To my parents and my family:

for their love

ACKNOWLEDGMENTS

First and foremost I want to thank my CEA supervisor, Mr. Olivier ASSERIN, for supporting me over the years, for believing in me and trusting me. He has been an inspiration to me. It has been an honour to work with him. He taught me, consciously and un-consciously, how good research is done. I appreciate all his contributions of time and ideas; I also appreciate his devotion to make my Ph.D. experience productive and stimulating. I am grateful to Mr. Han ZHAO my Ph. D. director, for his kindness and humour that made the work much less stressful. I am also grateful to Mr. Guilhem-Michel ROUX, who left me a great and ambitious Ph.D. subject and who put me on the right track with his guidance and implication all along the thesis. I want to express my gratitude to both Mr. Michel CORET and Mr. Assaad ZOUGHAIB of the reading committee, for accepting the onerous task of reporting my manuscript. I would also like to thank the other examiners of this thesis: Mr. Djimédo KONDO and Mr. Olivier HUBERT for their constructive comments and insightful questions.

I am particularly indebted to Jean-Christophe BRACHET for putting his expertise in our disposal. I am also indebted to Olivier ALLIX, Ahmed BENALLAL, Guillaume DE DINECHIN and Franc ¸ois HILD, for affording material and funding sources and for giving the opportunity to participate to national and international conferences.

Moreover, I would like to thank all my friends and colleagues of the LTA laboratory at CEA-Saclay with whom I have had the pleasure of working over the years. These include the permanents: Danièle AYRAULT, Michel BARRAS, Gaëtan CANNEAU, Franc ¸ois CASTILAN, Christophe CHAGNOT, Arnaud DELMAS, Olivier DOYEN, Laurent FOREST, Jacques GRALL, Arnaud MONNIER and Marc RONFARD-HARET; other past and present trainees: Laëtitia FER-NANDEZ, Yann GUILLEMOT, Cécile JARDIN, Marine JESSON, Michaël LIGOREAU, Valérie LOUBINOUX, Élodie ROUESNE, Alexandra SEBBEN and Itziar SERRANO, and finally the past and current Ph.D. colleagues: Alix BONAVENTURE, Michel BROCHARD, Fabien COR-PACE, Xiaofei KONG, Matthieu PETELET and Jean WISNIEWSKI.

The members of the LMT-Cachan laboratory have contributed immensely to my personal and professional time. The lab has been a source of friendships as well as good advice and collaboration. I am especially grateful for the "'Multiphysics"' group members I had the honour to work with: Ricardo AFONSO ANGELICO, Anne-Charlotte COCHEZ, Jean-Pierre CORDE-BOIS, Guillaume LATOUCHENT, Karine LAVERNHE, Saïd LAZREG, Jorge Enrique MUNOZ GARCIA, Nicolas GUILLEMOT, Olivier HUBERT, Anne MAYNADIER, Amir OUESLATI, Casius RIUL, Karl-Joseph RIZZO, Nicolas SCHMITT and Sébastien VINCENT. Without forgetting my colleagues with whom I shared the open space office: Benoît BLAYSAT, Chloé DUPLEIX, Flavien FR ÉMY, Carole HENNINGER, Louis KOVALEVSKI, Thi Thanh Huyen NGUYEN, Amir OUESLATI and Bastien TRANQUART.

3.18

Evolution of the thermo-mechanical response with tempering for the monotonic test at 400 • C for a respective tempering of X T = 0 and X T = 0.6 . . . . . . . . . 3.19 Evolution of the thermo-mechanical response with tempering for the cyclic test at 500 • C for a respective tempering of X T = 0 and X T = 0.6 . . . . . . . . . . . . 3.20 Evolution of the thermo-mechanical response with tempering for the monotonic test at 500 • C for a respective tempering of X T = 0 and X T = 0.6 . . . . . . . . . List of Tables

Introduction

X10CrMoVNb9-1 martensitic steel is considered as a candidate for some future Very High Temperature nuclear Reactors' components. The stability of this 9wt% chromium, 1wt% molybdenum martensitic steel -up to 750 • C without mechanical loading -is sought for various high temperature applications.

The assembly of some of these thick components is usually done using multipass Gas Tungsten Arc Welding (GTAW). Welding introduces unwanted residual stress in structures, leading to brittle fracture, hydrogen embrittlement, and a deterioration of fatigue life. During this process the Heat Affected Zone is subjected to successive and complex cyclic thermal loads.

Ensuring mechanical properties, such as the tensile strength and fracture toughness of the welded joint should also be considered in the fabrication of actual welded structures. Since the mechanical properties of welded joints are remarkably influenced by a complex thermal cycle due to welding, it is essential to clarify an appropriate procedure to control the thermal cycle in the welding of materials. The performance of the welded joint is affected by the coupling of heat cycles, microstructural changes, and macro-mechanics, such as non-uniform strain and stress fields. These phenomena can affect the quality and the mechanical strength of the welded component dramatically.

This thesis is following that of G.-M. Roux [START_REF] Roux | Prévision des contraintes résiduelles induites par le soudage TIG d'un acier martensitique (X10CrMoVNb9-1)[END_REF]. This work of 6 years aims to predict the microstructural state after welding in the HAZ. A first version of a Thermo-Metallurgical-Mechanical model (TMM) has been developed and validated in terms of residual stresses on single-pass welding simple tests.

In this thesis a new version of the Thermo-Metallurgical-Mechanical (TMM) model is proposed to improve the forecasts in terms of the microstructural state and residual stresses post welding in the HAZ. The main improvements of this model TMM its capabilities to take into account the successive heating and cooling cycles, characterizing the multi-pass welding process. This thesis is based on a detailed modeling approach for phase transformations, namely modeling the transformation base material -austenite during anisothermal complex loadings, modeling transformation austenite -martensite and finally the martensite tempering modeling. Not to mention the characterization of the mechanical behavior of each phase and the study of multiphase behavior. Numerical simulation with the finite element code Cast3M of several welding processes was used to compare models with experimental results.

Using numerical simulation to quantitative prediction of the microstructure and residual stresses induced by a welding operation requires the good knowledge of phase thermal, mechanical and metallurgical kinetics.

Focus in this thesis was made on the development and validation of numerical tools and models needed to perform a proper multipass welding simulation.

Hence, the first chapter of this manuscript is devoted to the knowledge of the material, to the description of the Thermal-Metallurgical-Mechanical model and to the determination of the temperature, temperature rate, deformation and deformation rate domains via a first thermal numerical simulation of the multipass welding process.

The second chapter is devoted to phase-phase transformation that material endures during a multipass welding process. Different models were developed, numerically implemented, identified and validated.

In the third chapter, the identification of the thermo-mechanical behaviour of each phase was done and the influence of the metallurgical state and the mechanical behaviour was studied.

The fourth and final chapter is devoted to the comparison of experimental results and numerical predictions obtained during different more or less welding cases. The multiphasic behaviour of the material is described and modelled in this chapter. The results were then used to implement the MUSICA project which is described in the first chapter.

The conclusions and perspectives of this manuscript are gathered at the end, before the bibliography and various appendices.

Chapter 1

Validity domain of the Thermo-Metallurgical-Mechanical model

In order to get a first idea about the temperature ranges, cooling and heating rates, maximum induced strain and strain rates encountered during a welding process, a first 2D monophasic simulation of a multipass welding test case is realized.

In this chapter, we begin first by presenting the material, then we introduce the issues related to multipass, we describe the MUSICA test case and we simulate it using the finite element CEA code Cast3M. The numerical results will then be used as test specifications for further studies detailed in the next chapters. The chromium-molybdenum (Cr-Mo) ferritic steels are an important group of alloys used as structural materials at elevated temperatures in the chemical, petrochemical and fossil-fired power generating industries. The properties of Cr-Mo steels are controlled by their microstructures, grain size, solid solution hardening and precipitation hardening [START_REF] Keehan | Effect of microstructure on mechanical Properties of High Strength steel weld metals[END_REF][START_REF] Adda | Elements de métallurgie physique[END_REF], which in turn is controlled by chemical composition and heat treatment used.

The heat treatments employed are usually simple and, therefore, these steels are economically attractive. Various grades of Cr-Mo steels are outlined in ASTM Standard ASTM A 213 [START_REF]Standard specification for seamless ferritic austenitic alloy steel boiler superheater and heat exchanger tubes[END_REF]. Among the ferritic steels, 2.25Cr-1Mo (T22) and 9Cr-1Mo (T9 or P9) steels have been extensively used for Steam Generator (SG) applications because they offer useful combination of high temperature mechanical properties and corrosion/oxidation resistance [START_REF] Bergquist | Consumables and welding modified 9 cr-1 mo steel[END_REF]. The advent of new generation power plants has increased the operating steam temperatures to achieve higher efficiency and better environmental protection. This has led to the development of modified versions of 9% Cr steels with excellent combination of creep strength and ductility [START_REF] Lundin | Microstructure and mechanical properties of a 10% chromium steel with improved creep resistance at 600 • c[END_REF].

The effect of the chemical composition

The 9Cr-1Mo-V steel was co-developed by Combustion Engineering and the Oak Ridge National Laboratory (ORNL) in the late 1970s. The modification consists of additions of vanadium, niobium, and nitrogen and the narrowing of the maximum compositional ranges for carbon, chromium, silicon, molybdenum, phosphorous, and sulphur [START_REF] Sikka | Modified 9cr-1mo steel -an improved alloy for steam generator application[END_REF]. Maximum contents are set for nickel and aluminium. The composition range for the steel, as provided in TAB. 1.1, is designed to produce a fully martensitic microstructure. The modified 9Cr1Mo was selected because of its [START_REF] Coudreuse | Fabrication d'une tôle en nuance x10crmovnb9-1[END_REF] high thermal conductivity and low thermal expansion coefficient. In addition, its improved resistance to stress-corrosion cracking ( [START_REF] Swindeman | Issues in replacing cr-mo steels and stainless steels with 9cr-1mo-v steel[END_REF][START_REF] Gieske | The influence of thermal aging on the microstructure and fatigue properties of modified 9cr-1mo steel[END_REF]) makes it a better candidate than usual austenitic steels for water-steam systems. The main physical properties of the material are given in TAB.1.2. In the

CNB = Cr + 6Si + 4Mo + 1.5W + 11V + 5Nb + 9Ti + 12Al -40C -30N -4Ni -2Mn -1Cu (1.1)
Where the elemental contents are in weight percent. If the CNB is less than 10, delta ferrite is not usually present. For CBN between 10 and 12, the presence of delta ferrite is not readily predicted. For CBN above 12, significant quantities of delta ferrite are present.

The alloy elements play an important role on the mechanical behaviour [START_REF] Michaud | The effect of the addition of alloying elements on carbide precipitation and mechanical properties in 5% chromium martensitic steels[END_REF] (see FIG. 1.1 and 1.2) of the material via precipitates such as carbides, which improve the alloys characteristics at low temperature stabilize the microstructure at high temperature (by inhibiting the growth of grain). In fact, Brachet [START_REF] Brachet | Alliages Martensitiques 9Cr-1Mo: effets de l'addition de l'azote du niobium et du vanadium sur la microstructure: les transformations de phases et les propriétés mécaniques[END_REF] shows that a controlled additions of niobium, vanadium and nitrogen improve the mechanical properties at low and high temperature as well as creep resistance. It was also shown that the addition of vanadium and niobium increases the weldability. It was also noticed that the addition of tungsten also improves resistance to high temperature [START_REF] Hong | The effects of tungsten addition on the microstructural stability of 9cr-mo steels[END_REF][START_REF] Alamo | Physical metallurgy and mechanical behaviour of fecrwtav low activation martensitic steels: effects of chemical composition[END_REF][START_REF] Abe | The effect of tungsten on dislocation recovery and precipitation behavior of low-activation martensitic 9cr steels[END_REF].

Carbon has a major effect on steel properties. Carbon forming carbides is the primary hardening element in steel. Hardness and tensile strength increase as carbon content increases, contrariwise ductility and weldability decrease with carbon increase.

Chromium is commonly added to steel to increase corrosion resistance at low and high temperatures or to improve high-temperature strength. As a hardening element, chromium is frequently used with a toughening element such as nickel to produce superior mechanical properties. Alphagenic element, it also contributes to microstructure stability and creep resistance at high temperature. The most common chromium carbides noted M 23 C 6 , M 7 C 3 and M 3 C, where M represents (Fe, Cr, Mo), are principally of Cr 23 C 6 and Cr 7 C 3 type and dominants regard the Cr 3 C 2 chromium carbides type (see FIG. 1.3). The binary iron-chromium equilibrium diagram (x% carbon) gives a practical presentation of phases at equilibrium in 9% Cr steels such as the X10CrMoVNb9-1 denoted as T91 (see FIG. 1.4). The austenitic transformation that takes place around 800 • C transforms the ferritic structure (Body Centered Cubic BCC) stable at low temperature in an austenitic structure (Face Centered Cubic CFC). During this transformation, the M 23 C 6 carbides disappear gradually. At higher (1300 • C and above) temperatures the δ ferrite appears giving a biphasic austenite/ferrite domain. Molybdenum may produce secondary hardening during the tempering of quenched steels. Molybdenum increases steel hardenability and enhances the creep strength of low-alloy steels at elevated temperatures. Like chromium, molybdenum is an alphagenic element which forms intermetallic precipitates (Laves) Fe 2 Mo (approximately 45 wt% of molybdenum). This precipitate has the peculiarity to grow in grain boundaries, unlike the M 23 C 6 which is generally intragranular. Molybdenum also has the remarkable property to foster the bainitic transformation. FIG. 1.5 represents the binary Fe-Mo phase diagram. Nickel is a gammagenic element, it promotes the gamma phase appearance at relative small temperatures as shown in the iron-chromium binary diagram for various nickel concentrations (see FIG. 1.6). Nickel has also a hardening effect, it increases the hardenability of alloys [START_REF] Vodarek | Effect of nickel on the precipitation processes in 12crmov steels during creep at 550 • c[END_REF]. Manganese has similar effects to those of nickel (see FIG. 1.7). Manganese encourages segregations during cooling, which limits its proportion in the alloy. In addition, it limits cracking risk during cooling (by trapping the sulphur in a precipitate form). Niobium and vanadium have a great affinity with carbon. The small and very stable formed carbides limit the creep, increase the hardness and intergranular corrosion resistance. The main precipitate is NbC carbide (or Nb 4 C 3 ). The binary Fe-Nb phase diagram Oxygen is a major factor in gas contamination of the weld metal as it is present in significant quantities in the electrode core wire and may also be present in large amounts in the electrode coating. Oxygen is also present in the atmosphere, in the parent metal and on the parent metal Figure 1.2: Variation of 10 5 h creep rupture strength with temperature for several materials [START_REF] Raj | Mechanical properties and non-destructive evaluation of chromium-molybdenum ferritic steels for steam generator application[END_REF] as oxide or scale. Oxygen may be absorbed into the weld, forming iron oxide (Fe 3 O 4 ) and other oxides such as that of silicon. Iron oxide may react with carbon in the steel to form CO resulting in blow holes. If iron oxide is present, oxidation of the weld will occur and that will produce a great increase in the grain size. Absorption of oxygen by weld reduces its tensile strength and ductility, and decreases its resistance to corrosion, therefore oxygen is harmful to both the base material and the welding area. Deoxidizers such as Mn, Si, AI, Ti, etc. may be added in electrodes to prevent formation of iron oxide.

The hydrogen is much more soluble than the other two previous gases, but disappears rapidly during cooling by diffusion due to it small atomic radius. Atomic hydrogen can also diffuse during operation at elevated temperatures. The main sources of Hydrogen are the electrode coating or the atmosphere. Hydrogen may be present in the coating as cellulose or moisture and in the atmosphere as moisture. Hydrogen begins to diffuse out of the weld metal immediately after the welding process, and continues to do so over a long period. The dissolved hydrogen diffuses into steel with the carbon in solid solution or with carbides to form methane [START_REF] Siquara | Effects of hydrogen on the mechanical properties of a 2.25cr-1mo steel[END_REF]. The presence of hydrogen reduces the tensile strength of the weld, makes the arc erratic and contributes to the formation of cooling cracks or fisheyes in welds. Underbead cracking is often attributed to the In metal arc welding, two methods are employed to prevent contamination of the weld metal

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016 In fact, coating provides a protective atmosphere of gas by decomposition in the heat of the arc, which restricts contamination of the weld metal by the air and also forms a low melting point slag, which protects the molten weld metal from contamination during solidification. Finally, in some coatings, additions of deoxidants are made, which reduce the available oxygen in the weld metal.

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016 Typically, a cooling rate of 6 • C/min or higher from the normalization temperature avoids the formation of proeutectoid ferrite. The martensite start (M S ) temperature is around 400 • C and the martensite finish (M f ) temperature is around 200 • C. When heat-treated to meet the requirements of the specifications for Grade 91, the as-tempered microstructure consists of a fine prior austenite grains containing a lath-like tempered martensite structure with a high dislocation density that is A thick (140 mm) steel plate Fe-9Cr grade91 (T91) was supplied by the CEA for studies conducted on gas-cooled reactors. Indeed, the Fe-9Cr (X10CrMoVNb9-1) steel was a candidate for the nuclear vessels for future gas cooled reactors. Specifications, the range of manufacturing and

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016

acceptance testing are detailed in the report [START_REF] Coudreuse | Fabrication d'une tôle en nuance x10crmovnb9-1[END_REF]. It corresponds to the order Industeel (Arcelor Group) 31428K following precise specifications. This cast is numbered 13330. The ferritic steels were preferred to austenitic steels because of their low expansion coefficient and high resistance to thermal shock ( [START_REF] Tavassoli | Materials design data for reduced activationmartensitic steel type eurofer[END_REF][START_REF] Antalffy | The potential for using high chromium ferritic alloys for hydroprocessing reactors[END_REF]). The T91 steel is part of the class said stainless alloys. The stainless steels were first developed in 1821 by Berthier when he introduced chromium as an addition element. The first industrial production of stainless steel was introduced by Goldschmidt in 1897. The chromium alloys with low carbon were manufactured industrially in France for the first time in 1904. The steel was produced in conventional oven. The final composition was performed in the hot ladle refining. An ingot 81.5 t (thick 1310 mm) was then obtained by casting. After casting, the ingot has undergone several hot-rolling:

• Rolling till 400 mm thickness: the ingot was preheated to 1300 • C and underwent a first pre-crushing and pressed to 700 mm of thickness, then undergone a unidirectional rolling to 400 mm thickness. The ingot obtained was then slowly cooled.

• finish rolling to 140 mm thickness: after pre-heating to 1250 • C, the ingot has suffered unidirectional rolling to a thickness of 140 mm. The sheet obtained has then slowly cooled.

• Before delivery, the sheet has undergone a heat treatment of austenitizing at 1071 • C for 4 h followed by a water quenching and an income to 757 • C for 5 h31. The delivered sheet is in a fully tempered martensite state.

MUSICA test case

MUSICA is a multipartner project (AREVA, CEA, CETIM, ESI Group, Institut de Soudure) for automation of welding numerical simulation (see FIG. 1.11). Different test cases corresponding to different welding situations (geometry, material, process...) are being studied to improve the welding Thermo-Metallurgical-Mechanical model. The CEA test case consists of a multipass welding in a narrow groove for a martensite X10CrMoVNb9-1 steel plate. A first preliminary experiment had been performed. This experience permitted us to identify the thermal boundary conditions and provided us with many useful information such as temperature evolutions, deformation changes, hardness profiles and macrographic cuts.

Test case presentation

The experimental temperature data of the preliminary test case are obtained through instrumentation of three thermocouples TC1, TC2, and TC3. The characteristics of the instrumentation and the locations of thermocouples are shown in FIG. 1.12. The welding characteristics of this experiment can be summarized as follows:

• Filler metal: a 1.2 mm diameter of Thyssen wire.

• Preheated plate to 210 • C.

• Inter-pass temperature fixed to 200 • C.

• A post weld heat treatment at 325 • C during 2 h 30'.

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016 The pre and post heating power is provided through a block that is screwed to the plate to be welded. This block is pierced with four holes where tubular resistance heaters of 16 mm diameter. The temperature is regulated via a thermocouple attached to the plate.

Multipass GTA-Welding simulation

Welding is defined by the American Welding Society (AWS) as a localized coalescence of metals or non-metals produced by either heating of the materials to a suitable temperature, with or without the application of pressure, or by the application of pressure alone, and with or without the use of A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016 Welding is an important process and has very important part in industry, especially in the automotive, maritime and energy. Although welding has many advantages the welding process has some disadvantages: thermal expansion and shrinkage, microstructural transformations and stresses. Arc welding is a widely used process in industry because of its versatility and low cost.

Resistance welding is also a low-cost, high-production process used first of all in the assembly of car bodies and similar parts made of thin metal sheets. Other welding processes to be named are solid-state welding, laser beam welding, induction welding, electroslag welding, electron beam welding, etc. Gas Tungsten Arc Welding (GTAW) can be applied to all weldable stainless steels, in both the wrought and cast forms, to clad products, such as stainless clad carbon steel and to dissimilar stainless alloys. Inert gas shielding eliminates the hazard of gas metal reactions.

