M Abid

C Kharif
email: kharif@irphe.univ-mrs.fr.

H.-C Hsu

Y.-Y Chen

Transverse instability of gravity-capillary solitary waves on deep water in the presence of constant vorticity

Keywords: NLS equation, gravity-capillary solitary waves, transverse instability, vorticity

The bifurcation of two-dimensional gravity-capillary waves into solitary waves when the phase velocity and group velocity are nearly equal is investigated in the presence of constant vorticity. We found that gravity-capillary solitary waves with decaying oscillatory tails exist in deep water in the presence of vorticity. Furthermore we found that the presence of vorticity influences strongly (i) the solitary wave properties and (ii) the growth rate of unstable transverse perturbations. The growth rate and bandwidth instability are given numerically and analytically as a function of the vorticity.

Introduction

We consider the bifurcation of gravity-capillary waves propagating at the free surface of a vertically sheared current of constant vorticity when the phase and group velocities are nearly equal. In the absence of vorticity, the existence of steadily propagating solitary gravity-capillary waves on deep water was proved numerically by [START_REF] Longuet-Higgins | Capillary-gravity waves of solitary type on deep water[END_REF]. [START_REF] Iooss | Bifurcation d'ondes solitaires en présence d'une faible tension superficielle[END_REF] investigated the problem mathematically and found it was a 1:1 resonance phenomenon. Later on, [START_REF] Vanden-Broeck | Gravity-capillary solitary waves on water of infinite depth and related free surface flows[END_REF] established the link between the numerical calculation of [START_REF] Longuet-Higgins | Capillary-gravity waves of solitary type on deep water[END_REF] and the rigorous analysis of [START_REF] Iooss | Bifurcation d'ondes solitaires en présence d'une faible tension superficielle[END_REF]. [START_REF] Dias | Numerical study of capillary-gravity solitary waves[END_REF] investigated numerically the behavior of these waves in arbitrary depth away from their bifurcation point. [START_REF] Longuet-Higgins | Capillary-gravity waves of solitary type and envelope solitons on deep water[END_REF] and [START_REF] Akylas | Envelope solitons with stationary crests[END_REF] demonstrated that gravity-capillary solitary waves occur when the phase velocity of the carrier wave is equal to the group velocity of the envelopesoliton of the NLS equation. Note that the linear dispersion relation of gravity-capillary waves has a minimum in phase velocity which is equal to the group velocity. For values of the phase velocity less than this extremum the NLS equation admits envelope-soliton solutions such that the wave crests are stationary in the frame of reference of the wave envelope. Later on, steady three-dimensional gravity-capillary solitary waves have been investigated by several authors, among them [START_REF] Parau | Nonlinear three-dimensional gravity-capillary solitary waves[END_REF], [START_REF] Kim | On gravity-capillary lumps[END_REF] and [START_REF] Milewski | Three-dimensional localized solitary gravity-capillary waves[END_REF]. Gravity-capillary solitary waves have been observed experimentally in wind waves generated in a wind wave facility by [START_REF] Zhang | Measuring the two-dimensional structure of a wavy water surface optically: a surface gradient detector[END_REF]. During laboratory experiments [START_REF] Zhang | Capillary-gravity and capillary waves generated in a wind-wave tank: observations and theories[END_REF]) and field experiments [START_REF] Zhang | Observations on waveforms of capillary and gravity-capillary waves[END_REF]) on wind waves on deep water, gravity-capillary solitary waves were observed and found to closely match numerical profiles obtained by [START_REF] Longuet-Higgins | Capillary-gravity waves of solitary type on deep water[END_REF]Vanden-Broeck &[START_REF] Vanden-Broeck | Gravity-capillary solitary waves on water of infinite depth and related free surface flows[END_REF]. [START_REF] Longuet-Higgins | Experiments on capillary-gravity waves of solitary type on deep water[END_REF] generated experimentally gravity-capillary solitary waves on deep water by the resonant excitation of surface waves at a velocity below the minimum of gravity-capillary waves. The mechanism of generation is due to a jet of air impinging on the surface of steady current. Forced three-dimensional gravity-capillary solitary waves on deep water (called lumps) have been investigated experimentally by several authors [START_REF] Dorio | Resonantly forced gravitycapillary lumps on deep water. part 1. experiments[END_REF], Masnadi & Duncan (2017a), Masnadi & Duncan (2017b) and [START_REF] Park | Experimental observation of gravity-capillary solitary waves generated by a moving air suction[END_REF]). To the best of our knowledge, there is no study regarding the effect of constant vorticity on the transverse instability of two-dimensional envelope solitons. On the contrary, many investigations on the transverse instability have been carried out in the absence of vorticity. [START_REF] Zakharov | Instability of wave guides ans solitons in nonlinear media[END_REF] and [START_REF] Saffman | Stability of a plane soliton to infinitesimal two-dimensional perturbations[END_REF] demonstrated that plane gravity envelope solitons are unstable to long wave transverse perturbations. Later on, [START_REF] Ablowitz | On the evolution of packets of water waves[END_REF] extended their analyses to show equivalent results in the case of finite depth. [START_REF] Deconinck | Transverse instabilities of deep-water solitary waves[END_REF] obtained a variety of transverse instabilities of gravity envelope solitons in infinite depth. Within the framework of the fully nonlinear potential flow equations [START_REF] Kim | Transverse instability of gravity-capillary solitary waves[END_REF] found that plane gravity-capillary solitary waves of depression are unstable to transverse perturbations. [START_REF] Kim | long-wave transverse instability of weakly nonlinear gravity-capillary solitary waves[END_REF] verified the long wave transverse instability within the framework of the weakly nonlinear third-order truncation model. The presence of transverse instability suggests the existence of solitary lumps which are the three-dimensional counterpart of plane gravity-capillary solitary waves. Our purpose is not to consider three-dimensional localized solitary gravity-capillary waves. [START_REF] Akers | A model equation for wavepacket solitary waves arising from capillary-gravity flows[END_REF], [START_REF] Akers | Dynamics of three-dimensional gravity-capillary solitary waves in deep water[END_REF], [START_REF] Wang | Dynamics of gravity-capillary solitary waves in deep water[END_REF] and [START_REF] Milewski | Transversally periodic gravity-capillary solitary waves[END_REF] used several model equations to investigate numerically the stability of two-dimensional and three-dimensional gravity-capillary solitary waves in deep water. Regarding two-dimensional waves they found that both gravity-capillary solitary waves of elevation and depression are unstable to transverse perturbations. The purpose of the present paper is twofold: (i) to revisit the problem of two-dimensional gravity-capillary waves of solitary type on deep water when the effect of an underlying vertically sheared current is taken into account and (ii) to investigate the stability of one-dimensional gravity-capillary solitons, solution of the NLS equation, to transverse perturbation in the presence of constant vorticity.

Section 2 is devoted to the computation of the bifurcated gravity-capillary solitary wave on deep water in the presence of constant vorticity. In section 3 the stability of its envelope to infinitesimal transverse perturbations is investigated.

Gravity-capillary solitary waves with constant vorticity

We consider envelope soliton bifurcation from a uniform gravity-capillary wave train in the presence of vorticity. Our study is confined to two-dimensional water waves propagating in infinite depth. The fluid is considered inviscid and incompressible. The waves move along the x-axis and the z-axis is oriented upwards opposite to the gravity g = (0, -g). The origin z = 0 is the undisturbed free surface. The waves are travelling at the surface of a vertically sheared current of constant vorticity. In the fixed frame the underlying current is given by U (z) = U 0 + Ωz where Ω is the shear intensity and U 0 is the current velocity at the surface. Note that the vorticity is -Ω. We choose a reference frame moving with the horizontal velocity U 0 . Consequently, in this frame of reference the current at the surface vanishes and the underlying current is given by U (z) = Ωz.

Very recently, [START_REF] Hsu | A nonlinear schödinger equation for gravity-capillary water waves on arbitrary depth with constant vorticity: Part i[END_REF] derived a nonlinear Schrödinger equation for the envelope of two-dimensional gravity-capillary waves propagating on arbitrary depth at the free surface of a vertically sheared current of constant vorticity -Ω. Herein, we consider the case of infinite depth. This NLS equation for the complex envelope of the elevation is written as follows

ia τ + αa ξξ = γ * | a | 2 a, (2.1)
where a is the complex envelope of the free surface elevation which depends on slow variables ξ = ε(x -c g t) and τ = ε 2 t. The group velocity of the carrier wave of wavenumber, k, and frequency, ω, is c g . We consider weakly nonlinear waves and consequently we set

ε 1. The dispersion coefficient α reads α = - ω k 2 (2 + Ω/ω) (ρ 2 - κ 1 + κ α 1 ),
with

α 1 = (1 + Ω/ω)(1 + 2ρ) + 2(ρ -2 κ 1 + κ (1 + Ω/ω)),
where

ρ = 1 + Ω/ω 2 + Ω/ω (1 + 2κ 1 + κ ).
The inverse of the Bond number is

κ = k 2 T ρ w g ,
where T is the surface tension coefficient and ρ w the water density.

The nonlinear coefficient γ * is

γ * = g 2 4ω 2 ( 1 + κ 1 + Ω/ω ) 2 γ,
where

γ = k 4 2ω(1 + Ω/ω)(2 + Ω/ω) - 3κ 1 + κ (1 + Ω/ω) 2 + (1 + Ω/ω)(8 + 6Ω/ω) + 1 + Ω/ω 1 -κ(2 + 3Ω/ω) γ 1 + 2 (1 + Ω/ω)(2 + Ω/ω) 2 ρΩ/ω -(1 + Ω/ω)/(1 + κ) (Ω/ω) 2 ,
and

γ 1 = 12κ + 10(1 + 4κ)Ω/ω + 6(3 + 7κ)(Ω/ω) 2 + 2(4 + 7κ)(Ω/ω) 3 + (1 + κ)(Ω/ω) 4 .
Equation (2.1) admits the following solution

a = ±a 0 sech{a 0 ( -γ * 2α ) 1/2 ξ} exp(- i 2 γ * a 2 0 τ ),
and the elevation of a weakly nonlinear wave train of envelope a(ξ, τ ) writes

ζ = ± 1 2 εa 0 sech{a 0 ( -γ * 2α ) 1/2 ξ} exp[ikx -i(ω + 1 2 γ * ε 2 a 2 0 )t] + c.c.,
where k and ω are the carrier wavenumber and carrier frequency, ε 1 and c.c. denotes the complex conjugation. The wave height at the origin is εa 0 .

The crests of the wave group travel at phase velocity

c = ω k + 1 2 γ * ε 2 a 2 0 /k,
whereas the envelope travels at group velocity c g .

