Associate Professor Simone Bertoli
email: simone.bertoli@udamail.fr

Associate Professor Michaël Goujon
email: michael.goujon@udamail.fr

Olivier Santoni
email: olivier.santoni@udamail.fr

Catherine Araujo Bonjean

Keywords: International trade, Bilateral maritime distances, Road distances

We describe the publicly available CERDI-seadistance database, which contains bilateral maritime distances between 227 countries and territories. The relevant port(s) for countries with access to the sea is defined as the coastal cell of a country that contains the highest number of shipping lines, and each landlocked country is associated to the (foreign) port with the shortest road distance to its capital city. The length of the existing shortest sea route between the two ports is then computed; this represents the bilateral maritime distance included in our database, which also contains a few additional ancillary variables.

Since the seminal contribution of [START_REF] Tinbergen | Shaping the world economy: suggestions for an international economic policy[END_REF], trade economists have relied on the great-circle or as the crow flies distance, which represents the length of the shortest path between two points on the surface of a sphere, to measure the distance between two trading countries.1 This distance is included in widely used datasets of bilateral and dyadic variables (see, for instance, [START_REF] Mayer | Notes on CEPII's distances measures: the GeoDist Database[END_REF]. The great-circle distance represents a very good proxy of the air distance between two countries, but a substantial-albeit declining-share of world trade still goes through the sea, and maritime and air distances can greatly differ for pairs of countries that are separated by huge land masses; for instance, the sea distance between Bremen (Germany) and Tokyo (Japan) via the Suez Canal stands at 11,399 nautical miles, while the great circle distance between the two cities stands at 4,898 nautical miles.

The CERDI-seadistance database provides a matrix of bilateral sea distances for 227 countries and territories. 2 The construction of the dataset has required to: (i ) identify the location of the relevant port(s) for each country, and (ii ) compute of the sea distance between the relevant ports of each one of the 227¢226 country pairs. We also computed and included in the dataset (iii ) the road distance between the capital of each country and its relevant port(s). For countries with access to the sea, the relevant port is located in the coastal cell of its territory that contains the highest number of shipping lines among the cells that include a major road.3 For a landlocked country, the relevant port is defined as the (foreign) relevant port with the shortest road distance to its capital city. We also include in the dataset (iv ) the road distance between the capital cities of two countries when this is shorter than the sum of the road distance connecting each of them to the corresponding relevant port.

The CERDI-seadistance database is closely related to [START_REF] Feyrer | Trade and Income -Exploiting Time Series in Geography[END_REF], but it differentiates from it in several respects; specifically, we describe explicitly which is the criterion used to identify the relevant port for each country, including also landlocked countries, and we rely on existing maritime routes rather than on an algorithm that finds the shortest path between two ports to define maritime distances. Last but not least, our database is made freely publicly available. Another closely related database is represented by [START_REF] Fugazza | Building A Dataset For Bilateral Maritime Connectivity[END_REF], who measure the shortest maritime linear shipping routes between any pair of countries for a sample of 178 countries over 2006-2012. The maritime distances between main container ports in each country are computed, and information about the need of container transshipments when countries are not connected directly, which could induce higher transportation costs.To the best of our knowledge, this database is not publicly available.

The paper is structured as follows: Section 2 describes the various analytical choices and the different steps that we followed for the construction of the database, while Section 3 describes the structure and content of the Excel and Stata data files that contain the CERDI-seadistance database.

Construction of the database

We provide here a detailed description of the different phases that have led to the construction of the CERDI-seadistance database.

Identification of the relevant port(s)

The identification of the relevant port(s) for each country goes through a series of intermediate steps. First, we divided the area of each country or territory into a grid of cells of 100 squared kilometers, and selected the coastal cells. Then, we relied on [START_REF] Halpern | A global map of human impact on marine ecosystems[END_REF], who provide information about the number of shipping lines that went through each squared kilometer of the sea in 2005 (see Figure 1), to count the number of shipping lines that go through each coastal cell using the Spatial Analyst Zonal statistics tool. 4 Finally, we relied on the RWDB2 database by the FAO to identify the coastal cells that include major land roads, and selected from this subset of cells the one with the highest number of shipping lines. This last step is meant to avoid identifying as the relevant port a coastal cell that is located in an unpopulated area, as it would have happened, for instance, in the case of Namibia, where the coastal cell with the highest number of shipping lines is within the Skeleton Coast National Park.

We have identified two relevant ports for three countries, namely Canada, Russia and the United States, following [START_REF] Feyrer | Trade and Income -Exploiting Time Series in Geography[END_REF]. The justification for this choice is related to that the bilateral sea distances of these countries with the other 226 countries in the database would be otherwise greatly sensitive with respect to the choice to locate the relevant port on their Western or Eastern coast. Specifically, the average distance between Canada and the relevant ports of the other countries with access to the sea stands at 8,126 kilometers, and the average of the absolute value of the difference in the distances measured from its relevant port on the Eastern or the Western coast stands at 7,333 kilometers, i.e., more than 90 percent of the average distance. 5 Canada is clearly not the only country that have ports located both on the Atlantic and on the Pacific Ocean; what differentiates, say, Canada from Colombia is that in the latter case the two ports are not separated by a huge land mass, as shipping lines that go through the Panama Canal, and this is why we have decided not to select two relevant ports for other countries. 42 out of 227 countries do not have access to the sea; 6 for each one of them, we selected the (foreign) relevant port with the minimal road distance to its capital city. 7,8 We also computed the road distance between the capital city of each country and its relevant port(s), adopting the adjustment for the slope of the terrain described above. 9 Figure 2 provides an example of our use of the (adjusted) data on the road network to identify the relevant port for Mali, which happens to be located in Guinea.