During last three decades many of the existing welding processes have been simulated by numerical methods, especially by the finite element methods. A numerical simulation of a welding process is not easy, as it is a complex process involving the interaction of thermal, mechanical, electrical and metallurgical phenomena. Simulation of multipass welding involves additional modelling aspects compared to single-pass welding (metallurgical aspect such as tempering and carbides precipitations, mechanical aspect such as softening in the Heat Affected Zone (HAZ) and it effect on creep resistance [START_REF] Cerjak | The effect of welding on the properties of advanced 9-12%cr steels[END_REF], morphology aspect like grain size and growth [2] (FIG. 2.1). Decision must be made whether the actual welding procedure should be simulated or if some simplifications should be introduced. Secondly, the addition of filler material poses some extra computational inconveniences. Despite the increase in problem size during the later years the computational task can be overwhelming when a good spatial resolution is required in the neighbourhood of the weld. The thermal and mechanical fields from the welding process have large gradients near the arc. Finite element (FE) simulation of welding has been performed for more than thirty years [START_REF] Goldak | Coupling heat transfer microstructure evolution and thermal stress analysis in weld mechanics[END_REF][START_REF] Radaj | Heat effects of welding: temperature field residual stress distortion[END_REF][START_REF] Ueda | Establishment of computational weld mechanics[END_REF][START_REF] Mackerle | Finite element analysis and simulation of welding: a bibliography (1976-1996[END_REF][START_REF] Lindgren | Finite element modeling and simulation of welding part 1: Increased complexity[END_REF][START_REF] Lindgren | Finite element modeling and simulation of welding part 2: Improved material modeling[END_REF]. The early models were only two-dimensional (2D). The first three-dimensional (3D) models were presented around 1990 [START_REF] Oddy | Transformation effects in the 3d finite element analysis of welds[END_REF][START_REF] Lindgren | Deformations and stresses in welding of shell structures[END_REF][START_REF] Friedman | Thermomechanical analysis of the welding process using the finite element method[END_REF][START_REF] Andersson | Thermal stresses in large butt-welded plates[END_REF][START_REF] Mackerle | Finite element analysis and simulation of welding-an addendum: a bibliography (1996-2001)[END_REF][START_REF] Duranton | 3d modelling of multipass welding of a 316l stainless steel pipe[END_REF][START_REF] Deng | Prediction of welding residual stress in multi-pass butt-welded modified 9cr-1mo steel pipe considering phase transformation effects[END_REF].

Identification of thermal boundary conditions

The welding conditions of the preliminary test case are given in ANNEX A. The temperatures acquisition frequency for the first 12 passes was fixed to the maximum Sefram machine capacity '50 Hz'. Temperature evolutions of the three thermocouples, are given in FIG. 1.15. Since the heat source is closer to the third thermocouple TC3 during the first two passes, recorded temperatures are higher than those recorded by the second thermocouple TC2. In this section, we present the optimization principle used, the boundary conditions selected, the thermal parameters used to identify the shape and strength of the equivalent heat source relative to each welding pass.

From a mathematical point of view, optimization consists in determining the minimum or maximum of a function. In our modelling case, optimization plays an important role. It enables us to determine the equivalent heat parameters source. The identification is done using the Cast3M LEVM operator. This operator uses the generalist Levenberg-Marquardt method, normally used for non-linearity cases.

Optimization background

Consider the n observables, Y i (i = 1, ..., n), for n number of sets of k variables, X i = X i 1 , ..., X i k . The Y i are for example the measured temperatures at different positions in the fusion zone or the heat affected zone. X i regroups the coordinates of the thermocouple measuring the temperature Y i as well as the time of the measurements. Let the vector A = (A 1 , ..., A p ) whose components are the p parameters of the function Y (X, A). We seek a parameter A which minimize the gap between the known solutions Y i and those given by the function Y (X i , A). In the case of adjustment using the least squares method, the function to minimize is the following:

χ 2 (A) = n ∑ i=1 Y i -Y (X i , A) σ i 2 (1.2)
Where the σ i (i = 1, ..., n), can permit to assign different weight factors to Y i to favor certain experimental measurements. If the Y function is linear respective to the A j parameters, the function χ 2 is extremal for the A j values, where the partial derivative of χ 2 respective to A j is zero. So the search for the p parameters A j resolving the following p equations:

∂χ 2 (A) ∂A j = - n ∑ i=1 2 Y i -Y (X i , A) σ 2 i ∂Y (X i , A) ∂A j = 0 (1.3)

Numerical simulation

The numerical simulation was performed using the environment of the finite element code Cast3M developed by CEA. Different optimization methods are available in Cast3M [START_REF] Charras | L'optimisation dans cast3m[END_REF]. The LEVM operator described earlier on was used to achieve this optimization study. This operator allows optimization using Levenberg-Marquardt method described above. The numerical simulations are based on Roux thesis works [START_REF] Roux | Prévision des contraintes résiduelles induites par le soudage TIG d'un acier martensitique (X10CrMoVNb9-1)[END_REF]. The mesh has been achieved basing on the macrography. The simulation were done in 2D assuming plane strain. The mesh elements related to the material addition of various passes are activated successively during the calculation. The thermal boundary conditions can be summarized as followed:

• Zero flux on the axis of symmetry.

• A heat source uniformly distributed in the volume of the activated elements.

• Convection between the outer surfaces of the plate and the external environment at 20 • C.

The mechanical boundary conditions can be summarized as followed:

• Zero displacement in the x section across the axis of symmetry.

• Unilateral support at 80 mm from the axis.

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016

q = -λ (T ) ----→ grad T (1.4)
Where q denotes the flux, λ the thermal conduction parameter and T the temperature. The heat equation:

div q + r = ρ C P(T ) ∂T ∂t (1.5)
Where r is the outer source, ρ the apparent density and C P the heat capacity of the material. In case of no outer source, (1.4) and (1.5) give:

∂ ∂x λ ∂T ∂x + ∂ ∂y λ ∂T ∂y + ∂ ∂z λ ∂T ∂z = ρ C P(T ) ∂T ∂t (1.6)
A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016

The heat flux losses due to convection φ Convection can be written as follows:

φ Convection = h S (T -T ext ) (1.7)
Where h denotes the convection coefficient, S the total surface area and T ext the external temperature. As for the heat flux losses due to radiation φ radiation are expressed by the following equation:

φ radiation = ε S σ T 4 (1.8)
Where ε denotes the thermal emissivity of the material, S the total surface area and σ the Stefan-Boltzmann constant. Roux [START_REF] Roux | Prévision des contraintes résiduelles induites par le soudage TIG d'un acier martensitique (X10CrMoVNb9-1)[END_REF] chooses to model both convection and radiation by a single equation:

φ Convection/radiation = h (T ) (T -T ext ) (1.9)
Where h denotes the convection/radiation coefficient and T ext the external temperature. Given the wide range of temperature to which the plate is subjected during welding (20 The variation of the thermal conductivity is given in TAB.1.4 and represented in FIG. 1.18. The convection/radiation parameter is reported in TAB. 1.4. Evolution of the apparent density (ρ in kg m -3 ) and thermal expansion (α in K -1 ) was given for martensite and austenite phases by Roux [START_REF] Roux | Prévision des contraintes résiduelles induites par le soudage TIG d'un acier martensitique (X10CrMoVNb9-1)[END_REF].

ρ M (T ) = 7732.1 -0.25918T -4.6272 10 -5 T 2 (1.10)
ρ AU (T ) = 7956.9 -0.48954T -9.7052 10 -5 T 2 (1.11)

α M (T ) = 11.1 10 -6 T + 2.35 10 -9 T 2 + 9.435 10 -14 T 3 (1.12)

α AU (T ) = 20.2 10 -6 T + 1.42 10 -9 T 2 (1.13)
The sudden drop of the specific heat and conductivity at a temperature of 830 • C (beginning of austenite) is expected. It is due to the phase change (α → γ). A sensibility analysis regarding the exchange convection/radiation coefficient was performed and result are detailed in ANNEX B. The heat source is replaced by an equivalent volumetric heat source [START_REF]Welding metallurgy[END_REF] uniformly distributed in the mesh of the added passes. The initial temperature of these added elements is equal to the melting temperature: 1475 • C. The evolution over time of this power flux density was determined by optimization. These calculations were done only for the first, fourth and seventh welding passes For more information on equivalent heat source modelling, please refer to ANNEX C. The problem was reduced to the determination of the maximum power Q max for each pass so that the simulated maximum temperatures are equal to those measured by the thermocouples. The optimization was then performed for a complete calculation with a triangular signal as Welding-pass N

T λ (FT M) λ (QM) λ (AU) h ( • C) (W/m) (W/m) (W/m) (W m -

The thermo-mechanical behaviour

A Chaboche type model [START_REF] Lemaitre | Mécanique des matériaux solides[END_REF] has been chosen to model the equivalent monophasic thermo-mechanical behaviour of the material. Standard strain partition is assumed so that:

ε = ε the + ε vp (1.14)
A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016 Where ε, ε the and ε vp respectively denote the total, thermoelastic and viscoplastic strain tensors.

The elastic domain is modelled by the Von-Mises yield function

f p = J 2 s -X -R -σ y (1.15)
Where s, X, R and σ y respectively denote the stress deviatoric tensor, the kinematic hardening tensor, the isotropic hardening and the initial yield stress. To take account of the behaviour of the material during the complex cyclic loading that may be induced by the welding process, kinematic hardening evolution is modelled by adding a linear Prager type term to a non-linear Armstrong-Frederick term so that:

Ẋ = Ẋ1 + Ẋ2 (1.16) with: Ẋ1 = C 1 εvp -γ 1 λ X 1 (1.17) Ẋ2 = C 2 εvp + Ṫ X 2 dC 2 dT (T ) C 2 (T ) (1.18)
Where λ denotes the cumulated plastic strain rate, whereas C 1 , C 2 and γ 1 are material parameters. Note that during this first simulation, kinematic hardening was not considered. Non-linear isotropic Voce type hardening is assumed so that:

Ṙ = b (T ) λ (R ∞ (T ) -R) + Ṫ R dR ∞ dT (T ) R ∞ (T ) - db dT (T ) b (T ) (1.19)
where b and R ∞ are material parameters. The same expression can also be written as followed:

Ṙ = a (T ) λ 1 - R a (T ) b * (T ) + Ṫ R dR ∞ dT (T ) R ∞ (T ) - db dT (T ) b (T ) (1.20)
A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016

where a = R ∞ b and b * = 1/b. To take account of the behaviour of the material at very different loading rates as induced by the welding process, the visco-plastic strain evolution law is chosen of a hyperbolic sine type so that:

εvp = 3 2 λ s -X J 2 s -X + εpt (1.21) with: λ = ∂Φ * V P ∂ f p = ε∞ (T ) sinh f p K(T ) N (1.22)
where K, N and ε∞ are material parameters. For this simulation the viscous stress was modelled using Norton equation, where K and N are material parameters:

σ v = K ṗ 1 N (1.23)
εpt denotes the so-called TRansformation Induced Plastic strain that is modelled with so-called Nancy (or Hamata) model [START_REF] Hamata | A model for nodular graphite cast iron coupling anisothermal elasto-viscoplasticity and phase transformation[END_REF]:

εpt = 3 2 J 2 s λ s J 2 s ẏ (1.24)
All the thermo-mechanical parameters are thermal dependent. The evolution of these parameters are taken from Roux [START_REF] Roux | Prévision des contraintes résiduelles induites par le soudage TIG d'un acier martensitique (X10CrMoVNb9-1)[END_REF] given in TAB. In the following, some results are shown after running the implemented TMM model with Cast3M.

FIG. 1.27 to 1.28 show the evolution of the hydrostatic stress in the deposit of the seventh pass. The stress concentration is located around the filled groove. The hydrostatic stress gives an idea about the location of deformed areas where we can find the highest triaxiality rate. In fact, the evolution of the deformation depends on the triaxiality rate σ eq σ H so greater the hydrostatic stress is, greater the deformation is.

Due to the thermal shock imposed, volume changes and stresses are generated. Volume changes are responsible of the residual stresses remaining in the structure at the end of the passes. 1.29 to 1.31 represent the Von-Mises stresses and displacements along the 'Y' axis, at the end of cooling after the first, sixth and twelfth filing passes. The figures show that the residual stresses A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016 A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016 Validity domain of the Thermo-Metallurgical-Mechanical model

Discussion

In this chapter we have presented the T91 grade steel, its composition as well as its properties.

In the second section, MUSICA test case was presented as well as a first numerical thermomechanical simulation. The results of this simulation will be used in the next chapter defining the experiments to be done to be representative to the case of multipass GTA Welding. Heating loads and heating rates will be used in the definition of dilatometric tests and deformations as well as deformation rates found herein will be used as input to the mechanical tests.

Chapter 2

Modelling of the solid-solid phase transformation

Welding process leads to some coupling between thermal, mechanical and metallurgical fields. In this chapter focus is made on the links between thermics and metallurgy. Important characteristics of diffusional and displacive transformations are summarized herein and metallurgical models existing in order to simulate allotropic transformations occurring during the welding process of X10CrMoVNb9-1 martensitic steel are also listed. These models are generally based on, differential equations describing the kinetics of phase transformations, phenomenological formulas with more or less physics behind and simple equations easily implementable in the FEM codes.

. . . 69

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016

Introduction to solid-solid phase transformation

The study of welding metallurgy is very important because the over all mechanical properties of a weldment are determined by the characteristic properties of the individual microstructure present in the weld deposit and the weld Heat Affected Zone (HAZ). It has long been recognized that one of the major problems associated with fabrication by welding arises from the inability to obtain uniform mechanical properties throughout the weldment. Even extensive postweld heat treatments fail to eliminate this weld induced variability in mechanical properties. Both chemical inhomogeneity and changes in metallurgical structures are known to result from welding operation because most fusion welding processes generate high rates of heating and cooling in the weld metal and parent metal adjacent to the weld. Weld metal zone is formed as the weld metal solidified from the molten state. This is a mixture of parent metal and electrode (or filler metal), the ratio depending upon the welding process used, the type of joint, the plate thickness, etc. Weld metal zone is cast metal (i.e., structure is as cast) of particular composition of the mixture that has cooled; its micro structure reflects the cooling rate in the weld. Due to relative very fast cooling and the chemical composition of the material, a martensite structure in the weld is obtained after cooling. Adjacent to the weld metal zone is the heat-affected zone that is composed of parent metal that did not melt but was heated to a high enough temperature for a sufficient period so that grain growth can occur. Heat affected zone is that portion of the base metal whose mechanical properties and microstructure have been altered by the heat of welding. The heat affected zone is subjected to a complex thermal cycle (sudden heating followed by rapid cooling) in which all temperatures from the melting range of the steel down to comparatively much lower temperatures are involved and HAZ therefore consists of a series of graded structures ringing the weld bead. HAZ, usually contains a variety of microstructures. The heat affected zone is often determined (i.e., visualized) thanks to hardness means or chemical etching. The width of HAZ varies according to the welding process and technique; in arc welds it extends only a few mm from the fusion boundary. The HAZ comprises three metallurgically distinguished regions.

• The grain growth region.

• The grain refined region.

• The transition region.

Grain growth region is immediately adjacent to the weld metal zone (fusion boundary). In this zone parent metal has been heated to a temperature well above the upper critical temperature. This resulted in grain growth or coarsening of the structure. The maximum grain size and the extent of this grain growth region increase as the cooling rate decreases. Grain growth kinetic is detailed in ANNEX D. Adjacent to the grain growth region is the grain refined zone. In this region, the parent metal has been heated to just above the critical temperature where grain refinement is completed and the finest grain structure exists. In the transition zone, a temperature range where partial allotropic recrystallization takes place. Outside the heat affected zone is the parent metal that was not heated sufficiently to initiate a change of microstructure. The process of fusion welding subjects the parent plate in the vicinity of the weld, known as the heat-affected zone (HAZ), to rapid localized heating and cooling. This induces a variety of microstructural changes, as illustrated in FIG. • Melting of electrode and parent (or base) metal

• Solidification of weld metal

• Gas absorption and gas metal reactions

• Slagmetal reactions • Surface phenomena • Solid state reactions
The purpose of this chapter is to provide the minimum knowledge of metals on the phase transformations in steel model in terms of thermodynamics, kinetics and diffusion. Focus was made on solid-solid phase transformation. Models will be then used in a welding application. The transformation to solid state can be obtained: solid-solid phase transformation

• By germination and growth; It is spread over very large distances compared to the internal atomic distances.

• Or without diffusion, as in the martensite. The displacement of each atom is less than the interatomic distance. The solid state processing is done by shearing the crystal lattice.

General metallurgical models

In the following general kinetic models from bibliography are reported. These models are often used to describe the solid-solid phase transformation induced during multipass welding. This section relies on previous studies listed here: [START_REF] Babu | Modeling and characterization of nonequilibrium weld microstructure evolution[END_REF][START_REF] Bhadeshia | Diffusional and displacive transformations[END_REF][START_REF] Muddle | Formation of bainite as a diffusional-displacive phase transformation[END_REF][START_REF] Mujahida | Coupled diffusional/displacive transformations: Effect of carbon concentration[END_REF][START_REF] De-Abreua | Deformation induced martensite in an aisi 301ln stainless steel: characterization and influence on pitting corrosion resistance[END_REF][START_REF] Kelly | The martensite transformation in steels with low stacking fault energy[END_REF][START_REF] Muddle | The shape strain of the (225) f martensite transformation[END_REF][START_REF] Hannnink | Metastability of the martensitic transformation in a 12 mol% ceria-zirconia alloy: I. deformation and fracture observations[END_REF][START_REF] Chang | Martensitic transformation induced by a tensile stress pulse in fe-22.5 wt% ni-4wt% mn alloy[END_REF][START_REF] Jacobus | Effect of stress state on the stress-induced martensitic transformation in polycrystalline ni-ti alloy[END_REF][START_REF] Koeda | Giant magnetostriction in fe-pd alloy single crystal exhibiting martensitic transformation[END_REF]. The desire to limit the identification to a CCT diagram, led Waeckel to limit the variables involved in phase transformations. The evolution law is written:

ż = f T, Ṫ , z, d (2.1)
Where z is a metallurgical parameter that can denote the new phase fraction as well as the martensite temperature M S and d is the austenite grain size. The phase formation must obey the metallurgical data provided by the CCT diagrams, for instance the formation of ferrite, pearlite or bainite cannot be performed at a temperature below Ms ( [START_REF] Waeckel | A thermometallurgical model for steel cooling behavior: proposition validation and comparison with sysweld model[END_REF][START_REF] Burlet | Microstructure and residual stresses issued from the bonding of an austenitic onto a ferritic steel by solid diffusion[END_REF]). The Ms temperature properly saying is not a metallurgical variable, but as it evolves during processing, it has been incorporated in the metallurgical variables. The grain size d has one unique value per CCT diagram, and it is difficult to study its influence. However Waeckel maintain the d in his formulation as a parameter that can be activated in case of using different CCT diagrams with different austenite grain sizes. Even if Waeckel is aware of the influence of the average carbon concentration and stress state on transformation, he does not take them into account since their effects do not appear in the CCT diagram.

The importance of this model is in taking into account the cooling rate which seems quite natural to model the anisothermal kinetics. Waeckel sought to give a theoretical justification (based on the thermodynamics of materials with internal variables) to the equation above, this justification can be found in his thesis [Waeckel94]. States relative to parameters (T, Ṫ , z, d) correspond to points on the CCT diagram. Apart from these points, the function is obtained by linear interpolation. The experimental data used are the temperature start and end of transformations and the final proportions of the phases. The transformations are assumed to be linear. The Waeckel model has important advantages: after identification, the implementation of the model is very fast and simple.

Kirkaldy model

Models have been established to link the temperature to the microstructure. Kirkaldy model proposes to couple the thermal phenomena to the chemical composition of the steel to predict the evolution of the microstructure.

Model establishment

Several parameters interfere during metallurgical kinetic calculations and depend on the chem-solid-solid phase transformation ical composition of the steel in question. Thus, the first step in the construction of this model is to include in the algorithm the chemical composition. In fact, the chemical composition has a significant influence on the borders of phases in equilibrium.

For low carbon steels, the liquidus and solidus temperatures can be approximated as functions of the carbon content only. A more precise approach to determine different temperature starts of various phase transformations such as the austenite-ferrite transformation start temperature A e3 was established by Kirkaldy and Baganis [START_REF] Kirkaldy | Thermodynamic prediction of the ae3 temperature of steels with additions of mn si ni cr mo cu[END_REF]. Here are some of these formulas:

Ae 3 ( • C) = 912 -203 √ C -15.2Ni + 44.7Si + 104V + 31.5Mo + 13.1W -30Mn -11Cr -20Cu + 700P + 400Al + 120As + 400Ti (2.2)
where the compositions are expressed in weight percent. The eutectoid temperature is expressed as followed:

PS( • C) = 723 -10.7Mn -16.9Ni + 29Si + 16.9Cr + 290As + 6.4W (2.3)
The temperature solubility of the ferrite in the critical region is:

T F( • C) = 912 -8486C (2.4)
The temperature start of bainite formation is:

BS( • C) = 656 -58C -35Mn -75Si -15Ni -34Cr -41Mo (2.5)
Finally, the martensite start temperature is written as:

MS( • C) = 561 -474C -33Mn -17Ni -17Cr -21Mo (2.6)
Phase transformations equations Kirkaldy has developed a set of equations that model the austenite transformation to each phase formed by diffusion. These equations assume that the decomposition of austenite is described by one continuous equation (the phases are formed successively and not simultaneously). The model describes both nucleation and growth:

∂y ∂t = B (d, T ) y m (1 -y) p (2.7)
Where y denotes the volumetric fraction of the new phase, B an equation coefficient, d the austenite grain size, m and p semi empirical coefficient insuring convergence. The B coefficient includes the austenite grain size d , alloying elements influence with their diffusion temperatures. Equation (2.7) can be factored as follows:

∂y i ∂t = F d F C F T F y i (2.8)
Where F d describes the effect of austenite grain size, F C the effect of alloy composition, F T the effect of temperature and F y i the effect of the newly formed phase proportion. F T can expressed as follows:

F T = (T CR -T ) n e -Q/
F y i = (y i ) 2 3 (1-y i ) (1 -y i ) 2 3 y i (2.10)
And in case of bainitic transformation F y i is expressed by:

F y i = (y i ) 2 3 (1-y i ) (1 -y i ) 2 3 y i e C R y i 2 (2.11)
Where the C R coefficient is delay coefficient due to the formation of alloy composition:

C R = 1.9C + 2.5Mn + 0.9Ni + 1.7Cr + 4Mo -2.6 (2.12)
The y i phase fractions varies between 0 and 1 and their sum is equal to 1. During heating, austenite is formed and for temperature exceeding the temperature solubility of carbides and nitrides T S , the austenite grain grows. During cooling, austenite grains continue to grow while the temperature is still greater than Ae 3 . The diffusivity of carbon, germination and coalescence of new phase continue till the temperature decreases below Ae 3 . The calculation of the phase proportions have to be normalized to obtain values between 0 and 1. The purpose of this normalization is to have a reference for comparing the phase obtained proportions. Let us consider that X FE is the proportion of phase equilibrium. It is calculated from the diagram iron-carbon at equilibrium for a given chemical composition. Kirkaldy defines a phase increment ∆ XF during normalization process of ferrite phase:

F d = 2 (1-d)/
ỹF = y F y FE ∆ ỹF → ∆ y F = ∆ ỹF X FE y F = ∑ ∆ y F (2.18)
As for bainite, the proportion of phase is normalized by the previously formed ferrite [START_REF] Watt | An algorithm for modelling microstructural development in weld heat affected zones (part a) reaction kinetics[END_REF]. In summary, this model is based on the kinetic theory which allows to reproduce quite accurately the CCT and TTT diagrams for a large number of steels. However, the kinetic factors used are not universally accepted. They should separate the phases of nucleation and growth from the additivity principle applicable only for growth. An incubation time can be easily added to the equations. This model remains one of the few which takes into account the alloy composition.

