FARHANG RADJAI Dietrich E Wolf

Sabine Dippel

Dissipation in granular materials

Fo r two simple sct-up:.;, we discuss how dissipati ve grain-grain interactions give rise to unexpected properties on large scales. ln the first example, a spherical pa rticlc rolling along a rough surface experiences an effective viscous friction on large lime scales. It is due to temporal correlations among collisions with incomplete normal restitution. The second cxample is a sheared granular packing:. There, spatial correlations among non-sliding and sliding contacts with C o ulomb frictio n suggest that the devialoric stress, altho ugh responsible for the d issip<nion , is Jocali zed on bonds that are non-sliding and bence nondissipative.

Granular media (such as sand) are classical many-particle systems with dissipative inleractions (for a recent review see Jager et al. ( 1996)). In the absence of external driving, the kinetic energy of the grains decreascs in each collision owing to the irreversible transfer of energy into the internai degrees of freedom of the grains!. This is called collisional cooling §. Oftcn the grains fonn lasting contacts within a finite time, which is called inelastic collapse (McNamara and Young 1994).

It happens if the relative velocity of the grains (i.e. the granular tempera ture) drops to zero. white the local pressure stays finite. An example is a heap o f sand. The weight of the grains provides a finite pressure in the pile, while the co llisional cooling eliminates ali relative motion. That this happens within a finite time can easily be illustrated with a single steel bail bouncing back from a p lane and losing a finite fraction of its kinetic energy with each co llision. ln the absence of a gravitational field the inelastic collapse may still happen in a finite region of the granular material if the outer regions provide enough pressure to compactify the inner part. ln order to keep granular media in a steady o r periodic motion, the translationa) and rotational degrees of freedom of the grains have to be agitated externally. One can regard it as the second characteristic pro pert y, besides the dissipa tive interactions, distinguishing the c1ass of granular materials from other many partic1e systems such as liquids or solids. tha t the typical agitation energy per degree of freedom has to be much larger than the thermal cnergy k 8 T /2. A sa nd pile is frozen into a mctastab\e configuration, for instance. In order to agitate any gra in, a minimal t e-mail:d.wolf(aluni-duisburg.de :::The partial regain of translational energy from internai degrees of freedom has bcen discussed by Giesc and Zippelius ( 1996) and leads to a stochastic res titution coefficient. This will not be co nsidered further in this paper. §The term 'cooling' herc refl!rs to the so-called granular temperature which is the mean squa re deviation of the velocities from their average. energy of the order of mgr is needed, where 111 and r denote tht! mass and radius respcctively of a grain and g is the gravitational acceleration. For r > l J.tm this becomes larger than the thermal encrgy of room temperature. Therefore. for cxample Brownian motion is unimportant for granular media. In so-called dry granular media, cohesion and hydrodynamic interactions can also be neglccted. This lS the case consrdered in this papcr.

Two types of irreversible interaction mode) are important in granular ma terials. Therc is on the one hand the incomplete normal restitution in head-on collisions. The corresponding phenomenological material parameter is the normal restitution coefficient

( l )
which is the ratio of the relative velocities after and bef ore the collision. lt is smaller than unity, because a fraction 1 -e~ of the kinetic energy is irreversibly lost to internai degrees of freedom of the grains.

On the other hand, therc is cnergy dissi pation during a sliding contact due to Co ulomb friction. at a rate F , t\. The friction force F 1 is proportional to the normal force Fn pressing the grains together. The dynamic friction coefficient is the ratio of th e two forces:

F, • 1' -d = T . Il (2)
Of course, th ese d issi pa tive grai n-grain interact ions characterized by equatio ns ( 1) and ( 2) are idealized (sec fo r example Schafer et al. (l996) and Baumbergcr and Rerthoud (1997) for refinements), but they arc Jegitimat.e starting points for eluciciating the dissipation phenomena on large scales, wh ich is the aim of this paper.

In general the d issipation in a granular matcrial is d ominated by o nly one of thesc two irreversible grain-grain interactions. For cxample, Coulomb friction is unimportant as a so urce of dissipation in a granular gas, where the dynamics are mostly due to binary collisions. On the other hand, the plastic deformation of granular packing involves ahnost exclusively sliding of particles with respect to cach other, so that the incomplete normal resti tution in the few collisions contributes only very little to the ovcra ll dissi pation.

Accordingly we present the following two examplcs: the fi rst in which the incomplete normal restitu tion is the microscopie source of dissipation, and the second in which the Coulomb frict ion is the microscopie source of dissipation. Jn both cxamples the dissipation on large scalcs shows qua litatively new fcatures which a re not obvious from the microscopie laws. The incomplete normal restitu tion givcs rise to a velocity-independent ('viscous') friction for a sphere rolling along a rough surface, and the Coulomb friction leads to a localization of dissipation on a small fraction (about 8°/tl) of contacts, when a granular packing is plastically deformed. assume thal the spheres of the incline are densely packed. More general situations have been investigated and do not lead to any important differences [START_REF] Dippel | [END_REF].

Veloâty-:fàrce diagrams

Let us first consider three simpler cases. A sphere rolling down a plane accclerates. It never reachcs a steady state. By contrast, a solid block shding down the tilted plane may reach a steady stale, albeit a trivial one. If the inclination is small enough, the block simply stops slid1ng. If the driving force F = mg sin () exceeds the Coulomb friction force, however, the block will accelerate for cver. Figure 2 shows the Coulomb graph. which allows one to read off the points (v, F) for which a steady statc with velocity v and driving force F (which of course is compensated hy the friction force in a steady state) cxists. Thcse points lie on the bold lines. Ali other points (v, F) do not correspond to a stcady state but evolve in time along the broken flow lines. For example, for v= 0 any driving force smaller than F_, will be compensated by static Coulomb friction. However, for a driving force F > Fs the block will start to slide and be accelerated with the force F -Fd. The Coulomb graph should be contrasted with the corresponding diagram for viscous friction (figure 3). A sphcre falling in a viscous medium experiences a frictional force wh ich is linear for laminar and quadratic fo r turbulent How. Here any driving force leads to a steady state. The veloci ty adjusts itself such that the viscous friction compensates the driving force .

F ~ .. ... . -----... ----. .. --• ' . ... ----. . ... , ... . ---• .. .. F s v
Having a rugged instcad of a flat incline, little as this change may scern, leads to a surprisingly rich velocity-force diagram (figure 4). There are at least three force intervals, scparated by with any velocity v will bccomc trapped, that is it stops rolling after passing a number of substrate spheres. If FAs is exceedcd and the initial velocity is large enough to go over the first little bump on the incline, the rolling spherc reaches a steady state. lf the driving force is larger than F 6 c, howcver, no stable state exists any longer, as indicated by the flow lines. The sphere starts to make larger and Jarger bounces. In the following we discuss the effective friction which guarantees a steadystate motion in the intcrmediate regime between FAR and Fsc• 2.2. The ej(ective ji-iction in the steady-state regime ln a steady state the driving force F can be identified (apart from the sign) wit h an effective frictional force which descrihes the dissipation of the energy input.