Computation of the sea distance between relevant ports

Distances between relevant ports have been calculated as the shortest path between them, using the raster dataset in [START_REF] Halpern | A global map of human impact on marine ecosystems[END_REF] and the Mollweide projection. From each relevant port, a raster distance map is created using the tool Coast Distance from Spatial Analyst. Then, with Spatial Analyst's Zonal statistics we get the distance from the home port to all other relevant ports, following actual shipping lines from [START_REF] Halpern | A global map of human impact on marine ecosystems[END_REF]. The computation is limited to 11,000 kilometers from the central meridian to avoid distortions due to map projection far away from it; to calculate all distances, the central meridian is moved eight times by 45 0 (Figure 3 provides an example).

Notice that the bilateral maritime distance can be equal to zero for two landlocked countries if they are both connected to the same (foreign) relevant port. For instance, this is the case for the bilateral distance between Austria, Hungary, Slovakia and Liechtenstein, as the relevant port for these four countries is the Slovenian one, or for Afghanistan, Kyrgyzstan, Tajikistan and Uzbekistan, which are all associated to a relevant port located in Pakistan.

relevant ports.

7 The data source for the road network is the GeoNetwork database by the FAO (http://www.fao.org/geonetwork/srv/en/); we adjusted the road distance through a function of its slope using the digital terrain model provided by ISciences-Terraviva (http://www.TerraViva.NET/DataSolutions/projects elevation.html), by multiplying the length of the road by a coefficient that is an increasing function of the slope of the terrain (notice that 86 percent of the cells have a zero slope, so that this adjustment has actually a limited empirical relevance).

8 We considered, but did not opt for, the option of taking two relevant ports also for some landlocked countries, e.g., Chad, for which the road distances between its capital city and two relevant ports located on two different coasts were very close to each other. 9 This distance was not calculated for small islands, small coastal territories, such as Macau, and for the Democratic Republic of the Congo and some islands such as Cuba for which information on the road network was not available.

Computation of the road distance for landlocked countries

For some pairs of landlocked countries, the road distance between their capitals is shorter than the sum of the road distances between them and the two relevant ports (for instance, Mali and Burkina Faso), so that merchandise trade is unlikely to go through the sea in this case. The database reports the direct road distance, and it includes a dummy when this is smaller than the sum of the distances to the two ports.

3 The CERDI-seadistance database in Excel and Stata

The CERDI-seadistance database, whcih is publicly available on Zenodo at the address http://dx.doi.org/10.5281/zenodo.46822, contains the following seven variables: 1. iso1: the alpha-3 ISO country code of the first country (string).

2. iso2: the alpha-3 ISO country code of the second country (string).

3. seadistance: bilateral sea distance in kilometers (numeric).

4. capitalport1: road distance between the capital of the first country and the relevant port in kilometers (numeric).

5. capitalport2: road distance between the capital of the second country and the relevant port in kilometers (numeric).

6. roaddistance: bilateral road distance between two capitals in kilometers (numeric).

7. short: a dummy variable which is equal to 1 if roaddistance is lower than capitalport1+capitalport2, and 0 otherwise (numeric).

Notice that the variables capitalport1 and capitalport2 are bilateral rather than country-specific as for six countries (Canada, Russia, the United States, plus Azerbaijan, Kazakhstan and Turkmenistan) we use two relevant ports.

Figure 1 :

 1 Figure 1: Shipping lines

Figure 2 :

 2 Figure 2: Identification of the (foreign) relevant port for Mali

Figure 3 :

 3 Figure 3: Area calculation with the central meridian at 180 0

[START_REF] Feyrer | Trade and Income -Exploiting Time Series in Geography[END_REF] represents a notable exception here, as discussed below.

The list of countries and territories comes from Gridded Population of the World Database by the SEDAC-CIESIN, Columbia University.

For some large countries, notably the United States, Canada and Russia, we have identified two relevant ports, one of the Eastern and one on the Western coast, as in[START_REF] Feyrer | Trade and Income -Exploiting Time Series in Geography[END_REF].

Specifically, we attribute to each coastal cell the highest number of shipping lines that go through the set of maritime cells that it contains.

Études et Documents n° 7, CERDI, 2016

The corresponding figures for Russia and the United States stand at 51 and 120 percent respectively.

The three countries bordering the Caspian Sea, namely Azerbaijan, Kazakhstan and Turkmenistan are considered as being landlocked, but we also computed the direct maritime distances between their (domestic)

Acknowledgment

The authors are grateful to Martine Bouchut for the technical help she provided for the treatment of the data; the geolocalized databases program led by the CERDI is supported by the Agence nationale de la recherche of the French Government through the program `Investissements d'Avenir' (ANR-10-LABX-14-01), and the IDGM+ initiative conducted by the FERDI (Fondation pour les études et recherches sur le développement international); the usual disclaimers apply.

This work was supported by the LABEX IDGM+ (ANR-10-LABX-14-01) within the program "Investissements d'Avenir" operated by the French National Research Agency (ANR). http://www.cerdi.org/

Users should cite the CERDI-seadistance database as follows:

Bertoli, S., M. Goujon, and O. Santoni (2016): "The CERDI-seadistance database,"

Études et Documents No. 7, CERDI, Clermont-Ferrand.


