Yano | Acoustic properties of Brazilian rosewood used for guitar back plate[END_REF] or increasing it [START_REF] Obataya | The effects of water-soluble extractives on the acoustic properties of reed (Arundo donax L.)[END_REF][START_REF] Sakai | Effects of impregnation of simple phenolics and natural polycyclic compounds on physical properties of wood[END_REF].

Resulting variations in vibrational properties are of practical importance, notably in musical instruments, where different functions are associated to preferred ranges of properties. For xylophones, where the primary vibrating body is the wood piece itself, a very low tanδ is the most important factor in the judgement of woods [START_REF] Aramaki | Sound quality assessment of wood for xylophone bars[END_REF][START_REF] Hase | A comparison between acoustic physical factors of Honduras rosewood for marimbas and xylophones and a sensory evaluation of these instruments[END_REF]. While for top plates of string instruments, a very high specific modulus E'/ρ predominates (associated to moderate values of tanδ). Axial to shear anisotropy of modules and of damping is also very important for the frequency response in flexural vibration [START_REF] Obataya | Vibrational properties of wood along the grain[END_REF][START_REF] Ono | The frequency dependance of the dynamic Young's modulus and internal friction of wood used for the soundboard of musical Instruments II. The dependance of the Young's modulus and internal friction on frequency, and the mechanical frequency dispersion (in Japanese)[END_REF].

Our work aimed at clarifying how these vibrational properties are influenced by, respectively, grain angle deviations and extractives, which are notable features of most tropical woods. It refers to the species Pterocarpus soyauxii Taub. (African Padauk) which was selected on the basis of its interlocked grain and predominant uses in xylophone-like instruments. We previously found that its extractives strongly affect damping (Brémaud et al. 2010a). In the present paper, after assessing local variations in grain and microfibril angles, the grain angle dependence of vibrational properties was analysed through a mechanical model. This served to identify mechanical parameters along the fibres and in shear for Padauk wood, and to compare their values with hardwoods average and with the same wood after extractives removal.

Material and methods

Material and preparation of specimens

Wood material of African Padauk (Pterocarpus soyauxii Taub.) from one tree (provenance Cameroon) was obtained from CIRAD sawmill. Sampling is schematized on Fig. 1. Two adjacent flat-sawn planks (12×80×800 mm, R×T×L) with contrasted orientations were selected in medium heartwood. They were then cut into series of specimens for vibrational tests (12×2×150 mm, R×T×L), all parallel to the main axis of the trunk. Total number of specimens was 158. Then, 6 representative specimens were cut, each one into 6 smaller specimens (1×2×20 mm ; R×T×L) for X-Ray measurements, successively in radial direction.

Measurement of surface grain angle (GA)

Grain angle has been recorded on each tangential plane of planks before specimen cutting, basically following vessels. Records were always viewed as "from outside the trunk". A grid was affixed to pictures of LT planes of planks, so as to materialise the location of specimens prepared from them. For each box of the grid, 12 measurements (accuracy ± 0.1°) of local angles were made using the image analysis software ImageJ (Rasband 1997(Rasband -2008)). Global pathways of vessels were also traced all along the LT planes, and their intersecting angle with the centre of the boxes/specimens measured. Both gave very similar results. Angles recorded on the LT planes were called α, and the mean of absolute values of angles for a given vibrational specimen was called θ.

Microfibril angle (MFA) and GA measured by X-Ray diffraction

Experiments were performed on a 4-circle diffractometer (Oxford Diffraction Gemini S) equipped with a 1024x1024 CCD camera. CuK_α radiation was generated by an X-ray generator operating at 50 kV, 25 mA. Specimens (R×T×L = 1×2×20 mm 3 ) were mounted perpendicular to the beam and with ≤ 0.6° positioning error for verticality. Images were recorded during 100 seconds and integrated along the whole 360° azimutal interval to plot the intensity diagram of the (200) plane. An automatic procedure detected (200) peaks and their inflexion points. The T parameter as defined by [START_REF] Cave | Theory of X-ray measurement of microfibril angle in wood[END_REF], is measured as the half distance between intersections of tangents at inflexion points with the baseline. The average MFA of each sample was estimated as MFA = 0.6×T [START_REF] Cave | Theory of X-ray measurement of microfibril angle in wood[END_REF]. As MFA were moderate, (200) peaks were monomodal. Thus the tilting angle of the fibre axis to the sample axis, i.e. the grain angle, was measured as the deviation angle of the maximum intensity peaks to the horizontal.

Vibrational properties

Samples were kept for at least 3 weeks in controlled conditions of 20±1°C and 65±2%RH, before being tested by non-contact forced vibrations of free-free bars (e.g. [START_REF] Obataya | Vibrational properties of wood along the grain[END_REF]. Specimens were made to vibrate through a tiny iron piece glued on one end, facing an electric magnet. Their displacement was measured using a laser sensor. Vibration emission and detection were computer-driven through a semi-automated interface that we developed using Labview® software [START_REF] Brémaud | Diversity of woods used or usable in musical instruments making. -Experimental study of vibrational properties in axial direction of contrasted wood types mainly tropical. -Relationships to features of microstructure and secondary chemical composition[END_REF]). E'/ρ was deduced from the first resonant frequency according to the Euler-Bernouilli equation. Damping coefficient tanδ was measured both through the quality factor and through logarithmic decrement. Measurement frequencies were in the range of 200-400 Hz. 3 repetitions were made for each specimen and mean error was ≤5%.

Removal of extractives

After first vibrational tests, 5 groups (26 specimens each) with not significantly differing ranges of grain angle and properties were defined. Each group has been exhaustively extracted in a given solvent and its properties after extraction measured again (see Brémaud et al. 2010a). Solvents were: diethyl-ether (ET), methylene dichloride (MD), acetone (AC), methanol (ME) and hot water (HW, c. 90°C).