Molecular dynamics simulations [START_REF] Wolf | Computationa/ Ph.ysics: Selected J \letlrods-Simple Exacises-Serious Applications[END_REF] and experiments (Ristow et al. 1994) show that this effective frictional force depends on the velocity approximately like

(3)

The offset vA B is a function of the ratio of the radius R of the rolling sphere to the radius r of the surface sphcres and approaches zero for increasing R/ r (figure 5).

Equation ( 3) means that the rolling particle e.ffcctively feels a viscous friction.

There is a seçond even more remarkable observation. The data for ditTerent restitution coefficients e 11 (see figure 5) and also for different friction coefficients /-id are indistinguishable. The cffecti ve friction depends very little on the mate rial coefficients charactcrizing the dissipation on the scale of one grain.

These two key observations will now be explained qualitatively. A quantitative analysis will be given in the next section. For the explanation of equation ( 3 ln this limiting case the motion becomes particularly simple. Because P•d ___, oo the sphere has no slip relative to the bumps on the incline. It must roll without dissipation. The only dissipation happens when the rolling spherc first hits a new bump. Recause en = 0, the kinetic encrgy stored in the motion perpendicular to the new bump surface is dissipated at once. The moving sphcre does not bounce back. For simplicity we consider only the case when it stays always in contact with the inclined surface (no detachment due to centrifugai force).

The kinetic encrgy at any point on bump k + 1 is related to that at the corresponding point on the preoeding bump by

The last term is the energy dissipated in one collision, and

tlEpnl = 2rF = 2rmg sin () (4) (5)
is the difference in potential energy. Obviously, for a steady state, one must have

'} nw• ')rF• - n -E' • . . ; . . --2 = d 1ss •
This explains why the effective friction has a quadratic vclocity dependcnce.

(6)

Simple geometry (figure 6) shows that the normal component of the velocity 1.' just before the collision with a new bump is

1'n = {} sin (2!:nax) ~ (7)
where 'Ymax = sin -1 [r /( R + r) ]. This explains why the steady-state velocity increases with incrcasing ratio R / r > 1 of the radii of the rolling sphere and those forming the incline (sce figure 5). The la rger the rolling sphere, the smaller is the normal component of its velocity, when it hits the new bump, and hence the Jess eflicient is the dissipa ti on .

With equations (7) and (6) and some elementary trigonometrie transformations the steady-state velocity v just before the collision with a new bump is ., ln the ncxt section we shall see that vis representative for the average velocity ii.

Fr v-- - -mc( l-c) ' with . . . , ( r ):! c = sm .. rm.tx = R + ,. (X)
The surprising rcsult that the effective friction force is nearly independent of the value of e 11 can be understood in the following way. We found by computer simulation thal a steady state requires essenlially that the moving sphere undergoes an inelastic collapse on each substrate particle. One can assume that it has formed a lasting contact when it reaches the next substrate partide, with almost the same tangentia1 ve1ocity as in the case en = O. Whether or not the inelastic collapsc can be completed bcforc the next surface bump is hit depends on the coefficient of restitution. Therefore Fac is a function of <'w lncreasing en tends to destabihze the steady state. The results are independent of the friction coefficient /l'ct' as long as the moving sphcre roUs whcn it is in contact with the surface.

Anal_vtic calculation <~lthe veloàty-:fàrce diagram

In order to show the stability of the steady state and to cvaluatc the average velocity, one necds to know the kinctic energy Ekin in equation ( 4). As the sphere is roll ing, it rotates about its centre of mass with angular frequency w = vf R. The kinetic energy has a translational and a rotational contribution:

(9) with an effective mass mcff = m( l + J / mR 1 ) , where J denotes the moment of inertia.

Hence equation ( 4) becomes As E ctisJ Ekin is smaller than unity, b.vJ.: converges to zero exponentially. Now we use these rcsuhs to determine the average velocity in the steady state. Knowing the vclocity v at 1ma. \ (see figure 6) any previous velocity u(! ) can he obtained from energy conservation for -'l'max < r < l'max:

~ ~ n • lcf(Pk 1-1 mcrrvf: . . . . . 2 ( ) ,., 2 = 2 + .!.rF -rnc l -c vi_:. ( 1
mef(02 -m~n•v2(1') . . ' . : . . 1 2 - 2 + mg (1 -t-R) [cos ((f + r) -cos(8+ "Yrnax)J. ( 12
)
Solving this for 'V 2 ('y ). one obtains ( 15 )

'u 2 (-( ) = ' vÔ[h -cos (fJ + !)] = (R + r)i (
The average velocity v is given by the arc lcngth 21'max (R + r) divided by the duration T of the contact with one bump:

-2Î'max(R + r) v = T -. (16)
lnserting equation ( 13) into

( 17)

the average veloci ty ( 16) is determined by

Vo l J& +-rmax d'"'( V = 2' Ymax B-1 1 nax (b -COS 1') 112 . ( 18
)
The integral in equation ( U~) is an elliptic integral of the first ki nd and cannot be solved in closed form. The theoretical curves in figure 5 are obtained by numerical evaluation of this formula and are in excellent agreement with the data.

Howevcr, one can gct additional insight by expanding ii for small r/ R (or small <.:.

(equation ( 8))), keeping the driving force F (i.e. th e inclination 8) fixed. This is done in the appendix. The result is

[ .., ] m ., R~ R m r -'lF = F ( -) + 2-+ 2 ( 1 --) + 0 (-) . r r r m• R etf ( 19)
This specifies the proportionality constant in equation ( 3) and shows tbat the offsets vanish in the limit r / R ~O.