The condition for the envelope-soliton to be a solitary wave is that the group velocity is equal to the phase velocity

c g = c c g = ω k + 1 2 γ * ε 2 a 2 0 /k. (2.2)
Whatever Ω, the nonlinear coefficient of the NLS equation is negative (see figure 1), therefore the solitary wave travels at a constant velocity which is a little less than the minimum phase velocity of linear gravity-capillary waves. Following Longuet-Higgins (1993), we set g = 1, T = 1 and ρ w = 1 which is equivalent to use (T /ρ w g) 1/2 and (T /ρ w g 3 ) 1/4 as unit of length and unit of time, respectively. For the sake of clarity the same symbols are used to define the dimensionless frequency, vorticity and wavenumber. Then, the nonlinear coefficient γ * becomes in dimensionless form

γ * (k) = ( 1 + k 2 ω + Ω ) 2 γ/4, where γ = k 4 2ω(1 + Ω/ω)(2 + Ω/ω) - 3k 2 1 + k 2 (1 + Ω/ω) 2 + (1 + Ω/ω)(8 + 6Ω/ω) + 1 + Ω/ω 1 -k 2 (2 + 3Ω/ω) γ 1 + 2 (1 + Ω/ω)(2 + Ω/ω) 2 ρΩ/ω -(1 + Ω/ω)/(1 + k 2 ) (Ω/ω) 2 ,
with

γ 1 = 12k 2 + 10(1 + 4k 2 )Ω/ω + 6(3 + 7k 2 )(Ω/ω) 2 + 2(4 + 7k 2 )(Ω/ω) 3 + (1 + k 2 )(Ω/ω) 4 ,
and

ρ = 1 + Ω/ω 2 + Ω/ω (1 + 2k 2 1 + k 2 ).
The dimensionless linear dispersion relation is

ω 2 + Ωω -k(1 + k 2 ) = 0.
The dimensionless linear phase velocity and group velocity are equal at the linear phase velocity minimum which occurs when k is solution of the following equation

k 2 + Ω √ 2 k 1/2 -1 = 0. (2.3)
For the case Ω = 0 which was considered by Longuet-Higgins (1993) we find k = 1, ω = √ 2 and γ * (k) = -11 √ 2/32. Then, the velocity of the solitary wave is [START_REF] Longuet-Higgins | Capillary-gravity waves of solitary type and envelope solitons on deep water[END_REF].

c g0 = √ 2(1 - 11 64 ε 2 a 2 0 ), which is the equation (3.19) of
Equation (2.3) admits the following aproximate solution

k = 1 - Ω √ 2 + O(Ω 2 ). (2.4) Then ω = √ 2 - 3 2 Ω + O(Ω 2 ).
Negative vorticity (Ω > 0) reduces k and ω whereas positive vorticity (Ω < 0) increases k and ω.

An approximate solution of equation (2.2) is

c g = √ 2 - Ω 2 -( 11 √ 2 64 + 31 64 Ω)ε 2 a 2 0 + O(Ω 2 , ε 3 a 3 0 ),
which can be rewritten as follows

c g = c g0 - Ω 2 (1 + 31 32 ε 2 a 2 0 ) + O(Ω 2 , ε 3 a 3 0 ). (2.5)
Solitary wave velocity is increased when the vorticity is positive (Ω < 0) and reduced when the vorticity is negative (Ω > 0).

To extend the work of [START_REF] Akylas | Envelope solitons with stationary crests[END_REF], we investigate the bifurcation of a train of periodic gravity-capillary waves into a solitary wave with decaying oscillatory tail in the presence of constant vorticity. Now, we use T /(ρ w c 2 ) and T /(ρ w c 3 ) as unit length and unit time where c is the phase velocity of the envelope-soliton, which means that c is the reference phase velocity scale. Note that these time and length units were used by Vanden-Broeck & [START_REF] Vanden-Broeck | Gravity-capillary solitary waves on water of infinite depth and related free surface flows[END_REF] and [START_REF] Akylas | Envelope solitons with stationary crests[END_REF]. Consequently, with these new scaled variables the dimensionless form of the NLS equation (2.1) becomes

iã τ + αã ξ ξ = β 2 γ * | ã | 2 ã, (2.6)
where β = gT /(ρ w c 4 ), τ = ρ w c 3 τ /T , ξ = ρ w c 2 ξ/T , ã = ρ w c 2 a/T , α = ρ w cα/T and γ * = T cγ * /(ρ w g 2 ).

Using

ω = T ω/(ρ w c 3 ), k = T k/(ρ w c 2 ), Ω = T Ω/(ρ w c 3 ) and κ = k2 /β the coefficient γ * is γ * = ωk4 (1 + k2 /β) 2 8(ω + Ω) 3 (2ω + Ω) {- 3 k2 (ω + Ω) 2 ω2 (β + k2 ) + (ω + Ω)(8ω + 6 Ω) ω2 + (ω + Ω) β ω -k2 (2ω + 3 Ω) (β γ1 ) + 2(ω + Ω)(2ω + Ω) 2 Ω2 ω4 (ρ Ω -β(ω + Ω)/(β + k2 )) },
where

β γ1 = β(10∆ + 18∆ 2 + 8∆ 3 + ∆ 4 ) + k2 (12 + 40∆ + 42∆ 2 + 14∆ 3 + ∆ 4 ), ∆ = Ω ω , and ρ = (ω + Ω)(1 + 2κ/(1 + κ)) 2ω + Ω .
Let μ = β 2 γ * . For Ω = 0, we have checked that -μ is equal to the expression derived by [START_REF] Akylas | Envelope solitons with stationary crests[END_REF]. In figure 1 is plotted the nonlinear coefficient μ as a function of Ω.

An enlargement is inserted in the figure in the vicinity of Ω ≈ -0.2 to show that this The dimensionless NLS equation (2.6) becomes, dropping the tildes

ia τ + αa ξξ = µ | a | 2 a, (2.7)
and the free surface elevation writes

ζ = ± 1 2 εa 0 sech{a 0 ( -µ 2α ) 1/2 ξ} exp[ikx -i(ω + 1 2 µ ε 2 a 2 0 )t] + c.c.
The condition for the envelope-soliton to be a solitary wave is, due to the chosen normalisation,

c g = c = 1 1 = ω k + 1 2 µ ε 2 a 2 0 /k. Hence, k = ω(k, Ω, β) + 1 2 µ ε 2 a 2 0 .
(2.8)

For infinite depth, the dimensionless linear gravity-capillary dispersion relation in the presence of a shear of intensity Ω is

ω 2 + Ω ω -βk -k 3 = 0, (2.9)
which has a minimum in phase velocity at k = k 0 . At this minimum the phase velocity is equal to the group velocity

k 0 = ω(k 0 , Ω, β 0 ) and ∂ω ∂k (k 0 , Ω, β 0 ) = 1.
Solving these two equations gives

k 0 = 1 2 ,
(2.10) and

β 0 = 1 4 + Ω.
For Ω = 0 the values of [START_REF] Akylas | Envelope solitons with stationary crests[END_REF] are rediscovered. Note that the shear has no effect on the dimensionless value of k 0 . Nevertheless, with the scaled variables used by Longuet-Higgins (1993) equation ( 2.10) reduces to equation ( 2.4) at O(Ω).

Herein, ω > 0 and consequently ω 0 = 1/2. From the dispersion relation (2.9), it can be demonstrated that we always have -Ω < ω or -ω < Ω. In our case the carrier wave frequency is close to ω 0 and as a result the considered values of Ω are larger than a value close to -1/2.

The wavenumber k and β are expanded about (k 0 , β 0 ) as follows

k = k 0 + k 1 ε 2 + O(ε 3 ), β = β 0 + β 1 ε 2 + O(ε 3 ).
Expanding ω(k, Ω, β) about (k 0 , β 0 ) and using equation (2.8) with ∂ω/∂k(k 0 , Ω, β 0 ) = 1 gives

β 1 = - 1 2 µ(k 0 , Ω, β 0 ) ∂ω/∂β(k 0 , Ω, β 0 ) a 2 0 ,
and

k 1 = - ∂ 2 ω ∂β∂k (k 0 , Ω, β 0 ) ∂ 2 ω ∂k 2 (k 0 , Ω, β 0 ) β 1 . Using ∂ 2 ω ∂k 2 (k 0 , Ω, β 0 ) = 2α(k 0 , Ω, β 0 ), we obtain k 1 = - ∂ 2 ω ∂β∂k (k 0 , Ω, β 0 ) 2α(k 0 , Ω, β 0 ) β 1 .
Let δβ be the difference between β and the critical value

β 0 δβ = β 1 ε 2 + O(ε 3 ), δβ = - 1 2 µ(k 0 , Ω, β 0 ) ∂ω/∂β(k 0 , Ω, β 0 ) a 2 0 ε 2 . Knowing that ∂ω/∂β(k 0 , Ω, β 0 ) = 1 2(1 + Ω) , then a 2 0 ε 2 = - δβ (1 + Ω)µ(k 0 , Ω, β 0 ) . (2.11)
From equation (2.11) and figure 1 we can see that the wave height decreases as positive values of Ω increases. For negative values of Ω the wave height presents a maximum in the vicinity of Ω ≈ -0.2. Finally,

a 2 0 ε 2 = 16 2 δβ 11 + 128Ω + 464Ω 2 + 512Ω 3 + 192Ω 4 .
(2.12)

Knowing that

∂ 2 ω ∂β∂k (k 0 , Ω, β 0 ) = Ω (1 + Ω) 2 , and ∂ 2 ω ∂k 2 (k 0 , Ω, β 0 ) = 2α(k 0 , Ω, β 0 ) = 1 1 + Ω , then k 1 = - Ω 1 + Ω β 1 ,
and

k = k 0 + k 1 ε 2 = k 0 - Ω 1 + Ω β 1 ε 2 = k 0 - Ω 1 + Ω δβ.
(2.13)

Note that in the absence of vorticity the dimensionless carrier wavenumber is unchanged at O(ε 2 ). Negative vorticity (Ω > 0) reduces k whereas positive vorticity (Ω < 0) increases k.

The expression of the solitary wave is

ζ(x, t) = ±16( δβ 11 + 128Ω + 464Ω 2 + 512Ω 3 + 192Ω 4 ) 1/2 sech{(δβ) 1/2 (x-t)} cos{k(x-t)}.
(2.14) The profile of the solitary wave is known once the bifurcation parameter δ β is fixed. Equations (2.12) and (2.13) allows the determination of the parameter εa 0 and k, respectively. Profiles of solitary waves of elevation and depression which bifurcate at β = β 0 from a linear uniform periodic wave train of frequency ω 0 and wavenumber k 0 , are plotted in figures 2, 3, 4 and 5. For negative vorticity (Ω > 0), an increase of Ω leads to a decrease of the wave height (see figures 2 and 4). For positive vorticity (Ω < 0), an increase of |Ω| may lead to an increase (see figures 3, 5, 6 and 7) of wave height.

Transverse instability of the envelope of gravity-capillary solitary waves with constant vorticity

The two-dimensional version of the NLS equation in deep water (2.1) is

ia τ + αa ξξ + λa ηη + 2Γ | a | 2 a = 0, (3.1)
where η is the transverse coordinate and Γ = -γ * /2.