Identification of anisotropic parameters

The grain angle dependence of vibrational properties was analyzed by a mechanical model based on transformation formula for elastic solids (i.e. Bodig and Jayne 1982) applied to complex compliances (Brémaud et al. 2010b). This allows expressing the storage (E') and loss (E'') Young's moduli along the specimen's axis (L) at a given grain angle θ, as a function of anisotropic properties in the fibre-related system of axis (3, 2, see Fig. 1):

( )

2 ' ' 3 2 1 ( ) 1 ' ' L u E E a u b u θ + ≈ + + and ( ) ( ) ( ) 2 2 " ' 3 33 2 2 1 1 " " ( ) tan 1 ' ' L u a u b u E E a u b u θ δ + + + ≈ + + (1)
With the parameters:

( )

tan ² u θ = and 3 32 32 3 2 ' ' / ' 2 ' ' ' / ' a E G b E E υ = - = and 32 3 2 3 tan " ' (1 ) tan tan " ' tan G a a q b b δ δ δ δ = + = (2)
Where E' 3 and E' 2 , and tanδ 3 and tanδ 2 , are respectively the storage modulus, and the damping coefficients, along, and tangential to, the fibre direction; G' 32 and tanδ G32 are storage modulus and damping coefficient in shear (longitudinal-tangential to the fibres); υ' 32 is Poisson's ratio (about 0.46), and the term q amounts to 0 to 8% depending on Poisson's loss factor (for which no data are available). Grain angle dependence of damping coefficients was given by: " ' tan ( ) ( ) / ( )

L L L E E δ θ θ θ = (3)
The radial gradient of GA inside specimens (see Fig. 1) was approached by linear extrapolation between the angles α on both tangential faces and 12 discrete, absolute values of angle were defined (i.e. every mm across the width). Apparent moduli E' L (θ) and E" L (θ) and damping coefficient tanδ L (θ) were calculated for each of these points using eq.1 and 3, and the properties along the specimen axis were obtained by integration over these 12 values.

The root of the mean square error (rmse) between calculation and experimental measurements of both E' L /ρ (expressed in GPa) and of tanδ L (in ‰) were calculated. Then the model parameters (eq.2) were fitted to minimize the sum of both rmse, using the solver of Microsoft Excel.

As, in the range of GA under study, parameters b' and b'' have secondary influence, they were assigned plausible values (11.5 for b' and 40 for b'', based on data on another species of the same genus (Caldersmith and Freeman 1990)) for the identification of other parameters. For analysis on extracted material, they were slightly adjusted by admitting an analogy with the results of [START_REF] Yano | Acoustic properties of Brazilian rosewood used for guitar back plate[END_REF].

Results and discussion

Grain angle measurements

Surface angles α ranged from -26° to +33° (mean +9°), with different profiles: about one half of specimens had only positive angles up to +33° and the other was cross-grained with a distribution from -20° to +20°. At a given longitudinal position, GA differences along 12 mm radial distance ranged from 1.5° to 30° (mean 15°). GA varied also axially: variations along 150mm ranged from 0.5° to 25° (median 5.5°). In tangential direction GA varied from 2° to 23° (median 8.5°). When comparing (Fig. 2) surface records (based on vessel alignments) and local XRD measurements (which should represent mainly fibres orientation) of GA inside vibrational specimens, overall trends were very similar. The relation between the two types of measurements was close to unity and had a high (87%) coefficient of determination, which tends to confirm the validity of XRD for GA assessment. However, where discrepancies occurred, surface GA showed bigger absolute values. This might be related to observations by [START_REF] Ogata | Macroscopic and anatomical investigation of interlocked grain in Acacia mangium[END_REF] on interlocked grain of Acacia mangium, where the orientation of vessels tends to show higher peak values than fibres.

Local variations of microfibril angle

The range in microfibril angle (MFA) was from 13° to 21° (mean 17.2°) according to [START_REF] Cave | Theory of X-ray measurement of microfibril angle in wood[END_REF] relation. On the whole set of samples, there was no apparent relationship between local values of MFA and of GA (both determined by XRD on the same small specimens) (Fig. 3). But within each group (coming from different locations), there was a tendency of MFA to decrease with increasing GA, which is significant for 4 of the 6 groups. In addition, according to Sarén and Serimaa (2006), in the XRD conditions that we used, actual MFA values for specimens with wide GA shall be lower than the apparent ones (shown in Fig. 3). This would then reinforce a tendency of MFA decreasing with increasing GA.

This might be related to a theory [START_REF] Schulgasser | The mechanism of spiral grain formation in trees[END_REF] about the mechanism of spiral grain formation. A model developed by these authors predicts leftward GA for very high MFA, progressing towards right-hand angles when MFA diminishes (to about 19°, which seems consistent with our results). However, this hypothesis is designed for the case of spiral grain and is difficult to extrapolate to more complex phenomena such as interlocked grain. This issue seems to remain open. Co-occurrences of oscillations in MFA and in GA are also observed in Spruce (Sarén et al. 2006), and in tension wood of an Acacia sp. [START_REF] Hillis | An unusual formation of tension wood in a natural forest Acacia sp[END_REF]), but variations in MFA and in GA are not clearly synchronous. Be it as it may, the range in MFA strongly overlapped between groups, and their average values were not statistically different. Also, the amplitude of variations in MFA (8°) was 5 times smaller than that for GA variations (40°). Thus, the main affecting factor for mechanical properties will be grain angle, in addition to which MFA variations would bring modulations of secondary importance.