Finally. we determine the value FAs of the driving force, bclow which v= 0 is the only steady state. The integral ( 18) is only finite if b > l. It diverges for h ---t 1 like lln (b-l )1, that is v vanishes like 1 /f ln (b -1) 1• We conclude that b = 1 implicitly determines F.I\B• Th is can also be seen from equation (13). While rolling over a substrate spherc, the highest point (~f = --fJ) is rcachcd with zero velocity, if h = !.

lnscrting equations ( 8) and ( 14) into equation ( 15), one finds that

b = a sin B + ( 1 -c) 1 /:2 cos () (20)
with the ab breviation

_ (meff 1 t) 1/ 2 a--- - c . 2m c( 1 -c) (2 1)
Suhstituting b = l and solving for BAB= sin-1 (FA 8 /mg) , one obtains (see appendix)

( ) ( 1 !/ ) . _ 1 1 _ 1 (1-c) '- OAB = sm If' -tan . ( a 2 + 1 -c ) - a ( 22
)
Figure 7 shows a comparison of equation ( 22) with simulation results. BAB vanishes for large R/r like cj2a, that is ) is the analytical result for e 11 = O. § 3. GRANULAR PACKINGS: DISSIPATION BY FRICTION As in the collisional regime, the quasistatic deformation of granular materials involves both gain (collision) and loss of contacts between particles, which result in the evolution of the contact network and the appearance of an induced anisotropy. However, the average lifetime of contacts is compara ble with the macroscopic time scalc associated with the global deformation of the system. Owing to the non-smooth character of the Coulomb friction law, at those 'lasting' contacts, the particles may roll upon eac h othcr. This is a non-dissipative microscopie mechanism of deformation. If ail particlcs could roll upon one anothcr, then a granular assembly would defonn without dissipa tion. This is, h owever, forbidden by the frustration of particle rotations (Radjai and Roux 1995). This means that it is impossible for ali particles to move and rota te without sliding at sorne contacts. We see that a granular medium in slow deformation is heterogeneous not on1y because of the heterogeneous distribution of contact forces~ but also owing to the sliding (dissipative) and non-sliding states of contacts. l t is our aim here to show that this heterogeneity due to dissipation is correlated with that of the force network.

The lt'eak and the strong network of/orees

The numerical study of the statistical distribution of forces in static pac kings using the contact dynamics approach (Radjai et al. 1996) shows tha t the probability distributions PF of normal or (absolute values of) tangential forces have two distinct parts separated by the average force. The forces lower than the average ('weak' forces) are power law distributed, whereas the higher forces ('strong' forces) have an exponentia Il y decreasing probability:

Pr ex { p-ü( f3F) exp -(F) , F < ( F)~ F > (F) . (25)
We observed the samc behaviour also in a simulated granular system under quasistatic; biaxial c;ompression (Radjai et al. 1997). The absence of a characteristic force in the weak part of the distrihution indicates that the weak subnetwork (supporting v.:eak forces) does not dircctly feel the influence of the deviatoric load. In contrast. the strong forces have a characteristic force set by the external Joad and the average coordination number.

The stress tem•ors

What are the respective contributions of the weak and the strong subnetwork to the transmission of forces through the system? The answer to this question needs the investigation of the stress tensor. In a granular system. the stress tensor involves both the contact forces and the contact orientations. It is obtained from (Christoffersen er al. 1981)

cr 11 = v-1 L< Ff"d)'.
where the summation is over ali contacts and Vis the volume of the system. Ft is the i component of the total force (normal plus tangential) at the contact c, and dj' is the j component of the vector connecting the centres of the contacting particles.

ln ordcr to separate the contributions of weak and strong forces to the stress tensor, th!! summïJtion is to be rcstricted to contacts in the weak or the strong subnetwork respectively. Figure 8 shows the principal axes and the corresponding eigenvalues of the weak stress tensor aw and the strong stress tensor <Jsat the shear peak for the wholc sample. The surprising phenomenon observed here is that the wcak subnetwork provides a negligibly small contribution to the dcviatoric part of the total slress tensor <J = aw + cl, while it represents 28% of the mean pressure. The whole deviatoric load is thus sustained by the strong subnetwork.

The absence of a deviatoric stress is the charactcristic property of fluids in static equilibriurn. The solids, by contrast. can bear a finite deviatoric stress. In this respect, the weak subnetwork in our system can be viewed as a liquid, whcreas the

.

  Figure l. A sphere rolling along a rugged surface experiences an effective viscous friction.

Figure 2 .

 2 Figure 2. T he points (v, F) on the bold li nes correspond tc steady stmes of a sol id block subject to Coulomb friction. For tixed driving force the velocity evolves along the broken lines.

FvFigure 3 .

 3 Figure3. Frictional force F of a sphere moving with velocity •1.' in a viscous medium. For fixed drivi ng force the velocity evolves along the broken linc.

  Fig ure 4. Schematic vclocity-force diagram of a sphere rolling along a ruggcd surface. Velocity v <.lveraged over dUI•arion of the contact with one surface sphere. The bold lines correspond to s teady states. Below a driving force FAn the sphere stops rolling.Betwee n FAB and Fsc there exists a stcady state wi th finite velocit y. Abovc FBc computer simul ations indicate a force intcrval, for which a steady-state vclocity can be reached from below. but not fr om ahove. For even larger driving force, no steady state is rcached.

Figure 5 .

 5 Figure 5. Simula tion results of f i as a function of the driving force F "-mg sin e. The ratio of the radius R of the rolling sphere to the radius r or surface spheres is R/ r = 1. 75 for the lowest curves ( e ), 2.25 in the middle (•) and 3 for the uppcrmost curves (.A.). The data for en = 0.7 (-----) and 0.5 (----•• ) arc indistinguishable from those ror e 0 = 0.1 (e , •• .A.). Also shown in the analytical prediction for e 11 = 0 ( --).

Figure 6 .

 6 Figure 6. At angle ; = 'Ym:~x the rolling sphere hits a new suhstrate sphere and looses the norma l component t'n of ils velocity. The kinetic energy of the tangential motion is redistrihuted between the rotational and translational degrees of frcedom.

  0)The stability of the steady-statc solution is now easily chccked. Let D.v~ = v 2 -11~ denote the distance from the steady-state value. Then the iteration has the simple form

  13) with the characteristic veloci ty Vo = ( 2m g(R + r)) l/2 111.;(( and the dimensionlcss constant b = (• u )

Figure 7 .

 7 Figure 7. Computation simulations data for F.l.\a = mg sin ()AB as a function of R/ r for t'n = 0.7 (-•••••)and 0. 1 (-----): (-) is the analytical result for e 11 = O.

Figure 8 .

 8 Figure 8. Principal axes and eigen values of the stress tensor. The contribu tion of weak contact fon:es to the stress tensor is isotropie. The deviatoric stress is transmitted by the strong forces.

sliding, and a rapid inspection shows that almost all of them are on the weak subnetwork. This means that the strong contacts are non-sliding, and the whole frictional dissipation takes place within the weak subnctwork.

The friction force at a non-sliding contact is basically a reaction force. It ranges from zero to a limit value given by the normal force times the coefficient of friction, and its actual value is a function of ali othcr forces acting on the two contacting particles. According to Coulomb's law of friction, the friction force cannot incrcase beyond its limit value. and sliding occurs as soon as this limit is reached. These two aspects of friction, reaction force on the one hand and dissipative force on the other hand. appcar as complementary behaviours in the strong and wcak subnetworks respective\ y.

APPENDIX

We derive eq uation ( 19) for fixed inclination()> BAB by expanding fJ (equation ( 16)) in the sm ali varia ble c = sin 2 l'max (equation ( 8)). First we ex pa nd equation ( 20):

In leading order the integral ( 18) can be replaced by the integrand at ~~ = rJ: Sub~ti tuting equation (A 2) and m , .