The transverse dispersion coefficient λ reads For ∂/∂η = 0, equation (3.1) admits the following solution

λ = ω 2k = g(1 + 3κ) 4k(ω + Ω/2) .
ā = A(ξ) exp(iΓ a 2 0 τ ), (3.2)
where Let us perturb with an infinitesimal perturbation the solution (3.2)

A(ξ) = a 0 sech{a 0 ( Γ α ) 1/2 ξ}.
a = ā + a exp(iΓ a 2 0 τ ).
Linearisation of equation (3.1) about ā gives

ia τ -Γ a 2 0 a + αa ξξ + λa ηη = -2Γ A 2 a * -4Γ A 2 a .
(3.3) Let us consider a = u + iv (with u and v reals) and substitute this expression into (3.3)

v τ = αu ξξ + λu ηη -(-6Γ A 2 + Γ a 2 0 )u, ( 3
σv = α d 2 û dξ 2 -λq 2 û -(-6Γ A 2 + Γ a 2 0 )û, σû = -α d 2 v dξ 2 + λq 2 v + (-2Γ A 2 + Γ a 2 0 )v.
The above system can be written as follows

σ 2 û = -(L 0 -λq 2 )(L 1 -λq 2 )û, (3.8) with L 0 = α d 2 dξ 2 + 2Γ A 2 -Γ a 2 0 ,
and

L 1 = α d 2 dξ 2 + 6Γ A 2 -Γ a 2 0 ,
and the eigenfunction û vanishes as |ξ| → ∞.

The numerical method to solve equation (3.8) is outlined in the Appendix. We set g = 1, T = 1 and ρ w = 1. For moderate and small values of the vorticity the coefficients of the NLS equation (3.1) are positive, Γ > 0, α > 0 and λ > 0. For comparison we use herein the analytic relation of the instability growth rate given by [START_REF] Rypdal | Stability of solitary structuresin the nonlinear schrödinger equation[END_REF] which writes in dimensionless form

σ 2 = 64 π 2 λΓ q 2 (1 - λ 6Γ q 2 ). (3.9)
The maximum growth rate is

σ 2 max =
96Γ 2 π 2 , and

q max = 3Γ λ .
The gravity-capillary solitary wave is marginally stable for q = q c with

q 2 c = 6Γ λ . (3.10)
For small values of the vorticity the instability growth rate and marginal wavenumber are

σ 2 = q 2 ( 11 π 2 - 16 3π 2 q 2 ) + q 2 ( 153 + 32q 2 6 √ 2 π 2 )Ω + O(Ω 2 ),
and

q c = 33 16 + 87 33 √ 2 Ω + O(Ω 2 ).
In figure 8 is shown the numerical and analytic instability growth rates for several values of the vorticity. The solid lines correspond to numerical solutions of equation (3.8).

Negative vorticity amplifies the rate of growth whereas positive vorticity reduces its value. Surface elevation of the envelope of gravity-capillary solitary waves perturbed by their most unstable modes corresponding to Ω = 0, q = 1.01 and Ω = 0.10, q = 1.08 are shown in figure 9. These waves which are localized in the direction of propagation and periodic in the transverse direction have been investigated by [START_REF] Milewski | Transversally periodic gravity-capillary solitary waves[END_REF]. This new solution found by the latter authors was called transversally periodic plane solitary wave.

Conclusion

Within the framework of the 1D-and 2D-NLS equations, the effect of constant vorticity on the properties and stability of gravity-capillary solitary waves in deep water has been investigated. These two-dimensional solitary waves of elevation and depression types bifurcate from linear uniform wavetrains at the phase speed minimum where the phase velocity equals the group velocity. We found that negative vorticity (Ω > 0) reduces the wave height whereas positive vorticity (Ω < 0) increases the wave height. Numerical and analytic instability growth rates of transverse perturbations have been computed and derived, respectively. We have shown that negative vorticity amplifies the rate of growth and bandwidth of transverse instability whereas positive vorticity diminishes both the rate of growth and bandwidth.

Appendix. Numerical method to solve equation (3.8)

The eigenfunctions û are computed numerically using the interpolant function [START_REF] Tang | The hermite spectral method for gaussian-type functions[END_REF]; [START_REF] Boyd | Chebyshev and Fourier spectral methods[END_REF]; [START_REF] Weideman | A matlab differentiation matrix suite[END_REF]). e (-x 2 j /2) φ j (x)û j , where φ j (x) = H N (x) H N (x j )(x -x j ) , and H N (x) is the Hermite polynomial of degree N . The grid points x 1 , . . . , x N are the roots of H N (x). Note that for continuous eigenfunctions on R vanishing at infinity they can be uniformly approximated by functions of the form exp(-x 2 )p(x) where p is a polynomial (see [START_REF] Lang | Real and functional analysis[END_REF] page 62). This spectral interpolant automatically enforces the vanishing of the eigenfunctions at ±∞. We used N = 160 grid points giving a domain half size L = 8.61 (x ∈ [-L, L]). With this choice, we obtained |û N (±L)| = O(10 -8 ).

Figure 1 .

 1 Figure 1. Dimensionless nonlinear coefficient, μ, of the NLS equation as a function of the vorticity.

Figure 2 .

 2 Figure 2. Dimensionless profiles of solitary waves of elevation: Solid line (Ω = 0) and dashed line (Ω = 0.10).

Figure 3 .

 3 Figure 3. Dimensionless profiles of solitary waves of elevation: Solid line (Ω = 0) and dashed line (Ω = -0.1).

Figure 4 .

 4 Figure 4. Dimensionless profiles of solitary waves of depression: Solid line (Ω = 0) and dashed line (Ω = 0.10).

Figure 5 .

 5 Figure 5. Dimensionless profiles of solitary waves of depression: Solid line (Ω = 0) and dashed line (Ω = -0.1).

Figure 6 .

 6 Figure 6. Dimensionless profiles of solitary waves of elevation: Solid line (Ω = -0.10) and dashed line (Ω = -0.4).

Figure 7 .

 7 Figure 7. Dimensionless profiles of solitary waves of depression: Solid line (Ω = -0.10) and dashed line (Ω = -0.4).

Figure 8 .

 8 Figure8. Square of the dimensionless growth rate of the transverse unstable mode of the envelope of gravity-capillary solitary waves in deep water as a function of the square of the dimensionless wavenumber for Ω = 0 (solid line), Ω = 0.1 (•) and Ω = -0.1 ( ). The dashed lines correspond to the analytic expression given by equation (3.9). a0 = √ 2.

Figure 9 .

 9 Figure 9. Surface elevations of perturbed envelopes of gravity-capillary solitary waves of depression in deep water for Ω = 0, κ = 1, qmax = 1.01 (left) and Ω = 0.10, κ = 0.86, qmax = 1.08 (right)


P Lauret

P Slangen

F Heymes

L Aprin

N Lecysyn

A Osmont

NATURAL BACKGROUND ORIENTED SCHLIEREN AND MULTISCALE VISUALIZATIONS OF OVERPRESSURE WAVE RESULTING FROM VAPOR CLOUD EXPLOSION

. Because of the light fluctuation due to the explosion, the reference image for a frame corresponds to the previous frame. Direct visualization shows the combustion of the mixture with increasing velocity. A transition from deflagration to detonation is identified while the combustion front reaches the middle of the envelope. Analysis of the BOS images shows the presence of two overpressure waves of decreasing velocity outside the envelope. Evolution of the global dynamics of the phenomenon is achieved through the use of synchronization of near and far field in a composite image where RGB image layers are respectively set with the BOS image, the global view and the flame view scaled to fit the far field view.

INTRODUCTION

In case of vapor cloud explosions, velocities of both the combustion front and the blast wave are essential to correctly understand the whole phenomenon, including when deflagration to detonation transition is involved. Trinitrotoluene (TNT) equivalency method is still widely used by industrials to determine consequences of vapor cloud explosions. However, TNT equivalence method assumes that the equivalent blast source is punctual. When a Vapor Cloud Explosion (VCE) occurs, it is important to determine the volume of the explosive cloud and thus the point source assumption may be discussed. Moreover, the dynamics of the phenomenon can vary as well. The turbulent flame moves at various velocities, depending on the turbulence produced and the nature of the incoming gas oxygen mixture.

Improvements in high speed imaging technology (frame rate, sensors sensitivity, resolution…) give the opportunity of better understanding fast phenomenon such as explosion. When interest in measuring overpressures arises, it is possible to combine pressure sensors couples with high frame rate recording. This leads to highly accurate measurements of peak overpressure and impulse at one discrete point, using e.g. piezoelectric. High speed imaging is complementary to these measurements and contributes to enhance the spatial and temporal resolution. In case of combustion, the phenomenon to investigate is visible and thus the overpressure front is relatively easy to capture. Visualization of the overpressure front without combustion required special techniques to be set up. In this study, the size of the phenomenon to investigate is large. It is therefore required to use easy to set up technique for large field experiment.

STATE OF THE ART

The literature is abundant of review of visualization techniques for fluid flows. Versluis [START_REF] Versluis | High-speed imaging in fluids[END_REF] explains different applications depending on the characteristic length and temporal scale and velocity. A list of available techniques is given in order to capture overpressure wave. While classical shadowgraphy and schlieren techniques require special devices such as mirrors and powerful lighting, some others techniques are easier to set up, except the need to high frame rate recording.

Background Oriented Schlieren requires only a high speed camera. This technique consists in recording the variation of the refractive index due to target phenomenon between the camera and a contrasted background [START_REF] Settles | A review of recent developments in schlieren and shadowgraph techniques[END_REF][START_REF] Richard | Principle and applications of the background oriented schlieren (BOS) method[END_REF][START_REF] Gojani | Assessment of some experimental and image analysis factors for background-oriented schlieren measurements[END_REF]. This background can be made from printer for example. Special pattern are drawn on a screen. The interested refractive index modification, located between the background and the camera, deviates the light rays recorded by the camera. In order to retrieve what have moved, it requires little numerical computing: the target image is subtracted to a reference one, without refractive index variation as in equation [START_REF] Heymes | An experimental study on vapor cloud explosion of propane-oxygen stoichiometric mixture[END_REF].

𝑛𝑒𝑤(𝑖, 𝑗) =

(𝑓𝑙𝑜𝑤(𝑖,𝑗)-𝑡𝑎𝑟𝑒(𝑖,𝑗)) 2 𝑓𝑙𝑜𝑤(𝑖,𝑗)+𝑡𝑎𝑟𝑒(𝑖,𝑗) 2 +1

(1)

tare is the reference image, flow is the considered image, new is the BOS image result, i and j represents respectively raw and columns index of the image. As mentioned by [START_REF] Hargather | Background-oriented schlieren diagnostics for large-scale explosive testing[END_REF], this method reveals the overpressure wave location by distorting the background. Once these calculations have been made, it is possible to use a crosscorrelation to measure the pixel shift between images, because the pixel shift is related to the magnitude of the density gradient.

Hargather [START_REF] Hargather | Naturalbackground-oriented schlieren imaging[END_REF] has compared several natural backgrounds that can be used to reveal refraction index modification. Generally, the background is not chosen and results from the selection of the test facility with safety criteria. In [START_REF] Mizukaki | Background-oriented schlieren with natural background for quantitative visualization of open-air explosions[END_REF], Mizukaki uses a background composed of soil, grass, trees and clouds. The soil, grass and the sky are not perfectly contrasted and do not provide qualitative results whereas the trees allow identifying shockwaves. Both subtraction and cross-correlation are used in these works. Subtraction gives qualitative information on the shockwave. The reference image can be a frame before the beginning of the phenomenon. Doing this, every modification in the brightness between the two images is clearly enlightened. Therefore, using reference before phenomenon can lead to soften the shock wave in the resulting image. Hargather showed that subtraction might be simpler and give higher resolution for the shock leading edge location in certain situation.