Vincent Villaret

Frédéric Deschaux-Beaume
email: frederic.deschaux-beaume@umontpellier.fr

Cyril Bordreuil

A solidification model for the columnar to equiaxed transition in welding of a Cr-Mo ferritic stainless steel with Ti as inoculant

Keywords: Solidification, Microstructure, Modeling, Ferritic stainless steel, Welding a

A solidification model for the columnar to equiaxed transition in welding of a Cr-Mo ferritic stainless steel with Ti as inoculant

Introduction

The welding remains the main joining process used for manufacturing structural metallic parts. Unfortunately, this joining process can have a detrimental effect on the material properties, specially in the melted zone, where both composition (when a filler wire is used) and microstructure are changed. In particular, the formation of a fully columnar grain structure in the melted zone during welding can be a major handicap, as it shown in a previous paper (Villaret et al., 2013a), because this type of structure has a very low strength. According to [START_REF] Flemings | Solidification processing[END_REF], it is possible however, under certain conditions, to modify the morphology of the grains in the melted zone from a columnar to an equiaxed structure, as evidenced by [START_REF] Greer | Grain refinement of aluminium alloys by inoculation[END_REF] in the case of aluminum alloys. This kind of structure significantly improves the ductility of the melted zones [START_REF] Petersen | Fined grain weld structure[END_REF] and decreases the transition temperature [START_REF] Lakshminarayanan | Effect of autogenous arc welding processes on tensile and impact properties of ferritic stainless steel joints[END_REF], especially if, as evidenced by [START_REF] Bayraktar | Effect of welding conditions on the formability characteristics of thin sheet steels: mechanical and metallurgical effects[END_REF], the obtained equiaxed grain is fine. This transition from columnar grains to equiaxed grains, so called CET (Columnar to Equiaxed Transition) depends on the temperature field and its evolution in the solidification area. Two parameters have a particular importance on the solidification morphology, the thermal gradient at the solidification front G, and the moving velocity of this front V. It is generally admitted [START_REF] Kurz | Fundamentals of Solidification[END_REF]) that a decrease in the G/V ratio promotes the CET.

The dominant theory for explaining the CET implies the nucleation and growth of new equiaxed grains in the undercooled liquid near the solidification front that stop the columnar grain growth, by mechanical interaction according to [START_REF] Kurz | Columnar to equiaxed transition in solidification processing[END_REF], or by solutal interaction according to [START_REF] Martorano | A solutal interaction mechanism for the columnar-to-equiaxed transition in alloy solidification[END_REF]. The cooling conditions promoting a high undercooling in the liquid near the solidification front, i.e., a low G/V ratio, favor the nucleation of new equiaxed grains and consequently the CET. In welding, an increase of the welding speed for instance can decrease the G/V ratio, thus promotes the equiaxed grain structure and, as evidenced by [START_REF] Bayraktar | Effect of welding conditions on the formability characteristics of thin sheet steels: mechanical and metallurgical effects[END_REF], improves the ductility of the melted zone. However, the number of nuclei giving equiaxed grains, formed in the undercooled liquid, is mainly controlled by the weld pool composition, and specially by the content of some elements favoring the formation of nucleation sites. According to [START_REF] Ostrowski | Precipitation of titanium carbonitrides in as-cast 17% chromium stainless steels[END_REF], titanium for instance has a very active role on the morphology of solidification grains for many alloys. In the case of ferritic stainless steels, it has been shown in a previous paper (Villaret et al., 2013b) that Ti forms refractory compounds in the liquid alloy that provide heterogeneous nucleation sites for equiaxed grains. The exact nature of these compounds is not always identified, and [START_REF] Villafuerte | Grain structures in gas tungsten-arc welds of austenitic stainless steels with ferrite primary phase[END_REF], [START_REF] Villafuerte | Mechanisms of equiaxed grain formation in ferritic stainless steel gas tungsten arc welds[END_REF] indicate it seems to vary depending on the composition and cooling conditions.

As the formed grain structure in the melting zone has a primary importance on the weld properties, some modeling have been developed in order to try to predict the microstructure formed after welding. The modeling based on Cellular Automaton, generally coupled to Finite Element (CAFE) methods, recently developed, were applied by [START_REF] Rappaz | Prediction of grain structures in various solidification processes[END_REF] and [START_REF] Carozzani | 3D CAFE simulation of a macrosegregation benchmark experiment[END_REF] to predict with a rather good accuracy the grain structure formed during ingot casting, or by [START_REF] Chiocca | Microstructure generation during CuNi bead on plate: comparison of experimental results and CA prediction[END_REF] for weld solidification, but needs very long computation time. At the opposite, the classical models based on Hunt's theory [START_REF] Hunt | Steady state columnar and equiaxed growth of dendrites and eutectic[END_REF] are more simple methods to predict if the CET occurs or not during the solidification, as demonstrated by [START_REF] Kurz | Columnar to equiaxed transition in solidification processing[END_REF], but require the preliminary determination of the equiaxed grain density, that is not predictable by the model, and that can change, as evidenced by [START_REF] Gaumann | Nucleation ahead of the advancing interface in directional solidification[END_REF] , according to the cooling rate.