Grong model

The previous presented models are based on kinetic and thermodynamic approach to appreciate the dimensions of phase in the HAZ in welding. The Grong model suggests an analytical approach of phase evolutions during welding. The principle lies on coupling differential equation expressing the microstructure evolution of the material with the thermal history. This model is based on an analytical approach directly taking into account the grain size, but the disadvantage of this model is the use of the nitrogen concentration of each concerned phase, which is not always a known data.

Model establishment

Let's first define what an isokinetic transformation is. A transformation is called isokinetic whether the metallurgical behaviour evolutions for infinitesimal time increments are additive. Mathematically, this translates into:

∂y ∂t = G (y) H (T ) (2.19)
Where G (y) and H (T ) are arbitrary functions of the state variable and temperature. This equation is an equation with separable variables and can therefore be integrated as follows:

y 0 ∂y G (y) = t 0 ∂t H (T ) (2.20)
If the state variable is a dimensionless parameter, then the function H (T ) becomes a time constant which includes the temperature dependence of the reaction, and solving this equation can be done using Scheil equation. The assumption is made for anisothermal transformations where the resolution is done by integration of successive isothermal steps.

Model parameters

In this model, the thermal history undergone by the material is not explicitly described. The proportions of transformed phase depend only on the instantaneous values of temperature, grain size or nucleation sites. In addition, the nucleation phase is assumed to be zero during the early formation of ferrite, bainite, perlite and austenite. A final assumption was established for this model consisting on the fact that the reactions occur in succession rather than simultaneously. Thus the input parameters of this model are:

• The material initial alloys weight percent

• The initial microstructure state of the material And outputs are the metallurgical state of the HAZ function of time and temperature.

Mathematical formulation

The formulation was first established by Hemmer and Grong for the austenite α to ferrite δ transformation case [START_REF] Hemmer | A process model for the heat-affected zone microstructure evolution in duplex stainless steel weldments: Part i. the model[END_REF]. Due to the formulation itself and the metallurgy, this formulation can be extended to the ferrite, bainite, perlite and to martensite. The delta ferrite has the same crystal structure as alpha-ferrite, and the formulation involves the diffusion coefficients of the parent phase to daughter phase. These coefficients are measured for each transformation.

For austenite dissolution, the austenite-ferrite interface is studied. The interface evolves according to Fick's first law given in equation (2.21):

(C γ -C δ,i ) ∂x i ∂t = D δ ∂C ∂x (2.21)
Where C γ is the initial nitrogen concentration in austenite, C δ,i the concentration of nitrogen at the austenite-ferrite interface and D δ the diffusion coefficient of nitrogen in the delta ferrite. In this equation, the concentration of nitrogen in austenite is assumed to be constant and independent of temperature whereas in reality this concentration is more important at the interface as it is shown in FIG. 2.4. Thus the concentration of delta ferrite is defined by:

∂C dx = - C δ,i -C δ (x 0 -x i ) (2.
∂x i ∂t = - D δ C δ,i -C δ (x 0 -x i ) C γ -C δ,i
(2.23)

(x 0 -x i )
value can be obtained by subtracting the A 1 and A 2 areas shown in FIG. 2.4.

x γ -x i C γ -C δ = 1 2 (x 0 -x i ) C δ,i -C δ (2.24)
Hence:

x 0 -x i = 2 x γ -x i C γ -C δ C δ,i -C δ (2.25)
The dissolution rate is then calculated by putting this expression in equation (2.21):

∂x i ∂t = - D δ C δ,i -C δ 2 2 x γ -x i C γ -C δ C γ -C δ,i (2.26)
The decomposition of ferrite delta is accompanied with austenite formation on the ex-ferrite grain boundaries (See FIG. 2.5). These grains are assumed to be spherical with a r δ radius. Thus, according to Fick's first law, one can write:

C γ -C δ,i ∂r i ∂t = D δ ∂C ∂r (2.27)
By analogy:

C (r) = C 0 -C 0 -C δ,i r 3 -r 3 0 r 3 i -r 3 0 (2.

28)

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016

Hence:

∂C ∂r = -C 0 -C δ,i 3 r 2 i r 3 i -r 3 0 (2.29)
With r 0 is the position from which the concentration of nitrogen in the ferrite grain begins to decline. By introducing now the average concentration of nitrogen in austenite, we get:

Cγ = r i ∑ r δ C γ,i dV 4π 3 r 3 δ -r 3 i (2.30)
Equation (2.29) implies that nitrogen cannot spread elsewhere than in the austenite during the cooling cycle, this assumption remains valid for high speed cooling. For simplicity, the average concentration of nitrogen in austenite is replaced by a variable concentration depending only on time. Once this variable is set, the conservation of material is respected only if the two areas of FIG. 2.5 noted V 1 and V 2 are equal. This means that the concentration gradient at the interface of the reaction may be related to the concentration of nitrogen and to the austenite proportion:

V 2 = Cγ -C 0 4π 3 r 3 δ -r 3 i (2.31)
And:

V 2 = V 1 = r i r=r 0 (C 0 -C (r)) 4πr 2 dr = C 0 -C δ,i r 3 i -r 3 0 r i r=r 0 r 3 -r 3 0 4πr 2 dr (2.32)
After integration:

V 2 = V 1 = 2π 3 C 0 -C δ,i r 3 i -r 3 0 (2.33)
By replacing in equation (2.27), we finally obtain:

dr i dt = - 3D δ C 0 -C δ,i 2 2 C γ,i -C δ,i Cγ -C 0 r 2 i r 3 δ -r 3 i (2.34)

Volumetric phase fraction

It is necessary to convert the r position distances to equivalent volume fractions. In practice, each ferrite grain boundary is shared by two grains, which means that the total area of grain boundaries per unit volume S V is in fact equal to 2/ d2 where 2/ d2 is the average grain size. Thus, the volume fraction of austenite formed is:

r δ = 2 1 2 d3 = 2 1 2 3 2 d2 = 3 2 d2 (2.

35)

Hence:

y = 4π 3 r 3 δ -r 3 i 4π 3 r 3 δ = r 3 δ -r 3 i r 3 δ (2.36)
Finally all these previously presented models are not easily adaptable to your case, thus we have chosen other more or less phenomenological models widely used in metallurgical studies of multipass welding. The majority of phase transformations occur in the solid state and are thermally activated. In the field of welding, it is particularly interesting to know the phase transformations induced by variations in temperature, such as eutectoid transformations. solid-solid phase transformation

Transformation at equilibrium

The theoretical determination of equilibrium diagrams is extremely complex and simplifying assumptions are often necessary for implementation dues. In practice, the equilibrium diagrams are plotted experimentally using thermal analysis.

The thermal analysis consists in measuring the temperature of a sample during it cooling from a temperature above it melting temperature till room temperature. The determination of equilibrium diagrams (FIG. 1.10) is then based on the fact that liquid -solid state changes and different allotropic transformations are characterized by and endothermic or exothermic reactions which permit the identification of the start and finish temperatures of a transformation.

Phases α and δ are the ferrite phase γ represents the austenite, Fe 3 C cementite. Perlite is a combination of ferrite and cementite. During thermodynamic equilibrium conditions, relative to very slow cooling rate, diffusion can take place. Solidification of the alloy or phase transformation occurs following a step by step process in the interval where both phases coexist. The so-called pro-eutectoid ferrite is formed during cooling of hypo-eutectoid steel. It comes in various morphologies classified according to their characteristics:

• Coarse ferrite precipitates located at grain boundaries are called alliotriomorphe (GBA:

Grain Boundary Alliotriomorphs). Similar precipitates are called idiomorphic ferrites when they are randomly distributed in the grain.

• Long needle ferrites aligned along preferential planes are called Widmanstätten ferrite.

• Thin lenticular ferrites generated inside the grain are called needle-shaped ferrites.

During rapid austenite to ferrite transformation, the ferrite may be in the form of elongated grains with a high density of dislocations that accommodate the deformation such as Widmanstätten ferrite case.

As the ferrite, the so called pro-eutectoid cementite is initially formed at the grain boundaries. It can developed as coarse precipitates, platelet Widmanstätten type or laths in case of richest carbon alloys. The precipitated phases are however very different. In case of ferrite, the precipitates give high ductility properties to the material as for case of cementite, precipitates give a rigid structure relatively fragile [START_REF] Durand-Charre | La microstructure des aciers et des fontes: Genèse et interprétation[END_REF].

One way to quantify the effect of alloying elements on the stability fields of phases α and γ is to draw two diagrams, depending on the content of alloying elements and the eutectoid Changes in the carbon content of the eutectoid. The effects of alloying elements are also translated by formulas linking the austenite transformation points A S and A f to the chemical composition. For example, here are the Andrews' formulas:

A S ( • C) = 723 -10.

Nonequilibrium Transformations

In nonequilibrium conditions, rapidly heating and cooling, the solid phase of germination is delayed resulting a few phase germs formation. The diffusion being incomplete, there is a delay of homogeneity. A number of significant residual liquid phase is trapped in the solid phase. It leads to the minor segregation resulting some impurities, solute and gas. By continuing to cool, the gas A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016 produces a bubble effect and could introduce porosities in the heart of the material. This phase is called the major segregated. Transformation diagrams are simply another form of roadmap to allow us to predict a steel's response to heat treating. Also known as isothermal transformation (IT) diagrams, TTT diagrams measure the rate of transformation at a constant (isothermal) temperature [START_REF] Xiao | Isothermal transformation of low-carbon microalloyed steels[END_REF] (See FIG. 2.3). In other words, once a part is austenitized, it is rapidly cooled to a lower temperature and held at that temperature while the rate of transformation is measured. The different types of microstructures produced (ferrite, pearlite, bainite, martensite) are then indicated on the diagram with theirs respective holding times required for each transformation.

TTT diagrams are useful in planning heat treatments and in determining the critical cooling rate on quenching, which is the cooling rate at which one just avoids the nose of the TTT curve. If the austenite-to-martensite transformation is incomplete, retained austenite usually transforms during tempering into the transformation product indicated on the TTT diagram. In general, TTT diagrams allow us to understand the influence of alloying elements on transformations during continuous cooling by comparing the temperatures at which the transformation products occur. These experimental diagrams can also be used to identify theoretical models of decomposition of austenite under isothermal conditions. The isothermal kinetics of these transformations is quite well described by the Johnson-Mehl-Avrami law:

y = 1 -exp -b (T ) t n(T ) (2.39)
Where y denotes the proportion of austenite transformed at time t, T the temperature of isothermal holding, b (T ) and n (T ) experimental parameters. The time t is counted from the beginning of the transformation.

However, most of the heat treatment, including welding, are associated with thermal cycles that do not correspond to those represented in the TTT diagrams. Therefore metallurgists sought to characterize experimentally the behaviour of steels in continuous cooling. That is the reason why we use the CCT (Continuous Cooling Transformations) diagrams.

Also known as cooling transformation (CT) diagrams, CCT diagrams measure the degree of transformation as a function of time for a constantly changing (decreasing) temperature (FIG. 2

.2).

In other words, a sample is austenitized and then cooled at a predetermined rate, and the degree of transformation is measured using such techniques as dilatometry, magnetic permeability or other physical methods.

The aim of CCT diagrams is to define temperatures start and end of transformation, the microstructural state (nature and composition) and the hardness of steel subjected to various cooling rates. As the TTT diagrams, CCT diagrams depend on austenitization conditions, FIG. 2.2 gives the CCT diagram of T91 for two different austenite conditions.

All transformations proceed by nucleation and growth. Discontinuities and grain boundaries represent a privileged site for germination. Usually, grain boundaries are the mainly germination site. The grain size of austenite plays an important role on the phase transformations during cooling, and the experimental measurements show that this parameter can not be neglected.

The grain size does not change the thermodynamic equilibrium but affects on the kinetics: it does not influence the transformation temperatures but change nucleation and growth rates. It can also change the morphology of the formed phases. On the other hand, phase transformations are related to the diffusion potential of carbon, so any changes in the average concentration of carbon influence the phase transformations kinetics. The carbon stabilizes the austenite and therefore a solid-solid phase transformation larger content carbon leads to a slower ferrite formation: the CCT and TTT diagrams are shifted towards longer times. Temperature below which the supersaturated ferrite can be formed decreases when the average concentration of carbon increases as the energy required for processing increases. So solid-solid phase transformations appears:

• During welding, in the heat affected zone.

• During cooling, in the heat affected zone as well as in the melted and solidified metal

The rapid heating during welding thermal cycles shifts the temperature range of the transformation. According to the maximum reached temperature, different mechanisms can take place:

• Partial annealing and recrystallization [START_REF] Murry | Soudage et soudabilité métallurgique des métaux[END_REF] All these mechanisms are complex and can be easily studied alone. During heating we will be focusing on martensite tempering and martensite to austenite transformation. The relative important cooling rates during the welding cycles lead to a significant shift, under equilibrium conditions, of the temperature ranges where phase transformations occur. It is therefore necessary to describe just what are these cooling conditions. Several parameters can be used:

• Coalescence of precipitates • Dissolution of precipitates • Allotropic transformation

• Average cooling rates between two temperatures

• Cooling rate at a given temperature

• Cooling time between two temperatures

Since the studied material is a self quenching material, the only transformation that will be modelled is austenite to martensite transformation.

Martensite transformation

During cooling, independently of the cooling rates (| Ṫ | > 0.2 K/s), austenized material transforms into quenched martensite. This is one of the advantages of the self-quenching steel as shown in CCT diagram FIG. 2.2. At the end of cooling the material has no residual austenite, as is the case for 5% Cr steel [START_REF] Caia | Microstructural change of a 5% cr steel weld metal during tempering[END_REF].

It should be point out that during martensitic transformation, alloying elements as well as austenite grain size play an important role on the transformation kinetic. The CCT diagram previously presented shows that for different austenitization temperatures (respectively different austenite grain sizes) transformation kinetics during cooling are not the same, we can also note that the martensitic start and finish temperatures differ. Similar studies can be found in [START_REF] Babu | Modeling and characterization of nonequilibrium weld microstructure evolution[END_REF][START_REF] Bhadeshia | Diffusional and displacive transformations[END_REF][START_REF] Muddle | Formation of bainite as a diffusional-displacive phase transformation[END_REF][START_REF] Mujahida | Coupled diffusional/displacive transformations: Effect of carbon concentration[END_REF][START_REF] De-Abreua | Deformation induced martensite in an aisi 301ln stainless steel: characterization and influence on pitting corrosion resistance[END_REF][START_REF] Kelly | The martensite transformation in steels with low stacking fault energy[END_REF][START_REF] Hannnink | Metastability of the martensitic transformation in a 12 mol% ceria-zirconia alloy: I. deformation and fracture observations[END_REF][START_REF] Chang | Martensitic transformation induced by a tensile stress pulse in fe-22.5 wt% ni-4wt% mn alloy[END_REF][START_REF] Jacobus | Effect of stress state on the stress-induced martensitic transformation in polycrystalline ni-ti alloy[END_REF][START_REF] Koeda | Giant magnetostriction in fe-pd alloy single crystal exhibiting martensitic transformation[END_REF].

But first, one should understand how the austenite-martensite transformation takes place. In 1924, Bain [START_REF] Bain | The nature of martensite[END_REF] proposes a structure for the austenite-martensite transformation occurring at homogeneous deformation FIG.

9). Standard Koistinen-

Marburger model [START_REF] Koistinen | A general equation prescribing the extent of the austenite-martensite transformation in pure iron-carbon alloys and plain carbon steels[END_REF] has been used to model this transformation so that the martensite fraction y Q during cooling is modelled by:

y Q = y QMax [1 -exp (-K S (M S -T ))] (2.40)
Where y QMax denotes the initial volume fraction of the austenite that depends on the heating loading faced by each point in the vicinity of the weld, whereas K S and M S are material coefficients. Identified parameters are given in TAB. 2.2. The martensitic temperature start M S can also be estimated from the steel composition. Béres et al. [START_REF] Béres | Proposed modification to schaeffler diagram for chrome equivalents and carbon for more accurate prediction of martensite content[END_REF][START_REF] Béres | Welding of martensitic creep-resistant steels[END_REF] propose the following formula for 9-12% This formula gives a value of M S = 375 • C , which correlates well with our results. Although this model does not take into account neither the dependency of the austenite grain size (and hence the dependency of the heating process) nor the influence of the mechanical stress [START_REF] Ning | Variation of martensite phase transformation mechanism in minor-stressed t91 ferritic steel[END_REF], its simplicity remains an advantage for numerical simulations.

Martensite tempering

Quenched structure is usually a combination of untempered martensite, retained austenite and carbides retained from austenitization. The goal of tempering is to improve the strength and toughness of the quenched die steel by stress relieving and atomic rearrangement as per these equations:

• Diffusion of carbon atoms to create stronger and tougher tempered martensite.

• Precipitation and dispersal of alloy carbides in the matrix ( [START_REF] Djebaili | Characterization of precipitates in a 7.9cr-1.65mo-1.25si-1.2v steel during tempering[END_REF][START_REF] Caia | Microstructural change of a 5% cr steel weld metal during tempering[END_REF]) so that they do not coarsen during hot working. This phenomena causes hardening and is termed secondary hardening.

Tempering of martensitic steels also involves the segregation of carbon, the decomposition of retained austenite, and the recovery and recrystallization of the martensitic structure. All the processes occur simultaneously and evolve independently as function of the tempering temperature [START_REF] Valencia Morales | Kinetic theory of the overlapping phase transformations: case of the dilatometric method[END_REF]. The most important parameters that modify these changes during tempering are:

• Number of tempers.

• Duration of temper.

• Tempering temperatures.

Tempering time and temperatures have similar effects on the microstructure [START_REF] Babu | Materials and surface engineering for precision forging dies[END_REF]. Usually the tempering curves from steel producers provide variation of hardness with tempering temperatures for a fixed time of one or two hours. Another approach to specify effect of tempering is using charts and functions that track hardness with a combination of time and temperature. Although this approach is better suited to evaluate softening of hot working dies during the service, few curves are available in the literature. Alloying elements have a strong effect on the dies resistance to thermal softening. Tool steels for hot work applications depend on strong carbide formers, like V, Mo, T, Cr, etc. to provide resistance to thermal softening. However, these elements have to be dissolved in the austenite during austenitization. On the other hand, the dissolution of primary carbides decreases wear resistance. There is hence, a trade-off between resistance to thermal softening and wear that needs to be understood by the forging designers.