Vibrational properties and grain angle dependence

The samples had homogeneous specific gravity (0.76 ± 0.02). The mean grain angle θ inside vibrational specimens ranged from 0 to about 25°. Resulting variations in specific modulus of elasticity E' L /ρ and in damping coefficient tanδ L were of a more than 2 fold order (8 to 18 GPa and 4.4 to 9.9 ‰ respectively). In contrast, the specific loss modulus E'' L /ρ varied only of about 25% (72 to 89 MPa). Calculations based on the fibre-scale related properties identified for Padauk (see table 1) described, respectively, 82% and 77% of the experimental variations in E' L /ρ and of tanδ L (along the trunk and samples axis) as a function of mean grain angle θ (Fig. 4). Residual dispersion was mainly linked to different initial locations, i.e. local variations in GA and MFA profiles should contribute. When comparing (Fig. 4), the grain angle dependence for Padauk wood (P) and for properties corresponding to a "mean hardwood" MH (Brémaud et al. 2010b), Padauk had a lower than average apparent specific storage modulus at small grain angles, yet correction (P corr ) for the contribution of extractives to specific gravity reduced the difference. Padauk E' L /ρ decreased slower than for MH over angles up to 15° but for bigger GA trends were similar. Loss properties of Padauk were more atypical. Specific loss modulus E" L /ρ of Padauk was nearly 2 times smaller over small angles, and it decreased very slightly over GA up to 25°, while for a mean hardwood a pronounced decrease starts from about 10° GA. Damping coefficient tanδ L of Padauk increased less steeply as a function of GA than for MH and trends evolved in parallel for both types of materials, Padauk values being much lower (about one third). When plotting (Fig. 4d) the evolutions in tanδ L against those in E' L /ρ, dispersion was very much reduced: uncertainties on GA profiles inside specimens are here sidestepped. For Padauk heartwood, tanδ along the grain was nearly half that of a mean hardwood of similar E'/ρ, and remained systematically lower with increasing GA, curves for both type of material being parallel. Identification of properties in the fibre-related system of axis (table 1), obtained by optimization of the model to fit the data, showed that Padauk had quite lower E' 3 /ρ (along the fibres) and elastic anisotropy than a "standard hardwood" of equivalent density, for which E' 3 /ρ would be about 21-22GPa, and a' about 13 (Guitard and ElAmri 1987). The relatively low value of E' 3 /ρ is partly linked to the contribution of extractives to specific gravity: it increased of 1.6 GPa after correction for this factor. Yet these lower values of E'/ρ along the fibre, and of anisotropy ratios, are also linked to the quite high microfibril angles. In this range of MFA, microstructural models predict a comparable decrease in E 3 /ρ and in 3-2 anisotropy [START_REF] Gachet | Influence relative de la morphologie cellulaire et de l'angle des microfibrilles sur l'anisotropie élastique tissulaire Longitudinale/Tangentielle du bois sans défaut des résineux[END_REF][START_REF] Obataya | Vibrational properties of wood along the grain[END_REF]. On the other hand, damping coefficient tanδ 3 for Padauk (<5‰) is much lower than those indicated in the literature (about 8‰) for similar MFA [START_REF] Norimoto | Specific Dynamic Young's Modulus and Internal Friction of Wood in the Longitudinal direction (in Japanese)[END_REF].

Table 1: Properties in the fibre-related system of axis obtained by optimization of model parameters to fit experimental data. rmse: root mean square error.

E' 3 /ρ E'' 3 /ρ tanδ 3 a' a" tandδ G32 32 3 tan tan G δ δ b' b" rmse E' L /ρ rmse tanδ L Padauk heartwood (Untreated) (GPa) (MPa) (‰) (see eq. 2) (‰) (see eq.2) (GPa) (‰)

Anisotropic vibrational properties as affected by extractives

Extractive removal (2.3, 3.1, 8.8, 13.4 and 0.6% respectively for solvents ET, MD, AC, ME and HW) affected global-scale storage modulus only marginally as compared to grain angle (Fig. 5a). On the contrary, for specific loss modulus E" L /ρ, methanol (ME) extraction caused variations as important -but opposite-as that for a rotation of 45°. Damping coefficient was strongly dependent both on orientation and on extractives. The variation after ME extraction for damping along the fibres was close in amplitude to that resulting from a rotation of 20°. Curves between calculated tanδ and E'/ρ (Fig. 5d) all had very similar shape, while differences in extractives content shifted their tanδ values.

The amplitudes of modifications (Fig. 6) of axial and shear properties (in the fibre-related system of axis) caused by extractive removal ranked as tanδ 3 ≈ tanδ G32 ≈ E" 3 /ρ ≥ G' 32 /ρ ≥ G'' 23 /ρ >> E' 3 /ρ. Anisotropic effects of Padauk extractives can be summarized as follows: in the direction of fibres, the presence of extractives in the cell walls of native wood decreased strongly the loss properties E" 3 /ρ and tanδ 3 , while they little affected -positively-the specific storage modulus. On the contrary, concerning shear properties, the storage modulus G' 32 /ρ was clearly higher (> 20 %) before extraction, indicating a role of extracts in reinforcing the matrix substances in native wood. The effect of extractives on damping coefficient was similar both along the fibres and in shear. This sounds consistent with the analyses of [START_REF] Obataya | Vibrational properties of wood along the grain[END_REF][START_REF] Norimoto | Specific Dynamic Young's Modulus and Internal Friction of Wood in the Longitudinal direction (in Japanese)[END_REF]) stating that the elasticity of matrix substances would be of little significance when the viscous matrix and the elastic reinforcement of the microfibrils are in parallel, while the influence of matrix properties would increase as the reinforcement makes a bigger angle. Perhaps, the slight effect of extractives observed here on storage modulus along the fibres would be barely noticeable in a wood with smaller MFA. On the other hand, loss coefficient is governed by matrix viscosity and as such can be affected by changes in extractives content of the cell-wall, whatever the orientation. Extractives soluble in ET (diethyl-ether) were mainly lumen located (Brémaud et al. 2010a) and had different effects: no change in tanδ 3 (along the fibre) and their presence slightly increased shear damping. This may by interpreted as: (i) lumen-located compounds could only increase the loss coefficient of wood, as was proposed by [START_REF] Akitsu | Uniaxial modelling of vibrational properties of chemically modified wood[END_REF], this being modulated by their characteristic properties. (ii) For rather small amounts of lumen filling or lining, such an effect would be barely noticeable in the fibre direction, as compared to the cell-wall properties, but may become more significant in shear or transverse directions.