In this paper, an alternative modeling for predicting solidification grain morphology in welding, based on a two step computation, is presented. The first step consists in a thermal numerical simulation of the welding process, in order to predict the thermal field evolution all around the solidification front. The second step consists in modeling, from this thermal field evolution, the grain solidification by the combination of the columnar growth model previously proposed by [START_REF] Niel | Modeling hot cracking in 6061 aluminum alloy weld with a microstructural based criterion[END_REF] , the nucleation model developed by [START_REF] Tronche | Modelling of inoculation of metallic melts: application to grain refinement of aluminium by Al-Ti-B[END_REF], and a CET criteria derived from the Hunt's criteria (1984). The two steps of the mod-eling are first described, and then applied to predict the CET during Gas Tungsten Arc Welding of a ferritic stainless steel. The model predictions are compared to experimental results.

Thermal modeling of welding

A simplified thermal modeling of the welding process is retained to predict the thermal fields in the solidification area, using the commercial finite element software Sysweld. The welded sample is considered as a solid submitted to conduction heat transfer, with boundary conditions to model heat transfers between the sample and the surrounding environment. The heat source of the welding process is modeled by a heat flux distribution located on the surface of the sample.

Geometry and mesh

The modeled samples are thin sheets of 160 × 60 × 1.8 mm 3 melted in their centerline using a Gas Tungsten Arc Welding (GTAW) process without filler metal. In order to reduce computation time, a two-dimensional simulation is chosen in the plane of the sample surface, as the melting zone is fully penetrated and have a rather similar width in back and upper side for a low sheet thickness. Due to the centerline symmetry, only half a sample is modeled and meshed with linear rectangular elements with 4 nodes. The elements around the symmetry axis, that are closed to the trajectory of the heat source center, suffer higher thermal gradients, and are more finely meshed to get a better computation accuracy. The center of the heat source is positioned at the initial time on the left of the symmetry axis of the sample, and is translated along the y-axis at a speed corresponding to the welding speed.

Heat equation

Conduction heat transfers in solids are governed by the heat equation:

Cp • ∂T ∂t -div ( • gradT ) = Q (x, y, z, t) (1)
with T the temperature (K), the density of the material (kg m -3 ), Cp the specific heat (J kg -1 K -1 ), the thermal conductivity (W m -1 K -1 ), and Q a potential source of external heat. Solving this equation in any point of the sample in a Lagrangian (O,x,y,z) coordinate system associated to the sample, and at each time step, allows to know the evolution of the temperature field in the sample under the effect of the heat input Q corresponding to the welding source.

To take into account the latent heat of fusion Lf (Lf = Hl -Hs), the specific heat is increased within the solidification range (between Ts and Tl) such as:

Tl Ts CpdT = Lf (2)
To take into account the increase in heat transfers in the weld pool due to convection phenomena, the thermal conductivity of the material is artificially increased from the liquidus temperature.

Boundary and initial conditions

The initial temperature of the specimen is taken equal to 20 • C for all the nodes.

The heat transfers between the sample and the outer environment by convection and radiation are modeled by a flow imposed on all outer surfaces (Neumann condition):

- ∂T ∂n = h (T -T 0 ) + ε T 4 -T 4 0 (3)
with n the normal to the outer surface, h the convection exchange coefficient, the emissivity of the material, and the Stephan-Boltzmann constant.

Heat source

The heat source is modeled by a heat flow density imposed on the sample surface interacting with the arc. A Gaussian distribution is chosen, centered on the arc axis, moving in translation along the y-axis at a speed equal to the welding speed Vs. The surface distribution of heat flow density is then given by:

Q (r) = 2Q 0 R s exp -3r 2 R 2 s (4)
with Q 0 the Gaussian amplitude parameter, Rs the width parameter, and r the distance to the center of the source, defined by:

r = (X -X o) 2 + (Y -Y 0 -V s t) 2 (5)
where X 0 and Y 0 are the coordinates of the initial position of the heat source, and t the time.

The total power transmitted to the sample by this distribution, obtained by integrating the relationship (4) for r values from 0 to infinity, must match the effective welding power. One can then test 1 (a), 2 (b) and3 (c). deduce the following relation between the distribution parameters and the process parameters:

Q 0 = Á • UI (6)
with U the welding voltage, I the welding current and the process efficiency.

Modeling of the CET

The modeling is based on the approach developed by [START_REF] Niel | Modeling hot cracking in 6061 aluminum alloy weld with a microstructural based criterion[END_REF] coupled to the "athermal" nucleation model proposed by Greer [START_REF] Tronche | Modelling of inoculation of metallic melts: application to grain refinement of aluminium by Al-Ti-B[END_REF]. It consists in compute the growth rate of columnar grains all around the solidification front, and the volume of nucleated equiaxed grains in the undercooled liquid near this front. When the volume fraction of equiaxed grains reaches a given limit at one point of the solidification front, then the CET occurs at this point.

Growth rate of columnar grains

The first step of the modeling consists in calculating the growth rate of columnar grains, all around the solidification front located on the rear border of the weld pool. It is supposed in this step the solidification front is on the liquidus isothermal line.

The growth direction of columnar grains is often supposed to be rather perpendicular to the liquidus isothermal line. However, experience indicates a divergence between heat flux direction and growth direction, specially for high welding speed, due to the preferential growth direction in crystalline materials.

In the modeling, the orientation of grains is given by the ␣ angle, defined as the angle between the welding direction and the growth direction. The variation of this angle is calculated from eq. ( 7):

˙= A (˛-˛n) (7)
where ␣n is the angle between welding direction and heat flux, and A is a constant chosen for a given material according to preliminary welding results.