METHODS

In order to reproduce the effects of a vapor cloud explosion in the near field at an intermediate size, a 5.5 m 3 envelope of stoichiometric mixture of methane and oxygen was used. The cloud geometry parameters are defined in table 1. The free vent distance 𝐿 𝐹𝑉 is defined as the minimum of the cloud height H and half the cloud width W. The cylindrical distance 𝐿 𝐶𝑦𝑙 is introduced to measure the maximum distance a cylindrical flame front can travel before reaching the edge of the cloud. The flame travel distance 𝐿 𝑓 is the distance from ignition to the flame front, with a maximum value equal to the distance to the edge of the cloud. As described by [START_REF] Geng | A study of the blast wave shape from elongated VCEs[END_REF], the cloud is considered as elongated if the ratio of the cloud length (𝐿 = 6,5𝑚) to the free vent distance 𝐿 𝐹𝑉 is larger than 10 and the ratio of the cylindrical distance to the free vent distance (𝐿 𝐶𝑦𝑙 𝐿 𝐹𝑉 ⁄ ) does not exceed 2. These parameters were respected in this study: (2)

The envelope was made of 200 µm thin polyethylene. Due to plastic welding, it was not a perfect cylinder as one can see on the following picture: Location of both the experimental apparatus and recording location were imposed for safety reasons. Therefore, the background pattern was composed of natural bushes and trees, by different orientation for example. As shown on figure 4, several zones can be noticed:

-The direct environment of the envelope was mostly composed of sand, soil and grass with few bushes. -As also seen on satellite view, bushes were mostly present in a second plan. Contrast was more important. -Due to meteorological conditions, the cloudy sky was not contrasted. Global lighting of the scene was influenced by the sunshine. As expected, the contour lines are enlightened. Envelope, sensors base, earth-sky interface have the best local contrast values (red). Globally, bushes and soil give better results than sky. Blast wave between the camera and the background should be less visible when the sky is reached, except at the earth-sky interface.

RESULTS AND ANALYSIS

The characterization of combustion regime requires the flame velocity measurement. Visualized images of the flame in the gas mixture are shown in figure 6. In this figure, each image is separated by 1.25 ms. Violet line points the combustion front of the mixture inside the envelope. On the left side, the blue line shows the split of the envelope due to expansion burning gases. In addition, an increase in light intensity is noticed when the flame front reached the middle of the envelope. It corresponds to creation of a hemispheric front, and a velocity of approximately 2390 m.s -1 . This corresponds to a detonation of the methane-oxygen mixture in stoichiometric quantities. After investigations, it appears that the location of the starting detonation corresponds to the location of the filling fitting connection to the envelope. This accelerating front is emphasized in figure 7: This reaction front creates an aerial blast wave as it is transmitted to the air outside the envelope. This blast wave is not visible with this camera because no luminous combustion process is then realized. BOS processing is done on the phantom V711 pictures as seen on figure 8. In addition to equation ( 1), image values are mapped in the whole accepted interval to increase contrast. As mentioned previously, best results are obtained in the bushes/soil area. The blast wave is more visible in the third BOS image at the cloud-bushes interface, as contrast is very high in this area. Initial deflagration wave is slightly visible on the left side of the envelope. Main difference with shock waves from explosives is the shape of the overpressure wave: no symmetry is reached. No classical hemispherical shape is retrieved.

Positions of the envelope and the different pressure sensors are lost with BOS image. Moreover, with the high light intensity generated by the combustion, it is impossible to see overpressure wave inside. Nevertheless, these images give useful information on the velocity of the overpressure wave in the near field of the cloud.

Understanding the whole phenomenon in the near field is made easier with by synchronizing all the data from image sensors and pressure gauges. The whole phenomenon is reconstructed by creating RGB composite image superposing original image in green layer, BOS image in red layer and dimensioned v2512 image on blue layer (figure 9). The last one is reduced in resolution. Because of the differences in recording frame rates, images of blue layer are displayed at 2.1 times the final frame rate. All the data can be displayed and discussed on x-t diagram (figure 10). On the first 13 ms, the combustion front is reported until the transition happens from v2512 images. Then, both left and right fronts are reported. It corresponds to detonation of the mixture, traveling at a speed of about 2390 m.s -1 . Limits of the envelope are reported in the green rectangle at bottom of figure 10. A slight gap in the measurements of the velocity is visible when using V711 images, due to light intensity of the combustion. Outside the envelope, the blast wave slows down rapidly (figure 11).

Figure 11: Blast waves velocity outside the envelope. Colors refer to corresponding rectangle in figure 10. In order to smooth uncertainties on blast wave position at each time step, velocity is computed between 30 images. Green lines illustrate geometrical the limits of the initial envelope.

Velocity of the blast wave decreases as traveling far from the envelope, from about 650 m.s -1 to 360 m.s -1 at the end of the visualization of the right side blast wave (end of the image). Gaps visualized at +/-13 m are due to the lower distance traveled by the blast waves, increasing uncertainties on the velocity.

CONCLUSION

Combustion dynamics on methane-oxygen stoichiometric mixture and ignited by an electric spark was investigated. Quantitative analyses of different overpressure wave velocities resulting of the combustion of the methaneoxygen mixture of the cloud was done, both by direct visualization and natural BOS algorithm. Quality of the background has been evaluated before recording. Local contrast map have been created and showed agreements with results in BOS processing. Near field and far field were characterized and recorded synchronously. The near field, recorded by direct visualization of the phenomenon shows a transition in the combustion front, at the middle of the envelope. The resulting detonation regime propagates at about 2390 m.s -1 while decelerating as combustion stops outside the envelope and the front becoming an aerial blast wave. These temporal data obtained by BOS imaging techniques are in perfect accordance with pressure synchronized sensors. BOS images also give the velocity decrease of the blast waves outside the envelope. Post processing consists in creating composite images of both the direct zoomed visualization of the combustion, the direct visualization of the aerial shock wave and the BOS processing, allowing better knowledge of the phenomenon on one solely video file. In a future work, algorithm like Optical flow or Particle Image Velocimetry (PIV) will be used in order to access information about the intensity of the overpressure captured by the cameras. Quality of the background has to be investigated in order to avoid important noise that can results in wrong values. Moreover, the intensity change due to the light generated by the mixture combustion in the background has to be taken into account.

ACKNOWLEDGEMENT

The authors would like to acknowledge the support of Camp des Garrigues military camp, Nîmes, France.

NOMENCLATURE

Figure 1 :

 1 Figure 1: polyethylene envelope on the test field The apparatus is unfolded on a military site due to safety requirements. 1.83 m 3 of methane and 3.67 m 3 of oxygen are premixed (T amb : 299 K ; P amb : 1013 mbar). The mixture filled the envelope of 5.5 m 3 . Fourteen PCB 137A23 blast pressure sensors were disposed around the envelop at 3, 5, 10.3 and 20 m from the center of the envelope at both axial and longitudinal directions as shown in figure 2. Data acquistion for blast sensors was achieved with HBM Genesis GEN7T at 200kHz.

Figure 2 :

 2 Figure 2 : Location of overpressure sensors Altitude of the sensors is respectively 0.88 m, 1.80 m, and 1.95 m for ○, + and □. As overpressure signals recorded by these sensors are punctual, visualizations techniques were added. Three high speed cameras were synchronously used. The first one, a Phantom V2512, was set to visualize the flame front inside the envelope. With the same direction of sight, another camera, Phantom V711, was set to capture the overpressure wave using natural BOS. Because of the 3D aspects expected from the overpressure wave, a last camera, Photron SA-3, was set with an angle of 45° with respect of the axis of the envelope. The cameras implementation is presented in figure 3.

Figure 3 :

 3 Figure 3: location of visualization Specifications of the viewing setup are listed in Table2:

Figure 4 :

 4 Figure 4: Natural background pattern, as seen by Phantom V711 Analysis of the general background can be done quantitatively. The normalized gradient of this image figures the local contrast. It is computed and displayed on figure 5:

Figure 5 :

 5 Figure 5: Normalized image gradient of the initial background before explosion Values near to zero (dark blue) represent low local contrast.As expected, the contour lines are enlightened. Envelope, sensors base, earth-sky interface have the best local contrast values (red). Globally, bushes and soil give better results than sky. Blast wave between the camera and the background should be less visible when the sky is reached, except at the earth-sky interface.

Figure 6 :

 6 Figure 6: Flame propagation from ignition (left) to right corner in the envelope from V2512.

Figure 7 :

 7 Figure 7: transition in the combustion front from V2512.Time step between images is 300 µs

Figure 8 :

 8 Figure 8: Blast wave in the free fieldleft: Original imagesright: BOS processing. 106 µs between images.

Figure 9 :

 9 Figure 9: Composite RGB image at t=25 ms with original image in green layer, BOS image in red layer and dimensioned v2512 image on blue layer. All images are compiled in video to visualize the whole phenomenon at once.

Figure 10 :

 10 Figure 10: x-t diagram of the explosion. X-axis represents the axial line of the envelope with the origin centered on the envelope. t = 0 corresponds to the ignition of the electric spark. Overpressure signals of B1, B4, B6 and A4 sensors are represented at their respective locations with the same dimensionless scale.

Table 1 .

 1 Cloud geometry parameters

	Parameter	Definition		length (m)
	Free vent distance 𝐿 𝐹𝑉 Cylindrical distance 𝐿 𝐶𝑦𝑙	𝑚𝑖𝑛 (𝐻, 𝑚𝑎𝑥(𝐻,	𝑊 2 𝑊 2	) )	0,6 1,2

Table 2 :Table 2 .

 22

		camera specifications
	Camera	V2512	V711	SA3
	Technique	Direct	NBOS	NBOS
		visualization
	Resolution	1024 * 704	1200*640	768*512
	Frequency	20 000	9 400	3 800
	(fps)
	Magnification	9.6	28	25
	(mm/pxl)
	Field of view	9.8 * 6.8	33.7*18	19.2*12.8
	(m.m)
	Exposure	0.4	20	167
	Time (µs)

  L FV Free vent distance, m 𝐿 𝐶𝑦𝑙 Cylindrical distance, m

	H	Cloud height, m
	W	Cloud width, m
	L	Cloud length, m
	𝐿𝑓	Flame travel distance, m


Denis Bourras

Rémi Cambra

Louis Marié

Marie-Noëlle Bouin

Lucio Baggio

Hubert Branger

Houda Beghoura

Gilles Reverdin

Boris Dewitte

Aurélien Paulmier

OCARINA (Ocean coupled to the atmosphere: instrumented research on an auxiliary ship)

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Theo Persenot
email: theo.persenot@insa-lyon.fr

Alexis Burr

Guilhem Martin

Jean-Yves Buffiere

Remy Dendievel

Eric Maire

Effect of build orientation on the fatigue properties of as-built Electron Beam Melted Ti-6Al-4V alloy

Keywords: Electron Beam Melting, Titanium alloys, Fatigue behavior, X-ray microtomography, As-built properties, Manufacturing orientation

As-built Ti-6Al-4V thin parts were manufactured in three different orientations using EBM and characterized by laboratory X-ray tomography. Fatigue tests were performed. The comparison with results for machined samples from the literature showed a large reduction of fatigue strength. SEM observations of the fracture surfaces showed that surface defects which were identified as notch-like defects on tomographic images caused the failure. Their impact on fatigue results was rationalized by Kitagawa-Takahashi diagrams. A build orientation impact on the fatigue properties was observed and linked to its effect on defects distributions and crack growth. The limits of roughness measurements were also discussed.