From a point of view of application, as compared to the same wood without extractives, native Padauk heartwood could be considered as a "natural chemical modification", and had: a E' 3 /ρ slightly higher (+7.5%), a tanδ 3 much lower (-36%), a lower ratio E' 3 /G' 32 (-13%) and a slightly higher ratio tanδ G32 /tanδ 3 (+5%). The combination of these latter two ratios was proposed by [START_REF] Obataya | Vibrational properties of wood along the grain[END_REF] as a descriptor of frequency response and "tone quality" of vibrating wooden pieces: higher E/G anisotropy results in higher loss and weaker radiation at high frequencies [START_REF] Ono | The frequency dependance of the dynamic Young's modulus and internal friction of wood used for the soundboard of musical Instruments II. The dependance of the Young's modulus and internal friction on frequency, and the mechanical frequency dispersion (in Japanese)[END_REF]. In the case of xylophone bars, for which Padauk is a favourite material, the best descriptor of sound qualification is reported to be a low damping of the first mode, associated to few spectral components [START_REF] Aramaki | Sound quality assessment of wood for xylophone bars[END_REF][START_REF] Hase | A comparison between acoustic physical factors of Honduras rosewood for marimbas and xylophones and a sensory evaluation of these instruments[END_REF]. Thus, Padauk extracts, as they decrease its damping along the fibres about 4 times more than they decrease the axial to shear viscoelastic anisotropy, must participate in making this wood suitable for xylophone-like instruments. Of course, it shall also be reminded that special care should be taken concerning local grain orientation when preparing pieces of wood from this species and others for which interlocked grain is a frequent feature.

Conclusion

This study aimed at clarifying the respective influence of the orientation of wood elements, and of extractives, on vibrational properties, referring to a selected tropical hardwood with interlocked grain. After assessing local variations in grain and microfibril angles, properties were analysed by a mechanical model. Mechanical parameters in the fibre-related system of axis [1,2,3] were identified for untreated and extracted Padauk wood. Results showed that: -Mean grain angles up to 25° naturally occur. Resulting variations in E'/ρ and in tanδ are well described by the model, yet with residual dispersion due to local profiles in grain and microfibril angle. -For native Padauk E' 3 /ρ and elastic anisotropy are quite low, which is partly due to the mass contribution of extracts, and mostly to rather high microfibril angle (13-21°). -Loss modulus E" 3 /ρ is much lower than average and its anisotropy very reduced; tanδ 3 is also very low, while its GA dependence is parallel to the general case. -Extractives affect mechanical parameters in the rank: tanδ 3 ≈ tanδ G32 ≈ E" 3 /ρ > G' 32 /ρ ≥ G'' 23 /ρ >> E' 3 /ρ. Extractives' effects are nearly isotropic on tanδ but clearly anisotropic on storage moduli: they significantly increase shear modulus, but only slightly along the grain. Damping along the grain is only affected by cell-wall located extracts, while lumen ones have a significant and opposite effect on shear damping. From a practical point of view, results also confirm that special caution should be taken when preparing specimens from the many woods with grain deviations such as interlocked grain.

Fig. 1

 1 Fig.1 Schematic view of the sampling procedure and of the systems of axis related to the trunk and specimens [R,T,L] and to the grain direction [1,2,3]

Fig. 2

 2 Fig.2 Trends in grain angle across the width (LR plane, R direction) of 6 vibrational specimens

Fig. 3

 3 Fig.3 Local values of microfibril angle plotted against local grain angles, for 6 groups = initial location. Trend lines are shown for groups where the relation is significant

Fig. 4

 4 Fig.4 Experimental (circles) and calculated properties for untreated Padauk heartwood (bold line: apparent values P; thin dashed line: values corrected for weight of extracts P corr ) and for a "mean hardwood" (gray curve, MH). Trends in specific storage modulus (a), specific loss modulus (b) and damping coefficient (c) at the global scale, as a function of grain angle. (d): evolution of tanδ plotted against that of E'/ρ

Fig. 5

 5 Fig.5Trends in mechanical properties as a function of grain angle, calculated with parameters in the fibre-related system of axis identified on Padauk heartwood before and after extractions (properties are corrected for the contribution of extractives to specific gravity)

Brémaud, I., P. Cabrolier, J. Gril, B. Clair, J. Gérard, K. Minato & B. Thibaut. Identification of anisotropic vibrational properties of Padauk wood with interlocked grain. Publié dans: Wood Science and Technology 44(3) :355-367 (2010)

Publié dans: Wood Science and Technology 44(3) :355-367 (2010)

Acknowledgements:

We are grateful to Yves ElKaïm, in LMGC, for setting up the interface for vibrational tests, and to Arie Van Der Lee (IEM Montpellier) for his help in XRD measurements. This work has been supported by a Fellowship from Japanese Society for the Promotion of Science.


Myriam Paire
email: paire@chimie-paristech.fr

Laurent Lombez
email: lombez@chimie-paristech.fr

Jean-François Guillemoles

Daniel Lincot

Measuring sheet resistance of CIGS solar cell's window layer by spatially resolved electroluminescence imaging

. The EL experiment consists in using solar cell as a light emitting device : a voltage is applied to the cell and its luminescence is detected. We develop an analytical and quantitative model to simulate the behavior of CIGS solar cells based on the spread sheet resistance effect in the window layer. We determine the repartition of the electric potential on the ZnO, for given cell's characteristics such as sheet resistance and contact geometries. Knowing the repartition of the potential, the EL intensity is measured and fitted against the model.

The procedure allows the determination of the window layer sheet resistance.

Introduction :

The EL spatially resolved imaging is a convenient characterization tool to analyse solar cells or modules. Camera-based EL experiments are widely used as online quality tests in the industry, but are also used in laboratories to gain more insight in loss mechanisms [1-3, 5, 6]. As it was previously suggested [START_REF] Helbig | 24th European Photovoltaic Solar Energy Conference[END_REF][START_REF] Helbig | [END_REF], it is indeed possible to extract the value of the window layer sheet resistance from an electroluminescence image.

The present paper focuses on the determination of the window layer sheet resistance in CIGS solar cells, but our approach could be transferred to other photovoltaic technologies, especially to thin films where the determination of sheet resistance is difficult on the completed device. An appropriate model of the sheet resistance effect, in terms of voltage drop on the cell's surface, is developed. From an EL spatially resolved image, the decay of the EL intensity with respect to the distance from the electrode is determined and fitted using our model. A value of the sheet resistance on the completed cell is therefore accessible.

Experiment

We conduct an EL experiment on a CIGS solar cell. The structure of the cell is Soda-lime glass/Mo/CIGS/CdS/i:ZnO/ZnO:Al, where the ZnO layers are deposited by sputtering. The EL experiment consists in using the cell as a light emitting device. We 3/12 use a two points probe configuration and a Keithley 2635A source meter. A tungsten probe is placed on the Mo back contact, the other one is centred on the cell. Under dark conditions, we apply a voltage on the cell. Once the applied voltage is sufficient, we detect a luminescence signal. The intensity of the luminescence is recorded by a CCD camera, which can provide a 2D spatially resolved image of the luminescence of the cell. We integrate the signal on an angular sector in order to account for material inhomogeneities and get an average value of the luminescence at a certain distance of the probe. We remark that the luminescence is decreasing with respect to the distance from the front contact probe. We propose to evaluate quantitatively the sheet resistance of the window layer from the variation of the luminescence signal.