In the literature, several techniques are used to determine the kinetic of tempering, among them the resistivity [START_REF] Speich | Tempering of low-carbon martensite[END_REF], the dilatometry ( [START_REF] Aubry | Modélisation et étude expérimentale des cinétiques de revenuautorevenu d'aciers trempés: Prévision des contraintes résiduelles de trempe dans un acier cémente en incluant l'autorevenu[END_REF][START_REF] Cheng | The tempering of iron-carbon martensite: dilatometric and calorimetric analysis[END_REF]), the calorimetry ( [START_REF] Cheng | The tempering of iron-carbon martensite: dilatometric and calorimetric analysis[END_REF][START_REF] Cheng | Tempering of iron-carbon-nitrogen martensites[END_REF]), the Transmission Electron Microscopy (TEM) ( [START_REF] Pesicka | The evolution of dislocation density during heat treatmentand creep of tempered martensite ferritic steels[END_REF][START_REF] Ennis | Microstructural stability and creep rupture strength of the martensitic steel p92 for advanced power plan[END_REF][START_REF] Agamennone | Evolution of microstructure and deformation resistance in creep of tempered martensitic 9-12%cr-2%w-5%co steels[END_REF]), the Energy Filtered Transmission Electron Microscopy (EFTEM) ( [START_REF] Warbichler | On the application of energy-filtering tem in materials science: Iii. precipitates in steel[END_REF][START_REF] Thomson | Carbide precipitation in martensite during the early stages of tempering cr-and mo-containing low alloy steels[END_REF][START_REF] Shtansky | Phase transformation in fe-mo-c and fe-wc steels-i: The structural evolution during tempering at 700c[END_REF][START_REF] Shtansky | Phase transformation in fe-mo-c and fe-w-c steels-ii: Eutectoid reaction of m23c6 carbide decomposition during austenitization[END_REF][START_REF] Hättestrand | Precipitation hardening in a 12%cr-9%ni-4%mo-2%cu stainless steel[END_REF]), the X-rays ( [START_REF] Cheng | Tempering of iron-carbon-nitrogen martensites[END_REF][START_REF] Pesicka | The evolution of dislocation density during heat treatmentand creep of tempered martensite ferritic steels[END_REF][START_REF] Agamennone | Evolution of microstructure and deformation resistance in creep of tempered martensitic 9-12%cr-2%w-5%co steels[END_REF][START_REF] Jemian | Characterization of 9cr-1movnb steel by anomalous small-angle x-ray scattering[END_REF]), Atom Probe Field Ion Microscopy (APFIM) ( [START_REF] Thomson | Carbide precipitation in martensite during the early stages of tempering cr-and mo-containing low alloy steels[END_REF][START_REF] Hättestrand | Influence of strain on precipitation reactions during creep of an advanced 9% chromium steel[END_REF]), and the Electron Spectroscopic Imaging (ESI) [START_REF] Hofer | Imaging of nanometer-sized precipitates in solids by electron spectroscopic imaging[END_REF]. Since carbides have only limited effect, hardly measurable at the global scale (e.g. with dimensional and thermal analysis), the Thermo-Electric Power (TEP) ( [START_REF] Hofer | Imaging of nanometer-sized precipitates in solids by electron spectroscopic imaging[END_REF][START_REF] Perez | Microstructural evolution of martensitic 100cr6 bearing steel during tempering: From thermoelectric power measurements to the prediction of dimensional changes[END_REF][START_REF] Brachet | Some examples of thermoelectric power (tep) measurement application on 9-12cr(mo;w) ferritic-martensitic steels and on zr-base alloys[END_REF][START_REF] Houze | Influence des traitements thermiques sur le Pouvoir Thermoélectriquedes aciers de cuve: Effet des évolutions microstructurales des zones fortement ségrégées[END_REF]) a less common technique has been investigated. solid-solid phase transformation

ThermoElectric Power and hardness measurements

To investigate the microstructure evolution occurring during the tempering of martensite, TEP measurements were performed after isothermal ageing treatments. In order to have the same initial microstructure, X10CrMoVNb9-1 martensitic steel parallelepiped (2 × 2 × 12 mm 3 ) samples were austenitized for [START_REF] Muddle | Formation of bainite as a diffusional-displacive phase transformation[END_REF] Dimensional changes occurring during tempering are due to: (i) the precipitation of e-carbides, (ii) the decomposition of retained austenite, (iii) the precipitation of cementite and (iv) the recovery of the dislocation structure and coarsening of martensite structure. However, due to their small volume fraction, carbides have only a limited effect, hardly measurable at the global scale (e.g. with dimensional and thermal analysis). It is therefore interesting to use alternative techniques which can provide new insights into this subject. The measurement of TEP could be one such technique. The high sensitivity of TEP to crystal defects such as dislocations and solute atoms provides a powerful experimental tool for characterizing the kinetics of structural transformations taking place in various alloys.

The principle of TEP measurements ( [START_REF] Houze | Influence des traitements thermiques sur le Pouvoir Thermoélectriquedes aciers de cuve: Effet des évolutions microstructurales des zones fortement ségrégées[END_REF][START_REF] Brachet | Correlation between thermoelectric power (tep) and martensite start temperature (ms) measurements of 9cr-w-v-(ta) martensitic steels[END_REF][START_REF] Blatt | Thermoelectric power of metals[END_REF]) is to establish a temperature gradient ∆T between the junctions of the studied alloy with two blocks of pure copper (see FIG. An assumption was made that consists on relating the tempering state parameter x T to TEP measurements by the following linear expression:

x T = T EP -T EP Q T EP AR -T EP Q (2.

43)

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016 solid-solid phase transformation

- → J = -D --→ grad C → 1D → J = -D ∂C ∂x (2.44)
On the other hand Fick's second law or Green-Ostrogradsky law known as the law of conservation of species is given:

∂C ∂t = -div - → J → 1D → ∂C ∂t = - ∂J ∂x → ∂C ∂t = D ∂ 2 C ∂ 2 x (2.45)
In isothermal case, the diffusion front is proportional to the square root of time

x = √ Dt (2.46)
By expressing D as a temperature dependant, a replacing the square root by a 1/n factor, a generalised formula follows:

x = D 0 exp - ∆H R T t 1 n (2.47)
By derivation, the expression becomes:

ẋ = 1 n D 0 exp - ∆H R T x 1 -n (2.

48)

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016

By replacing now x by a tempering state parameter x T and multiplying by braking expression and a heaviside function, the previous formula becomes:

ẋ T = 1 n D 0 exp - ∆H R T x T 1 -n (1 -x T ) H [T -T T T h ] (2.49)
Where T T T h is the threshold tempering temperature under which no carbides precipitation cannot take place. FIG. 2.15 shows the identified model prediction in comparison with experimental data. The "high"-temperature activation energy ∆H = 278 kJ/mol is close to that for activation diffusion energy of Chrome. The analysis of TEP measurements at different times and temperatures (from 500 to 750 • C) shows that all processes occurring during tempering are related to (i) the self-diffusion activation energy of Chromium (activation energy of 3.51 eV = 339 kJ/mol [START_REF] Campbell | Positron trapping and self-diffusion activation energies in chromium steels[END_REF]) driving the kinetics of carbide precipitation; and (ii) the mobility of dislocations and iron atoms (activation energy of 190 kJ/mol), driving the kinetics of recovery and coarsening of martensite lathes.

Hardness evolution

The same parallelepiped samples were subjected to hardness measurements. The samples were wrapped with a hard resin to facilitate the measurements. Measurements were made using the CEA hardness tester machine 'Duramin-A300' (See FIG. 2.16). A test load of 1 kgf was used.

For each sample, series of ten measurements were performed to verify the homogeneity of the material. The starting point and test series were easily selected using the macro option of the observation camera. The fully automatic evaluation of hardness ensures the repeatability and solid-solid phase transformation

Model validation

A real narrow groove gas tungsten arc multipass welding experiment was simulated using the finite element CEA code Cast εM. The above models were implemented under Matlab c and the thermal results of the previous simulation were used as inputs. The hardness profiles across (FIG. 2. [START_REF] Raj | Mechanical properties and non-destructive evaluation of chromium-molybdenum ferritic steels for steam generator application[END_REF]) and through the different welding passes (FIG. 2.20) were computed and compared to real measurements.

After successive passes, the resultant microstructure is very heterogeneous: different more or less tempered martensite phases can co-exit. This induced us to use a simplified homogenisation rule to calculate the equivalent tempered martensite:

x T eq = n ∑ 1=1 x T i y Q i y Q T = n ∑ 1=1 y Q T i (2.51)
Where x T eq denotes the equivalent tempering state of the total tempered martensite phases y Q T and Q T i denotes a martensite phase that has been partially tempered x T i . This assumption seems logical since tempering is a cumulative problem. The hardness of the resultant mixture of quenched and tempered martensite is then assessed by the following expression:

HV = y Q × HV Q + (1 -y Q ) × HV -(HV Q -HV AR ) exp x T -1 x 0 (2.

52)

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016 solid-solid phase transformation where y Q represents the quenched martensite fraction. Small scatters can be observed between the hardness measured profiles of the two different trials. One can note that the general trend is found and the computation results are very close to experimental results.

Discussion

TEP has been successfully used, to characterize the interaction between interstitial and substitutional atoms in low alloyed steels and to study the ageing of ultra-low carbon alloys [START_REF] Lavaire | Etude des phénomènes à l'origine du vieillissement des aciers pour emballage à Ultra Bas Carbone (ULC) : Apport du Pouvoir ThermoElectrique à la caractérisation des états microstructuraux[END_REF]. TEP has also been used to characterize the precipitation sequence (carbides and cementite) occurring in ferrite. However, few authors have used TEP to characterize microstructural evolution occurring during the tempering of martensite. In this section, TEP has been showed to be an alternative method to study martensite steel tempering.

Martensite dissolution

Austenitization is an inevitable occurrence during arc welding. Despite this consideration, less attention has been paid so far to the study of the formation of austenite as compared with the vast amount of research on its decomposition. That is because the steel properties depend basically on the transformation processes following austenitization. However, the initial austenitic condition is important to the development of the final microstructure and its mechanical properties ([98, 99, 100, 101, 102, 103, 71, 104]). We should mention that austenitization is one of the phase transformations that are responsible for the complex post-welding microstructural and mechanical state of the material in the HAZ induced by multipass welding process.

In this section, we present three different approaches that may be used to model the phase transformations by diffusion during heating. These approaches were firstly applied to model the austenitization of the X10CrMoVNb9-1 martensitic steel fully tempered at 757 • C for 5 h30. At the end of this section we show the effect of tempering on austenite transformation and we illustrate the improvement brought to the models by taking into account this effect.

The corresponding models are first identified from standard dilatometry tests performed over a large range of heating rates, viz. [0.1 • C/s, 100 • C/s]. Finally, the response of these models, and therefore, their predicting capabilities, are compared to the experimental evidence for various complex heating-cooling loadings that have been designed to be representative of the temperature history in different points of a multipass welding Heat Affected Zone.

Pre-study

A first version of a Thermo-Metallurgical-Mechanical (TMM) model for X10CrMoVNb9-1 steel has been validated in terms of residual stresses on simple single-pass spot-welding tests ( [START_REF] Roux | Prévision des contraintes résiduelles induites par le soudage TIG d'un acier martensitique (X10CrMoVNb9-1)[END_REF][START_REF] Roux | Identification of thermal boundary conditions and thermometallurgical behaviour of x10crmovnb9-1 steel during simple tig welding tests[END_REF]). After implementation in Cast εM finite element code, this TMM model has been used to analyse different welding tests including a 16-pass narrow groove GTA-Welding test (as presented in Chapter1).

Predicted temperature and temperature rate evolutions in the HAZ, in a plane perpendicular to the weld where steady state can be assumed, have been favourably compared to the responses of several K-type thermocouples located at different depths and distances from the groove. Typical results are illustrated by the plots given in FIG. 2

.21 and FIG. 2.22.

The pseudo-binary iron-chromium (0.1 wt% carbon) equilibrium phase diagram given in FIG. 1.4, shows that martensite to austenite transformation is activated once the temperature exceeds 800 • C. The continuous cooling transformation diagram given in FIG. 2.2 shows that austenite always transforms to martensite during cooling down to room temperature -except for very slow cooling rates that are out of the scope of welding applications.

The initial microstructure of the as-received material studied herein corresponds to tempered martensite as obtained after ingot casting, hot rolling and successive heat treatments including final water quenching and tempering at 757 • C for 5h30 (see FIG. 2.23). Of course, during a multipass welding process, depending on the heating-cooling cycles imposed by the successive passes, this material may be either partially or totally austenitized. solid-solid phase transformation

Three different approaches to model austenitization

The results of dilatometry tests that were performed on X10CrMoVNb9-1 at constant heating rates within the range [0.1 • C/s, 100 • C/s] exhibit the standard anomaly related to the volume change associated to the transformation of BCC tempered martensite to FCC austenite (see FIG. 2

.24).

It is a common practice to derive from such standard dilatometry plot the so-called Continuous Heating Transformation diagram given in FIG. 2.25. At a given constant heating rate ( Ṫ ), the macroscopic effects of the austenite transformation -such as the anomaly on the dilatometry curves -starts (resp. finishes) at so-called austenite start (resp. finish) temperature denoted by A S (resp. A f ), so that the fraction of formed austenite y γ ∼ = 0 (resp.1) at A S (resp. A f ). The transformation that is associated to diffusion processes is clearly dependent on the heating rate. At the microscopic scale, the transformation starts by random nucleation at temperatures T ≥ A eq where A eq ≤ A s

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016 A first class of models to describe the out of equilibrium transformation is based on a splitting hypothesis such that (nucleation-)growth(-coalescence) phenomenon -from A S to A f -is uncoupled from so-called incubation phenomenon -from A eq to A S . A second class of models assumes that the transformation is associated to coupled nucleation and growth(-coalescence) phenomena ([106, 1]) -from A eq to A f .

Brachet type model

First approach discussed herein corresponds to a slight modification of a model initially proposed by Brachet et al. [START_REF] Brachet | Modelling of phase transformations occurring in low activation martensitic steels[END_REF]. In this model, incubation and growth-coalescence of austenite are uncoupled. Following for instance [START_REF] Aeby-Gautier | Transformations perlitique et martensitique sous contrainte de traction dans les aciers[END_REF], incubation is modelled using Scheil additive rule such A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016 solid-solid phase transformation

(T ) = n ∑ i ∆t i τ (T i ) = 1 (2.53)
where incubation time τ (T i ) at a given constant temperature T i is modelled by the empirical law proposed in [START_REF] Roux | Prévision des contraintes résiduelles induites par le soudage TIG d'un acier martensitique (X10CrMoVNb9-1)[END_REF] such that

τ(T ) = A(A Sat -T ) exp C T -A eq (2.54)
where A, A sat , C and A eq are material parameters. Once incubation is complete, the evolution of the austenite fraction, y γB , is modelled by the differential equation initially proposed by Holt et al.

[109] and modified by Brachet et al. [START_REF] Brachet | Modelling of phase transformations occurring in low activation martensitic steels[END_REF] by adding the term 1y γB such that:

ẏγB = K B exp - W B R T T -T eq (y γB ) n B + 1 -y γB (2.55)
where K B , W B and n B are material parameters, and where T eq (y γB ) denotes the so-called equilibrium transition temperature which corresponds to infinitely slow transformations. Following ( [START_REF] Duthilleul | Etude des évolutions microstructurales lors d'un traitement thermique rapide à haute température d'un acier martensitique de type 9cr1mo[END_REF][START_REF] Danon | Heterogeneous austenite grain growth in 9cr martensitic steels: influence of the heating rate and the austenization temperature[END_REF][START_REF] Brachet | Modelling of phase transformations occurring in low activation martensitic steels[END_REF]), this equilibrium kinetic is modelled as:

y eq (T ) = 1 -exp [-(K eq (T -A eq )) m eq ] (2.56)
Where K eq and m eq are material parameters, or, conversely, T eq y γ = A eq + 1 K eq ln 1 1y eq 1 m eq (2.57)

JMA type model

The second approach is based on the probabilistic theory originally proposed by Kolmogorov and Johnson-Mehl-Avrami ([110, 111, 112, 113]) assuming that the evolution -from time to nucleation, τ, to present time, t ≥ τ -of the so-called extended fraction of the austenite phase, is the result of a convolution between its random nucleation at rate dλ(τ)/dτ, and its growth -without interaction -at rate Z, so that: T (t) -A eq ≥ 0

y γ (t) = 1 -exp - t 0 ẏγex (τ,t) dτ = 1 -exp - t 0 Z(t -τ) dλ(τ) dτ dτ (2.58)
At constant temperature (T ≥ A eq ), equation (2.57) leads to so-called Avrami equation [START_REF] Avrami | Kinetics of phase change. ii-transformation-time relations for random distribution of nuclei[END_REF]:

y γ (t) = 1 -exp (-K(T ) t n ) (2.59)
where n and K(T ) are material parameters depending on the geometrical and time characteristics of the nucleation and growth laws. A particular form of equation (2.58) for constant temperature rates and for T (τ) -A eq ≥ 0, valid for iso-kinetic transformations -i.e. transformations for which the ratio of the nucleation and growth rates is constant at any temperature -, corresponds to socalled Ozawa equation [START_REF] Ozawa | Kinetics of non-isothermal crystallization[END_REF] such that

y γ (t) = 1 -exp K Ṫ n (2.60)
where n and K are material parameters. The following forms for the nucleation and growth laws, are assumed:

λ (T ) = λ 0 T (τ) -A eq T 0 m H
for T (τ) -A eq ≥ 0 (2.61)

Z (t-τ) = Z 0 t -τ t 0 3n H (2.62)
another closed form solution of equation (2.58) can be derived [START_REF] Roux | Prévision des contraintes résiduelles induites par le soudage TIG d'un acier martensitique (X10CrMoVNb9-1)[END_REF] for constant temperature rates, so that:

y γH (t) = 1 -exp - K JMA m H T 0 m H t 3n H 0 Ṫ 3n H (T (τ) -A eq ) m H +3n H B(m H , 3n H + 1) (2.63)
at constant Ṫ and for T (τ) -A eq ≥ 0 with B(x, y) = t 0 t x-1 (1t ) y-1 dt . Where K JMA , λ 0 , Z 0 , T 0 , t 0 , m H , n H and A eq are material parameters. Herein, it is proposed to model the effect of temperature on the growth law by substituting in equation (2.62) the constant parameter t 0 by the following expression:

t 0 = t ∞ exp -Q R T (2.64)
so that equation (2.63) becomes

y γJMA (t) = 1- exp   - K JMA m JMA T 0 m JMA t 3n JMA ∞ t 0 (T -A eq ) m JMA -1 (t -τ) 3n JMA exp 3 n JMA Q R T dT dτ dτ   (2.65)
where t ∞ , Q, m JMA , n JMA are material parameters.

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016 solid-solid phase transformation

Leblond type model

Avrami equation (2.59) can be generalised as following:

y γ (t) = y eq (T ) [1 -exp (-K(T ) t n )] (2.66)
where y eq (T ) corresponds to the kinetics at equilibrium, so that: ẏγ (t) = n K 1/n (y eq (T )y) ln y eq (T ) y eq (T )y The response of this popular phase transformation model can be improved by adding an extra term f Ṫ , function of the temperature rate, Ṫ , so that the evolution law for the austenite fraction is given by: ẏγL = y eq (T ) -

y γL τ(T ) f ( Ṫ ) (2.69)
where characteristic time τ (T ) and function f Ṫ may take different forms. Herein, following [START_REF] Cavallo | Contribution à la validation expérimentale de modèles décrivant la Z.A.T. lors d'une opération de soudage[END_REF], characteristic time t (T ) is chosen such that:

τ(T ) =    τ L1 T < A s τ L1 + T -A s A f -A s (τ L3 -τ L1 ) for A s ≤ T ≤ A f τ L3 T > A f    (2.70)
where t L1 , t L2 and t L3 are material parameters. Besides, function f Ṫ is chosen as

f ( Ṫ ) = A L exp -B L Ṫ (2.71)
where A L and B L are material parameters. An extension of this model for n phases is given in ANNEX E.

Parameters identification

The usual assumption is to consider that the austenite fraction is proportional to the anomaly of the dilatometry curve. The evolution with temperature of the austenite fraction -at constant heating rates within the range [0.1 • C/s, 100 C/s heating rate). Furthermore, the equilibrium kinetics was extrapolated from these tests at different constant heating rates following the procedure described in [START_REF] Duthilleul | Etude des évolutions microstructurales lors d'un traitement thermique rapide à haute température d'un acier martensitique de type 9cr1mo[END_REF]. According to Zhu and Devletian [START_REF] Zhu | Determination of equilibrium solidphase transtion temperature using dta[END_REF], during heating at a slow heating rate Ṫ , the temperature T corresponding to a given austenite fraction y γ can be modelled by the following relationship:

T (y γ ) = C ZD Ṫ T exp Q ZD R T 1 / 3 + T eq (y γ ) (2.

72)

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016 ) and the values of material parameters A eq , K eq and m eq -identified by least square minimisation -are valid for both Brachet and Leblond type models, austenite temperature at equilibrium A eq being also valid for JMA type model. Material parameters A, A Sat , C and A eq appearing in equation (2.54) were directly derived from austenite start A S curve. Then, Brachet type model identification was completed by the identification of material parameters K B , W B and n B appearing in equation (2.55). For JMA type model, closed-form equation (2.63) with beta function -corresponding to constant t 0 in growth law equation (2.62) -is very useful to identify material parameters K JMA = λ 0 Z 0 , T 0 , t 0 , m H and n H . After introducing through equation (2.64) a temperature dependence in growth law equation (2.62), material parameters t ∞ , Q, m JMA and n JMA had to be identified while parameters K JMA = λ 0 Z 0 , T 0 , (and A eq ) were kept constant. It is worth noticing that this refinement of the model does not increase significantly its capability to reproduce the response of the material at constant heating rates. The values of the material parameters for each model are given in TAB. 2.6 to TAB. 2.9.

A = 0.044 A Sat = 869 • C C =56.47 • C -1 K B = 104 10 15 s -1 W B = 485 kJ/mol n B = 2.52 A eq = 802 • C K eq = 0.056 • C -1 m eq = 1.151

Validation of the models

The response of these models were compared to the results of complex dilatometry tests performed at various heating-cooling loadings that were designed to be representative of the temperature history in different points of a multipass GTAW HAZ. The first series of tests correspond to an solid-solid phase transformation The third series of tests correspond to a partial austenitization at a constant heating rate of 50 • C/s up to 50% of the austenite transformation, followed by quenching down to 500 • C i.e. below A eq (but above martensite start temperature) and then, reheating at the same constant heating rate of 50

Tempering influence on austenitization kinetics

The influence of tempering on the microstructure has been earlier discussed. In the following focus is made on determining the influence of these microstructural changes on austenitization kinetics. A set of dilatometry tests was performed on more or less tempered X10CrMoVNb91 steel at constant heating rate of 50 • C/s. From dilatometry curves, the evolution with temperature of the austenite fraction can be deduced as shown in FIG. 2.38. From these evolutions, we can deduce the austenite start temperature A S function of the tempering factor x T . This evolution can be modelled as follows:

A S = A S0 -C A S exp 1 -x t x A S (2.

74)

Where A S0 , C A S and x A S are material parameters given in TAB. 2.10. The problem now is to take into account this evolution in modelling the partial austenitization and re-austenitization kinetics.

A S0 C A S x A S 877 • C 2.67 0.386

Rewriting Brachet type model

The Brachet type model permits to deal with austenitization as an uncoupled incubation/growth problem. The incubation as proposed by Roux et al. [START_REF] Roux | Prévision des contraintes résiduelles induites par le soudage TIG d'un acier martensitique (X10CrMoVNb9-1)[END_REF] is written as follows:

τ(T ) = A(A sat -T ) exp C T -A eq (2.75)
where A, A Sat , C and A eq are material parameters. And the growth is modelled by the following differential equation proposed by Brachet et al.