F2m


Michel Bellet
email: michel.bellet@mines-paristech.fr

Gonghao Qiu
email: gonghao.qiu@edf.fr

Jean-Michel Carpreau
email: jean-michel.carpreau@edf.fr

COMPARISON OF TWO HOT TEARING CRITERIA IN NUMERICAL MODELLING OF ARC WELDING OF STAINLESS STEEL AISI 321

Keywords: Welding, Hot cracking, Hot tearing, Solidification cracking, Numerical modelling, Hot cracking criteria

Two hot cracking criteria have been tested: the RDG criterion, based on the prediction of liquid cavitation as a precursor of crack formation, and a strain-based solid mechanics criterion. Both criteria have been implemented in a finite element thermomechanical simulation of gas tungsten arc welding. After comparison with experimental results obtained in a test campaign on stainless steel AISI 321, both criteria have shown good ability to predict crack occurrence. Yet, the best response in terms of cracking prediction was obtained with the strain-based solid mechanics criterion.

INTRODUCTION

Hot cracking is a common defect occurring during solidification and welding of metallic alloys. In industrial welding practice, a lot of prototype tests are realized to define operating parameters (welding heat input, chemical composition) in order to avoid hot cracking. Such tests are expansive and time consuming. In addition, as they do not cover all real industrial welding configurations, they do not guarantee absence of cracking. Alternatively, it might be expected that the progress in the numerical modelling of welding processes could be used in hot cracking sensitivity prediction. The aim of this contribution is to study the capacity of two hot cracking criteria to predict such a defect. In a first section, we will proceed to a brief reminder of what is hot cracking and what are the main physical phenomena involved. The two main classes of hot cracking criteria that can be found in the literature will be introduced.

In a second part, the main equations governing the thermomechanical analysis of welding will be presented. A third section will present the experimental results obtained by GTA (gas tungsten arc) welding on rectangular plates of stainless steel AISI 321. The numerical thermal analysis of those tests will be discussed in a fourth section, showing the accurate calibration of the thermal model with respect to experimental records. Then, the thermomechanical analysis will be presented and finally the response of the two types of hot cracking criteria will be analyzed and discussed.

Hot cracking and prediction criteria

Hot cracking (also named hot tearing or solidification cracking) is a well-known welding defect occurring at the end of solidification. Readers can refer to [START_REF] Campbell | Castings[END_REF] which gave a comprehensive and synthetic description of this defect in his book on castings. During solidification, low melting point liquid exists between dendrites. At the end of solidification (high solid fraction) the stresses initiated by solidification shrinkage or by thermal gradients in the surrounding solid may be sufficient to open interdendritic spaces not yet completely solidified and create intergranular cracks that liquid feeding cannot fill. Among many authors, [START_REF] Eskin | A quest for a new hot tearing criterion[END_REF] studied and established a comprehensive review of the different physical phenomena leading to this defect. Alloys are considered the most vulnerable in the so-called Brittleness Temperature Range (BTR). This temperature interval -the name of which expresses that hot cracking may be seen as a ductility loss -is between the coherency and the coalescence temperature. The coherency temperature can be defined when dendrites begin to transmit forces but with low rigidity. Above coherency temperature, any deformation or displacement of solid dendrites can be easily compensated by the circulating liquid phase.

The coalescence temperature is reached when dendrites are strong enough to accommodate strain, while numerous solid bridges limit the risk of crack propagation. In between these characteristic temperaturesor equivalently characteristic solid fractionsthat is when liquid cannot flow any more through interdendritic spaces, the mushy zone is susceptible to hot cracking. [START_REF] Farup | In situ observation of hot tearing formation in succinonitrile-acetone[END_REF] provided spectacular evidences of the combination of phenomena leading to hot cracking: liquid cavitation, lack of liquid feeding, and plastic deformation of solid bridges.

From the previous considerations, it can be deduced that a given alloy will be more or less prone to solidification cracking depending on its thermophysical and rheological properties and on local specific solidification conditions determining for instance dendritic spacing and, as a consequence, mushy zone permeability. In addition, [START_REF] Cross | Weld solidification cracking: critical conditions for crack initiation and growth[END_REF] showed like other researchers the existence of a critical strain, below which no solidification crack occurs, and the influence of strain-rate. Regarding the prediction of hot cracking, models can be grouped into two great families that are presented in the next two sections.

Liquid cavitation approach: RDG criterion

The so-called RDG criterion (so-called from the initials of their authors) was proposed by [START_REF] Rappaz | A new hot tearing criterion[END_REF]. The authors considered the mass balance between solidification shrinkage and liquid feeding through the permeable solid dendritic network. Hot tearing was supposed to occur due to a deficit of liquid feeding through the mushy zone, leading to cavitation in the interdendritic liquid, and then to crack initiation. Assuming a thermal gradient oriented along x direction, also the direction of columnar dendritic growth, the mass balance equation was approached as

                          x l s s zz yy g x p x t f f       ) )( 1 (   (1) where 1 /   l s   
is the solidification shrinkage coefficient, is the solid fraction, yy  and zz  are solid strain rate components perpendicular to the growth direction. The two terms summed in the left hand side of Eq. ( 1) are associated with the mechanical deformation and the solidification shrinkage, respectively. The deformation parallel to the dendrites was ignored, as it could not induce hot tearing, and the densities of the two phases, liquid and solid, were assumed constant. The right hand side term expresses the liquid feeding governed by Darcy law, where K is the permeability, the liquid viscosity and p the liquid pressure.

The permeability K may be expressed by the Carman-Kozeny equation:

2 3 2 2 180 ) 1 ( s s f f     (2)
in which  2 denotes the secondary dendrite arm spacing (SDAS). This value is often determined experimentally by metallurgical expertise. It increases with solidification time (the time spent in the solidification interval) by dendrites maturing. This phenomenon, which is due to the diffusion of atoms from high to low curvature zones in order to minimize the chemical potential, decreases the number of secondary branches and thus increases the interdendritic space. This explains why the distance between dendrites depends strongly on the cooling rate. The SDAS can be estimated by power law type expressions as a function of the solidification time t f , such as introduced by [START_REF] Kurz | Fundamentals of Solidification[END_REF]:

s f  3 / 1 2 f Mt   (3)
where M is a material constant. [START_REF] Rappaz | A new hot tearing criterion[END_REF] proceeded to the integration of the mass balance equation along the length of the mushy zone, which lead to a relationship between the tensile strain rate  applied perpendicular to the solidification direction on one hand, and the liquid pressure drop through the mushy zone

p p p m   
on the other hand, p m denoting the metallostatic pressure in the liquid pool, at dendrite tips. The liquid pressure p continuously decreases in the mushy zone, from the tips to the roots of primary columnar dendrites. Close to dendrite roots, that is below the coalescence temperature, near the full solid region, the liquid pressure may reach the cavitation pressure. The comparison of this pressure drop to its critical value

C m crit p p p   
, where p C is the cavitation pressure, leads to a critical strain rate  .