The model

The luminescence intensity Φ is related to the voltage V that is applied between the back and front contact of a cell:

) / exp( 0 kT qV φ φ = (1.)
where kT/q is the thermal voltage and Φ 0 a calibration factor that depends on, among others factors, the camera calibration settings, the device optics, the diffusion lengths and surface recombination velocities. Therefore when the front contact sheet resistance cannot be neglected, the voltage on the cell surface is a function of the distance from the contact V=V(r) and, as a consequence, so is the luminescence Φ =Φ(r). In order to quantify this sheet resistance effect from the electroluminescence experiment, we develop an analytical model.

4/12

The detailed description of the model is given elsewhere [7]. We limit our study at the window layer of the cell, which is supposed to be resistive with a certain sheet resistance R . We suppose that the cell characteristics are well described by a one-diode model. Therefore the current density coming from the p-n junction is: Jz(z=0) =-J0(exp(q ψ(z=0)/nkT) -1) -ψ(z=0)/Rsh (2.) where ψ is the electric potential in the window layer and z the altitude, which is set to 0 at the bottom of the window layer and equals t at the surface, J 0 , n, kT/q, R sh are respectively the diode saturation current density, the diode ideality factor, the thermal voltage, and the shunt resistance. The series resistance of the junction itself is neglected as its effect is smaller than that of the sheet resistance. In order to get the values of those parameters for the cell studied, we apply a simple one-diode fit on a dark currentvoltage measurements that is made in-situ before the beginning of the electroluminescence experiment.

Our goal is to get the repartition of the potential on the cell surface with respect to the distance from the electrical contacts. For the sake of simplicity, we solve the problem in a cylindrical symmetric situation, where the probe contact is a tip, placed at the center of a circular cell, but the results can easily be extended to other geometries such as parallel grid lines. We show that the potential ψ on the surface of the cell is the solution of the following one-dimensional equation [7]:

0 / )) 1 ) / (exp( ( / / 1 / 0 [] 2 2 = - + - ∂ ∂ × + ∂ ∂ sh R nkT q J R r r r ψ ψ ψ ψ (3.)
where r is the radial distance from the probe tip, R = ρ/t is the sheet resistance (Ohm) and ρ the resistivity of the window layer. This differential equation can easily be solved by a standard solver software. The boundary conditions that we used to solve equation 5/12 (3.) were the following. As the probe tip is an equipotential at the applied voltage V, ψ(b) = V where b is the probe tip radius. In the absence of recombination current at the cell perimeter, we have the Neumann condition ∂ψ/∂r(a) = 0, where a is the cell's radius.

For a given sheet resistance R of the window layer, the potential repartition on the cell front surface is determined by solving equation (3.), and the corresponding electroluminescence intensity is deduced from equation (1.). Note that our method differs from the one proposed in [START_REF] Helbig | [END_REF], where the potential is deduced from the electroluminescence signal.

In order to compare the luminescence signal from our simulated data with the experimental ones, we normalize both luminescence signals by their maximum value.

We fit the experimental data with our model by adjusting the value of R , which is the single fit parameter. Therefore we extract from the luminescence decay the value of the sheet resistance of the window layer of the cell under study.

.

Results

We study CIGS solar cells with the classical structure Soda-lime glass/Mo/CIGS/CdS/i:ZnO/ZnO:Al, where the ZnO:Al layer is 400 nm thick. We analyze square solar cells which were mechanically scribed from the same larger substrate of two different areas : 0,1 and 0,3 cm². We simulate these cells by circular 6/12 cells of the same area. We record the EL image for different applied voltages. We fit the EL intensity signal by our model and give the results below.

In order to fit the EL signal we need to know the effective applied voltage at the ZnO surface. Therefore we need a tool to determine the contact resistance between the probe tip and the ZnO surface, as this resistance was found very important in our experimental setup. We record several EL images at different reference applied voltages. Then we fit these signals in order to extract two parameters, the effective applied voltage and the sheet resistance. We validate our model by verifying that the difference between the reference and effective applied voltages is due to a contact resistance that we evaluate. One can see that the difference between the reference voltage and the effective voltage is due to a contact resistance effect. The contact resistance, in our contacting geometry with a tungsten probe is approximately 79 Ohm. Therefore it is an important effect that has to be taken into account.

8/12

The fit of our EL images gives us access to the effective voltage applied on the ZnO surface but more importantly to the sheet resistance of the window layer of the cell. We studied cells of various sizes to test the validity of our model. We give the results found for two different cells (0.1cm² and 0.3cm²). On figure 1 and2, one can notice that for distances from the probe tip below 50 µm, the electroluminescence signal is dimmed. This is due to the fact that the probe creates a shadow which attenuates the signal. The data used to simulate the EL signal are extracted from a current-voltage measurement done prior to the EL experiment.

From a simple one-diode fit, we found for example for the 0.1 cm² cell I0 = 2.974 10 -9 9/12 A, n = 1,815. As the shunt resistance is 2 10 5 Ohm, we decided to neglect its influence and we consider an infinite shunt resistance. The best fit is obtained for a sheet resistance of 30 Ohm/square +/-1 Ohm/square for the different voltages (0.71V, 0.73V, 0.76V) that we applied on this 0.1 cm² solar cell. Therefore our model is able to take into account variations in the EL intensity due to different applied voltages. We proceed in the same manner for the 0.3 cm² solar cell. The data extracted from the currentvoltage measurement are I0 = 5.137 10 -10 A, n = 1,718. We fit the EL luminescence signal at three applied voltage (0.72V, 0.74V, 0.75V) and found that the sheet resistance is again 30 Ohm/square +/-1 Ohm/square. This is coherent with the results of the 0.1 cm² cell as the two cells were mechanically scribed on the same substrate, and therefore have the same window layer.