ẏγB = K B exp - W B R T T -T eq (y γB ) n B + 1 -y γB (2.76)
Since it is an uncoupled model, the choice was made to work on the incubation equation by rewriting A Sat as function of x t . To do so a first simple sensitivity analysis to this parameter was solid-solid phase transformation done. FIG. 2.40 shows the influence of this parameter on the austenite start temperature. This can be modeled using the following equation:

A Sat = A Sat0 -C A Sat exp 1 -x t x A Sat (2.

77)

Where A Sat0 , C A Sat and x A Sat are material parameters given in TAB. 2.11. model predictions for x t = 0, 0.3, 0.6 and 1

A Sat0 C A Sat x A Sat 873 • C 0.816 0.225

Rewriting Avrami type model

The Avrami type model is a coupled nucleation/growth model written as follows:

y γJMA (t) = 1- exp   - K JMA m JMA T 0 m JMA t 3n JMA ∞ t 0 (T -A eq ) m JMA -1 (t -τ) 3n JMA exp 3 n JMA Q R T dT dτ dτ   (2.78)
The choice was made to rewrite K JMA and m JMA as function of the tempering factor x t . By reidentifying these two parameters for each dilatometry test of different tempered martensite, we get the evolution of these parameters function of the tempering factor as shown in FIG. 2.43. These two parameters can be modelled as follows:

K JMA = K JMA0 + (K JMA1 -K JMA0 ) exp - x t x K JMA (2.79) m JMA = m JMA0 -(m JMA1 -m JMA0 ) exp - x t x m JMA (2.80) (2.81)
Where K JMA0 , K JMA1 , x K JMA , m JMA0 , m JMA1 and x m JMA are material parameters given in TAB. 2.12.

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016 solid-solid phase transformation model predictions for x t = 0, 0.3, 0.6 and 1

K JMA0 K JMA1 x K JMA m JMA0 m JMA1 x m JMA

Discussion

In this chapter an effort was made in modelling the metallurgical transformations that this material can undergo. For austenitization, the three approaches presented herein can be compared in terms of efforts required for identification and numerical implementation in a finite element code, and in terms of predictive capabilities. Concerning numerical implementation, it can be concluded that

• Leblond type model popularity in the computational mechanics community is justified by the smaller implementation and computation efforts it requires when compared to other approaches

• the fundamental drawback of JMA type model is linked to its integral form to be compared to the differential forms of Brachet and Leblond type models Concerning the efforts required for the identification, it can be noticed that

• an efficient multistep identification strategy can be developed for the three approaches

• the closed form expression of JMA type model for constant heating rates (see (2.63) with beta function) corresponding to the simple nucleation and growth laws given by (2.61) and (2.63) may be of great help Last, both JMA and Brachet type models exhibit excellent prediction capabilities even for complex realistic thermal loadings.

It is worth noticing that the excellent quality of the results obtained with the JMA and Brachet type models is probably due to the fine microstructure of the tempered martensite considered herein, and to the fact that its austenitization -which fundamentally corresponds to the crystallographic changes of the iron matrix from bcc to fcc -is fully decoupled from other metallurgical phenomena such as the dissolution of the carbides that are initially present in the as-received material. This last point is indeed confirmed by the study of the quenched martensite tempering.

To model martensite transformation the simple Koistinen-Marburger model was employed. Although he does not take into account neither the influence of the austenite grain size nor the influence of cooling rate, the results remain satisfying.

For tempering modelling, a simple differential model was developed. The results of identification have shown a coherence with Chromium bulk diffusion and carbides growth-coalescence. On the other hand, this model can not be used to model the secondary ageing (embrittlement) so it can not be applicable to holding times exceeding 10 hours and temperatures inferior to 475 • C.

Influence of tempering on austenite kinetic has also be modelled. The result of such modelling is quite satisfying.

Chapter 3

Thermo-mechanical behaviour of each phase

The thermo-mechanical behaviour of X10CrMoVNb9-1 alloy is identified in order to carried out numerical simulations of the welding process. The proposed model is elasto-viscoplastic with isotropic and kinematic hardening.

The prediction of microstructure and mechanical properties evolution during welding requires an accurate knowledge of phase transformations kinetics [START_REF] Schneider | Simulation of the kinetics of precipitation reactions in ferritic steels[END_REF]. Many previous works focus on carbides hardening contribution and their influence on creep resistance ( [START_REF] Ennis | Microstructural stability and creep rupture strength of the martensitic steel p92 for advanced power plan[END_REF][START_REF] Agamennone | Evolution of microstructure and deformation resistance in creep of tempered martensitic 9-12%cr-2%w-5%co steels[END_REF][START_REF] Thomson | Carbide precipitation in martensite during the early stages of tempering cr-and mo-containing low alloy steels[END_REF][START_REF] Kostka | On the contribution of carbides and micrograin boundaries to the creep strength of tempered martensite ferritic steels[END_REF][START_REF] Aubry | Modélisation et étude expérimentale des cinétiques de revenuautorevenu d'aciers trempés: Prévision des contraintes résiduelles de trempe dans un acier cémente en incluant l'autorevenu[END_REF]) without mentioning the tempering impact of mechanical behaviour.

In this work the thermo-mechanical behaviour of the material at different tempering stage was studied according to several mechanical tests at different temperature and strain rate. The parameters of the elasto-viscoplastic model were then rewritten taking account a parameter x T for the tempering. Thermo-mechanical behaviour of each phase

Thermo-mechanical model of the material

To evaluate the stress and deformation in a structure induced by a temperature field and their evolution over time, it is necessary to describe the material behaviour by a thermo-mechanical model.

A thermo-mechanical constitutive model is a mathematical expression that connects quantities such stress, strain, strain rate, temperature and microstructure of a material. In literature, constitutive laws that take into account the multiphase aspect of the material are divided in two categories: those that refer to the crystal plasticity at micro scale and those who are purely macroscopic:

• The first category attempts to model the mechanisms of deformation at a microscopic scale, and get the macroscopic behaviour of the elementary volume by homogenization of local behaviours, through scale change rules that describe the transfer from micro to macro scale.

• The second category of models, sometimes referred to as "phenomenological" study a homogeneous volume element [START_REF] Lemaitre | Mécanique des matériaux solides[END_REF], relatively large compared to the grain scale. The thermodynamic approaches help to rationalize the selection of variables and to clearly distinguish the influence of stresses, temperature, strain and internal variables associated to the strain hardening phenomena.

In this study, a mixed approach has been chosen: a phenomenological model is used for each phase and coupled to an explicit scaling change between each phase critical variables and the macroscopic variables (stress, strain). The thermo-mechanical behaviour is modeled by an elastoviscoplastic law including a mixed isotropic/kinematic strain hardening as already presented in Chapter1. This model is based on a classical approach, introduced in the years 70-80 by Chaboche and Lemaitre [START_REF] Lemaitre | Mécanique des matériaux solides[END_REF]. During their motion, the dislocations get together and expand. Their density increases, thereby increasing the barriers that prevent their displacement. The result is a hardening of the material by cold working. This phenomenon of plasticity generates different mechanisms of hardening including kinematic and isotropic hardening. The Bauschinger effect is the main manifestation of the kinematic hardening. It results in a lowering of the yield strength in compression after a tensile pre-stress. In this study, it has been highlighted by the tension-compression tests. On the other hand, the isotropic hardening induces in the case of a cyclic deformation an increase (or decrease) of the stress amplitude; it results in hardening (or softening) of the material.

In the case of a periodic stress, a stabilized behaviour can be reached for which the stress does not evolve anymore. The viscoplastic behaviour is usually observed for materials working at high temperature (T > 0.4 T M ). The viscoplastic strain has the following origins:

• Deformation resulting from dislocation motion at the crystalline scale. Temperature is a driving force which permits their dislocation to acquire new degrees of freedom. In fact, when the temperature reaches a third or half of the melting temperature, the dislocations can climb or move out of their slip planes. It is the movement of these dislocations which produces what is called the stress-dislocation.

• Distortion caused by atomic rearrangement at the grain boundary without involving any dislocations. At high temperatures, this diffusion occurs mainly in the volume of the crystal itself. For a lower temperature, the diffusion of grain boundaries becomes dominant.

Let us recall the main equations of the chosen model:

εp = 3 2 ṗ s -X J 2 (s -X) (3.1)
f p = J 2 (s -X) -R -σ y (3.2) ṗ = ε∞ sinh f p K N (3.3) Ẋ = [C 1 εp -γ 1 ṗX 1 ] +C 2 εp (3.4) Ṙ = b (R ∞ -R) ṗ (3.5)
Where E, σ y , ε∞ , K, N,C 1 , γ 1 ,C 2 , b and R ∞ are the thermo-mechanical model parameters to identify at each temperature for each phase. We should point that, the classical scalar isotropic hardening It results from the short distance obstacles such as network friction forces and small precipitates sheared by dislocations. It is directly linked to the increase dislocation density. X describes usually the long distance interactions such as inter-or transgranular stresses induced by incompatibility of plastic deformation from one grain to another or caused by interactions between dislocations and precipitates.

Mechanical tests presentation

An experimental campaign has been realized to characterize the thermo-mechanical behaviour of each phase of the material that occur during a welding operation, ie the base material (fully tempered martensite) austenite, partially tempered and quenched martensite. These mechanical tests permit to identify:

• The elasto-viscoplastic behaviour of each phase of the material as a function of temperature.

• The thermal deformation behaviour of each phase and the volume change associated with the phase transformation. This characterizes the thermal expansion of the material.

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016

• Transformation plasticity, which appears during a phase transformation under applied stress.

This characterization test campaign was carried out in LMT-Cachan using the experimental devices presented below.

Test specimens

The shape of the test specimens (FIG. 3.2) was chosen to minimize the risk of buckling in compression with a useful part of length (8 mm) compatible with the extensometer gauge. Unfortunately, an error was made during the first machining where grooves where machined on the specimen instead of points (see FIG.

Heating system

A high frequency aperiodic generator and an electromagnetic inductor were used to heat the test specimens (see FIG. 3.4). An inductor heating system was choosen for the simplicity during mounting especially when an extensometer and thermocouples are used, and the good temperature distribution provided by the system. The temperature variation in the useful part of the specimen is approximately 10 • C at 1000 • C. During testing, hot docking heads and the inductor is cooled by a water circulation system at a constant rate, the extensometer is cooled by a fan. To measure temperature uniformity during testing, two thermocouples 'type K' (Chromel-Alumel) are welded near the stems of the extensometer and in the middle of the useful part of the specimen. The average strain of the useful part of the specimen is measured thanks to an extensometer with alumina rods. The rods, positioned in two conical stamps (see FIG. • Monotonic tensile test with variable strain rates

• Cyclic test (or tension-compression test for different amplitudes of loading) at a constant strain rate of 10 -4 s -1

The monotonic tensile test is presented in FIG. 3.6 is carried out at different strain rates 10 -5 , 10 -4 and 10 -3 s -1 , and followed by a stress relaxation. This test is used to properly determinate the effects of viscosity at different temperatures. The cyclic test corresponds to a successive tension/compression loading at different amplitudes (FIG. 3.7). The value of the amplitudes depends on the temperature and associated sensitivity to buckling; thus the amplitudes were lowered for higher temperatures. Relaxations were also included during these tests giving the possibility Thermo-mechanical behaviour of the austenitic phase 77

Identification method

The parameters of the elasto-viscoplastic model were identified from various isothermal mechanical tests for quenched, partially tempered (40 and 60%), fully tempered martensite (As-Received material) and austenitized material. Identification is carried out by solving an inverse problem.

The identification was firstly carried out on each test individually in 1D thanks to Matlab c using a Runge-Kutta algorithm to solve differential equation and the Simplex algorithm for minimisation. Some parameters such as the Young modulus E and the apparent (0.2%) yield strength σ y were easily estimated thanks to a linear regression on the elastic part of the curve (FIG. 3.1).

The identified parameters on each curve were then used as initial parameters to a second identification round. This time, the identification was carried out on all the isothermal tests of each phase of the material. In this second identification we imposed a monotonous evolution of the material parameters respective to temperatures. After this identification, the parameters that vary slightly with the material state (quenched, partially tempered (40 and 60%) or fully tempered martensite) were averaged and fixed for the third global identification round which will be presented later on.

Thermo-mechanical behaviour of the austenitic phase

For the austenitic phase, monotonic tensile tests with variable strain rate at five different temperatures (500, 600, 700, 800 and 900 • C). Before applying the mechanical load, each test specimen had undergone a homogenization heat treatment. This heat treatment plays an important role on the mechanical behaviour of the austenitic phase due to the more or less complete carbides dissolution and grain size enlargement. We have chosen a heat treatment that ensures the complete dissolution of carbides and no austenite grain growth, ie 1045 • C for 10 minutes (FIG. 3.8). The

Thermo-mechanical behaviour of martensite

Many works on this material concern one of the key properties: the high-temperature creep resistance and its correlation to the microstructural evolution ( [START_REF] Hättestrand | Influence of strain on precipitation reactions during creep of an advanced 9% chromium steel[END_REF], [START_REF] Dimmiler | Quantification of the laves phase in advanced 9-12% cr steels using a standard sem[END_REF], [START_REF] Polcik | On the microstructural development of the tempered martensitic cr-steel p91 during long-term creep -a comparison of data[END_REF], [START_REF] Vodarek | Effect of nickel on the precipitation processes in 12crmov steels during creep at 550 • c[END_REF], [START_REF] Vodarek | Minor-phase composition changes in 12crmovnb steels during long-term exposure[END_REF]). One important parameter for the creep resistance of these steels is the formation and interaction of different precipitate populations like NbC, VN, M 23 C 6 , Lave phases and modified Z-phase. Few works involve the direct influence of tempering on the mechanical properties on the creep resistance of the material.

During tempering, two simultaneous processes seem to take place. First, the recovery that causes some reduction of dislocation density after austenitization and the formation of subgrains and dislocation networks, secondly, the precipitation of carbides and carbonitrides in prior austenite grain boundaries, on subgrain boundaries and within the martensite laths. These processes change the mechanical behaviour of the material. The purpose of this section is to establish a direct link between tempering and thermo-mechanical parameters.

A total of 24 test samples were previously austenitized (45 min at 1020 • C) and quenched in an atmosphere controlled oven leading to a homogeneous quenched martensite state. Two third of this test samples were tempered at 550 • C. Half were tempered for 6 minutes and the other half for one hour leading to 40% (x T = 0.4) and 60% (x T = 0.6) tempering state. A resume of mechanical test specifications is given in TAB. and 500 • C). The phenomenon of dynamic strain aging (DSA), manifested by a negative sensitivity to strain rate, is correlated with the plastic flow, this is known as so-called Portevin-Le-Chatelier effect. This phenomenon is well known in the aluminum alloys ( [START_REF] Kubin | Dynamic strain ageing and the mechanical response of alloys[END_REF], [START_REF] Benallal | Dynamic strain aging and related instabilities: experimental theoretical and numerical aspects[END_REF]), but also present in some steels ( [START_REF] Sachdev | Dynamic strain aging of various steels[END_REF]). Physically, the dynamic strain aging is due to the rapid diffusion of atoms in solution in the matrix coupled to the movement of dislocations, which led to some instabilities. The aging dynamics have been studied for the 9Cr1-Mo alloys ([129], [START_REF] Kishore | Effect of dynamic strain ageing on the tensile properties of a modified 9cr1mo steel[END_REF]), appearing typically in the temperature range 200-450 • C. Some have considered this phenomenon in the mechanical models ( [START_REF] Ho | The modeling of unusual rate sensitivities inside and outside the dynamic strain aging regime[END_REF], [START_REF] Yaguchi | Unified inelastic constitutive model for modified 9cr-1mo steel incorporating dynamic strain aging effect[END_REF]). There is an increase of tensile strength accompanied by a decrease in duc- tility. According to [START_REF] Kishore | Effect of dynamic strain ageing on the tensile properties of a modified 9cr1mo steel[END_REF], carburization obtained after normalization treatment (quenching and tempering), leads to a greater or lesser sensitivity to dynamic strain aging. All the rest of the identifications and comparisons with experimental curves are shown in ANNEX F.

Discussion

In sum, it has been shown that tempering has a no negligible effect on thermo-mechanical behaviour. Since the isotropic hardening depends on dislocations density, and the yield strength can be considered as the initial value of this hardening, the precipitates act both on the yield strength of the material and on its hardening, so the variation of properties with tempering was expected.

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016 σ y (MPa) On the contrary, the variation of the young modulus with tempering seems difficult to explain. Although the number of defects increases with tempering, the kinematic hardening does not vary significantly. This can be explained by the large number of dislocations already existing in the as received material.

x T =
Finally, we have been able to rewrite the Young modulus 'E', the yield strength 'σ y ' and

Transformation Induced Plasticity

TRansformation Induced Plasticity (TRIP), in its 'classical' definition, can be explained as the "significantly increased plasticity during a phase change. For an externally applied load for which the corresponding equivalent stress is small compared to the normal yield stress of the material, plastic deformation occurs". This definition can be found in the review work by Fischer [START_REF] Fischer | A new view on transformation induced plasticity (trip)[END_REF].

Explicit models can represent the behaviour of the different phases by seeking a constitutive model only at the macroscopic scale. Thus, Inoue et al. [START_REF] Inoue | Coupling between stress temperature and metallic structures during processes involving phases transformations[END_REF], Hamata et al. [START_REF] Hamata | A model for nodular graphite cast iron coupling anisothermal elasto-viscoplasticity and phase transformation[END_REF] and Aliaga et al. [START_REF] Inoue | Coupling between stress temperature and metallic structures during processes involving phases transformations[END_REF] assume only internal variables on the macroscopic scale and infer the ratios of the different phases from an energy-based mixing law. The behaviour can be plastic or viscoplastic and TRIP is viewed as an additional dissipative term.

Ėt = Ė + Ė pt E = ε i ∀i (4.1)
In this model, classical plasticity and transformation plasticity are assumed to be uncoupled, which is true for small strains. Thus, the homogenization law for stresses is:

∑ = n ∑ i=1 y i σ i (4.2)
This modeling scheme provides great flexibility in the calculation. Arbitrary constitutive laws, as well as different models of transformation plasticity rates, can be selected for each phase.

Mechanisms related to the TRIP

TRIP can be defined as the anomalous plastic strain observed when metallurgical transformation occurs under an external stress much lower than the yield limit. From a technological point of view it plays an essential role in many problems, in particular for the understanding of residual stresses resulting from welding operations and for their prediction in practical cases. From an experimental point of view, it is usually characterized thanks to a TRIP test (called also sometimes creep test or dilatometric test FIG. 4.1) where a constant stress is applied during transformation under prescribed cooling conditions as presented by Taleb et al. [START_REF] Taleb | A micromechanical modeling of the greenwood-johnson mechanism in transformation induced plasticity[END_REF]. During these tests, the applying load leads to irreversible deformation even for loading levels below the limit of elasticity of the material at the considered temperatures. The first to illustrate this phenomenon was Porter in late fifties [START_REF] Porter | Effect of applied tensile stress on phase transformations in steel[END_REF]. Desalos [START_REF] Desalos | Comportement dilatométrique et mécanique de l'austénite métastable d'un acier a533[END_REF] showed an effect of anisotropy of the material. He rode that for the specimens taken transversely to the rolling direction, the transformation plasticity deformation is higher than in the case of specimens taken in the rolling direction. Even for free dilatometry test(without any external load), the residual strains are not zero and depend on the direction collection of specimen. Desalos attributed these differences to the presence of internal stresses within the material due to delayed changes in segregated zones oriented along the rolling direction.

From a microstructural point of view, two mechanisms are usually considered to explain TRIP, the Magee [START_REF] Magee | Transformation Kinetics: Microplasticity and Ageing of Martensite in Fe-3l-Ni[END_REF] mechanism corresponds to the formation of selected martensitic variants resulting from the applied stress, the Greenwood-Johnson [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF] mechanism corresponds to the micromechanical plastic strain arising in the parent phase from the expansion of the new phase.

In fact, the martensite grows as platelets, oriented randomly in the absence of external load which leads to a global and isotropic volume change. In the case of a non-zero applied stress, the platelets are developed along a preferred direction (FIG. 4.2). The Greewood-Johnson macha- Finally, TRIP depends on the orientation of the platelets, on the local stress resulting from the internal and external stresses and on the difference of compactness between phases (free deformation). The platelets orientations play an crucial role in the beginning of the transformation where the internal stresses are still low and the accommodation of transformation strains is essentially elastic.

Modelling of TRIP phenomenon

In most cases as well as in the case of X10CrMoVNb9-1 steel welding, Magee mechanism seems not to be significant. Most existing models for TRIP are based on the Greenwood-Johnson mechanism. Moreover, Magee developed a model based only on platelets orientations, that is practically Many models have been proposed to describe the transformation plasticity. Detailed descriptions of these models are available in [START_REF] Denis | Modélisation des interactions contrainte-transformation de phase et calcul par éléments finis de la genèse des contraintes internes au cours de la trempe des aciers[END_REF] or [START_REF] Petit-Grostabussiat | Conséquences mécaniques des transformations structurales dans les alliages ferreux[END_REF]. Various authors ( [START_REF] Greenwood | The deformation of metals under small stresses during phase transformations[END_REF][START_REF] Aeby-Gautier | Transformations perlitique et martensitique sous contrainte de traction dans les aciers[END_REF][START_REF] Desalos | Comportement dilatométrique et mécanique de l'austénite métastable d'un acier a533[END_REF][START_REF] Simon | Effet des sollicitations thermomécaniques sur les transformations de phases dans l'état solide. aspects métallurgique et mécanique[END_REF][START_REF] Cherkaoui | Micromechanical modeling of martensitic transformation induced plasticity (trip) in austenitic single crystals[END_REF][START_REF] Fischer | A new view on transformation induced plasticity (trip)[END_REF][START_REF] Nagayama | Transformation induced plasticity in maraging steel: An experimental study[END_REF][START_REF] Coret | Experimental study of the phase transformation plasticity of 16mnd5 low carbon steel induced by proportional and nonproportional biaxial loading paths[END_REF][START_REF] Valance | Strain simulation of steel during a heatingcooling cycle including solid-solid phase change[END_REF][START_REF] Roux | Prévision des contraintes résiduelles induites par le soudage TIG d'un acier martensitique (X10CrMoVNb9-1)[END_REF]) observed a linear relationship between the level of the applied load and deformation of transformation plasticity and used a Greenwood-Johnson formulation to model it.