For each time step, the growth direction changes according to Eq. ( 8):

˛(t + dt) = ˛(t) + ˙dt (8)
The growth rate of columnar grains V is given by:

V = V s cos ˛n cos (˛-˛n) (9)
where Vs is the welding speed.

Undercooling

As suggested by [START_REF] Dantzig | Solidification[END_REF], the curvature undercooling as well the undercooling associated with the attachment kinetics are neglected. The undercooling T s then results only on constitutional undercooling T c and thermal undercooling T T [START_REF] Dantzig | Solidification[END_REF]: (10) with m l the liquidus slope, C 0 the solute concentration in the liquid, k 0 the partition coefficient of solute, and Iv3D the 3D Ivantsov function given by:

Ts = TT + T C = L f cp Iv3D (PeT ) -m l C0 (1 -k0) Iv3D (PeC ) 1 -(1 -k0) Iv3D (PeC )
Iv 3D (Pe T ) = Pe T exp (Pe T ) ∞ Pe T e -s s ds (11)
and Pe T and Pe C respectively the thermal and solutal Peclet numbers, given by:

Pe T = VR tip 2˛l (12) Pe C = VR tip 2D l (13)
with R tip the dendrite tip radius, ␣ l the thermal diffusivity, and D l the diffusion coefficient of solute. Supposing thermal and solutal Peclet numbers are below to 1, one can use the approximation of Ivantsov fonction proposed by [START_REF] Dantzig | Solidification[END_REF] :

Iv 3D Pe ≈ 1.5Pe 0.8 (14)
The undercooling is then given according to the growth rate of columnar grains by the simplified expression:

T s = T T + T c = 1.5 0.8 sl 0.4 L f D l 0.8 -m l C 0 1 -k 0 C p ˛l0.8 C p 0.6 ˛l0.4 D l 0.4 L f D l -m l C 0 1 -k 0 C p ˛l 0.4 V 0.4 (15)
with sl the Gibbs Thomson coefficient.

Nucleation and growth of equiaxed grains in the undercooling region

The width ␦ of the undercooling region near the columnar solidification front, supposed very small, is calculated from the undercooling T s and the maximal thermal gradient G (normal to the isothermal surface) on each point of the liquidus. Supposing a linear temperature variation in this area, the undercooling width ␦ is given by:

ı = T s G (16)
The nucleation of equiaxed grains is supposed to be possible only in this area. The undercooling on each point of this zone then varies between Ts at the columnar front to 0 at the distance ␦ from this front.

Number of formed nuclei

The athermal nucleation model developed by Greer to predict solidification during casting of aluminum alloys [START_REF] Tronche | Modelling of inoculation of metallic melts: application to grain refinement of aluminium by Al-Ti-B[END_REF] has been retained. This model is well suited for heterogeneous nucleation from small solid particles [START_REF] Greer | Grain refinement of aluminium alloys by inoculation[END_REF], as in ferritic stainless steels inoculated with Ti, in which ferritic grains nucleate from Ti-rich refractory compounds formed in the weld pool (Villaret et al., 2013b).

It is supposed the size distribution of Ti-rich precipitates can be approximated using a lognormal distribution:

dn d = n max √ 2 1 exp - 1 2 ln -ln 0 2 (17)
with ˚the precipitate diameter, n max the precipitates number density, ˚ and ln˚0 the distribution parameters, corresponding respectively to the standard deviation and the mean.

According the Greer nucleation model, each precipitate of diameter is associated to a nucleation undercooling T given by relation ( 18):

T = 4 sl (18)
One can deduce the nucleation undercooling distribution, i.e., the number of nuclei "activated" according to the undercooling T:

dn d T = n max T √ 2 1 T exp - 1 2 ln T -ln T 0 T 2 ( 19
)
with T = ˚ and T 0 = 4 sl /˚0.

The number of nuclei formed in each elementary volume of the undercooling zone, associated to the undercooling Ti (between 0 and Ts), is then given by relation (20):

dn( T i ) = n max T √ 2 1 T i exp - 1 2 ln T i -ln T 0 T 2 d T i (20)

Growth of equiaxed grains

Once nuclei are formed, equiaxed grains will grow at a growth rate depending on the undercooling T i in the surrounding liquid volume, supposed to be at the same temperature than the nuclei. The growth rate of equiaxed grains v e follows the same growth law than columnar grains:

ve = A T i 2.5 (21) with A = C p 0.6 ˛l0.4 D l 0.4 L f D l -m l C 0 (1 -k 0 ) C p ˛l 0.4 1, 5 0.8 sl 0.4 L f D l 0.8 -m l C 0 (1 -k 0 ) C p ˛l0.8 2.5
During the time between the moment where the nuclei is formed, at time t i , and the moment t s where the columnar front arrives (at the speed V), the equiaxed grain reaches a radius R given by:

R = ts t i v e dt = A GV Ts T i T 2.5 d T = A 3.5GV T s 3.5 -T i 3.5 (22)
The volume of equiaxed grains formed at the undercooling Ti is then equal to: The total volume of equiaxed grains is equal to the sum of the volumes of grains formed at all undercooling between 0 and T s :

dV ( T i ) = nmax T √ 2 1 T i exp - 1 2 ln T i -ln T0 T 2 4 3 A 3.5GV Ts 3.5 -T i 3.5 3 d T i (23)
V ge ( T s) = Ts 0 dV ( T i ) (24)
To take into account the decrease of the growth rate due to the possible blocking between adjacent grains, one considers the volume fraction of equiaxed grains follows an Avrami law:

f ge = 1 -exp -V ge ( T s) (25)