According to Rappaz et al., hot cracking was then supposed to initiate when:

                 A v p G B G T crit       1 1 180 2 2  ( 4
)
where G is the norm of the temperature gradient in the mushy zone, T v is the speed of the solidification front, A and B are integrals that depend on the solidification path of the alloy, the bounds of the integrals hereunder being the bounds of the BTR, upper bound T coh , lower bound T coal :

     coh coal T T s s dT f f A 2 2 ) 1 (             coh coal coal T T T T s s s dT dT f f f B 3 2 ) 1 ( (5)
In the context of welding, note that the metallostatic pressure can be neglected, so that

C crit p p   
. It can be seen that using this model, the critical strain rate can be determined by thermal analysis. A central parameter for the application of the RDG criterion is the liquid cavitation pressure, corresponding to the critical cavity size leading to hot cracking.

1.2 Mechanical approach: strain-based criteria

In mechanical criteria, the physical phenomena associated with cracking are not explicitly described. Such models are based on a mechanical load limit, which is generally expressed in terms of a critical strain cumulated through the BTR. Below this limit, welding is assumed to be free from hot cracking. In practice, the Hot Cracking susceptibility Index (HCI) is often considered and is expressed as the difference between the cumulated strain BTR  in the brittle temperature range and a reference value crit  . Yamanaka (1990) was the first one to express such a simple mechanical criterion:

crit BTR HCI     with   BTR BTR t d    (6)
Hence, positive values of HCI indicate hot cracking risk. In the summation,   denotes a scalar measure of the strain rate components in the direction perpendicular to the thermal gradient. Generally, compressive strain rate components are not taken into account because they do not have significant effect on hot cracking. This is why [START_REF] Bellet | Modeling of hot tearing during solidification of steels: assessment and improvement of macroscopic criteria through the analysis of two hot tearing tests[END_REF] proposed two different expressions for this measure (von Mises type or largest positive eigen value).

Note that these remarks also apply to Eq. (4) when using the RDG criterion.

It can be seen from Eq. ( 6) that, using this simple approach, the associated criterion has only three parameters which are on one hand the upper and lower limits of the BTR, expressed in terms of temperature or corresponding solid fraction, and on the other hand the strain limit crit  . [START_REF] Won | A new criterion for internal crack formation in continuously cast steels[END_REF] proposed another mechanical criterion, which is an extension of the previous one and includes strain-rate influence. It is expressed as follows, the critical strain crit  being expressed as a function of the strain rate itself and of the BTR:

n m crit BTR      (7)
Here , n and m are material constants, extending the number of parameters to 5.

Based on the work of Won et al., Bellet et al. (2009) proposed to express the critical strain as a function of the chemical concentration of specific elements to reflect the increase in the solidification range with segregation together with the influence of such chemical elements on phenomena such as wetting of the solid phase by the liquid phase, that is known to impact the sensitivity to hot tearing. The parameter  now depends itself on the mass concentration of carbon, manganese, sulphur and phosphorus:

                 0 0 ] P [ ] S [ ] Mn [ ]) C [ exp( 0 0 0 d c b a   (8)
This involves again additional material parameters: 9 in total.

In the present study, the analysis is restricted to the choice of the simplest mechanical criterion, the one of Yamanaka et al., because of its limited number of parameters. The use of alternative criteria would have required a more extended experimental data base, which was out the scope of this work.

Experimental results

The studied steel is a stainless steel, grade AISI 321 (titanium stabilized austenitic stainless steel), the chemical composition of which is indicated in Table 1. The values of thermo-physical properties and constitutive parameters can be found in [START_REF] Kerrouault | Fissuration à chaud en soudage d'un acier inoxydable austénitique (Hot tearing during welding of an austenitic stainless steel)[END_REF] who carried out various studies on welding of this steel grade. mm, thickness 2 mm, width 20, 25 and 40 mm), which is clamped at one end (Figure 1). The other end is free. Through-thickness welding is started at 5 mm from this free end and is done without any filler material, so that the weld bead in question is in fact a simple straight fusion line, as indicated on the figure. At the beginning of the test, the fusion zone passes through the width reduction (notch in the coupon). At this moment the 25 mm long appendix detaches while an overheated zone and an enlarged fusion zone are generated temporarily, which favors the initiation of an axial crack. Then, depending on welding conditions, either the crack stops immediately or it progresses in the axial direction in the wake of the torch, up to the end of welding. Such an axial crack is shown in Figure 1c. The test is then discriminant between cracked and non-cracked specimens which will be denoted hereafter "C" and "NC" respectively. Protection against oxidation is assured by argon gas on the upper side of the plate and by disposal of very thin copper foils maintained by adhesives along the lower face of the plate (Figure 1b). 2).

Six process conditions, corresponding to different welding parameters and plate geometry have been tested, as detailed in Table 2. Welding arc voltage is 10 V and welding heat input (energy) is about 235 J/mm, with variable arc current, welding speed and coupon width. The occurrence of a crack is indicated in the last column of the table. When crack is started through a notch at the coupon edge, like with this kind of specimen, the closure or continuation of the crack is often correlated with the coupon width. [START_REF] Cross | The effect of restraint on weld solidification cracking in aluminium[END_REF] analyzed this relation and showed the influence of the coupon width on the amplitude of transverse tensile stress and strain, leading to cracking. However, in the present case, it can be seen that there is no direct apparent relation between crack occurrence and one of the operating or geometrical parameters: coupon width, arc current, welding speed or heat input. Metallurgical investigations by scanning electronic microscopy (SEM, Figure 2) demonstrate that cracks are interdendritic, confirming a hot cracking mechanism. At the bottom of a section perpendicular to the welding direction, dendrites are found to grow perpendicular to the lower face of the specimen, that is perpendicular to isothermal surfaces that are parallel to this face. At mid-thickness the dendritic growth direction is out of the observation plane, which is consistent with the fact that the temperature gradient tends to align itself along the welding direction at the rear of the welding pool. From the observations, the secondary dendrite arm spacing can be estimated between 4 and 7 µm. Micro-probe analysis has not detected any sulphur or phosphorus segregation.

Numerical modelling

The numerical simulation consists of a non-steady-state thermal-mechanical finite element calculation, which is performed in two separate steps. First, the evolution of the temperature field is calculated. Second, this time dependent temperature distribution is used on the same finite element mesh in a mechanical resolution to determine stress and strain fields. The numerical code used to solve the non-linear equations for energy and momentum conservation is the open source code [START_REF] Code_Aster | [END_REF].