The Greenwood and Johnson model was developed in 1965 to model TRIP effect in case of constant uni-axial load σ. The TRIP expression was written as follows:

ε pt = Kσ (4.3)
Where K = 5 6σ y min ∆V V , ∆V V is the volume difference related to the compactness difference of the two phases and σ y min is the yield strength of the softer phase. Mitter [START_REF] Mitter | Umwandlungsplastizität und ihre berücksichtigung bei der berechnung von eigenspannungen[END_REF] modified the Greenwood-Johnson model to obtain the following expression:

ε pt = ∆V V α σ y min + β σ y max σ (4.4)
Where σ y max is the yield strength of the hardest phase, α and β are coefficients depending on the order of the metallurgical transformation. Another expression given by Abrassart [START_REF] Abrassart | Influence des transformations martensitiques sur les propriétés mécaniques des alliages du système Fe-Ni-Cr-C[END_REF] does not always satisfy the experimental results:

ε pt = K f (y) σ (4.5)
Where

K = 1 4σ y min ∆V V , f (y) = 3y -2y 3 2
, with y is the fraction of new phase formed during cooling. We should point out that at the end of the transformation f (y) = 1. Desalos [START_REF] Desalos | Comportement dilatométrique et mécanique de l'austénite métastable d'un acier a533[END_REF] modified Abrassart's expression to apply it to a similar case of an isothermal and an-isothermal bainitic transformation and found K = 10 -4 and f (y) = y (2y).

In case of multiaxial load, Giusti [START_REF] Giusti | Contraintes et déformations résiduelles d'origine thermique. Application au soudage et à la trempe des aciers[END_REF] developed the following expression which was rewritten by Leblond. This model was obtained from a micromechanical analysis based on the determination of the plastic strain induced in a spherical parent phase by the growth of a spherical product A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016

phase core. This model is very crude:

εpt i j = 3 2 K S i j f (y) ẏ (4.6)
Where K is a material parameter, f (y) the derivation of a normalized function, y the fraction of the new phase and S i j is the deviatoric component of the stress tensor. Hamata [START_REF] Hamata | Modélisation du couplage entre l'élasto-viscoplasticité anisotherme et la transformation de phase d'une fonte GS ferritique[END_REF] was the first to express the TRIP as function of S -X:

εpt = 3 2   J 2 S -X λ   N S J 2 S -X ẏ (4.7)
We should note that all these models must be identified for all possible temperatures and phase ratios.

Identification of TRIP models

Local/Global homogenization method

Another way to deal with multi-phasic behaviour problem is the use of a homogenization model. Homogenization is the art of relating the physical properties of different phases of a material to the physical properties of the equivalent homogeneous material. A good homogenization response to the following criterion: the overall macroscopic response of the real material and the equivalent homogeneous material to any thermo-mechanical loading should be the same. A summary of homogenization theories and techniques can be found in [START_REF] Schmitt | Homogénéisation des matériaux hétérogènes. revue de quelques méthodes pour les milieux élastiques à phases parfaitement dispersées[END_REF] and [START_REF] Huynh | Estimation des propriétés poromécaniques effectives des argilites: apport des méthodes d'homogénéisation[END_REF].

In this section, we will only use a self-consistent scheme ( [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion and related problems[END_REF][START_REF] Kröner | Zur plastischen verformung des vielkristalls[END_REF][START_REF] Budiansky | Theoretical prediction of plastic strains of polycrystals[END_REF]), the as-known 'β -model. In fact, this is an unique way to bypass the difficulties of modelling the elastoviscoplasticity behaviour of a multi-phasic material.

Homogenization background

The homogenization of multiphase materials is a key point in our study. It is used to estimate the macroscopic behaviour of a heterogeneous material from the properties of the different phases. The idea is to use the thermo-mechanical behaviour for each phase as modeled in the previous chapter and assemble the different phase behaviours using rule for the change scales. Doing so we should at any time retrieve the redistribution of local stresses related to the heterogeneity of the plastic deformation. The concept of the scales change is presented in FIG. 4.6. Classically, in any method of homogenization, we define three stages: representation, localization and homogenization. Representation is a crucial preliminary step: it defines the characteristic size of heterogeneity and specifies the scale of work and the parameters describing the microstructure. In our case, we can consider working at the grain scale. But for reasonable reasons, we will not be considering all the crystallographic orientations of each phase, which would lead to a very large number of A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016 variables (several hundred). We chose to represent preferentially the contrast between phases, that will lead us to consider only two phases, martensite and austenite.

Localization allows the inclusion of heterogeneities that generate differences in behaviour to macroscopic and microscopic scales. Localization consists in determining the local load fields σ i , ε i , ε P i from mechanic loading conditions Σ, E applied to the Representative Volume Element (RVE).

The β -scale transition rule

For scale change in sense of homogenization, the self-consistent approach is an efficient tool to derive the global mechanical behaviour of phases. The self-consistent scheme allows the determination of the overall stress Σ and strain E from stresses σ i and strains ε i fields of each phase i. A rigourous treatment of the self-consistent scheme to describe the elastoplastic behaviours of polycrystals has been initially proposed by Hill [START_REF] Hill | Continuum micro-mechanics of elastoplastic polycrystals[END_REF]. Berveiller and Zaoui [START_REF] Berveiller | An extension of the self-consistent scheme to plasticallyflowing polycrystals[END_REF] deduced an explicit transition rule from the previous approach for the case of a monotonic loading path:

σ i = Σ + 2µ (1 -β) α E P -ε P i with 1 α ≈ 3 2 µ 2 3 E P : E P J 2 Σ and β = 2 15 4 -5ν 1 -ν (4.8)
Where µ is Coulomb's modulus, J 2 Σ and E P are, respectively, the overall equivalent stress and the plastic part of overall strain in uni-axial tension test. In this formulation β is close to 1 2 . For α = 2 (respectively = 1), we get the so-called Lin model (respectively Kröner model) very close to Taylor model where the strains are assumed equal in each phase. The idea behind all these approaches is to introduce a corrective term depending on plastic strains, to compute local residual stresses. Nevertheless, a linear dependency of this term with respect to plastic strains gives too large stresses. The non-linear accommodation can be obtained by means of a phenomenological approach, where the Kröner solution with the difference between the local and global plastic strains is replaced by the difference between local and global non-linear hardening

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016

variables. The concept of β-rule, proposed by Cailletaud and Pilvin [START_REF] Cailletaud | Utilisation de modèles polycristallins pour le calcul par éléments finis[END_REF], introduced a variable β i and its volume average β presents a non-linear evolution with respect to plastic strain. When applied to material presenting uniform isotropic elasticity, the expression of the local stress is:

σ i = Σ + µ β -β i with β = 2 ∑ i=1 y i β i (4.9)
Where β i is a local non-linear kinematic variable. This variable tends to reduce internal stresses and to give a suitable description of cyclic test in elastoviscoplastic framework. Its evolution law is given by:

βi = εP i -D i β i -δ i ε P i 2 3 εP i : εP i (4.10)
Where D i and δ i are calibration parameters to be identified for each phase i.

Identification and first application

As for TRIP models, the β-scale transition rule model was identified on a series of dilatometry tests at various applied loads. TAB.

Numerical simulation

This section is dedicated to the numerical simulation of different welding operations using the TMM model previously discussed. The aim of these simulations is to verify the prediction capacities of the model in terms of residual stresses and distortions. Finite element simulations of the thermal and mechanical behaviour during welding dates back to the 1970s ( [START_REF] Ueda | Analysis of thermal elastic-plastic stress and strain during welding by finite element method[END_REF][START_REF] Hibbitt | A numerical thermo-mechanical model for the welding and subsequent loading of a fabricated structure[END_REF][START_REF] Friedman | Thermomechanical analysis of the welding process using the finite element method[END_REF][START_REF] Rybicki | A finite-element model for residual stresses and deflections in girth-butt welded pipes[END_REF][START_REF] Andersson | Thermal stresses in a submerged-arc welded joint considering phase transformations[END_REF]). The need to accommodate the effect of the evolving microstructure in the simulations was early recognised. Computational Welding Mechanics is therefore concerned with the thermomechanical response as well as the changes in material properties due to the thermal cycles in welding.

Herein, three different welding cases are simulated, The first simulation concerns the Satoh test which is frequently used to evaluate the effect of phase transformation on the generation of thermal stress. This simulation allowed us to compare the two different approaches used to model the multiphasic behaviour. Disk spot welding is simulated because of its simplicity and its relatively short required computation time. Resulting residual stresses are compared to the experimentally measured ones.

Finally, a narrow gap multi-pass welding operation is simulated and results are compared to the experimental ones.

Satoh test

As seen before, welding yields thermal, metallurgical and mechanical phenomena and induces non-uniform strain and stress fields. To quantify these phenomena, the simple so-called 'Satoh' test can be used. Uni-axial tests of the 'Satoh' type were performed for the first time, as implied by its name, by Kunihiko Satoh nearly 40 years ago ( [START_REF] Satoh | Transient thermal stresses of weld heat-affected zone by both-ends-fixed bar analogy[END_REF][START_REF] Satoh | Thermal stresses developed in high-strength steels subjected to thermal cycles simulating weld heat-affected zone[END_REF]). Improvements were then brought in the 90 t h ( [START_REF] Itoh | A study of cyclic thermal straining in a welded joint using a closed-loop servo-controlled testing machine[END_REF][START_REF] Dupas | Comparaison des modèles de comportement mécanique tenant compte des effets métallurgiques implantés dans le code aster et sysweld[END_REF]) where these type of tests were developed. Then many authors ( [START_REF] Dong | Welding residual stresses and effects on fracture in pressure vessel and piping components: A millennium review and beyond[END_REF][START_REF] Börjesson | Simulation of multipass welding with simultaneous computation of material properties[END_REF][START_REF] Long | Numerical simulation of residual stresses in a spot welded joint[END_REF][START_REF] Depradeux | Numerical simulations of thermomechanical phenomena during tig welding and experimental validation on analytical tests of increasing complexity[END_REF][START_REF] Adib | Three-dimensional finite element analysis of tensileshear spot-welded joints in tensile and compressive loading conditions[END_REF][START_REF] Mochizuki | Fundamental study of thermal stress generation during welding heat cycles. studies on numerical simulation of temperature microstructure and thermal stress histories during welding and their application to welded structures[END_REF][START_REF] Vincent | Thermo-mechanical consequences of phase transformations in the heat-affected zone using a cyclic uniaxial test[END_REF][START_REF] Lindgren | Numerical modelling of welding[END_REF][START_REF] Mochizuki | Strategy of considering microstructure effect on weld residual stress analysis[END_REF][START_REF] Dai | Prediction of residual stress distributions for single weld beads deposited on to sa508 steel including phase transformation effects[END_REF]) used these tests to compare the experimental results to the numerical simulations.

In our case the Satoh test has been simulated in order to show the mixture rules and to compare the two different approaches described earlier in this chapter. Carrying out tests of the Satoh type required the design of a specific experimental device. The geometry and dimensions of specimens should be defined such that the temperature field is homogeneous over the whole measured zone. This zone should be compatible with the heating and cooling systems and present a sufficient volume to represent correctuly the macroscopic behaviour.

The experience have been carried out by Roux ([3]). The test specimens were heated by induction. The temperature measurements were performed using micro-welded thermocouples on the external surface of the test piece. Several thermocouples were used to ensure that the temperature in the useful zone was uniform along the axis. The test specimen is restrained between two rigid In fact, since martensite has bigger volume and higher yield strength, the tensile stress is reduced due to the expansion caused by the martensite that results in a compressive stress.

The two modelling approaches have been simulated and results are reported on the same graph. The two approaches lead to similar prediction capacities with an overestimation of stress at the end of the modelling heating cycle. To be able to discriminate the two approaches, the test must include a tempering heat treatment. This heat treatment is automatically implemented in the β -scale transition rule method.

Disk spot test

Due to its experimental simplicity the 'simple Disk-Spot welding' is used to validate the numerical models developed in this study. This test is derived from the INZAT test implemented for the first time at INSA Lyon in 1998 ( [START_REF] Cavallo | Contribution à la validation expérimentale de modèles décrivant la Z.A.T. lors d'une opération de soudage[END_REF]) and used in several other studies ([180, 143, 158] on 16MND5 steel or [START_REF] Depradeux | Simulation numérique du soudage -acier 316L : validation sur cas tests de complexité croissante[END_REF] on 316L steel).

The heater used in these studies is the laser source (CO2) or GTA torch ( [START_REF] Depradeux | Simulation numérique du soudage -acier 316L : validation sur cas tests de complexité croissante[END_REF]). The principe of the experimental procedure is very simple. It consists in a thermal loading at the center of a disc, corresponding to either a single heating cycle with a holding time of a given duration or a more complex loading such as several cycles with a decrease of holding times [START_REF] Depradeux | Simulation numérique du soudage -acier 316L : validation sur cas tests de complexité croissante[END_REF] in order to understand the consequences of multi-pass welding. The cooling in this case is achieved by natural convection. During the test period, it is possible to measure changes over time in temperature (thanks to thermocouple, pyrometer, infrared camera) and displacement (thanks to displacement sensors or image correlation with camera operating in the visible spectrum ( [START_REF] Petit-Grostabussiat | Conséquences mécaniques des transformations structurales dans les alliages ferreux[END_REF][START_REF] Depradeux | Simulation numérique du soudage -acier 316L : validation sur cas tests de complexité croissante[END_REF]).

In our case, we have chosen to use as a heat GTA source to be representative of the studied process. In order to understand the effects of multi-pass welding, the discs were subjected to two or three successive heating, cooled to room temperature after each heating. The power supply is the same for each heating. The welding tests were realized during Roux' thesis [START_REF] Roux | Prévision des contraintes résiduelles induites par le soudage TIG d'un acier martensitique (X10CrMoVNb9-1)[END_REF] at the LTA (Laboratoire des Technologies d'Assemblage) of the CEA-Saclay (DEN, DANS, DM2S, SEMT, LTA) with the help of Mr. O. Blanchot. The experimental device was mounted on a modular table allowing the adaption of the mounting to any test specimens geometry (See FIG. 4.13. The welding machine is a Polysoude c machine coupled to an orbital torch (rotational movement of the torch is possible) all mounted on a motorized slide (horizontal translational movement is possible) and controlled by a computer. The positioning of the disk is provided by three ceramic pins disposed at an angle of 120 • C around the disc (See FIG. 4.14). All the welding parameters (Voltage, current, frequency, torch diameter, type of electrode...) are reported in Roux' thesis [START_REF] Roux | Prévision des contraintes résiduelles induites par le soudage TIG d'un acier martensitique (X10CrMoVNb9-1)[END_REF]. In this thesis one can also found a full description of this assembly. Eight displacement sensors were used to measure the displacement of different points and in different directions as shown in FIG. 4.15. Temperature measurements were done using thermocouples whose positions were optimized after a sensitivity analysis. FIG. 4.16 to 4.19 show simultaneously a numerical simulation of the temperature fields, the metallurgical composition, the Von-Mises stresses and the deflection obtained at the end of heating, respectively at the end of the cooling, using the TMM model presented in the previous chapters.

After welding, residual stresses measurements were performed. One way to categorize residual stresses is the scale over which they self-equilibrate. The macro-stresses are of type I because they vary continuously over large distances. This is in contrast to residual stresses which vary over the grain scale (type II or intergranular stresses) or at the atomic scale (type III).

In our case, two techniques were used: neutron diffraction (to get residual stresses measurements in depth) and X-rays diffraction (to get residual stresses measurements at the surface). A summary of measurements techniques can be found in [START_REF] Withers | Residual stress part 1 -measurement techniques[END_REF]. We can add to this summary the Ra- [START_REF] Constable | Raman microscopic studies of residual and applied stress in pvd hard ceramic coatings and correlation with x-ray diffraction (xrd) measurements[END_REF][START_REF] Tanaka | Measurement of residual stress in air plasma-sprayed y2o3zro2 thermal barrier coating system using micro-raman spectroscopy[END_REF]), the digital mixed-method numerical/experimental as the 'Bead Flush method' (see [START_REF] Lee | Utilization of thermo-elasto-plastic analysis of welding eigenstrain for improvement of the bead flush method[END_REF]) and various methods being developed through the development of tomography techniques.

Neutron diffraction measurements were performed at LLB (Laboratoire Léon Brillouin) of the CEA-Saclay with the help of Mr. Klosek [START_REF] Klosek | Détermination des contraintes résiduelles par diffraction de neutrons au llb[END_REF]. The X-rays diffraction measurements were realized at the ENSAM of Angers by Mr. Fischer. ). It was found that both methods give similar results, due to the fact that during the test no tempering heat treatment was considered. In the following section related to multi-pass simulation test where tempering effect is naturally induced. Large differences between the two methods can be observed.

Predicted stresses seem to give stratifying results, qualitatively and quantitatively, specially for residual surface stresses. show the results of these calculations is term of Von-Mises predicted residual stresses. The differences between the two strategies are more than obvious: the predicted residual stresses are not the same in both simulations. This leads to conclude that neglecting tempering can induce to a very bad estimation of results especially for prediction of residual stresses. These results have to be compared to experimental ones in order to see their real prediction activities.

Discussion

To gain a quantitative understanding of the way phase transformation changes thermal stress histories and residual stresses through application of the proposed technique for numerical simulations of temperature, microstructure, and thermal stress histories during welding, simple to more complex simulated welding tests were realized and were compared to experimental results. Thermal elasto-viscoplastic numerical simulations were used with consideration of the effects of phase ratios that change over time depending on the temperature and cooling rate, phase transformation properties, and the corresponding changes in the mechanical properties of constituent phases, and a fundamental investigation is performed to determine the way in which differences in the temperature histories and material properties affect the thermal stress histories and final residual stresses. The results obtained may be summarized as follows:

• The results obtained during numerical simulations of the temperature histories and thermal stress histories show good qualitative agreement with the experimental results.

• The thermal stress histories during welding heat cycles, self-evidently including final residual stresses, are shown to be strongly susceptible to the effects of the cooling rate and phase transformation properties such as martensite transformation.

• The relationship between the material properties, such as the phase transformation behaviour, the maximum residual stress, and the maximum heating temperature at different cooling rates, is quantitatively clarified.

• The temperature, microstructure, and thermal stress histories during welding as well as the residual stress can be determined in detail by means of numerical simulations considering phase transformation.

• The β-scale transition rule seems to be more realistic than the TRIP homogenization method especially when dealing with multi-pass operations of heat treatments that includes material tempering.

Finally, more improvements are yet to come, especially concerning a better description of the liquid/solid transformation and molten zone geometry prediction. Future improvements could come from the δ ferrite phase modelling and the filler material characterization. Because without any doubt, these elements have significant influence on the induced residual stresses.

Conclusion and perspectives

Martensitic stainless steels such as X10CrMoVNb9-1 may be used to manufacture many components, pressure vessels and piping. Many of these components are assembled by welding and more particular by multipass GTA Welding when these components are thick. The effect of phase transformations on the mechanical behaviour of the material during welding is particularly important in these high-strength steels.

Welding process induces residual stresses and/or distortions which may contribute to different failure mechanisms (brittle fracture, fatigue, buckling deformation, and stress-corrosion cracking) and hence influence the service life of welded structures.

Numerical simulation of welding is a very complex problem which involves a large number of physical phenomena (thermal, mechanical, metallurgical transformations). Coupling these phenomena is a great scientific challenge but is essential to predict the final state of the material (residual stresses, microstructure, defects, etc....) after the multipass welding process.

The first chapter of this dissertation is devoted to the definition of the thermal and mechanical domains observed in the HAZ and the development of models which are able to describe what really happens in these zones. The results of a first series of numerical simulations were used to define the tests performed to identify the models discussed in the following chapters as well as the tests performed to validate these models.

The main scientific contribution of this work consists in the description, improvement and validation of the models used to describe the phase transformations the material undergoes during the multipass welding process.

Whereas the simple Koistinen-Marburger was employed to model martensite transformation neglecting both the influence of the austenite grain size and the influence of cooling rate with however satisfying results, a big effort was made in modeling tempering and austenitization.

For austenitization, three different approaches were studied, identified and validated. Leblond type model requires small implementation and computation efforts but is unable to give good responses to complex anisothermal loadings.

On the other hand JMA and Brachet type model require more implementation and computation efforts but exhibit excellent prediction capabilities even for complex realistic thermal loadings. These two models were then improved to take into account the thermal history of the material and its more or less tempered state.

Tempering was described by the means of a simple differential model inspired from standard diffusion models. The results of the identification have shown that this model is coherent with chromium bulk diffusion and carbides growth-coalescence. The model was also used to predict hardness as a simple post-processor of thermal analyses of the welding process. The results are very satisfying when compared to experimental hardness profiles.

The austenite to δ-ferrite transformation was not discussed in this work but seems to have negligible effect on the overall results. The solidification process was not included in this work although its proficiency is required to describe the geometrical changes.

Another big effort in this study is related to the identification of the mechanical behaviour of each metallurgical phase. A large thermo-mechanical experimental campaign was performed and led to the characterization of thermo-mechanical behaviour of each of the considered solid phases (base-metal-fully tempered martensite, austenite, quenched martensite and more and less tempered martensite) throughout a large temperature range.