Introduction

Combining good mechanical properties with interesting thermal or acoustic properties is of great interest and, to this end, architectured materials are good candidates to replace monolithic materials. In particular, for industries where weight saving is also considered such as aeronautics industry, lattice structures are extremely promising, see e.g. [START_REF] Fleck | Micro-architectured materials: past, present and future[END_REF], [START_REF] Ashby | Designing architectured materials[END_REF], [START_REF] Ashby | Designing hybrid materials[END_REF], [START_REF] Gibson | Cellular solids: structure and properties[END_REF], [START_REF] Evans | Multifunctionality of cellular metal systems[END_REF], [START_REF] Evans | The topological design of multifunctional cellular metals[END_REF], [START_REF] Schaedler | Ultralight 345 metallic microlattices[END_REF], [START_REF] Wadley | Multifunctional periodic cellular metals[END_REF]. However, until recently, it was very challenging to build such metallic lattice structures with conventional processes as multiple steps were required, see e.g. [START_REF] Kooistra | Compressive behavior of age hardenable tetrahedral lattice truss structures made from aluminium[END_REF], [START_REF] Kooistra | Lattice truss structures from expanded metal sheet[END_REF]. With the emergence of powder-bed based additive manufacturing techniques such as selective laser melting (SLM) and electron beam melting (EBM), producing such structures is becoming easier.

If series production is to be considered, one needs to determine reliable mechanical properties for the manufactured lattices, see e.g. [START_REF] Park | Effective mechanical properties of lattice material fabricated by 360 material extrusion additive manufacturing[END_REF], [START_REF] Hernndez-Nava | The effect of density and feature size on mechanical properties of isostructural metallic foams produced by additive[END_REF], [START_REF] Yang | Modeling of uniaxial compression in a 3d periodic re-entrant lattice structure[END_REF], [START_REF] Park | Duty. comparing mechanical and geometrical properties of lattice structure fabricated using electron beam melting[END_REF], [START_REF] Lewandowski | Metal additive manufacturing: A review of mechanical properties[END_REF]. As damage caused by cyclic mechanical loading is the main cause of failure 10 for industrial parts, determining the fatigue properties of additively manufactured samples is mandatory. It has been demonstrated that the fatigue properties of Ti-6Al-4V EBM samples can be equivalent to those of the wrought materials geometries are considered (thin parts or lattice structures), conventional machining techniques are no longer usable and other appropriate surface treatments have to be considered. Some have already been investigated in the literature: chemical polishing [START_REF] De Formanoir | Improving the mechanical efficiency of electron beam melted titanium lattice structures by chemical etching[END_REF], [START_REF] Lhuissier | Geometrical control of lattice structures produced 405 by ebm through chemical etching: Investigations at the scale of individual struts[END_REF], [START_REF] Hooreweder | Improving the fatigue performance of porous metallic biomaterials produced by selective laser melting[END_REF], electrochemical polishing [START_REF] Grzegorz | Surface modification of ti6al4v open porous structures produced by additive manufacturing[END_REF], [START_REF] Urlea | Electropolishing and electropolishing-related allowances for powder bed selectively lasermelted ti-6al-4v alloy components[END_REF], shot peening [START_REF] Benedetti | The effect of post-sintering treatments on the fatigue and biological behavior of ti-6al-4v eli parts made by selective laser melting[END_REF], [START_REF] Calignano | Influence of process parameters on surface roughness of aluminum parts produced by dmls[END_REF] or abrasive flow machining [START_REF] Mohammadian | Surface finish control of additively-manufactured inconel 625 compo-430 nents using combined chemical-abrasive flow polishing[END_REF] but, in most cases, the obtained surface finish does not match that of machined surfaces. This can partly explain why the development of lattice structures in the industry is still limited. [START_REF] Ackelid | Additive manufacturing of dense metal parts by electron beam melting[END_REF] In addition, due to the layer by layer nature of the additive manufacturing processes, the orientation of one sample in the building chamber can affect its microstructure and mechanical properties, see e.g. [START_REF] Lewandowski | Metal additive manufacturing: A review of mechanical properties[END_REF] for a recent review. As a lattice structure is composed of struts built with different orientations, the impact of the build orientation on the defects and on the fatigue properties becomes a key issue.

Furthermore, for obvious geometrical considerations, the stress/strain distribution within lattice structures are complex; two types of lattices can be distinguished: the bending dominated structures and the stretching dominated ones [START_REF] Ashby | The properties of foams and lattices[END_REF]. For the latter, uniaxial tension and compression are the predominant loading modes. Consequently, the fatigue behaviour of individual struts in any stretching dominated structure can be analyzed based on the results of uniaxial fatigue tests. For both kinds of lattices, complex local loading conditions are known to occur at the vicinity of the nodes, see e.g. [START_REF] Huynh | Fatigue behavior of in718 microtrusses produced via additive manufacturing[END_REF], [START_REF] Tancogne-Dejean | Additively-manufactured metallic micro-lattice materials for high specific energy absorption under static and dynamic loading[END_REF], [START_REF] Lietaert | Fatigue life of additively manufactured ti6al4v scaffolds under 445 tension-tension, tension-compression and compression-compression fatigue load[END_REF]. This critical point however falls outside the scope of this paper which mainly aims at characterizing the effect of the build orientation on the fatigue performance of as-built Ti-6Al-4V thin parts produced by EBM. To mimic the behaviour of struts, thin fatigue tensile specimens with a nominal diameter of 2 mm have been used. A method combining 3D X-ray microtomography imaging before and after the fatigue test with SEM observations of the fracture surfaces was developed in order to identify the critical defects. Correlations between the build orientation, the defects and the fatigue properties were finally established.

Experimental procedures

Samples manufacturing

Cylindrical fatigue tensile specimens with a 10 mm gauge length and a 2 mm nominal diameter were produced using an ARCAM A1 EBM machine and Ti-6Al-4V ELI powder. The chemical composition of the initial powder batch is given in Table 1 (Ti-6Al-4V grade 23). The EBM machine operated at 60 kV accelerating voltage under a controlled pressure of He of 2.10-3 mbar. 50 µm thick powder layers were deposited on a stainless steel plate, then slightly sintered with a defocused beam and finally selectively melted according to the input CAD geometry (Figure 1).

The small nominal diameter of the samples was chosen so as to be representative of a single strut of a lattice structure (large surface to volume ratio). These thin specimens were manufactured in three differents orientations: their tensile axis was either perpendicular (vertical sample), tilted at 45 • or parallel (horizontal sample) to the build plate, see Figure 1. Throughout this paper, the samples manufactured in these three orientations will be respectively referred to as 90 • samples, 45 • samples and 0 • samples. These three orientations have been chosen because they are representative of struts in any lattice structure, whatever its build orientation and unit cell. From a thermal point of view, 90 • samples are manufactured in optimized conditions (for each layer, the area to be melted stands over bulk material) whereas the bottom of the 0 • samples is directly manufactured on unmelted powder particles (limited thermal dissipation). The case 50 of the 45 • samples stands in between these two opposite situations and those samples are representative of most struts with a tilted axis.

A preheating temperature of ∼ 730 • C was maintained throughout the build. The standard "Melt" build theme recommended by ARCAM and defined in the EBM Control 3.2 software was used. The melting strategy consists in scanning the area defined by the CAD starting by the sample contours (2 contours). It is then followed by the hatching 55 of the inner part of the CAD section with a snake-like pattern (line offset set to 0.1 mm and line order set to 1). The scanning direction was rotated by 90 after every layer.

X-ray tomography

The full 3D characterization of the gauge length of the fatigue samples was obtained using laboratory X-ray tomography.

The fatigue specimens were inspected in the as-built condition with a cone beam phoenix| x-ray V| tome| x laboratory 60 tomograph using a 2.5 µm voxel size (spatial resolution of ∼ 4 µm). Detailed information on the setup can be found in [START_REF] Buffiere | In situ experiments with x ray tomography: an attractive tool for experimental mechanics[END_REF].

5 scans were required at different heights along the sample axis to image the whole gauge length. The following scanning parameters were used: voltage V=90 kV, current I=240 µA, exposure time=333 ms, 720 projections per scan. No filter was needed. The resulting scanning time per sample (over its entire length) was roughly 1h. The reconstruction of the 3D images was performed using a standard filtered back projection method (phoenix datos x software). The reconstructed volumes were then post-treated and analyzed using Fiji software. The pore volume fraction was obtained by thresholding the 8 bits gray scale images. 1 .

Following the method initially suggested by Suard et al. [START_REF] Suard | Mechanical equivalent 450 diameter of single struts for the stiffness prediction of lattice structures produced by electron beam melting[END_REF], the tomographic images were used to evaluate the samples roughness. A series of profiles of the surface was extracted from a radial slice every 10 • around the sample circumference and used to compute an average measurement of the arithmetic (Ra) as well as the maximum (Rt) roughnesses (see [START_REF] Suard | Mechanical equivalent 450 diameter of single struts for the stiffness prediction of lattice structures produced by electron beam melting[END_REF] and [START_REF] Persenot | Enhancing the tensile properties of ebm as-built thin parts: Effect of hip and chemical etching[END_REF] for details). The values obtained with this method have been compared to measurements performed with an INTRA2 Taylor Hobson profilometer.

Fatigue testing

As-built fatigue tensile specimens were tested at room temperature using a servo-hydraulic INSTRON 8516 fatigue testing machine equipped with a 5kN cell force. Constant amplitude stress controlled uni-axial tests were carried out with a frequency f=10Hz (sinusoidal wave form) and a stress ratio R=0.1 to comply with the aircraft industry requirements [START_REF]Airbus Industrie Test Method standards[END_REF]). Because as-built sample cross section varies along the gauge length, the cyclic maximum and mean stress levels were calculated based on the average section along the gauge length determined from the tomographic scans of the samples.

This definition was found to be more appropriate for stress calculation in this type of samples [START_REF] Persenot | Enhancing the tensile properties of ebm as-built thin parts: Effect of hip and chemical etching[END_REF]. The fatigue tests were performed until failure or stopped after 10 7 cycles. The fracture surfaces of the samples were observed using scanning electron microscopy under secondary electron contrast and the broken samples were imaged by X-ray tomography.