This sheet resistance value of 30 Ohm/square +/-1 Ohm/square is also coherent with the one we found with a classic current-voltage experiment under AM1.5

illumination. We use a solar simulator to illuminate the cell, and we measure a currentvoltage curve. We adapt our model for the interpretation of experiment under illumination by introducing a photocurrent density term. The detailed equations of our model for the illuminated case are given elsewhere [7]. We are able to extract from the current-voltage curve the value of the sheet resistance, which is 34 Ohm/square +/-2 Ohm/square (see figure 3) for the 0,1 cm² cell. The coherence between the EL experiment and the illuminated current-voltage measurement confirms that the value of the sheet resistance extracted from the EL is valid and accounts for the sheet resistance effect visible on well-known current-voltage curve. The difference between the two 10/12 experiments can originate from the two different experimental setups, as well as the photocurrent not being strictly constant with the voltage, as supposed in our model. We then compare those values of window layer sheet resistance with the one available from a ZnO:Al layer deposited on glass under the same conditions. With a four point probe experiment on a ZnO:Al layer of 420nm thickness on glass we found that the sheet resistance is 14,7 Ohm/square +/-0,3 Ohm/square. The cells we study in the EL experiment have a ZnO:Al layer that is deposited in the same conditions, in terms of time and sputtering characteristics. Therefore it appears that the data of our fit gives a value of the sheet resistance that is higher than the one measured from ZnO:Al on glass. This can stem from different considerations. First, the ZnO :Al layer that is grown on the cell differs from the one grown on glass in terms on material properties, as the nucleation on ZnO:i and on glass is different resulting in different material properties, such as crystallites sizes and grain boundary properties for example.

11/12

Therefore the lateral conductivity of the ZnO:Al should differ in the two layers. Second, the cell is a rougher substrate than the glass. If the surface is not flat the average distance to the electrode is increased by the surface roughness. This can lead to an apparent sheet resistance that is more important than the one of a flat cell of the same area. Therefore the apparent sheet resistance of the window layer on the cell is higher than the one of the same layer on a flat glass surface. Third, in our model we neglect the internal series resistance of the solar cell. If this resistance is not negligible compared to the sheet resistance, it can artificially increase our estimated sheet resistance. Therefore more experiments, with window layers of different sheet resistance, will be conducted in order to determine the reasons for the difference between the sheet resistance of a window layer on the cell and on glass.

Conclusion

We developed a model that is able to estimate the voltage at the cell surface once given the sheet resistance of the window layer, the cell's diode characteristics and the applied voltage. Therefore we are able to interpret the decay of EL signal with the distance from the electrode as a sheet resistance effect, and we can give a quantitative estimation of the sheet resistance value. If this work has been done considering circular cells for the sake of simplicity, a development towards other geometries such as parallel grid lines can easily be made. Further studies will also concentrate on new contacting methods where the contact resistance will be accurately known and on the study of cells covered with window layers of different properties in order to gain more insight in the 12/12 link between the properties of the window layer deposited on glass and on the cell. In addition to the estimation of series and shunt, it is thought that EL imaging can be developed to estimate the sheet resistance for completed cells or modules.

Figure 1 :Figure 1

 11 Figure 1 : EL data (dots) and fit (solid line) at three different reference voltages

Figure 2 :

 2 Figure 2 : a-EL spatially resolved image of the 0.1cm² cell under 0.76V. b-

Figure 3 :

 3 Figure 3 : Current-voltage under AM1.5 for the 0.1 cm² cell. Dots are the

Table 1 : Reference voltage, effective applied voltage, intensity at the reference voltage and the calculated contact resistance from the three EL images on the 0.3 cm² solar cell.

 1

	1.1 +/-0.1%	0.72 +/-5%	4,81 +/-0.1 %	79 +/-5%
	1.2 +/-0.1%	0.74 +/-5%	5,89 +/-0.1%	78+/-5%
	1.3 +/-0.1%	0.75 +/-5%	6,94 +/-0.1%	79 +/-5%


Estelle Delpech

Lingua Et Machina

Béatrice Daille

Dealing with lexicon acquired from comparable corpora : validation and exchange

Keywords: terminology management, terminology extraction, comparable corpora, term-oriented terminology, TBX

Introduction

Recent years have seen a surge in research in multilingual terminology extraction. Secondgeneration tools have emerged, trying to tackle the drawbacks of term alignment from parallel corpora by using comparable corpora. There are two reasons for using comparable corpora instead of parallel corpora : (i) parallel corpora are scarce whereas comparable corpora are easily available, (ii) comparable corpora provide a way to observe language in use, contrarily to parallel corpora which are translations and bears influence from the original source text.

Techniques for the acquisition of terminology from parallel corpora were first introduced by [START_REF] Rapp | Identify Word Translations in Non-Parallel Texts[END_REF][START_REF] Fung | Finding Terminology Translations from Non parallel Corpora[END_REF]). These techniques rely on distributional semantics whose hypothesis is that words that are semantically close will tend to appear in the same contexts. Identification of term translations in comparable corpora requires three phases. The first phase consists in computing the context of each term in the source and target corpora. The context of a term T is represented by a vector indicating the number of times T co-occurs with each word W with in a given contextual window 1 . In the second phase, words in the source context vectors are translated into the target language by using a bootstrap bilingual dictionary 2 . In the third phase, the source and target vectors are compared 3 : the most similar the vectors, the most likely the 1 For instance : three words on its left and three words on its right. 2 The vectors are only partly translated due to the small coverage of the dictionary. 3 Using a similiarity measure such as the Cosine similarity, see [START_REF] Rapp | Identify Word Translations in Non-Parallel Texts[END_REF] and [START_REF] Fung | Finding Terminology Translations from Non parallel Corpora[END_REF]) for more details.

target and source terms are translations of each other. Finally, the output of the alignment algorithm is a list of one-to-many alignments : each source term is associated with an ordered list of candidate translations. The candidate translations are ordered from most to least probable.

Results are evaluated by examining the best candidate translation (Top1), the ten best candidate translations (Top10) or the twenty best candidate translations (Top20).