Since the results show that tempering has no negligible effect on thermo-mechanical behaviour, the Young modulus 'E', the yield strength 'σ y ' and isotropic hardening parameter 'R ∞ ' were rewritten as functions of the tempered rate 'x T '.

It would have been interesting to study the influence of austenitic grain size and hardening of austenite on the thermal-mechanical behaviour of the martensite. Besides it would have been interesting to perform a sensitivity analysis on the influence of the constitutive equations on the results of the multipass welding simulation.

In the final chapter, two approaches were confronted to see which one is the best to describe the multiphasic behaviour of the material. The first modelling taking into account the TRIP effect as well as the second modelling based on a local/global approach 'the β-scale transition rule' gave satisfying results in terms of prediction of residual stresses.

Finally, several numerical simulations describing simple and more complex welding were realized and were compared to experimental results. These numerical simulations used the improved Thermo-Metallurgical-Mechanical modelling based on a thermal elasto-viscoplastic model considering the effects of constituent phase percentages that change over time depending on the temperature and cooling rate, the phase transformation properties, and the corresponding changes in the mechanical properties of constituent phases, and a fundamental investigation is performed to determine the way in which differences in the temperature histories and material properties change the thermal stress histories and the final residual stresses.

The natural extension of this work is the modelling of specific phenomena related to the deposition of material and the improvement of numerical implementations in terms of computational time to be able one day to simulate a complete 3D multipass welding operation.

Parameters and welding conditions of the preliminary test case are briefly detailed in this annex.

• Welding torch perpendicular to the plate;

• Protective gas (70% He, 30% Ar), flow (20 l/min);

• Shielding gas nozzle (inner diameter 19 mm);

• Tungsten electrode (diameter 3.2 mm);

• Pre-gas time (15 s.);

• Direct current;

• Intensity (230 A) and voltage (12 V);

• Welding speed (9 cm/min.);

• Filler wire in front of the weld pool and inclined 40 • to the horizontal;

• Wire feed speed of the filler wire (77 cm/min.);

• Postgaz time (20 s.);

• Number of passes per layer (1).

The clamping of the plate is provided by the preheating block. The welding experiments are carried out in flat welding position. The plate is fixe; the displacement of the welding torch is carried by the longitudinal chassis. As far as filling in the groove, the output of the tungsten electrode respective to the gas shield nozzle is adjusted:

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016

Sensitivity analysis of the heat exchange convection/radiation coefficient An analysis on influence of the heat convection/radiation coefficient on the temperature field was carried out. Numerical computations of the first two passes were realized using a heat convection/radiation exchange coefficient that varies with temperature according to the curve h = h (T) given in FIG. We can also note that the temperature curves for different virtual thermocouples are not the same in case of h = 20 and 252 W/(m 2 K). The max calculated temperature is higher in case of lower h. In summary, the heat convection coefficient h has non negligible influence on temperature fields; an increase by a factor of ten of h carries more than 20% of variations on the temperatures fields.

C.4 Goldak heat source: (numerical model)

Goldak [START_REF] Goldak | Coupling heat transfer microstructure evolution and thermal stress analysis in weld mechanics[END_REF] suggests to replace the term of the heat source of the heat equation by a volumetric heat source. This volumetric source of heat is distributed in two quarts of half-ellipsoid, in front and rearward the torch. The thermal power density provided by welding process at the front of the torch is defined by the following mathematical relationship:

P f (x, y, z) = f f 6 √ 3 Q m a f × b × c × π 2 e -3(x S /a f ) 2 e -3(y S /b) 2 e -3(z S /c) 2 (C.4)
And that provided at the rear of the torch:

P r (x, y, z) = f r 6 √ 3 Q m a r × b × c × π 3/2 e -3(x S /a r ) 2 e -3(y S /b) 2 e -3(z S /c) 2 (C.5)
Where: a f is the distance along 'x' axis of the half ellipsoid at the front of the torch. a r is the distance along 'x' axis of the half ellipsoid at the rear of the torch. a = a f + a r is the dimension of the axis of the ellipsoid along 'x' axis. b is the dimension of the axis of the ellipsoid along 'y' axis. c is the dimension of the axis of the ellipsoid along 'z' axis. f f is the percentage of the total energy applied to the front of the torch. f r is the percentage of the total energy applied to the rear of the torch. 'r' is defined as the ratio of a r over a f . In this model, we have five independent parameters: a, b, c, r and Q m The usefulness of this source is it capacity to separate the front from In order to determine the influence of austenite grain size on the thermo-mechanical behaviour of the X10CrMoVNb9-1 steel, modelling of the growth kinetic of austenite grain size has been achieved. Approaches and results are detailed in this annex.

D.1 Modelling of austenite grain growth kinetic

In steels, during a thermal cycle of welding, the austenite grain size varies. This can have a significant effect on the metallurgical structure of the weld and its characteristics. Indeed, the grain size of austenite is likely to shift the CCT diagram to longer times. Thus, more grains of austenite are larger and less hardener is the weld and higher is the risk of cold cracking [2], hence the importance of taking into account the austenite grain size in the metal models.

In this section, we present the grain growth models used in the literature and we also show the influence of austenite grain size on the thermo-mechanical behaviour.

D.1.1 Influence of austenite grain size

Polycrystalline materials are almost all mechanical properties highly dependent on austenite grain size. It has been observed that the hardness decreases with the growth of the austenite grain. It is well known that the apparent yield strength σ y in tension of polycrystalline metals tends to increase with decreasing grain size [START_REF] Jaoul | Étude de la plasticité et application aux métaux[END_REF]. This relationship is well known as the Hall-Petch law ( [189] [190]).

σ y = σ 0 + K HP d -1/2 g (D.1)
Where d is the average grains diameter, σ 0 is the monocrystal yield strength and K HP is the Hall-Petch constant. This law is relatively realistic for austenitic grain sizes ranging from few nanometers to several hundred micrometers. The 1/2 value of the exponent is sometimes subject to discussion (See [START_REF] Sevillano | Large strain work hardening and textures[END_REF]). The distribution of grain size also plays an important role on mechanics [START_REF] Berbenni | Micromacro modelling of the effects of the grain size distribution on the plastic flow stress of heterogeneous materials[END_REF].

The extrapolation for plastic domains requires some modifications to the original law [START_REF] Hansen | Hall-petch relation and boundary strengthening[END_REF]. Grain size also influences the creep, reduction of the secondary creep rate is non-negligible ( [START_REF] Petry | Caractérisation et modélisation sur une grande plage de température du comportement et de l'endommagement d'alliages métalliques[END_REF], [START_REF] Frost | Deformation-Mechanism Maps: The Plasticity and Creep of Metals and Ceramics[END_REF]). The influence of grain size on the plasticity can be taken into account in the expression of the plastic strain rate, for example Petry [START_REF] Petry | Caractérisation et modélisation sur une grande plage de température du comportement et de l'endommagement d'alliages métalliques[END_REF] Where d 0 and γ are material parameters. At very high temperatures the strain hardening is negligible and the influence of grain size on creep mechanisms may be due either to an effect of the viscosity law or to damage mechanisms by grain boundary diffusion (See [START_REF] Vereecke | Une analyse probabiliste du comportement d'une famille d'aciers pour cuve de REP en cas d'accident grave[END_REF] and [START_REF] Petry | Caractérisation et modélisation sur une grande plage de température du comportement et de l'endommagement d'alliages métalliques[END_REF]).

Ṙ = b R ∞ k 1 + k 2 d g -R ṙ and σ y = σ 0 y k 1 + k 2 d g (D.3)
Where ṗ = f (σ)

K(d g ) n .
In the case of low carbon steels, it appears that the influence of grain size in tempered martensitic phase is very limited which is not the case for quenched martensite ( [START_REF] Swarr | The effect of structure on the deformation of as-quenched and tempered martensite in an fe-0.2%c alloy[END_REF] A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016 [START_REF] Bonadé | Plastic flow of martensitic model alloys[END_REF]). Where one sees clearly that the evolution of the hardening r depends on the size of present grains d g . To identify the parameters k 1 , k 2 et α, we set α at each temperature and then find k 1 et k 2 .

D.1.2 Grain growth model

The grain growth occurs in order to minimize the surface energy between grains. This growth can be delayed by the second phase particles or impurities in solution. The grain growth is generally homogeneous (with constant distribution of the shape and size of the grains), but it can happen that it is not ( [199] [200] as the case for some 9Cr1Mo grades [START_REF] Danon | Heterogeneous austenite grain growth in 9cr martensitic steels: influence of the heating rate and the austenization temperature[END_REF]). literature to describe the austenite grain growth (see the bibliography devoted to this phenomenon in the PhD thesis of Sestier [START_REF] Sestier | Evolution microstructurale et comportement mécanique à haute température de l'acier de cuve 16MnD5 en situation d'accident grave[END_REF]). Among these models:

D.1.2.

Beck model

Beck et al. [START_REF] Beck | Grain growth in high-purity aluminium and in an aluminium-magnesium alloy[END_REF] propose:

d g = k t n (D.4)
Where d g is the average grain size, K and n are material parameters that have a temperature dependences.

D.1.2.2 Hillert model

The Hillert model ( [START_REF] Hillert | On the theory of normal and abnormal grain growth[END_REF]) assumes a conservation of the total length of grains during growth (thus causing a loss of some grains to other). Assuming that growth is based on the boundary migration, we obtain:

dd g 2 dt = 1 2 α M γ (D.5)
Where M is a parameter characterizing the mobility of the grain boundary, a and g denotes the properties of interfaces. Integrating this equation for isothermal loading leads to the parabolic A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016 Where f (d g ) = d g 1-a with a an order factor.

D.1.2.6 Vereecke model

To take into account in an anisothermal model a saturation value of the grain size, Vereecke ([196]) proposed a phenomenological model of grain growth thermally activated described by the follow- It is this model which does not need to postulate an initial grain size that will be use this study. It requires the identification of four parameters b 0 , Q, d g 0 and W .

D.2 Identification

In this section we present the chosen modelization as well as the identification made. The heat treatments carried out were inspired Bracelet thesis [START_REF] Brachet | Alliages Martensitiques 9Cr-1Mo: effets de l'addition de l'azote du niobium et du vanadium sur la microstructure: les transformations de phases et les propriétés mécaniques[END_REF]. The estimation of austenitic average grain size diameter is very difficult due to the complexity of the martensitic structure and to the low reproducibility of specific metallographic states. The heat treatment applied are described in the above diagram. It involves heating the sample to a specific holding temperature, a maintain for a certain period of time and a first interrupted quenching at 630 • C with a holding time of 10h. This annealing allows to visualize the precipitation at grain boundaries, making them much more visible in optical microscope. Finally, a second quenching to room temperature is achieved. Note that the heating and cooling rates are set to 10 • C/min. Our samples were subjected to thermal treatment in an ATG " Thermo-Gravimetric Analyzer " under an neutral gas environment. These samples were then polished and chemically attacked in order to reveal the microstructure and to highlight the contours of grain during microscopic visualization. FIG.D.5 to D.6 show some resultant micrographics: Above a certain temperature T = 1275 • C , decarburization occurs and δ ferrite is formed, so grain size can't any longer be correctly estimated.

D.2.1 Heat treatments and resulting microstructures

D.2.2 Grain size estimation

The grain size can be characterized from the microscope photos by the mean of three conventional methods:

1. Counting method.

2. Assimilation method based on comparison with standard images.

3. Numerical method using image analyser codes.

In our study, we used both counting and assimilation methods.

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016 2. Assimilation method using comparison with standard images. The principle is simple, it involves comparing the photo to standard images which have a known grain sizes. These images allow us to obtain a conventional index G which can be a positive, zero or negative number. From this index, we can deduce m and then calculate the average grain size d g =

D.2.3 Identification of the grain growth model

The model that we decided to identify is the proposed by Vereecke [START_REF] Vereecke | Une analyse probabiliste du comportement d'une famille d'aciers pour cuve de REP en cas d'accident grave[END_REF] We recall that this model does not require knowledge of the initial grain size value. The four parameters that need to be identified are b 0 , Q, d g 0 and W . TAB.D.1 shows the grain size values found for the X10CrMoVNb9-1 for different holding times and temperatures. We assume that max grain size can be reached for holding times exceeding 10h which seems reasonable in light of similar work obtained in [START_REF] Vereecke | Une analyse probabiliste du comportement d'une famille d'aciers pour cuve de REP en cas d'accident grave[END_REF] and [START_REF] Sestier | Evolution microstructurale et comportement mécanique à haute température de l'acier de cuve 16MnD5 en situation d'accident grave[END_REF]. The max grain size for each temperature are reported in red in TAB.D.1. The indentification were done using optimisation methods and operators available T = 1000 • C T = 1100 in Matlab. Identifiaction were done in two steps: Knowing three max grain size values, so three known values for d g lim in equation (D.13), d g 0 and W were first identified using the least squares method to solve the system of three equations with two unknowns. We proceed in the same way to

D.3 Grain Size Control

As we seen before, grain size is a very important factor in considering the properties of polycrystalline materials because they affect the area and the length of the grain boundaries. Fine grained materials generally exhibit greater yield stresses than coarse grained materials at low temperatures whereas at high temperatures grain boundaries become weak and sliding occurs. The basic principle of grain size control is, In steel, grain refinement occurs only upon heating within or above the critical temperature range. Once an austenitic grain size has been established during heating, it cannot be refined during cooling, regardless of the cooling rate.It can be refined, however, by subsequent reheating.

In the case of pure metals and non-heat-treatable alloys, suitable grain refinement can be brought about by peening the weld, whilst still hot, and so bring about recrystallization. Vari- • Annealing time.

• Rate of heating. Slow heating will form few nuclei, favour grain growth and result in coarse grain.

• Degree of prior deformation. Heavy (prior) deformation favours the production of many nuclei on recrystallization and consequently the grain size will be small conversely the light deformation will give rise to few nuclei and result in large grain size.

• Insoluble impurities. The greater the amount and the finer the distribution of insoluble impurities the finer is the final grain size.

• Alloying elements such as nickel, limits grain growth during annealing and other heattreatment processes in steels and in some non-ferrous alloys.

All these factors are subject of several studies conducted in CEA and elsewhere.

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016 are the accelerating terms which are maximum in the temperature range where transformations take place according to the CCT diagram. The coefficients l i j (T ) are braking terms which play an important role for the temperatures where the transformation at equilibrium is incomplete. The coefficients h i j Ṫ are amplifying terms and are maximal in the range of important cooling rates.

Leblond emphasizes the influence of austenite grain size on the phase transformations and proposes a calculation of austenite grain growth, including the period of austenite.

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016 and identification results

In this annex we show all the comparison photos of the simulated and experimental curves. We illustrate first the results of identification on the austenitic phase mechanical test.

vi List of Figures 1 .

 1 32 Temperature field and corresponding generated stress field (Pa) at the end of the second welding pass (Point D) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.33 Temperature field and corresponding generated stress field (Pa) at the end of the third welding pass (Point D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.34 Temperature field and corresponding generated Von-Mises stress field (Pa) during the second welding pass (Point B) . . . . . . . . . . . . . . . . . . . . . . . . . 1.35 Temperature field and corresponding generated Von-Mises stress field (Pa) during the second welding pass (Point C) . . . . . . . . . . . . . . . . . . . . . . . . .

3. 1

 1 Representation of hardening in case of uniaxial loading (Test at 200 • C on a tempered material X T = 0.6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Test specimen layout [121] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Details of machining (a) and (b) . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Overview of the MTS test machine and the heating system . . . . . . . . . . . . 3.5 Rods of the extensometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Example of mechanical loading for the monotonic tensile test . . . . . . . . . . . 3.7 Example of mechanical loading for the cyclic test . . . . . . . . . . . . . . . . . 3.8 Austenitization heat treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Comparisons between experimental and identified curves for the austenitic phase 3.10 Evolution of the hyperbolic sinus viscous F and K parameters . . . . . . . . . . 3.11 Evolution of the isotropic hardening b and X ∞ parameters . . . . . . . . . . . . . 3.12 Evolution of the kinematic hardening C 1 and γ 1 parameters . . . . . . . . . . . . 3.13 Evolution of the kinematic hardening C 2 parameter . . . . . . . . . . . . . . . . 3.14 Evolution of the Young modulus with respect to T * and x T . . . . . . . . . . . . 3.15 Evolution of the yield strength with respect to T * and x T . . . . . . . . . . . . . 3.16 Evolution of isotropic hardening parameter R ∞ with respect to T * and x T . . . . . 3.17 Evolution of the thermo-mechanical response with tempering for the cyclic test at 400 • C for a respective tempering of X T = 0 and X T = 0.6 . . . . . . . . . . . . viii List of Figures

4. 1 F. 1

 11 Illustration of TRIP observed during dilatometric test under mechanical loading . 4.2 Illustration of Magee mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Determination of the compactness between the two phases . . . . . . . . . . . . 4.4 Dilatometric tests for various applied stresses . . . . . . . . . . . . . . . . . . . 4.5 Comparison between Hamata model, Desalos model and experimental results . . 4.6 Scheme presenting the localization/homogenization rule [158] . . . . . . . . . . 4.7 Dialtometry curves obtained using the β-model . . . . . . . . . . . . . . . . . . 4.8 Simulation of monotonic tensile test for various phases' proportions . . . . . . . 4.9 Simulation of monotonic tensile test for various proportions of quenched and fully tempered martensite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Schematic illustration of welding heat cycle test (Satoh test) . . . . . . . . . . . 4.11 Illustration of evolution of the imposed temperature for the Satoh test . . . . . . 4.12 Experimental stress evolution and comparison with both TRIP and β-scale transition rule simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13 Overall view of the Disk-Spot test . . . . . . . . . . . . . . . . . . . . . . . . . 4.14 View of the ceramic pins and displacement sensors locations . . . . . . . . . . . 4.15 Schematic showing the displacement sensors locations . . . . . . . . . . . . . . 4.16 Temperature fields at the end of heating, respectively at the end of cooling . . . . 4.17 Metallurgical state at the end of heating, respectively at the end of cooling . . . . 4.18 Von-Mises stress fields at the end of heating, respectively at the end of cooling . . 4.19 Deflection at the end of heating, respectively at the end of cooling . . . . . . . . 4.20 Predicted and measured radial stresses at the mid-section of the disk . . . . . . . 4.21 Predicted and measured radial stresses at the top surface . . . . . . . . . . . . . 4.22 Predicted and measured radial stresses at the bottom surface . . . . . . . . . . . 4.23 Predicted and measured tangential stresses at the mid-section of the disk . . . . . 4.24 Predicted and measured tangential stresses at the top surface . . . . . . . . . . . 4.25 Predicted and measured tangential stresses at the bottom surface . . . . . . . . . 4.26 Predicted and measured normal stresses at the mid-section of the disk . . . . . . 4.27 Comparison of the normal predicted stresses after deposing the first welding pass for the two different simulations, β model on the right . . . . . . . . . . . . . . . 4.28 Comparison of the normal predicted stresses after deposing the ninth welding pass for the two different simulations, β model on the right . . . . . . . . . . . . . . . 4.29 Comparison of the normal predicted residual stresses after deposing at the end of the welding operation for the two different simulations, β model on the right . . . 4.30 Comparison of the Von-Mises predicted stresses after deposing the first welding pass for the two different simulations, β model on the right . . . . . . . . . . . . 4.31 Comparison of the Von-Mises predicted stresses after deposing the ninth welding pass for the two different simulations, β model on the right . . . . . . . . . . . . List of Figures ix 4.32 Comparison of the Von-Mises predicted residual stresses after deposing at the end of the welding operation for the two different simulations, β model on the right . B.1 Heat exchange convection/radiation coefficient evolution . . . . . . . . . . . . . B.2 Fisrt pass, temperature field at t = 13 s and t = 50 s for h = 20 W/(m 2 K) . . . . . B.3 Fisrt pass, temperature field at t = 13 s and t = 50 s for h = 252 W/(m 2 K) . . . . B.4 Fisrt pass, temperature field at t = 13 s and t = 50 s for h = h(T) . . . . . . . . . . B.5 Second pass, temperature field at t = 505 s for h = 20 and 252 W/(m 2 K) . . . . . B.6 Second pass, temperature field at t = 585 s for h = 20 and 252 W/(m 2 K) . . . . . B.7 Calculated temperature evolutions of three virtual thermocouples for h = 20 and 252 W/(m 2 K) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.1 Double ellipsoid heat source configuration . . . . . . . . . . . . . . . . . . . . . D.1 Austenite grain size function of austenitizing conditions [1] Austenitized at 1050 • C with a heating rate of 10 • C/s (a) and 50 • C/s (b) . . . . . . . . . . . . . . . . . D.2 Example of Ashby and Easterling prediction diagram of grain size in HAZ [2] . . D.3 Austenitic heat treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.4 ThermoGravimetric Analysis (TGA) machine optical microscope equipped with a digital camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.5 Heat treatments at 1000 • C for corresponding holding times 20min and 10h respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.6 Heat treatments at 1200 • C (20 h) and 1300 • C (20 min) respectively . . . . . . . D.7 Grain size estimation using the counting method . . . . . . . . . . . . . . . . . . D.8 Grain growth evolution respect to holding time . . . . . . . . . . . . . . . . . . D.9 Maximum grain size function of holding temperature . . . . . . . . . . . . . . . Comparison between experimental and simulated curve during a monotonic tensile test on austenitic phase at 500 • C . . . . . . . . . . . . . . . . . . . . . . . . . . F.2 Comparison between experimental and simulated curve during a monotonic tensile test on austenitic phase at 600 • C . . . . . . . . . . . . . . . . . . . . . . . . . . F.3 Comparison between experimental and simulated curve during a monotonic tensile test on austenitic phase at 700 • C . . . . . . . . . . . . . . . . . . . . . . . . . . F.4 Comparison between experimental and simulated curve during a monotonic tensile test on austenitic phase at 800 • C . . . . . . . . . . . . . . . . . . . . . . . . . . F.5 Comparison between experimental and simulated curve during a monotonic tensile test on austenitic phase at 900 • C . . . . . . . . . . . . . . . . . . . . . . . . . . F.6 Comparison between experimental and simulated curve during a cyclic test on quenched martensite x T = 0 at 20 • C . . . . . . . . . . . . . . . . . . . . . . . . F.7 Comparison between experimental and simulated curve during a cyclic test on quenched martensite x T = 0 at 200 • C . . . . . . . . . . . . . . . . . . . . . . . F.8 Comparison between experimental and simulated curve during a cyclic test on quenched martensite x T = 0 at 300 • C . . . . . . . . . . . . . . . . . . . . . . . F.9 Comparison between experimental and simulated curve during a cyclic test on quenched martensite x T = 0 at 400 • C . . . . . . . . . . . . . . . . . . . . . . . F.10 Comparison between experimental and simulated curve during a cyclic test on quenched martensite x T = 0 at 500 • C . . . . . . . . . . . . . . . . . . . . . . . x List of Figures F.11 Comparison between experimental and simulated curve during a monotonic tensile test on quenched martensite x T = 0 at 200 • C . . . . . . . . . . . . . . . . . . . F.12 Comparison between experimental and simulated curve during a monotonic tensile test on quenched martensite x T = 0 at 500 • C . . . . . . . . . . . . . . . . . . . F.13 Comparison between experimental and simulated curve during a cyclic test on partially tempered martensite x T = 0.4 at 20 • C . . . . . . . . . . . . . . . . . . F.14 Comparison between experimental and simulated curve during a cyclic test on partially tempered martensite x T = 0.4 at 200 • C . . . . . . . . . . . . . . . . . F.15 Comparison between experimental and simulated curve during a cyclic test on partially tempered martensite x T = 0.4 at 300 • C . . . . . . . . . . . . . . . . . F.16 Comparison between experimental and simulated curve during a cyclic test on partially tempered martensite x T = 0.4 at 400 • C . . . . . . . . . . . . . . . . . F.17 Comparison between experimental and simulated curve during a cyclic test on partially tempered martensite x T = 0.4 at 500 • C . . . . . . . . . . . . . . . . . F.18 Comparison between experimental and simulated curve during a monotonic tensile test on partially tempered martensite x T = 0.4 at 200 • C . . . . . . . . . . . . . . F.19 Comparison between experimental and simulated curve during a monotonic tensile test on partially tempered martensite x T = 0.4 at 500 • C . . . . . . . . . . . . . . F.20 Comparison between experimental and simulated curve during a cyclic test on partially tempered martensite x T = 0.6 at 20 • C . . . . . . . . . . . . . . . . . . F.21 Comparison between experimental and simulated curve during a cyclic test on partially tempered martensite x T = 0.6 at 200 • C . . . . . . . . . . . . . . . . . F.22 Comparison between experimental and simulated curve during a cyclic test on partially tempered martensite x T = 0.6 at 300 • C . . . . . . . . . . . . . . . . . F.23 Comparison between experimental and simulated curve during a cyclic test on partially tempered martensite x T = 0.6 at 400 • C . . . . . . . . . . . . . . . . . F.24 Comparison between experimental and simulated curve during a cyclic test on partially tempered martensite x T = 0.6 at 500 • C . . . . . . . . . . . . . . . . . F.25 Comparison between experimental and simulated curve during a monotonic tensile test on partially tempered martensite x T = 0.6 at 200 • C . . . . . . . . . . . . . . F.26 Comparison between experimental and simulated curve during a monotonic tensile test on partially tempered martensite x T = 0.6 at 500 • C . . . . . . . . . . . . . . F.27 Comparison between experimental and simulated curve during a cyclic test on fully tempered martensite x T = 1 at 20 • C . . . . . . . . . . . . . . . . . . . . . F.28 Comparison between experimental and simulated curve during a cyclic test on fully tempered martensite x T = 1 at 200 • C . . . . . . . . . . . . . . . . . . . . F.29 Comparison between experimental and simulated curve during a cyclic test on fully tempered martensite x T = 1 at 300 • C . . . . . . . . . . . . . . . . . . . . F.30 Comparison between experimental and simulated curve during a cyclic test on fully tempered martensite x T = 1 at 400 • C . . . . . . . . . . . . . . . . . . . . F.31 Comparison between experimental and simulated curve during a cyclic test on fully tempered martensite x T = 1 at 500 • C . . . . . . . . . . . . . . . . . . . . F.32 Comparison between experimental and simulated curve during a monotonic tensile test on fully tempered martensite x T = 1 at 200