Results

Fatigue properties

The S-N curves of the as-built samples are presented in Figure 2 where results from the literature for machined samples have also been added (hollow symbols) [START_REF] Arcam | Documentation from Arcam training level[END_REF]. The build parameters (powder, machine, build theme, layer thickness, ...) for 85 these machined samples are the same as the ones used in this study but the samples geometry differs: ARCAM data were obtained on larger cylindrical tensile specimens that were machined from bulk materials. It is likely that this machining step has removed sub-surface pores as shown in [START_REF] Yadollahi | Additive manufacturing of fatigue resistant materials: Challenges and opportunities[END_REF]. For the 90 • samples, several stress levels were investigated in order to obtain fatigue lives ranging between 10 4 and 10 7 cycles. A clear reduction of the fatigue properties can be observed for the as-built samples tested in this work compared to machined samples. For instance, the fatigue limit σ f obtained 90 at 10 7 cycles is equal to σ f =140MPa whereas for machined 90 • samples, this value is of the order of σ f =400MPa [START_REF] Arcam | Documentation from Arcam training level[END_REF].

It can be seen from Figure 2 that the build orientation also has a significant impact on the measured fatigue properties.

The 45 • and 0 • samples have been tested at a maximum stress level σ max =290MPa corresponding to fatigue lives around 10 5 cycles since this value is classically used in aeronautical design [START_REF]Airbus Industrie Test Method standards[END_REF]. At this stress level, the average fatigue lives of the 90 • , 45 • and 0 • samples are respectively 50 000, 155 000 and 250 000 cycles. This effect of build orientation was not 95 reported in the literature for the machined samples (red and blue hollow symbols in Figure 2).

In order to identify the failure mechanisms, the fracture surfaces of samples with different build orientations have been systematically examined by SEM. Three representative fracture surfaces are shown on Figure 3. The relatively large flat zone corresponds to the stable crack propagation regime, its size a st scales with the fatigue lives (a st,90

• < a st,45 • < a st,0 • ).
For the three specimens shown on Figure 3, the crack leading to failure initiates from a surface defect. The Defects at the sample surface are therefore critical for the fatigue resistance, as already reported, see e.g. [START_REF] Yadollahi | Additive manufacturing of fatigue resistant materials: Challenges and opportunities[END_REF]. A detailed characterisation of those defects but also of those present in the bulk is presented in the next section.

Characterization of defects produced by the EBM process

Porosity 105

The pore volume fraction in as-built samples estimated from the 3D images is reported in Table 2 for each build orientation. A low value of internal porosity (< 0.05%) was measured in the gauge volume of all the samples, confirming the ability of EBM to produce almost fully dense bulk samples. The vast majority of the pores exhibited a nearly spherical morphology with an average diameter of ∼ 20 µm. Tammas-Williams et al. [START_REF] Tammas-Williams | Xct analysis of the influence of melt strategies on defect population in ti6al4v components manufactured by selective electron beam melting[END_REF] showed that such pores, usually named gas pores, are already present in the initial powder particles. Very few internal lack-of-fusion defects were observed.

110

Table 2: Effect of the build orientation on the as-built EBM samples properties: average section, internal porosity and roughnesses. The average values and the standard deviations are reported. For comparison, the nominal section is 3.14 mm 2 . Figure 4 shows projected views of the internal pores for the three differents orientations. A sub-surface circular region with a higher density of pores can be observed for the three orientations. Its width decreases when the build orientation goes from 90 • to 0 • . In the central part of the latter, some horizontal patterning with a higher density of pores can also be identified.

Material

Roughness measurements and surface irregularities 115

The roughnesses values calculated for samples manufactured in the three different orientations are reported in Table 2. The lowest average roughness (Ra) is obtained for the horizontal samples (0 • ) whereas the vertical samples (90 • )

show the highest values (30.6 ± 2.24 µm and 44.0 ± 3.6 µm respectively). Regarding the maximum roughness (Rt), the lowest and highest values are obtained respectively for the 45 • samples and 0 • samples (350 ± 45.5 µm and 515 ± 30.5

µm respectively). The build orientation has clearly an impact on the surface roughness of as-built samples. Additional 120 roughness measurements performed on some 90 • samples with a profilometer show a good agreement between the two different methods: the average value of Ra obtained by profilometry is 36.9 µm (v.s. 44.0 µm from tomographic images).

The 3D rendering of the sample surfaces shown in Figure 5 illustrates the effect of the build orientation at a macroscopic scale (visual aspect of the sample surface). In the case of 0 • (Figure 5b and5d) and 45 • samples (Figure 5c and Figure 5e), a clear difference between the upper and lower sample surfaces can be observed. In both cases, the upper surface is 125 smoother than the lower one. This tendency is more pronounced for the horizontally manufactured samples: the upper surface of the 0 • sample looks quite smooth while the successive manufacturing layers are visible on the 45 • sample upper surface (Figure 5g and 5i respectively). As there is no upper nor lower surface for the gauge length of 90 • samples, the surface state of those samples shows no evolution along its circumference (Figure 5f).

The average roughnesses of the upper and lower surfaces were computed for the 0 • and 45 • samples (Table 2). As 130 could be expected, in both cases, the upper and lower surfaces exhibit respectively lower and higher Ra values than the average ones. It is worth pointing out that for both build orientations, the roughness of the lower surfaces meets (45

•
samples) or exceeds (0 • samples) the average value of 90 • samples which show the highest average Ra value.

Surface defects and critical defect identification

The surface irregularities generated by the EBM process can be observed on radial tomographic slices as shown in 135 Figure 6. Two different types of surface defects can be observed: unmelted powders stuck to the struts and irregularities in the layer stacking, the so-called "plate-pile" like stacking defects described in [START_REF] Suard | Mechanical equivalent 450 diameter of single struts for the stiffness prediction of lattice structures produced by electron beam melting[END_REF], [START_REF] Lhuissier | Geometrical control of lattice structures produced 405 by ebm through chemical etching: Investigations at the scale of individual struts[END_REF], [START_REF] Körner | Fundamental consolidation mechanisms during selective beam melting of powders[END_REF]. The latter are observed in the 45 • and 90 • samples and they can sometimes lead to notch-like defects [START_REF] Persenot | Enhancing the tensile properties of ebm as-built thin parts: Effect of hip and chemical etching[END_REF]. These defects (red arrows in Figure 6d and 6f) are always oriented perpendicularly to the building direction. There are fewer defects of this type in the 45

•
samples than in the 90 • whereas hardly any can be observed in the 0 • samples (for example, no notch-like defect can be 140 found in the radial slice in Figure 6a). A schematic illustration of the different types of defects observed in EBM as-built samples is shown on Figure 7.

In order to identify the defect that lead to the fatal crack, tomographic scans performed before and after failure were acquired and compared. This enables to determine the tomograpic slice at which failure occured, see Figure 8. Then, from the SEM image of the fracture surface and a transverse reconstructed slice, the critical defect can be identified and one can get access to its exact geometry (size and shape) on the tomographic images before the fatigue test. For all investigated samples, and regardless of their build orientation, the crack was found to initiate from a "plate-pile" like stacking defect.

In the case of the 45 • and 90 • samples, notch-like defects were observed at the level of the stacking irregularity at which the cracks initiated. None were detected for the 0 • samples. Example of defects leading to the failure are shown in Figure 9 for the three build orientations.

Discussion

Impact of as-built state on fatigue properties

The fatigue results presented in Figure 2 show that, for every build orientation considered, the fatigue resistance of as-built samples is considerably lower than that of the machined samples [START_REF] Arcam | Documentation from Arcam training level[END_REF]. The as-built samples were manufactured using the same type of machine, the same powder and the same standard "Melt" build theme with the same layer thickness 155 than ARCAM used in reference [START_REF] Arcam | Documentation from Arcam training level[END_REF]. The microstructure of the as-built samples was reported in a previous study [START_REF] Persenot | Enhancing the tensile properties of ebm as-built thin parts: Effect of hip and chemical etching[END_REF] and was found to be similar to the one reported by ARCAM and other literature sources (see for example [START_REF] Ackelid | Additive manufacturing of dense metal parts by electron beam melting[END_REF], [START_REF] Galarraga | Effects of the microstructure and porosity on properties of Ti-6Al-4V ELI alloy fabricated by electron beam melting (EBM)[END_REF] and [START_REF] Facchini | Microstructure and mechanical properties of ti6al4v produced[END_REF]) despite the size difference between both kinds of samples: fine α-laths (∼ 1 µm width) separated by some residual

β phase.
For both SLM and EBM Ti-6Al-4V alloys machined samples, the reported crack initiation sites were either internal, 160 sub-surface or surface defects, see for example [START_REF] Li | Critical assessment of the fatigue performance of additively manufactured ti6al4v and perspective for future research[END_REF], [START_REF] Zhao | Comparison of the microstructures and mechanical properties of ti6al4v fabricated by selective laser melting and electron beam melting[END_REF], [START_REF] Gong | Influence of defects on mechanical properties of ti6al4v components produced by selective laser melting and electron beam melting[END_REF] or [START_REF] Leuders | On the fatigue properties of metals manufactured by selective laser melting the role of ductility[END_REF]. Those defects were often gas pores which were internal in the as-built state and brought to the surface or sub-surface during machining ( [START_REF] Yadollahi | Additive manufacturing of fatigue resistant materials: Challenges and opportunities[END_REF], [START_REF] Gong | Influence of defects on mechanical properties of ti6al4v components produced by selective laser melting and electron beam melting[END_REF]). Such internal or sub-surface defects were also observed in the as-buit samples of this study (Figure 4) but they were never found to be responsible for the sample failure. Instead, the crack leading to failure initiated from stacking irregularities (Figure 3) which were notch-like defects in the case of 45 • and 90 • samples (radial slices in Figure 9). Similar initiation sites were reported for 165 as-built SLM manufactured samples [START_REF] Kasperovich | Improvement of fatigue resistance and ductility of tial6v4 processed by selective 490 laser melting[END_REF].

The ability of a defect to initiate a fatigue crack can be assessed through its stress concentration factor K t [START_REF] Schijve | Fatigue of structures and materials[END_REF]. A larger local stress results in an increased level of plasticity which leads to crack initiation because of slip irreversibility.

In what follows, K t is used to compare the harmfulness of pores (internal or sub-surface) and that of "plate-pile" like stacking defects. A detailed analysis of the problem would theorically require the calculation of the local plastic strain 170 concentrations induced by the defects. However, such calculations are computationally intensive specially in the cyclic case.

Besides, Tammas-Williams et al. [START_REF] Tammas-Williams | The influence of porosity on fatigue crack initiation in 495 additively manufactured titanium components[END_REF] have shown by FE calculations that, in the neighborhood of a pore, the localizations of plastic strain concentrations correspond with a good accuracy to the zones where elastic strain concentrations occur.

In the rest of the discussion, therefore, only elastic strain concentrations are considered. With this assumption, the stress concentration factor K t of an internal spherical pore in an infinite elastic material is lower than 2 (perfect sphere). If this 175 pore approaches the surface, its K t can reach values of the order of 5 [START_REF] Borbély | A finite element modelling study of strain localization in the vicinity of near-surface cavities as a cause of subsurface fatigue crack initiation[END_REF].