Identification of data required for computation

The modeling described in the previous sections has been employed to study the behavior during GTAW of the melted zone of a ferritic stainless steel containing Ti as inoculant. Previous experimental results validated the assumption of an heterogeneous nucleation mechanism promoting the CET during Gas Metal Arc Welding of such stainless steels, and then confirms the choice of a nucleation theory based on the Greer model. In a previous paper (Villaret et al., 2013b), it has been shown the CET is favored by a high Ti content in the filler metal (upper to 0.15 wt%), that forms refractory compounds in the weld pool acting as heterogeneous nucleation sites. These refractory compounds can be detected by Scanning Electron Microscopy (SEM) and Energy Dispersive Spectroscopy (EDS) analyses, and their distribution are directly correlated to the Ti content. However the CET, that occurs when the columnar growth is stopped by the equiaxed grains formed in the undercooling zone, is controlled for a given composition (i.e., for a given content of inoculants as Ti), by thermal conditions during solidification, that determine the undercooling. In order to test the ability of the proposed modeling to predict the CET, a stainless steel having a controlled Ti content of 0.16 wt%, namely the K44 alloy supplied by Arcelor Mittal (Table 1), is used.

Thermal simulation

The physical characteristics of the material (, Cp, ) vary with temperature, and are given in Table 2 up to 800 or 1000 • C. These characteristics are assumed to be constant beyond these temperatures up to the solidus temperature of the material.

The heat source parameters are chosen using literature data and experimental considerations. The width parameter Rs is chosen to be rather similar to the arc radius, and de process efficiency is chosen according to literature data and validated by comparison between preliminary simulations and previous welding tests results (Table 3). Radiation and convection heat transfer coefficients are chosen according to previous instrumented welding tests, using comparisons between numerical and experimental cooling rates (Table 3).

CET modeling

Physical properties of the material

The physical properties required for the computation are gathered in Table 4. Formulations expressing the relationship between undercooling and growth rate ( 15) are given for binary alloys. In the case of multi-component alloys, one can consider an equivalent pseudo-system. m l , C 0 , and k 0 parameters are then replaced by the following expressions (Mingbo [START_REF] Yang | Analysis about forming mechanism of equiaxed crystal zone for 1Cr18Ni9Ti stainless steel twin-roll thin strip[END_REF]:

C 0 = i C i m l = i m i C i C 0 k 0 = i m i C i k i m l C 0 ( 26
)
where C i is the concentration of element i, m i and k i respectively the liquidus slope and the partition coefficient in the binary diagram Fe-element i.

Only Cr and Mo elements are took into account, that have the higher contents in the material, to compute these parameters.

Taking account of convection in the weld pool

Using values of Table 4, the undercooling at the columnar solidification front, composed of dendrite tips, given by eq. 15, becomes:

T s = 105.3V 0.4 (27)
with V in m/s. Eq. ( 15) is established using a "marginal stability criteria" that suppose the dendrite tip radius Rtip is equal to the minimal wavelength of disturbance min producing the instability of a planar solidification front [START_REF] Dantzig | Solidification[END_REF]. According to this criteria, the dendrite tip radius is given by:

R tip = min = 2 sl m l G c -G (28)
where Gc and G are respectively the solute gradient and the thermal gradient in the liquid at the solidification front, given, according to the Ivantsov analyze [START_REF] Dantzig | Solidification[END_REF], by:

G = - 2Pe T L f CpR tip (29) G c = - 2Pe C C l (1 -k 0 ) R tip ( 30
)
The dendrite tip radius calculated with the values of Table 4, of about 0.5 m, is very low compared to that generally observed after welding of these ferritic stainless steel, which is about 10 m. That seems to prove the convection phenomena due to fluid flow acting in the weld pool have a significant effect on the solidification mechanism, by promoting the thermal and solute transfers, and then by increasing the thermal and solute limit boundary layers, given by ı T = 2˛l/V and ı C = 2D l /V for pure diffusive mechanisms. If one considers thermal and solute boundary layers have the same thickness due to the convection, the effect of fluid flow can be artificially traduced by an increase of the thermal and solute diffusivity coefficients. The "equivalent diffusivity" Deq corresponding to a dendrite tip radius of 10 m is 2.3 10 -5 m 2 s -1 for a moving velocity of the [START_REF] Kagawa | Coefficients for equilibrium partition of a third element between solid and liquid in iron-carbon base ternary alloys and their relation to graphitization during iron-carbon eutectic solidification[END_REF] sl (Gibbs Thomson coefficient, ␥sl/ S f ) 1.9 10 -7 m K Dantzig and Rappaz (2009) ␥ sl (solid-liquid surface tension) 0.204 J m -2 Dantzig and Rappaz (2009) S f (volume entropy of fusion)

1.07 10 6 J m -3 K -1 [START_REF] Dantzig | Solidification[END_REF] ␣l (thermal diffusivity in liquid steel) 6.10 -6 m 2 s -1 Nishi et al. ( 2003) Cp (specific heat) 820 J kg -1 K -1 Dantzig and Rappaz (2009) L f (latent heat of fusion) 2.76 10 5 J kg -1 [START_REF] Dantzig | Solidification[END_REF] solidification front of 0.005 ms -1 , corresponding to a representative value in welding. By replacing ␣ l and D l by this new coefficient, Eqs. ( 15) and ( 27) become:

T s = T T + T c = 1.5 0.8 sl 0.4 C p 0.6 D eq 0.4 L f -m l C 0 (1 -k 0 ) C p 0.6 V 0.4 = 18.55V 0.4 (31)
with V in m/s.