The numerical simulation is not supposed to address the high complexity of the transient start of the weld especially when passing through the width reduction (notch) initiating cracks. The objective is limited here to the non-steady state modelling of the thermomechanical conditions undergone at the rear of the fusion zone once a quasi-steady state regime has been established in the plate.

The finite element meshes are composed of hexahedral and tetrahedral elements with linear interpolation, integrated with 8 and 6 Gauss points, respectively. The mesh size is coarser in the base metal, far from the welding direction (3 mm) and refined in the region of interest, that is at the rear of the mushy zone around x-locations along the welding line where crack occurrence has been experimentally observed (mesh size decreased up to 190 (x) x 220 (y) x 400 (z) µm). An illustration is given Figure 3. The mesh covers the zone of interest in the specimen for potential crack formation: the domain between the notch and the clamping zone (100 mm in length). The clamped region of the plate is not modelled: the displacement of the nodes at the end of the mesh is imposed to be null. In the two next sections, the main assumptions of the non-steady-state thermal-mechanical calculations are listed and the governing equations are presented.

Assumptions

The main assumptions are:

1. The transient welding process is treated by an equivalent thermal loading. There is neither plasma nor welding pool flow modelling.

2. The mechanical behaviour of the material is modelled on the whole temperature range using an elastic-viscoplastic constitutive law. Material properties are temperature dependent and adapted over solidus temperature to reflect the decrease in strength up to a given coherency temperature, over which strain and stress are considered null.

3. Solidification shrinkage is ignored, as it is supposed to essentially affect liquid flow, which is disregarded in the present modelling approach.

4. Energy associated with plastic deformation is ignored, due to very low strain rates, making it insignificant in front of heat sources arising from the welding process.

5. Chemical segregation due to solidification, fluid flow or diffusion in the solid state is ignored. The solidification path is given by the Gulliver-Scheil equation.

Governing equations

The equation for energy conservation is:

q T k t H        ) ( (9)
in which k is the thermal conductivity and H the enthalpy per unit of volume:

  T T p du u c T H 0 ) ( ) (  (10)
denotes the density, c p the specific heat and T 0 a reference temperature. The right hand side scalar q in Eq. ( 9) denotes the local heat input per unit of volume, which is associated with the welding process and will be detailed in the next section.

The thermal boundary condition applied to all faces of the welded plate consists of a mixed convection-radiation condition, for which the heat flux is defined as:

) ( ) ( 4 4 air r r air c T T T T h T k          n (11)
where h c is the convective heat transfer coefficient, σ r the Stefan-Boltzmann constant, ε r the material emissivity, T air the temperature of surrounding air, and n the unit outward normal vector.



The equation for momentum conservation is:

0     f σ (12)
in which inertia effects are neglected, f is the body force per unit of volume and σ is the Cauchy stress tensor. The deviatoric stress tensor and the von Mises equivalent stress are respectively defined as:

I σ σ σ ) ( tr 3 1   dev dev dev σ σ : 2 3   ( 13
)
where I is the identity tensor. The strain rate is divided into three components, elastic, thermal and viscoplastic as:

th vp el ε ε ε ε        (14)
In the framework of small elastic strains, the material being assumed isotropic, the elastic strain rate tensor el ε  and the thermal strain rate tensor th ε  are respectively given by:

I σ σ ε ) ( tr 1    E E el      I ε t T th d d    (15)
where is the Young modulus,  the Poisson coefficient, and the one-dimensional thermal expansion coefficient. The viscoplastic strain rate is expressed by means of a viscoplastic law including kinematic hardening. Denoting F the plasticity criterion and R the yield stress, we have:

R R F dev dev        ) ( : ) ( 2 3 ) ( ) , ( X σ X σ X σ X σ ( 16
)
where R is supposed, in the present study, to depend on temperature only. The tensor X is the kinematic hardening stress tensor, whose expression is given hereunder. The viscoplastic E  strain rate tensor is then expressed by

dev vp F ) ( ) ( 2 3 X σ X σ σ ε            (17)
from which it can easily be shown that the multiplier   is simply the generalized viscoplastic strain rate   :

vp vp ε ε     : 3 2     (18)
The kinematic hardening tensor X is given by

α X C 3 2  with α ε α        vp (19)
where C and  are material parameters. Note that X is a deviatoric tensor. Finally, the viscous flow is described by a power law:

n K F            (20)
in which the Macauley bracket  expresses that there is a non-zero viscoplastic strain rate

only if F is positive: otherwise 0 ; 0 if    F F F F
. In total, the mechanical constitutive model has seven material parameters, possibly temperature dependent: E, , R, C, γ, K, and n.

All parameter values for the considered steel AISI 321 can be found in [START_REF] Kerrouault | Fissuration à chaud en soudage d'un acier inoxydable austénitique (Hot tearing during welding of an austenitic stainless steel)[END_REF], together with its thermophysical properties.

Thermal analysis

The heat source term associated with the welding process and appearing as the right hand side of the heat equation (Eq. ( 9)) is:

  ) H( ) ( exp 3 ) , , , ( 2 2 2 2 2 s z r z e f y Vt x er UI t z y x q b p b                  (21)
where x is the spatial coordinate in the welding direction (y in the transverse direction and z in the thickness direction), U and I are respectively the arc voltage and current,  a process efficiency parameter, V the welding speed, e the plate thickness, r b the characteristic radius of the arc source, f p and s are factors defining the attenuation of q in the thickness direction z, H being the Heaviside function.

The convective heat transfer coefficient h

c has been taken equal to 15 W m -2 K -1 . Other parameters such as the thermal emissivity  r and the heat source parameters (, r b , f p and s)

have been identified by an inverse method, which consists in minimizing the difference between measured temperatures with thermocouples and calculated temperatures calculated by the finite element method. Details can be found in [START_REF] Qiu | Etude de la modélisation de la fissuration à chaud en soudage d'un acier inoxydable austénitique (Study of hot tearing during welding of an austenitic stainless steel[END_REF] who proceeded to this minimization by coupling the software Code_Aster and Matlab. The maximum difference between measured and simulated temperatures after optimization is 30 °C (Figure 4), which is acceptable in consideration of thermal gradients near mushy zone of the order of 50 °C/mm. Moreover, weld pool size and shape have been compared. Experimental depth and half-width are measured on post-mortem metallographic transverse sections, such as the one shown

Model results and discussion

All six tests described in Table 2 have been subject to thermal-mechanical simulation. In the present section we present and discuss the performance of the two hot cracking criteria, in comparison with the experimental occurrence of cracks in the specimens.

Interpretation through the mechanical criterion

The coherency temperature of the alloy (which is generally supposed as being the upper bound of the BTR) is chosen as T coh = 1400 °C, corresponding to a solid fraction f s = 0.8.