The drawback of acquiring terminology from comparable corpora is that the acquired lexicons are not as reliable as those acquired from parallel texts. Lexicon acquisition from parallel texts outputs a one-to-one term alignment with high accuracy scores. For example, recent work on the matter [START_REF] Lefever | Language-independent bilingual terminology extraction from a multilingual parallel corpus[END_REF] showed scores running from 85% to 90% accuracy. Conversely, systems that acquire lexicons from comparable corpora output one-to-many alignements : a source term associated to the set of its most probable target translations. As a consequence, the lexicons need to be post-editted before being injected in a termbase or in any other language processing module. For example, [START_REF] Fung | A statistical view on bilingual lexicon extraction : From parallel corpora to nonparallel corpora[END_REF] shows a 80% precision on the Top20 candidates for single words alignments computed from large general language corpora (hundred millions of words or more). Dejean et al. (2002) find a 60% precision on the Top 20 candidates for single word terms using specialized language corpora of small size. [START_REF] Morin | Bilingual Terminology Mining --Using Brain, not brawn comparable corpora[END_REF] indicate a 42% precision on the Top20 candidates for multi-word terms.

To our knowledge, existing term-alignment validation tools do not deal with this type of results.

For example, the iView application from the iTools suite [START_REF] Merkel | Terminology Extraction and Term Ranking for Standardizing Term Banks[END_REF] presents its user with a list of one-to-one term alignements that have to be validated as correct alignments and as being domain specific. Side-information on term pairs consists of sample context sentences and some statistical data. Similarly, the commercial product Araya Bilingual Term Extractor [START_REF] Waldhör | Araya Bilingual Term Extraction[END_REF]) also displays a list of one-to-one alignments with some statistical data. The

Xerox Terminology Suite® provides quasi-exhaustive terminological records but no confidence score to help sort the candidate translations.

Terminology extraction from comparable corpora has raised the need for a new kind of terminology validation tool. This new kind of tool should be able to deal with one-to-many alignements or even many-to-many alignments if the source language term is a set of term-like sequences reflecting several variants of the same term. On top of that, we believe that sample sentences and statistical data is important but not sufficient information to help the annotator or professional translator validate a term alignment. Last but not least, there should exist a data exchange format fitted for the exchange of automatically generated lexicon.

The paper is organized as follows : part 2 A term-oriented annotation tool describes the theoretical background that ruled the conception of our tool for the annotation of lexicon acquired from comparable corpora, part 3 The term-alignment validation interfaceintroduces the validation tool's interface and part 4 A TBX variant for automatically generated bilingual lexicons proposes a TBX (Term Base eXchange) variant for the exchange of automatically extracted lexicons.

A term-oriented annotation tool

Literature on the theory of terminology [START_REF] Bourigault | Pour une terminologie textuelle[END_REF][START_REF] Cabré | Theories of Terminology[END_REF]L'Homme 2004) shows that there are two main conceptions of this field of research. One conception is said "concept-oriented" while the other is said "term-oriented". The difference between those two standpoints can be very briefly summarized as follows :

"Concept-oriented" terminology may be considered as the main stream and historical terminology theory. It is the theoretical standpoint represented in the International Organization

for Standardization terminological norms. The aim of concept-oriented terminology is to modelize the knowledge of a domain by discovering its concepts and their relations; terms are considered as the mere linguistic expression of these concepts. The relation between a term and a concept is often seen as unequivocal and stable. This approach is well suited for prescriptive goals such as the creation of controlled-languages or language planning.

"Term-oriented" terminology rose in the 90's as a critic of concept-oriented terminology. It is deeply grounded in social sciences and linguistics. Its object of study is the term. This approach has shown that the definition of what is a term or not is rather subjective and highly depends on the final use of the terminology. This approach has also highlighted the fact that the term/concept relation is not always unequivocal and that terms do vary. Heavily relying on textual data, this approach seeks to describe terms usage and variation rather than prescribe it.

Contrarily to terminology management tools like OpenTerminologyManager (Waldhör 2002) or

Terminae [START_REF] Biébow | TERMINAE : a method and a tool to build a domain ontology[END_REF], the annotation tool presented in this paper manipulates no such things as "concepts" and can be considered as a term-oriented and text-driven tool. It deals only with pairs of lexical units extracted from a corpus that are supposed to be correct translations of each other and thus considered as terms from a translation-aid point of view. It also provides the user with raw information extracted from a domain-specific corpus. This information can be extracted during the term alignment process and come as a side product of it or it can be retrieved from online public ressources like Wikipedia or Wiktionary. This information is intended to help the annotator grasp the in-vivo linguistic behaviour of the terms she or he has to annotate.

Such information includes :

 normalized form of the term (with the least flexionnal morphemes -usually the result of lemmatization)

 part-of-speech

 frequency or number of occurrences in the corpus

 definition  collocations
 "soft" variants such as acronyms and orthographic variants

 stronger variants such as syntactic or morphosyntactic variants  terms that have the same stem  terms that appear in similar contexts  contexts in which the term occurs : a sample paragraph, with a link leading to the original document Although this kind of information would be of least interest in the construction of terminologies for localization or controlled-languages, we believe it is crucial in translation-aid applications.

For example, concept-oriented approaches to terminology frequently seek to minimize (if not deny) term variation in order to control the use of terms. In translation-aid applications, the detection of term variation and the harvesting of variants is very useful. While translating a text, a translator may stumble upon the variant of a term instead of the "authorized" canonical form of the term4 . For that reason, term variants need to be taken into account during the validation process.

Another example is the use of definition to help the annotator or the translator understand the meaning of a term. While a definition will always remain useful, the meaning of a term can also be inferred from its contexts or by relating it to terms that appear in similar contexts. We see these two kinds of information (definition vs contexts and neighbouring terms) as complementary and think both should appear in a term-alignment validation tool.

In the annotation tool described in section 3, what is called a "term" is a lexical unit associated with an information record which parallels the terminological records found in most terminology management tools. It is characterized by a normalized form, a part-of-speech and a frequency. A term may have variants or may be in relation with other terms (either morphologically or semantically). A term variant is a plain lexical unit with no information record. The definition, collocations, relations to other terms, contexts and validated translations of a given term fully apply to its variants.