Notch-like defects (see Figure 7) can be modelled as elliptical notches at the sample surface with a radius of curvature at the tip ρ and a depth a (Figure 7) for which the following classical formula can be used:

K t = 1 + 2 a/ρ (1)
The radius of curvature ρ at the tip of the elliptical notch is equal to b/a where b is the "thickness" of the notch. From Figure 6, typical values of a/b larger than 10 can be measured, leading to very large values of K t (for a/b = 10, K t = 21).

180

This impact of the aspect ratio of the defect on the K t values was also reported by Tammas-Williams [START_REF] Tammas-Williams | The influence of porosity on fatigue crack initiation in 495 additively manufactured titanium components[END_REF]. Such large K t values clearly explain why cracks systematically initiate at notch-like defects and not at internal or sub-surface pores.

In order to check that the important loss of fatigue performance observed for the as-built state is not related to the presence of internal and sub-surface pores, a HIP treatment (920 • C, 100MPa for 2 hours) was applied to as-built 90 • samples. X-ray tomography scans were performed before and after the treatment: with the spatial resolution of laboratory tomography (∼ 4 µm), no internal pore were detected after the HIP treatment as shown in Figure 10. No impact on the fatigue properties was observed: the HIP treated samples had the same fatigue lives than the as-built samples (pink v.s. blue dots in Figure 2). This demonstrates that, as far as as-built samples are concerned, the internal pores (and their removal) have no impact on the fatigue life. The systematic HIP treatment performed on machined samples (25% improvement of the fatigue limit σ f , see [START_REF] Li | Critical assessment of the fatigue performance of additively manufactured ti6al4v and perspective for future research[END_REF], [START_REF] Facchini | Microstructure and mechanical properties of ti6al4v produced[END_REF]) does not seem necessary for as-built samples.

190

If notch-like defects are more likely to initiate a crack than internal pores, this does not always mean that such a crack will propagate and lead to the failure of the sample. Consequently, there is a need to consider the ability of such cracks to propagate, which will be the focus of the next section.

Fatigue properties and as-built state: impact of build orientation

The comparison of the fatigue lives of as-built samples submitted to medium stress levels (σ max =290 MPa) shows an 195 impact of the build orientation (Figure 2) whereas no significant difference in fatigue properties can be found between the two orientations (90 • and 0 • ) of the machined samples (Figure 2). This impact on the fatigue properties is thus most likely due to the influence of the build orientation on the surface defects.

A traditional approach for designing components with materials containing defects is the damage tolerance approach.

The Kitagawa-Takahashi diagram [START_REF] Kitagawa | Applicability of fracture mechanics to very small cracks or the cracks in the early stage[END_REF] can be used to relate the fatigue properties of a material depending on the size 200 of the defects that it contains. Beretta and Romano [START_REF] Beretta | A comparison of fatigue strength sensitivity to defects for materials manufactured by am or traditional processes[END_REF] have recently shown that the Kitagawa-Takahashi diagram can reasonably well describe the fatigue properties of Ti-based and Al-based alloys fabricated by AM. The defects are considered as cracks and the stress intensity factor that they generate is compared to the crack growth threshold of the material in order to assess their ability to propagate. It has been shown above that, for the material studied here, the cracks leading to failure initiate from notch-like surface defects with a crack-like shape. Several approaches can be considered 205 for building the Kitagawa-Takahashi diagram (see for example [START_REF] Rotella | Influence of natural defects on the fatigue limit of a cast al-si alloy[END_REF]). The formulation based on the El Haddad's empirical parameter a 0 [START_REF] Haddad | Prediction of non propagating cracks[END_REF] is being used here:

a 0 = 1 π ∆K th Y * ∆σ f 2 (2)
where Y depends on the crack location (typically Y = 0.65 for a surface crack), ∆σ f is the stress range at the fatigue limit σ f and ∆K th is the crack growth threshold of the material (more details can be found in [START_REF] Kruzic | Kitagawa-takahashi diagrams define the limiting conditions for cyclic fatigue failure in human 515 dentin[END_REF]).

The fatigue limit σ f considered here is the one obtained for samples free of surface defects, i.e. machined samples.

210

Published results of standard fatigue crack growth tests on Ti-6Al-4V samples additively manufactured by EBM are used for the ∆K th values. Several studies ( [START_REF] Seifi | Evaluation of orientation dependence of fracture toughness and fatigue crack propagation behavior of as-deposited arcam ebm ti-6al-4v[END_REF], [START_REF] Seifi | Defect distribution and microstructure heterogeneity effects on fracture resistance and fatigue behavior of ebm ti6al4v[END_REF], [START_REF] Greitemeier | Fatigue performance of additive manufactured tial6v4 using electron and laser beam melting[END_REF], [START_REF] Zhai | Microstructure, static properties, and fatigue crack growth mechanisms in ti-6al-the International Conference on Structural Integrity[END_REF], [START_REF] Greitemeier | Effect of surface roughness on fatigue performance of additive manufactured ti-6al-4v[END_REF]) have been compared and the minimum, average and maximum values obtained from them have been reported in Table 3. For EBM machined samples, the ∆K th values provided by these studies were found to be dependent on the orientation of the notch with respect to the building direction, i.e. perpendicular (horizontal) or parallel (vertical). This dependency has been linked to the microstructural features such 215 as basket-weave and Widmanstätten microstructures which impact the nucleation and the short crack propagation [START_REF] Beretta | A comparison of fatigue strength sensitivity to defects for materials manufactured by am or traditional processes[END_REF], [START_REF] Collins | Microstructural control of additively manufactured metallic materials[END_REF]. The values used to plot the Kitagawa diagram displayed in Figure 11 are reported in Table 3. The fatigue results obtained for some as-built samples have then been reported in Figure 11. The defect size a was determined using the approach developed by Murakami [START_REF] Murakami | Analysis of stress intensity factors of modes i, ii and iii for inclined surface cracks of arbitrary shape[END_REF] which considers the projected area of the defect perpendicular to the load axis area so that: a = √ area. The resulting points are represented in blue, green and red symboles for 220 respectively the 90 • , 45 • and 0 • samples. As shown in Figure 6, the notches responsible for the samples failure are perpendicular to the building direction for 45 • and 90 • samples whereas they are parallel to it for the 0 • samples. The red and blue curves (crack growth respectively parallel and perpendicular to the build direction) will therefore be used respectively for the 0 • samples (red) and the 45 • and 90 • samples (blue).

For samples with an important surface roughness, it is quite difficult, based on X-ray tomography images, to distinguish between the beginning of the surface defect and the surface variations in the vicinity of the defect. As a result, the SEM images of the fracture surfaces (Figure 3) were used to obtain the projected size of the crack which has lead to fracture in order to check wether the Kitagawa-Takahashi approach is appropriate. Figure 2 shows that for 90 • samples, the knee region of the Wöhler curve correspond to a number of cycles of the order of ∼ 2 to 3.10 5 cycles. Broken samples with a fatigue life larger than this number of cycles can be placed directly on the Kitagawa-Takahashi diagram. As explained, however, lots of points on the Wöhler curve correspond to specimens with fatigue lives of the order of 10 5 cycles, for the three orientations. The points corresponding to those specimens cannot therefore be directly inserted on the Kitagawa-Takahashi diagram. Two different approaches have been tested to overcome this problem.

The first one suggested by Beretta and Romano [START_REF] Beretta | A comparison of fatigue strength sensitivity to defects for materials manufactured by am or traditional processes[END_REF] consists in extrapolating the fatigue lives from 10 5 cycles to larger fatigue lives corresponding to the knee region of the Wöhler curve. This has been done for the three orientations and the results are shown on Figure 11a. On this figure, the number of cycles used for extrapolation for 0 • and 45 • samples varies from 3 to 6.10 5 cycles which corresponds to the scatter of the data obtained at the knee-point for the Wöhler curve of the 90 • samples.

The second approach follows the work of Ciavarella and Monno who proposed a method to extend the Kitagawa-Takahashi diagram to finite lives prediction. The procedure can be briefly summarized as follow (for details, see [START_REF] Ciavarella | On the possible generalizations of the kitagawatakahashi diagram and of the el haddad equation to finite life[END_REF]):

when the fatigue resistance at a given number of cycles N f l is sought, both the left and the right part of the diagram should be modified. For the left part, the fatigue limit is defined as the stress amplitude corresponding to N f l on the Wöhler curve of the machined samples (Figure 2). For the right part of the diagram, a propagation law is assumed and integrated for each N f l . Here, a propagation law in the Paris regime taken from the work of Seifi et al. [START_REF] Seifi | Defect distribution and microstructure heterogeneity effects on fracture resistance and fatigue behavior of ebm ti6al4v[END_REF] was used for both 0 • and 90 • orientations. The results of this second approach are shown on Figure 11b (90 • samples) and Figure 11c 245 (0 • and 45 • samples).

Figure 11a shows that once the extrapolation is being carried out on the Wöhler curve, the experimental points fall reasonably well on the Kitagawa-Takahashi curves for the 3 investigated orientations. For 0 • samples, the predictions are no longer conservative once the extrapolated number of cycles used exceeds 4.10 5 cycles whereas, for 45 • samples, the predictions are always conservative and are improving when the number of cycles used for extrapolation increases.

250

The results from the second approach which aims at calculating pseudo Kitagawa-Takahashi diagrams for a given number of cycles are contrasted: good predictions are obtained for 0 • and 45 • samples (with very limited date available, see Figure 11c) but for 90 • samples, the fatigue resistance is clearly overestimated for fatigue lives shorter than the knee point of the Wöhler curve (Figure 11b). One reason might be that the cracks emanating from the notch-like defects might have a typical short crack behaviour with a faster growth rate than the one predicted by the classical Paris law established for long cracks (for a more detailed discussion on that point, see [START_REF] Ciavarella | On the possible generalizations of the kitagawatakahashi diagram and of the el haddad equation to finite life[END_REF]).

The Kitagawa approach shows that the critical parameters for assessing the fatigue strength are the crack growth threshold ∆K th and the stress intensity factor range ∆K which has been calculated through the √ area parameter.

As mentioned earlier, the ∆K th values found in the literature for 0 • samples are higher than the ones found for 90 • samples. Furthermore, Figure 3 shows that the build orientation has a direct effect on the surface defects size:

√ area 90 • 260 > √ area 45 • > √ area 0 • .
Thus, 0 • samples contain smaller critical surface defects as well as a microstructure which is more resistant to crack growth. Both factors can account for the longer fatigue lives measured for these samples.

In addition, even in the case of similar √ area values, some authors have shown that there is an influence of the defect orientation on the fatigue strength. In a recent paper, Lorenzino [START_REF] Lorenzino | Effect of small defect orientation on fatigue limit of carbon 545 steels[END_REF] demonstrated that the √ area parameter tends to be less predictive of the defect harmfulness when the angle between the load axis and the defect major axis is decreased 265 from 90 • up to 30 • , the harmfulness of the latter being slightly overestimated. It was also shown in [START_REF] Lorenzino | Effect of small defect orientation on fatigue limit of carbon 545 steels[END_REF] that this tendency was exacerbated when the yield strength of the material was high which is typically the case of the EBM Ti-6Al-4V alloy studied here (yield strength = 850MPa, [START_REF] Persenot | Enhancing the tensile properties of ebm as-built thin parts: Effect of hip and chemical etching[END_REF], [START_REF] Facchini | Microstructure and mechanical properties of ti6al4v produced[END_REF]). As shown in Figure 6, the notch-like defects of 90 • and 45 • samples are respectively perpendicular and tilted at 45 • with respect to the loading direction. This could explain the discrepancy in terms of fatigue lives between 90 • and 45 • samples: the defects found in the 90 • samples present the most harmful tilt.