Precipitates distribution in the melting zone

The nucleation model defines in Eq. ( 17) a precipitates distribution allowing the heterogeneous nucleation of equiaxed grains in the undercooling zone. In ferritic stainless steel, these precipitates are supposed to be titanium-based refractory compounds (Villaret et al., 2013b). One of the limitation of this model is it requires the identification of this precipitates distribution. In order to study this distribution, a methodology has been developed and tested on several stainless steels. The precipitates distribution analysis is carried out using a SEM equipped with INCA analysis setup developed by OXFORD © (www.oxford-instruments.com). This tool allows to automatically detect and analyze by EDS, in a predefined field of the sample, particles using the grey level differences compared to the matrix. This tool gives for each detected particle its position, size, grey level and composition. The magnification of SEM images for the analyze is fixed at 700 to detect particles of less than 0.05 m in diameter. The analysis field consists in 9 SEM images representing a total surface area of about 0.5 mm 2 . A threshold of grey level is chosen to separate Ti-rich particles, that appear in dark grey on SEM images, and the matrix. The main elements of the steel, namely chromium and iron, plus Ti and Nb additive elements are analyzed by EDS on each detected particle, in order to confirm it is Ti-rich particle. That allows to eliminate from the distribution for instance porosities, that appears also in dark contrast in the SEM images. Finally, a dedicated Python routine is developed to select, according to the EDS analyses, the Ti-reach particles, and to compute their size distribution.

This analysis method has been used to compare the Ti-rich precipitates distribution of the melting zone of ferritic stainless steels containing various Ti contents, and revealed significant changes between the compositions. Fig. 1 shows the differences between 2 steels containing various Ti contents, and then various Ti-rich particles distributions. The first one, containing 0.07 wt% titanium, exhibit a columnar grain structure in the melting zone, except in the center-part and the top, where some coarse equiaxed grains are formed. Conversely, the second one, containing 0.15% titanium, exhibit a quasi-complete equiaxed grains structure in the melting zone. As a consequence of a higher Ti content, a greater number of Ti-rich precipitates is formed, and CET is promoted in the melting zone. On the other hand, when a given composition is melted by GTAW process with various welding parameters, there is only minor changes in the precipitates distribution, that shows this distribution is mainly controlled by the steel composition. It is an interesting result because that means the precipitates distribution for a given steel composition has to be characterized only one time, whatever the welding parameters used.

Fig. 2 shows the size distribution of Ti-rich precipitates obtained by analyzing the melting zone of the K44 steel, containing 0.16% titanium, and the associated lognormal law. The parameters of the lognormal law are computed according to relations:

ln 0 = n i=1 n i n T ln i (32) = n i=1 ln i -ln 0 2 n i n T (33)
n max = n i=1 n i (34)
with n i the number of particles per volume unit of diameter i , n T the total number of particles per volume unit (m -3 ), and the width of the diameter classes.

The size distribution parameters identified from this analyze are given in Fig. 2. One can deduced from these parameters the distribution parameters of the "nucleation undercooling" distribution of Eq. ( 19).

Comparison with experimental results

In order to test the ability of the proposed modeling to predict the CET, bead on plate tests using GTAW process on K44 steel are carried out and analyzed for comparison to the simulation predictions.

Experimental details

Three GTAW welding tests have been achieved with various welding parameters chosen in order to produce increasing undercooling, and then to promote CET (Table 5). The welding speed and current have been simultaneously increased to obtain for each test a rather constant width of melting zone. A GTAW source Sincosalt AC/DC 400A, associated with an automatic water-cooled torch MEC4SAF-FRO equipped with a thoriated tungsten electrode (W-2% Th) of 2.4 mm diameter with a filing angle of 45 • , has been used. The shielding gas is a 99.5% pure argon (ARCAL 1 Air Liquide) with a flow rate of 15 l/min.

The samples are stainless steel sheets of 160 × 60 × 1.8 mm 3 . Six K-type sheathed thermocouples of 0.5 mm diameter were placed on each test specimen (Fig. 3), on the back face in order to protect them against the arc radiation, that can distort measurements. To keep thermocouples in position, they have been embedded in 0.5 mm diameter holes of 0.5 mm depth machined using hemispherical machining tools to improve contact surface between the sample and the thermocouples. Thermal acquisitions are made at a frequency of 20 Hz for 120 s.

Thermal modeling

Fig. 4 compares the thermal cycle measured by thermocouple T3 with those computed by numerical simulation at the nodes located near the location of thermocouple T3. The center position of the thermocouple does not correspond exactly to a node position, so several thermal cycles corresponding to neighboring nodes are shown. Comparison between simulated and experimental temperature curves shows good agreement, excepts for the thermocouples closest to the melting zone (T4 and T6), where the computed maximum temperature is about 200 • higher than that measured. This can be explained in particular by the thermal inertia of the sheathed thermocouples and the imperfect contact with the sample.

Fig. 5 compares the computed isothermal lines corresponding to the liquidus temperature of the alloy with the boundary of the weld pool observed at the end of the fusion line. One can see that despite a comparable width, the length of the experimental weld pool is always greater than the simulated one. That can be due to the fact that the physical phenomena related to the fluid flow in the weld pool are not taken into account by the simulation. Note also that in the numerical simulation, we consider the boundary of the weld pool is given by the liquidus isotherm, while the real boundary at the rear of the weld pool may be at lower temperature due to the undercooling, and may not correspond to an isothermal surface. However, despite the simplification adopted in the thermal modeling, the weld pool boundary and the thermal field are globally well predicted for each welding condition.