First, we examine the build-up of stress below this coherency temperature, as shown Figure 8. Figure 9 presents the calculated cumulated plastic strain between temperature T coh and the solidus temperature (1355 °C), for a material point at the rear of the weld pool, and for the six cases studied. As mentioned above, the cumulated plastic strain results from a time integration of the highest tensile component of the strain-rate tensor (here the yy  component) while the material point is found between coherency and solidus temperatures. Unlike stress levels, it can be seen that the highest cumulated strains are found for the specimens in which effective cracking is observed. The agreement with experimental findings is much better than when considering stress levels. In addition, and in the perspective of expressing the criterion by use of a cumulated deformation on the BTR [T coal , T coh ], it is interesting to notice that the choice of the lower bound for the BTRthe coalescence temperatureclose to solidus temperature, appears not to be crucial, as there is no marked strain increase near solidus.

However, due to the fact that the value of the solidus temperature is generally difficult to determine experimentally and that liquid fraction changes slowly with temperature near solidus, it might be recommended to fix the lower bound at a solid fraction of 0.95. That would correspond for the considered alloy to a temperature of 1360 °C.

From Figure 9 it can be seen that the cumulated strain for cracking specimens is higher than 0.02, while it is lower than 0.015 for non-cracking specimens. Therefore a critical plastic strain of 1.5 % in a BTR ranging from 1400 °C (f s = 0.8) to 1360 °C (f s = 0.95) or to the solidus could be recommended to estimate a risk of cracking. It can be seen that in those two additional calculations, the distinction between cracking and non-cracking tests is still possible. More precisely, when using a larger BTR (higher T coh ), the distinction is clear and the critical strain level is about the same as previously, around 0.015. However, when using a smaller BTR (lower T coh ), the distinction is not so clear and the critical strain level can be estimated around 0.0075, that is half the previous value.

Hence, the critical strain appears to be dependent of the extension of the BTR, which can be easily understood. In addition, it should be noted that such strains build-ups also depend on the definition of constitutive coefficients within the solidification interval, in a region where there is an evident lack of reliable data.

As a conclusion for the strain-based solid mechanics criterion, it has been found to provide an excellent discrimination between cracked and non-cracked specimens. The critical strain depends more on the choice of the upper bound of the BTR (coherency temperature)

than on the choice of the lower bound (coalescence temperature). A critical value of 1.5% for the cumulated strain through the BTR defined between 1400 °C (f s = 0.80) and 1360 °C (f s = 0.95) appears appropriate. The effect of strain rate on the value of critical strain, previously mentioned by [START_REF] Won | A new criterion for internal crack formation in continuously cast steels[END_REF] and by [START_REF] Bellet | Modeling of hot tearing during solidification of steels: assessment and improvement of macroscopic criteria through the analysis of two hot tearing tests[END_REF], was not been studied, due to the restricted number of welding tests. That would require additional tests.

Interpretation through the RDG criterion

In view of applying the RDG criterion, a preliminary task consists in determining the different parameters involved in Eq. ( 4). The liquid viscosity  is taken as 10 -3 Pa s. The value of the secondary dendrite arm spacing  2 has been estimated, from SEM observations (see Section 3), to 7 m. From data available in [START_REF] Kerrouault | Fissuration à chaud en soudage d'un acier inoxydable austénitique (Hot tearing during welding of an austenitic stainless steel)[END_REF], the shrinkage ratio  is taken as 0.086 and integrals A and B can be calculated as functions of the solidification path f s (T). Note that the bounds of the BTR used in Eq. ( 5) are the same as for the strain-based criterion: T coh = 1400 °C (f s = 0.80) and T coal = 1360 °C (f s = 0.95). These material data are supplied in the first five columns of Table 3. As for the process parameters, it can be noticed first that initiation of axial cracks at the rear of the mushy zone occurs in a quasi steady-state regime, during which the speed of the solidification front T v can be considered equal to the welding speed V. The thermal gradient G has been estimated from the numerical calculations performed for the six test cases: its value has been found in the range 97 to 113 K/mm. Similarly the strain rate has been found in the range 0.020 to 0.046 s -1 . Those values are gathered in the last two columns of Table 3.

From Eq. ( 4), cracking should occur if

          T crit v A G B G p       ) 1 ( 180 2 2 (22)
On the basis of material and process parameters values of Table 3, the right hand side of this equation, which is the pressure drop undergone during welding, ) ,

(  G p 
, can be calculated for the six test cases. Results can be found in Table 4. Ranking now the tests from the lower to the higher pressure drops, we get Figure 11. It can be seen that there is a rather good agreement between calculated pressure drop and cracking occurrence. However, the correlation is not as good as for the strain-based criterion, due to test 4 for which a crack has formed despite a rather low calculated pressure drop: 96 kPa. From these results, it appears that the transition between cracking and non-cracking specimens would take place for calculated pressure drops in the range 90 to 120 kPa. A critical pressure drop value which could be retained to form a conservative RDG criterion (somewhat overestimating the cracking risk) would be kPa 90   crit p . This value is much higher than the value suggested by [START_REF] Rappaz | A new hot tearing criterion[END_REF] in their initial article: 2 kPa.

However, it is of the same order as the limits determined experimentally by [START_REF] Wisniewski | Modélisation thermomécanique de la fissuration à chaud en soudage par faisceau d'électrons d'un alliage CuCrZr (Thermo-mechanical modelling of hot tearing in electron beam welding of a CuCrZr alloy[END_REF] for copper alloys (50 kPa), and by [START_REF] Carlson | Modeling of porosity formation in aluminum alloys[END_REF] for aluminium alloys (160 kPa). In view of such results, it is interesting to test the sensitivity of the calculated pressure drop values with respect to the different involved quantities. Supposing here that material data are known, our attention is focused on three other quantities: strain rate, thermal gradient and secondary dendrite arm spacing. The two first quantities are determined by analysis of the results of finite element modelling, whereas the last one is deduced either from Eq. (3), or from direct observation. From the expression of the pressure drop, Eq. ( 4), the following sensitivity coefficients can be derived: nor the pressure limit for the RDG criterion should be applied stricto sensu in thermomechanical analysis of welding for crack prediction. The reason for this is that the values that have been identified in the present study are obviously too dependent on the uncertainties regarding the thermomechanical response of the mushy zone, or its microstructural characteristics, especially the values of dendritic spacing, coherency temperature and the constitutive parameters around it. As such, the cracking criteria should be used as hot cracking qualitative indicators only, to serve in comparing different potential welding conditions and determine the best suited with respect to the risk of crack formation.