The term-alignment validation interface

As shown in Illustration 1, candidate term-alignments are displayed in the upper part of the interface. Because the accuracy of a term-alignment is not always clear-cut ( two terms may be the exact translations of each other or vague equivalents), a candidate term alignment can receive one of these four labels : correct, rather correct, rather incorrect, incorrect or remain unannotated. The value of the annotation is expressed via a colour code, the darkest the colour, the more correct the alignment is. The use of a colour code helps catch at a glance the correctness of each alignment. If the right translation is not present among the candidate translations, the annotator can create an alignment with a new term.

To help the annotator make a decision about the correctness of an alignment, the information records of the aligned terms are displayed in the lower part of the inteface. Each information record is divided into four sections :

 Tab 1 -Primary information : normalized form, part-of-speech, frequency, definition, collocations  Tab 2 -Related terms : terms that share the same stem or that occur in similar contexts  Tab 3 -Contexts : example of occurrences of the term and its variants in the corpus  Tab 4 -Variants : soft (acronyms, orthographic variants) and strong variants (syntactic and morpho-syntactic variants).

The manual annotation process being a long and tedious task, all main actions are accessible via keyboard short cuts.

The annotation tool was developped using PHP/MySql and Ajax. It is available online and can be freely tested at [http://62.193.49.219/Metricc/InterfaceValidation/]. First beta-testers were quite happy with it and formulated several suggestions. Next improvements will be :

 implementation of TBX import / export functions  hypertext navigation inside the information record : for example, a click on a related term should display its information record  advanced search options (boolean operators, regular expressions)

 term filtering and term ordering functionalities

A TBX variant for automatically generated bilingual lexicons

This section adresses the issue of data exchange. Currently, there exists no standard for the The TBX format perfectly fits with our goal because it is modular. It includes two modules : (i) a core XML structure, defined by a DTD and (ii) an XML formalism which identifies a set of data-categories and their constraints. The data-categories and constrainsts can be customized to create a TBX variant.

The core module is an XML structure compliant with the TMF (Terminological Markup

Framework) meta-model [START_REF] Iso | ISO norm 16642 : Computer applications in terminology Terminological markup framework[END_REF]. This meta-model organizes a termbase on three levels : The XML formalism used to identify a set of data-categories and their constraints is called XCS (eXtensible Constraint Specification). There exists a default TBX terminological markup language that uses the data-categories and constraints defined in the ISO norm 1260 (ISO, 1999). Obviously, the TBX default data-categories and constraints were not designed for the exchange of automatically generated lexicon. The TBX variant proposed here uses a subset of the default data-categories (partOfSpeech, frequency, usageNote, corpusTrace, termType, reliabilityCode) and three additional data-categories : termDefinition, relatedTerm and termReference.

Illustration 2 shows the TBX encoding of a term and its information record. The term is phonème (phoneme) and has one variant phon. which is an abbrevation. Two data categories had to be added in order to encode the whole information record :

 termDefinition : this data-category is used to encode the definition of each term. It appears at the term level (<ntig> or <tig> tag).

The default TBX data category definition could not be used for that purpose because its definition 5 in the ISO norm 12620 (ISO, 1999) states that it applies only to concepts, not to terms. Now, the annotation tool presented here is designed following a term-oriented approach. This means that we consider that even if two terms are correct translations of each other, they do not automatically fall under the same definition, as they may be mere equivalents or the meaning of a term in one language may be richer than the meaning of its translation.

 relatedTerm : this data-category is used to indicate terms that appear in the same contexts or that share the same stem. Again, the ISO norm 12620 (ibid.) provides data categories for relations between concepts, not between terms. This is why this category was added.

Illustration 3 shows the source term phonème (phoneme) with its set of candidate target translations. The data category reliabilityCode is used to indicate the score the extraction program gave to the alignment of the source and target term.

Like source terms, each target term comes with its information record. However, a target term might be repeated inside the TBX document because it may have been considered by the extraction program as a potential translation for several source terms. Repeating the whole information record each time a target terms appears in the document would be redundant and heavy to process. This is why a third data category has been added.This data category is called termReference and only occurs at the term level with a term target. It is used to refer to a term that has already occurred in the TBX document without having to repeat the whole information record again.

5 ISO12620-A0501: " Description: A statement that describes a concept and permits its differentiation from other concepts within a system of concepts."

5.

Conclusion

We have described an interface specialized in the annotation of lexicons automatically extracted from comparable corpora. The need for such a tool is justified by the fact that bilingual lexicons extractors that deal with comparable corpora output one-to-many alignments that are not processed by traditional term-alignments validators. We have also argued that a validation interface should provide its user with more information than sample sentences and statistical data, especially in the field of assisted translation. This additional information should be corpus driven, so as to enable the annotator to understand the in-vivo linguistic behaviour of the terms he or she has to align. Finally, we have suggested a TBX variants that enables the encoding and exchange of one-to-many alignments.

6.

  exchange of automatically generated bilingual lexicons. Such a format should allow the encoding one-to-many alignments. Well-known formats for the exchange of terminological data include OLIF (Open Lexicon Interchange Format) by Lieske (2001), Geneter (GENEric model for TERminology) by Le Meur (1998) and TBX (TermBase eXchange) developped by the International Standards Organization (2008).

  1. a concept-level, materialized by the xml tag <termEntry> 2. a language level, materialized by the xml tag <langSet> 3. a term-level, materialized by the xml tag <ntig> or <tig> A <termEntry> includes one or several <langSet>, which in turn includes one or severel <ntig> or <tig> where the terms are encoded along with their linguistic information (part-of-speech, frequency, etc). Terms belonging to the same <termEntry> are considered as synonyms. As a consequence, terms belonging to different <langSet> but to the same <termEntry> can be considered as translations of one another. This is how term alignments will be encoded in TBX format : as two terms (<ntig>) under the same <termEntry> and belonging to different <langSet> (see Illustration 3 for an example).

For example, aménagement de la forêt and aménagement forestier are variations of the same term and both translate into English as forest management(Morin et al. 2004).

Acknowledgements

This work was supported by the French National Research Agency (ANR), funding n° ANR-08-CORD-009. We also would like to thank the company Lingua et Machina for its support (http://www.lingua-et-machina.com).

available online at [http://www.heartsome.de/en/xliffug_en.pdf], accessed March 9 th , 2010. Waldhör, K. (2002)