Prediction of the CET

Fig. 6 shows in top view the grain structure of the melting zone formed for the three welding tests. One can see the fraction of equiaxed grains in the center part of the melting zone increases with the welding speed, as expected. Fig. 7 shows the evolution of the equiaxed grain fraction in the melting zone computed from the thermal simulation of test 1, using the "diffusive model" (Eq. ( 27)) or the "convective model" (Eq. ( 31)) for undercooling computation. The content of equiaxed grains increases from the fusion line boundary to the centerline. One can see that if the correction proposed for taking into account the effects of convection phenomena is not applied, the fraction of equiaxed grains reaches high values, traducing a CET that promote the formation of equiaxed grains in the major part of the melting zone, what is not in agreement with the experimental results. The model taking into account the convection phenomena seems then more realistic. Comparing this first numerical result with experiment, the fraction of equiaxed grains producing the CET is chosen equal to 20%. One can then compare the width of the columnar grain zone computed for the three welding tests with experiment. Fig. 8 shows the general tendency, that involves an increase of the width of the equiaxed grains zone with the increase of the welding speed, is well predicted by the modeling, despite a overestimation of the width of the columnar grain zone obtained with the highest welding speed.

Conclusions

-A modeling has been proposed to predict the columnar to equiaxed transition in the weld pool of ferritic stainless steels inoculated with Ti, according to welding parameters. -The first step of the modeling consists to a thermal numerical simulation of the welding process from which thermal fields all around the weld pool boundary are extracted. -In a second step, the thermal data are post-treated in order to compute the undercolling, taking into account the effect of the convection phenomena, and the fraction of equiaxed grains formed in the undercooling zone all around the weld pool boundary. -The modeling requires to know the size distribution of Ti-rich refractory compounds in the weld pool promoting the heterogeneous nucleation of equiaxed grains, that have been determined using a specific method based on SEM and EDS analyzes. -The confrontation of the model to experimental results shows it can reproduce in a realistic manner the columnar and equiaxed zone in a ferritic stainless steel containing Ti as inoculant.

Fig. 1 .

 1 Fig. 1. Ti-rich precipitates size distribution and grain morphology in melting zones containing 0.07% Ti (a), and 0.15%Ti (b).

Fig. 2 .

 2 Fig. 2. Size distribution of Ti-rich particles in K44 steel melting zone, and distribution parameters of the associated lognormal law.

Fig. 3 .

 3 Fig. 3. Sample geometry for welding tests and thermocouples location (T1 to T6).

Fig. 4 .

 4 Fig. 4. Comparison between thermal cycle measured with thermocouple T3 and computed thermal cycles for several nodes located near the center position of thermocouple T3.

Fig. 5 .

 5 Fig. 5. Comparison between numerical and experimental weld pool boundaries for welding tests 1 (a) and 2 (b).

Fig. 6 .

 6 Fig. 6. Micrograph in top view of the melting zone corresponding to the welding test 1 (a), 2 (b) and 3 (c).

Fig. 7 .

 7 Fig. 7. Evolution of the volume fraction of equiaxed grains in the melting zone according to the distance to the centerline of the weld.

Fig. 8 .

 8 Fig. 8. Comparison of the width of the columnar grains zone computed and experimentally observed for the three welding tests.

Table 1

 1 Composition of the ferritic stainless steel K44 (Arcelor Mittal).

	Element C	Si	Mn	Cr Mo	Al	Nb	Ti	V	Fe
	Weight% 0.02 0.45 0.39 18 2.09	0.007 0.29 0.16 0.1 79.9
	Table 2
	Material properties used for thermal numerical simulation.
	Temperature	Specific heat Cp	Thermal Conductivity		Density
	( • C)	(J kg -1 K -1 )		(W m -1 K -1 )			(kg m -3 )
	23	452			19.7				7675
	200	522			21.0				7632
	400	608			21.9				7581
	600	840			22.8				7526
	800	612			23.0				7467
	1000	-			-				7392

Table 3

 3 Heat source parameters and heat transfer coefficients used for thermal simulation.

	Rs (mm) Process efficiency Emissivity Convection coefficient h (W/m 2 )
	2	0.5	0.05	7.5

Table 4

 4 Physical properties used for the solidification modeling.

	Physical property	Value	References
	D l (diffusion coefficient of solute into the liquid)	3.10 -9 m 2 s -1	Dantzig and Rappaz(2009)
	m l (liquidus slope)	-186	Kagawa and Okamoto(1984)
	k0 (partition coefficient of solute ms/ml)	0.61	Kagawa and Okamoto (1984)
	C0 (solute concentration in the liquid)	0.209

Table 5

 5 Welding parameters for GTAW tests.

		Voltage (V)	Curent (A)	Welding speed (mm/s)
	Test 1	1 3	155	4.9
	Test 2	13.5	253.5	9.5
	Test 3	14.5	303.5	12.9

Acknowledgements

The authors wish to thank Air Liquide, Saint-Ouen l'Aumône, France for its financial support for this study, and Bertrand Petit, Laurent Faivre and Pierre-Olivier Santacreu, from Arcelor Mittal, Isbergue, France, for supplying the K44 steel.