                                        T T v A G B G p v A G B G G p B G p                   ) 1 ( 180 2 ) ( ) 1 ( 2 180 ) ( ) 1 ( 180 ) (

CONCLUSIONS

In this work, experimental and numerical analyses of solidification cracking in GTA welding of an austenitic stainless steel AISI 321 were carried out. After implementation of two crack criteria in the thermomechanical finite element modelling and comparison with experimental observations and measurements, the following conclusions could be drawn:

 The values of calculated internal stresses were not found discriminant between cracked and non-cracked specimens. This tends to exclude the use of stress-based crack criteria.

 The RDG criterion -based on the prediction of liquid cavitation as a precursor of crack opening -showed an acceptable ability to predict crack occurrence, at the price of a very high value (although steel admissible) for the liquid cavitation pressure.

 The best response in terms of cracking prediction was obtained with the strain-based solid mechanics criterion, using a critical value of 1.5% for the cumulated strain through the BTR defined between solid fractions 0.80 and 0.95.

  Hot cracking test: a) geometry of plate specimens ; b) experimental set-up showing clamping and protection against oxidation on the lower side of the plate ; c) example of a crack fully developed throughout the weld bead (test 4 of Table

  test 2, a crack formed but stopped after 30 mm propagation; this is the reason for the indication "C-NC".

Figure 2 :

 2 Figure 2: SEM observation of primary (left) and secondary (right) dendrites in the crack propagation zone.

Figure 3 :

 3 Figure 3: Finite element meshes for coupons of width 20 and 40 mm (top). The bottom picture shows a zoom in the refined region.

Figure 5 .

 5 Figure 5. The calculated dimensions are determined by the isothermal surface corresponding

Figure 4 :

 4 Figure 4: Comparison between simulated and measured temperature evolutions for test 2 (coupon width 20 mm, arc voltage 10 V, arc current 102 A, welding speed 4.33 mm/s). Thermocouples 1, 2 and 3 are welded on the upper face of the sheet, at a lateral distance of 5, 12 and 10 mm from the welding line, respectively.

Figure 5 :

 5 Figure 5: Comparison between simulated and measured weld pool size (test 2).

Figure 6 :Figure 7 :

 67 Figure 6: Representation of the calculated mushy zone and liquid weld pool for test 3 (coupon width 25 mm, arc voltage 10 V, arc current 102 A, welding speed 4.33 mm/s) after a steady state has been reached in the transient resolution. 1355 °C is the solidus temperature and 1440 °C is the liquidus temperature.

  In this figure, for each welding test the transverse stress component  yy is plotted for a material point located on the weld center line in the region of interest during the crossing of the lower part of the solidification interval, below T coh . Clearly, calculated stress levels do not correlate with the cracking status (C or NC) of specimens during tests. It is worth noting that contrary to stress-based hot tearing criteria, such as the one proposed by Lahaie and Bouchard (2001) who derived a stress limit deduced from the stress necessary to separate plates bonded by capillary force. Conversely, in the following paragraph, it will be shown that strain levels indeed correlate much better with cracking status.

Figure 8 :

 8 Figure 8: Calculated transverse stress  yy as a function of temperature during the stress build-up below the coherency temperature (1400 °C, f s = 0.8) for the six tests studied.

Figure 9 :

 9 Figure 9: Calculated cumulated plastic strain as a function of temperature below the coherency temperature (1400 °C, f s = 0.8) for the six tests studied.

Figure 10 :

 10 Figure 10: Influence of the choice of coherency temperature T coh on the build-up of plastic strain. Top: T coh = 1415 °C (f s = 0.60), bottom: T coh = 1392 °C (f s = 0.85).

Figure 11 :

 11 Figure 11: Comparison between calculated pressure drop and observed cracking status.

Table 1 : Measured chemical composition of steel AISI 321 (weight%).

 1

	C	Si	Mn	P	Cr	Ni	Ti	S	N
	0.029	0.46	0.94	0.0.21 17.15	9.02	0.310	<0.01	0.011
	Several types of hot cracking tests in welding conditions have been developed in the past,

essentially consisting of a straight welding line performed on a plate, in order to create a possible axial crack in the wake of the welding torch. Tests differ by the plate geometry and boundary conditions. The hot cracking test considered here is of rectangular shape, as in the work of

[START_REF] Shibahara | Finite element method for hot cracking analysis using temperature dependent interface element[END_REF]

who developed this test in the Joining and Welding Research Institute in Osaka. In the literature, this hot cracking test tends to be named JWRI test. It consists of a straight TIG (GTA) weld bead on a flat rectangular plate specimen (length 140

Table 2 : Welding test configurations.

 2

	Test #	Coupon width [mm]	Arc voltage U [V]	Arc current I [A]	Welding speed V [mm/s]	Welding heat input (UI/V) [J/mm]	Result: crack / no-crack
	1	20	10	109	4.67	233.6	C
	2	20	10	102	4.33	235.4	C-NC
	3	25	10	102	4.33	235.4	NC
	4	25	10	109	4.67	233.6	C
	5	40	10	109	4.67	233.6	NC
	6	40	10	117	5.00	234.0	NC

Table 3 : Material and process parameters used in RDG criterion.

 3

	Secondary dendrite	Liquid	Shrinkage	A	B	Strain rate	Norm of the
	arm spacing  2 [µm]	viscosity [Pa s]	coefficient 	[K]	[K 2 ]	[s -1 ]	thermal gradient G [K/mm]
	7	0.001	0.086	4837	3054000 0.020 to 0.046	97 to 113

Table 4 : Process parameters and calculated pressure drop for the six test cases.

 4

	Test #	Strain rate [s -1 ]	Thermal gradient G [K/mm]	Pressure drop p [kPa]
	1	0.046	97	133
	2	0.040	106	106
	3	0.023	113	81
	4	0.030	109	96
	5	0.020	111	84
	6	0.027	103	105

In Table 5 the impact of uncertainties on values of strain rate, thermal gradient and SDAS has been quantified, by choosing typical uncertainty values. It can be seen that effects of  and G are limited. Conversely, the effect of SDAS is strikingly high. A small variation by 1 m makes the pressure drop p vary considerably with respect to the critical value, previously identified around 90 kPa. This appears as a serious issue in the use of RDG criterion. Indeed, as neither formula such as Eq. ( 3), nor direct measurements can claim to be accurate enough, it is then extremely difficult to compare calculated pressure drops to a given critical threshold that would distinguish cracking and non-cracking states.

To summarize with the use of the RDG criterion, the consistency between calculated crack prediction and experimental observations is obtained at the price of a choice of a high value for the liquid cavitation pressure: 90 kPa. This value is among the highest ones found in the literature. Moreover, a sensitivity study has shown that the calculated pressure drop through the mushy zoneto be compared with this critical valueis dramatically dependent on the secondary dendrite arm spacing. This is an additional issue which might affect the predictability of this criterion.

In line with these last remarks, and before drawing final conclusions, it is worth drawing the attention on the fact that neither the critical strain threshold for the strain based criterion,