270

Both effects could therefore account for the observed dependence of the fatigue strength on the build orientation.

Another factor that can impact the fatigue properties is the number of defects which can initiate a crack. Figure 6 shows the differences between the density of surface defects within the gauge length of the three types of samples. Hardly any notch-like defects can be identified in the 0 • samples whereas many can be observed in the 90 • samples, 45 • samples being somewhere between those two cases (Figure 6). Even though the fatigue strength should be linked to the extreme 275 values of the defects distribution [START_REF] Romano | Qualification of am parts: Extreme value statistics applied to tomographic measurements[END_REF], a larger number of defects increases the probability of having one that is critical and could also explain the influence of the build orientation on fatigue lives. A thorough study of the defects distributions for each build orientation would be required in order to confirm this tendency.

Notch-like defects detection and fatigue life prediction: is roughness the appropriate index ?

From an industrial point of view, it might be interesting-or even critical-to be able to detect the presence of the 280 notch-like surface defects by non-destructive tests (NDT) and to link fatigue properties to standard parameters such as the surface roughness.

On the one hand, the notches leading to failure are visible on the X-ray tomographic images (see Figure 6 and Figure 9) when a low voxel size is used (∼ 2.5 µm in this case). However, only parts with a size of the order of 1 cm could be calculation (see e.g. [START_REF] Suard | Mechanical equivalent 450 diameter of single struts for the stiffness prediction of lattice structures produced by electron beam melting[END_REF], [START_REF] Persenot | Enhancing the tensile properties of ebm as-built thin parts: Effect of hip and chemical etching[END_REF]) extracts surface profiles every 10 • along the circumference of the gauge length. Very local surface irregularities such as notch-like defects are thus more likely to be overlooked. Second, the Rt value considers the distance between the highest and the lowest point of the surface but does not describe the shape of the corresponding defects. A large Rt value can be caused by a deep stacking defect (45 • and 90 • samples) with a large radius of curvature or even by some large protuberances caused by the thermal process, justifying why no correlation can be made between 305 the Rt values and the fatigue lives.

Conclusion

For additively manufactured lattice structures, conventional machining cannot be used to improve the surface state.

Furthermore, the struts forming the lattice are built in many different orientations. Determining the impact of as-built state and of build orientation on the fatigue properties is mandatory and was the aim of the work presented here. The 310 main conclusions can be drawn as follows:

❼ The detrimental effect of the as-built surface state on the fatigue resistance of individual struts was clearly shown; designing lattice structures against fatigue using data obtained on machined samples is unsafe and can lead to erroneous conclusions.

❼ The build orientation impacts the fatigue properties of as-built samples because of its influence on the shape, size 315 and number of notch-like defects from which fatal cracks initiate.

❼ Horizontal and vertical samples were shown to be respectively the best and worst samples in terms of fatigue lives.

❼ Considering the fabrication of lattice structures, it might be interesting to limit the number of vertical struts and to build struts as much as possible in the horizontal direction.

❼ Using roughness measurements of as-built AM samples to infer information regarding the fatigue properties is unsafe 320 since these measurements do not reveal the thin and deep notch-like defects. lives following the approach of Ciavarella and Monno [START_REF] Ciavarella | On the possible generalizations of the kitagawatakahashi diagram and of the el haddad equation to finite life[END_REF]. Each curve corresponds to a given number of cycles N f l and relies on the stress amplitude corresponding to N f l as well as a propagation law. The experimental stress range was directly used for the blue dots, green crosses and red squares.

35

 35

Figure 1 :

 1 Figure 1: Geometry of the fatigue specimens representative of a single strut of any lattice structure (here, an octet truss). (a) 3D isometric view of the Computer Assisted Design input. (b-d) Photographs of a lattice unit cell and of the fatigue specimens manufactured in three different orientations before removal of the manufacturing supports (b,c and d for 90 • , 0 • and 45 • samples respectively).

Figure 2 :

 2 Figure 2: S-N curves for as-built fatigue samples fabricated by EBM with 3 different orientations: 0 • (red squares), 45• (green crosses) and 90 • (blue dots). Hollow symbols correspond to data from ARCAM[START_REF] Arcam | Documentation from Arcam training level[END_REF] on machined samples obtained by EBM.

100 fracture surfaces of Figure 3

 3 are also representative of what was observed at higher and lower stresses for the 90 • samples.

Figure 3 :

 3 Figure 3: SEM micrographs of the fracture surfaces of as-built EBM fatigue samples. (a,d,f) Fracture surfaces showing the stable crack growth and sudden failure zones of 90 • , 45 • and 0 • samples respectively. (b,c,e,g) Higher magnification images showing the crack inititiation sites.

Figure 4 :

 4 Figure 4: Projected views of all the internal pores within the gauge length of three samples (10mm) as detected by X-ray lab tomography (2.5µm voxel size). (a) 0 • sample, (b) 45 • sample and (c) 90 • sample. The corresponding volume fraction pores are respectively 0.035%, 0.040% and 0.025% for the 0 • , 45 • and 90 • samples.

Figure 5 :

 5 Figure 5: 3D rendering of as-built samples manufactured in the three different orientations. Blue arrows indicate the building direction. (a-c) Side views of 90 • , 45 • and 0 • samples respectively. (d-f) Axial views of 0 • , 45 • and 90 • samples respectively. (g,h) Respectively upper and lower surface of a 0 • sample. (i,j) Respectively upper and lower surface of a 45 • sample.

Figure 6 :

 6 Figure 6: Impact of the build orientation on the surface of the samples gauge length. (a,c,e) Radial slices of samples manufactured respectively at 0 • , 45 • and 90 • . (b,d and f) Magnification around surface defects. The slices are oriented in order to mimic the sample orientation in the build chamber, the building direction being vertical (blue arrow).

Figure 7 :

 7 Figure 7: Schematic illustration of the defects that can be found in EBM as-built samples. Geometrical parameters describing a surface notch are also shown. a, b and ρ represent respectively the depth of the notch, its thickness and the radius of curvature at its tip.

Figure 8 :

 8 Figure 8: Schematic illustration of the methodology used to identify the defect responsible for the failure of a fatigue sample. (a) Radial slices obtained by X-ray tomography of a sample before and after fatigue failure. Their comparison enables us to identify the critical cross section, i.e. the height in the gauge length at which the failure occurred. (b) SEM observation of the fracture surface of the fatigue sample. (c) Slice view of the critical cross section of the sample before fatigue test. Comparison of (b) and (c) leads to the critical defect identification.

Figure 9 :

 9 Figure 9: Examples of surface defects responsible for the failure of as-built samples. (a,d,f) Axial slices of critical defects for respectively 90 • , 45 • and 0 • samples. (b,c,e,g) Radial slices of critical defects for respectively 90 • , 45 • and 0 • samples. Arrows and ellipses identify the critical defects. The dotted lines on the axial slices indicate the corresponding radial slices.

Figure 10 :

 10 Figure 10: X-ray tomography images showing the same radial slice of a sample gauge length before (a) and after (b) HIP treatment. After HIP, no remaining pores can be detected with a voxel size of 2.5µm.

Figure 11 :

 11 Figure 11: Modified Kitagawa-Takahashi diagrams of EBM Ti-6Al-4V alloy. (a) Method suggested by Beretta and Romano [52]. The blue and red lines correspond respectively to notches perpendicular and parallel to the building direction. The continuous lines were plotted using the average K th value from the literature while minimum and maximum values were considered for the dotted lines. The blue dots, green crosses and red squares correspond respectively to the 90 • , 45 • and 0 • samples for which the size of the defect responsible for the failure was measured. Their stress range was extrapolated to larger fatigue lives corresponding to the knee-point of the Wöhler curve. (b) Diagram extended to finite

Table 1 :

 1 Chemical composition of the initial Ti-6Al-4V powder batch used for this investigation.

	Elements	Al	V	C	Fe	O	N	H	Ti
	wt%	6.47	3.93	0.01	0.22	0.09	0.01	0.001	Bal.

Table 3 :

 3 Intrinsic crack size a 0 depending on the ∆K th values found in the literature.

			crack perpendicular to the building direction	crack parallel to the building direction
			∆K th,min	∆K th,av	∆K th,max	∆K th,min	∆K th,av	∆K th,max
	∆K th (MPa	√ m)	3.4	3.6	3.8	4.2	4.9	5.7
	a 0 (µm)		53	60	66	81	110	149

Pores with a volume smaller than 8 voxels (cube of

2*2*2 voxels) were disregarded in the analysis of the binary images

manufacturing, Acta Materialia 85 (Supplement C) (2015) 387 -395. doi:https://doi.org/10.1016/j.actamat.

by electron beam melting of prealloyed powders, Rapid Prototyping Journal 15 (3) (2009) 171-178. arXiv:https:

4v fabricated by additive manufacturing: Lens and ebm, Engineering Failure Analysis 69 (2016) 3 -14, special issue on

Acknowledgements

The authors would like to thank the French Agence Nationale de la Recherche and the French Fondation de Recherche pour l'Aéronautique et l'Espace for the financial support of this project entitled "Fabrication Additive et FAtigue de Structures Cellulaires Intgres en AEronautique" (FA2SCINAE-ANR-15-CE08-0038-03).

This size is likely to be too small for practical application of the lattice in macroscopic parts.

On the other hand, such thin and deep defects are systematically overlooked by classical profilometry as shown by the measurements performed on 90 • samples: compared to the Ra value based on tomography scans where crack-like notches are detected, the average value of Ra obtained by profilometry is 16% smaller (36.9 µm v.s. 44 µm). This technique is therefore not suited for the detection of notches and, as a result, only X-ray tomography based roughness measurements 290 might be used for fatigue life prediction.

Vertically (respectively horizontally) manufactured samples with their large (respectively limited) population of stacking defects (Figure 6) present the highest (respectively the lowest) Ra value, see Table 2. A good correlation seems to appear between Ra and the fatigue life, larger Ra values giving shorter fatigue lives (Table 2 and Figure 2). Nevertheless, fatigue failure is always triggered by notch-like defects (Figure 9) which are seldom present at the surface. Their impact 295 on the Ra measurements is thus limited. This means that, in spite of the apparent correlation reported above, Ra cannot be the relevant parameter for predicting the fatigue life in the case as-built EBM samples.

The Rt parameter which describes the local roughness variations could potentially be a more relevant parameter.

However, the Rt ranking differs from the fatigue resistance ranking: 0 • samples have the longest fatigue lives but also the highest Rt values, see Table 2. Two facts can explain this observation. First, the method used for the roughnesses