Contents 1 . 1 4 1. 1 . 1 4 1. 1 . 2 10 1. 1 . 3

 114114121013 The X10CrMoVNb9-1 steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The effect of the chemical composition . . . . . . . . . . . . . . . . . . . . . . Metallurgy of X10CrMoVNb9-1 . . . . . . . . . . . . . . . . . . . . . . . . . . Elaboration of X10CrMoVNb9-1 . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2 MUSICA test case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.1 Test case presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.2 Multipass GTA-Welding simulation . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2.3 Identification of thermal boundary conditions . . . . . . . . . . . . . . . . . . . 15 1.2.4 Optimization background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.2.5 Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.2.6 The thermo-mechanical behaviour . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016 1.1 The X10CrMoVNb9-1 steel

Figure 1 . 1 :

 11 Figure 1.1: Variation of conventional yield strength at 0.2% regard to the temperature for five different ferritic alloys [15]

Figure 1 . 3 :

 13 Figure 1.3: Fe-Cr-C phase diagrams[START_REF] Durand-Charre | La microstructure des aciers et des fontes: Genèse et interprétation[END_REF]

Figure 1 . 4 :

 14 Figure 1.4: Binary Fe-Cr phase diagram for C=0.1%wt[START_REF] Durand-Charre | La microstructure des aciers et des fontes: Genèse et interprétation[END_REF]

Figure 1 . 5 :Figure 1 . 6 :

 1516 Figure 1.5: Binary Fe-Mo phase diagram

Figure 1 . 7 :Figure 1 . 8 :

 1718 Figure 1.7: Binary Fe-Mn phase diagram

Figure 1 . 9 :

 19 Figure 1.9: Binary Fe-Si phase diagram

Figure 1 . 10 :

 110 Figure 1.10: Metastable Iron-Carbon equilibrium diagram

Figure 1 . 11 :

 111 Figure 1.11: Illustration of the multipartner MUSICA project

Figure 1 . 12 :

 112 Figure 1.12: Vertical section and top view of the plate

Figure 1 . 13 :

 113 Figure 1.13: View of the welding torch and the narrow groove

Figure 1 . 14 :

 114 Figure 1.14: Welding machine

Figure 1 . 15 :

 115 Figure 1.15: Experimental thermocouple temperatures for the first five passes

Figure 1 . 16 :Figure 1 . 17 :

 116117 Figure 1.16: Micrography cross section and equivalent mesh

Figure 1 . 18 :

 118 Figure 1.18: Evolution of the conductivity

Figure 1 . 19 :Figure 1 . 20 :

 119120 Figure 1.19: Evolution of the power flux density

Figure 1 . 21 :Figure 1 . 22 :

 121122 Figure 1.21: Evolution of the experimental measured temperatures during the first eleven passes

Figure 1 . 23 :

 123 Figure 1.23: Welding machine efficiency

Figure 1 . 24 :Figure 1 . 25 :Figure 1 . 26 :

 124125126 Figure 1.24: Evolution of the Young's modulus and the yield strength

Figure 1 . 27 :Figure 1 . 28 :Figure 1 . 29 :

 127128129 Figure 1.27: Hydrostatic stress (Pa) evolution during the seventh filling pass at point A and B

Figure 1 . 30 :Figure 1 . 31 :Figure 1 . 32 :

 130131132 Figure 1.30: Von-Mises stress (Pa) and displacement profiles at the end of the sixth pass (Point D)

Figure 1 . 33 :Figure 1 . 34 :Figure 1 . 35 :

 133134135 Figure 1.33: Temperature field and corresponding generated stress field (Pa) at the end of the third welding pass (Point D)

Contents 2 . 1

 21 Introduction to solid-solid phase transformation . . . . . . . . . . . . . . . . . . . . 30 2.1.1 General metallurgical models . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.1.2 Transformation at equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.1.3 Nonequilibrium Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.2 Martensite transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.3 Martensite tempering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.3.1 ThermoElectric Power and hardness measurements . . . . . . . . . . . . . . . . 46 2.3.2 Tempering modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.3.3 Hardness evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.3.4 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.4 Martensite dissolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.4.1 Pre-study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.4.2 Three different approaches to model austenitization . . . . . . . . . . . . . . . . 54 2.4.3 Parameters identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.4.4 Validation of the models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.4.5 Tempering influence on austenitization kinetics . . . . . . . . . . . . . . . . . . 64 2.4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 . 1 .

 21 As a result, the mechanical properties of HAZ metal are invariably different from and often worse than those of the parent plate. Temperature change and change in microstructures introduce volume changes in the area surrounding the weld and hence cause

Figure 2 . 1 :

 21 Figure 2.1: Schematic diagram for the various subzones of the heat affected area

2. 1 . 1 . 1

 111 Waeckel model Waeckel wanted to write a model easily identifiable from the CCT Continuous Cooling Transformation and TTT Time Temperature Transformation diagrams (FIG. 2.2 and FIG. 2.3) data and able of reproducing complex thermal histories such as that encountered during welding[START_REF] Waeckel | Modélisation du comportement thermo-métallurgique des aciers[END_REF]. This model has been implemented in the EDF FEM code 'Aster' and in the CEA FEM code Cast εM, as part of Martinez thesis work[START_REF] Martinez | Jonction 16MND5-Inconel 690-316LN par soudage-diffusion : élaboration et calcul des contraintes résiduelles de procédé[END_REF].

Figure 2 . 2 :

 22 Figure 2.2: X10CrMoVNb9-1 CCT diagram[START_REF] Duthilleul | Etude des évolutions microstructurales lors d'un traitement thermique rapide à haute température d'un acier martensitique de type 9cr1mo[END_REF]

Figure 2 . 3 :

 23 Figure 2.3: TTT or IT (Isothermal Transformation) diagram

Figure 2 . 4 :

 24 Figure 2.4: Schematic representation of the nitrogen concentration during austenite dissolution

Figure 2 . 5 :

 25 Figure 2.5: Schematic representation of the nitrogen concentration during the delta-ferrite decomposition

  2.6. From two face centered cubic austenitic lattices, one centeredA thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016

Figure 2 . 6 :Figure 2 . 7 :

 2627 Figure 2.6: Illustration of Bain transformation [65]

Figure 2 . 8 :Figure 2 . 9 :

 2829 Figure 2.8: Measured deformation during a dilatometry test

  min at 1020 • C under controlled atmosphere of argon (FIG. 2.10 a). The samples were then cooled in the oven to ambient temperature at a speed rate exceeding 600 • C/s enough to obtain a 100% homogeneous quenched martensite. Samples were then tempered at different temperatures and times tempering (FIG. 2.10 b), the all tempering heat treatments are resumed in TAB. 2.3.

Figure 2 . 10 :

 210 Figure 2.10: (a) Austenization heat treatment (b) Tempering heat treatment

  2.11). A Potential voltage difference ∆V induced by the Seebeck effect can be then measured. Blocks'

Figure 2 . 11 :

 211 Figure 2.11: TEP instrument illustration

Figure 2 . 12 :

 212 Figure 2.12: TEP values for different tempering heat treatments

Figure 2 . 13 :Figure 2 . 14 :

 213214 Figure 2.13: Micrography of different tempered samples (a) T = 500 • C, t = 60 s (b) T = 500 • C, t = 1 h (c) T = 750 • C, t = 5 h30

Figure 2 . 15 :

 215 Figure 2.15: Comparison between model and experiment after optimization Identification was done using Matlab c environment and the identified parameters are given in TAB. 2.4. D 0 ∆H n T T T h 2.4 10 13 278 kJ/mol 3.61 486 • CTable 2.4: Identified tempering model parameters

Figure 2 . 16 :

 216 Figure 2.16: CEA fully automatic hardness tester machine 'Duramin-A300'

Figure 2 . 17 :Table 2 . 5 :Figure 2 . 18 :

 21725218 Figure 2.17: Hardness measurements resultsto the one of the carbon trapped in the martensite matrix. The following phenomenological model relating to hardness to the tempering state x T was identified:HV T M = HV Q -(HV Q -HV AR ) exp x T -1 x 0 (2.50)Where HV T M denotes the Tempered Martensite Vickers Hardness, HV AR the As Received material (fully tempered) Vickers Hardness, HV Q the Quenched martensite Vickers Hardness and x 0 a

Figure 2 . 19 :

 219 Figure 2.19: Simulated and measured hardness profile across the seventh welding pass

Figure 2 . 20 :

 220 Figure 2.20: Simulated and measured hardness profile through the different welding passes

Figure 2 . 21 :Figure 2 . 22 :

 221222 Figure 2.21: Predicted temperature vs. time evolution at a point located in the HAZ

Figure 2 . 23 :

 223 Figure 2.23: As-received X10CrMoVNb9-1 tempered martensitic fine microstructure

Figure 2 . 24 :

 224 Figure 2.24: X10CrMoVNb9-1 dilatometry curves at different heating rates Ṫ within the range [0.1 • C/s, 100 • C/s]

Figure 2 . 25 :

 225 Figure 2.25: X10CrMoVNb9-1 Continuous Heating Transformation diagram

Assuming n = 1 ,

 1 Leblond et al. ([115,[START_REF] Leblond | Mathematical models of anisothermal phase transformations in steels and predicted plastic behaviour[END_REF]) derived from this equation a phenomenological model based on the following simple differential equation ẏγL = y eq (T )y γL τ(T ) (2.68)

Figure 2 . 26 :

 226 Figure 2.26: Standard strategy to derive austenite fraction evolution from a dilatometry curve

Figure 2 . 27 :

 227 Figure 2.27: Extrapolation of T eq (y γi = 0.9; Q ZD j = 50 kJ) from 0.1, 1 and 10 • C/s dilatometry curves A robust strategy was developed to identify from this database the material parameters for each above-described model, viz. (3 + 3 + 3) parameters for Brachet model, (1 + 5) or (1 + 2 + 4) parameters for JMA type model and (3 + 4) parameters for Leblond type model. The equilibrium transition curve derived from equation (2.73) -and plotted on FIG. 2.29 -was modelled

Figure 2 . 29 :

 229 Figure 2.29: Austenite fraction vs. temperature: experimental results vs. models predictions uncoupled incubation and (nucleation-)growth Brachet type model

Figure 2 . 30 :

 230 Figure 2.30: Austenite fraction vs. temperature: experimental results vs. models predictions coupled nucleation-and growth JMA type model without temperature dependence of the growth law

Figure 2 . 31 :Figure 2 . 32 :

 231232 Figure 2.31: Austenite fraction vs. temperature: experimental results vs. models predictions coupled nucleation-and growth JMA type model with temperature dependence of the growth law

Figure 2 . 33 :Figure 2 . 34 :

 233234 Figure 2.33: Complex dilatometry tests : isothermal loadings after heating at 10 • C/s up to 20 or 50% of austenite transformation, and variable heating rate, viz. shifting from 50 to 10 • C/s at 20% of austenite transformation

Figure 2 . 35 :Figure 2 . 36 :Figure 2 . 37 :

 235236237 Figure 2.35: Austenite fraction vs. time: experimental results vs. models predictions: isothermal loadings after heating at 10 • C/s up to 20 or 50% of austenite transformation

Figure 2 . 38 :Figure 2 . 39 :

 238239 Figure 2.38: Result of dilatometry tests on more or less tempered martensite

Figure 2 .

 2 Figure 2.40: A Sat parameter influence on Austenite Start Temperature A S

Figure 2 . 41 :

 241 Figure 2.41: A Sat vs tempering factor x t

Figure 2 . 42 :

 242 Figure 2.42: Brachet type model: Austenite fraction vs. temperature: experimental results vs. model predictions for x t = 0, 0.3, 0.6 and 1

Figure 2 .

 2 Figure 2.43: K JMA and m JMA function of x t : experimental results vs. models

Figure 2 . 44 :

 244 Figure 2.44: JMA type model: Austenite proportion vs. temperature: experimental results vs. model predictions for x t = 0, 0.3, 0.6 and 1

Contents 3 . 1

 31 Thermo-mechanical model of the material . . . . . . . . . . . . . . . . . . . . . . . 72 3.2 Mechanical tests presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.2.1 Test specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.2.2 Heating system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.2.3 Identification method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.3 Thermo-mechanical behaviour of the austenitic phase . . . . . . . . . . . . . . . . . 77 3.4 Thermo-mechanical behaviour of martensite . . . . . . . . . . . . . . . . . . . . . . 78 3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016

Figure 3 . 1 :

 31 Figure 3.1: Representation of hardening in case of uniaxial loading (Test at 200 • C on a tempered material X T = 0.6) variable R gives the elastic domain range and the kinematic hardening tensor X describes the elasticity field translation (FIG. 3.1). R is the change in yield stress induced by plastic deformation.It results from the short distance obstacles such as network friction forces and small precipitates sheared by dislocations. It is directly linked to the increase dislocation density. X describes usually the long distance interactions such as inter-or transgranular stresses induced by incompatibility of plastic deformation from one grain to another or caused by interactions between dislocations and precipitates.

  3.3 (a)). The test specimen were another time machined and another error was made during the machining creating two lines instead of points, in order to use this test specimens without reducing their diameter which increases considerably the buckling risk, another machining was applied on the other side of the specimens as shown in FIG.3.3 (b).

Figure 3 . 2 :

 32 Figure 3.2: Test specimen layout[START_REF] Vincent | Caractérisation et modélisation du comportement et de l'endommagement de deux alliages de cuivre sous chargements thermo-mécaniques sévères[END_REF]

  3.5), transmit mechanically the

Figure 3 . 3 :Figure 3 . 4 :

 3334 Figure 3.3: Details of machining (a) and (b)

Figure 3 . 5 :Figure 3 . 6 :

 3536 Figure 3.5: Rods of the extensometer

Figure 3 . 7 :

 37 Figure 3.7: Example of mechanical loading for the cyclic test

Figure 3 . 8 :

 38 Figure 3.8: Austenitization heat treatment

  hardening is essentially isotropic. For high temperatures 800 • C and 900 • C, the overall hardening is close to zero. When the temperature increases, the component R ∞ of the elastoplastic stress decreases, while its viscous component increases, with a global trend decreasing the overall stress with temperature. FIG. shows the comparison between experimental and simulated curves after

Figure 3 . 9 :

 39 Figure 3.9: Comparisons between experimental and identified curves for the austenitic phase

3 . 2 .Table 3 . 2 :Figure 3 . 10 :Figure 3 . 11 :Figure 3 . 12 :

 3232310311312 Figure 3.10: Evolution of the hyperbolic sinus viscous F and K parameters

Figure 3 . 13 :

 313 Figure 3.13: Evolution of the kinematic hardening C 2 parameter

Figure 3 . 14 :Figure 3 . 15 :

 314315 Figure 3.14: Evolution of the Young modulus with respect to T * and x T

Figure 3 . 16 :

 316 Figure 3.16: Evolution of isotropic hardening parameter R ∞ with respect to T * and x T 20 • C 200 • C 300 • C 400 • C 500 • C

Figure 3 . 17 : 6 Figure 3 . 18 : 6 Figure 3 . 19 :

 31763186319 Figure 3.17: Evolution of the thermo-mechanical response with tempering for the cyclic test at 400 • C for a respective tempering of X T = 0 and X T = 0.6

Figure 4 . 1 :

 41 Figure 4.1: Illustration of TRIP observed during dilatometric test under mechanical loading

Figure 4 . 2 :

 42 Figure 4.2: Illustration of Magee mechanism nism describes the difference of compactness δ and expansion coefficient between the initial and the new phase. This difference creates some plastic deformation at the microscopic level. The determination of the compactness parameter δ can be easily done using a free dilatometry test as shown in FIG. 4.3. The relative importance of these two mechanisms depends on the material and the transformation under consideration. Note that, both mechanisms are generally present in diffusional and diffusionless transformations.Finally, TRIP depends on the orientation of the platelets, on the local stress resulting from the internal and external stresses and on the difference of compactness between phases (free deformation). The platelets orientations play an crucial role in the beginning of the transformation where the internal stresses are still low and the accommodation of transformation strains is essentially elastic.

Figure 4 . 3 :

 43 Figure 4.3: Determination of the compactness between the two phases

Figure 4 . 4 :

 44 Figure 4.4: Dilatometric tests for various applied stresses

Figure 4 . 5 :

 45 Figure 4.5: Comparison between Hamata model, Desalos model and experimental results

Figure 4 . 6 :

 46 Figure 4.6: Scheme presenting the localization/homogenization rule[START_REF] Coret | Etude expérimentale et simulation de la plasticité de transformation et du comportement multiphase de l'acier de cuve 16MND5 sous chargement multiaxial anisotherme[END_REF]

Table 4 . 2 :

 42 4.2 gives the values of identified parameters and FIG. 4.7 represents the dilatometry curves prediction from the identified β-model. Identified parameters of the β-model

Figure 4 . 7 :

 47 Figure 4.7: Dialtometry curves obtained using the β-model

Figure 4 . 8 :

 48 Figure 4.8: Simulation of monotonic tensile test for various phases' proportions

Figure 4 . 9 :

 49 Figure 4.9: Simulation of monotonic tensile test for various proportions of quenched and fully tempered martensite

Figure 4 . 10 :Figure 4 . 11 :

 410411 Figure 4.10: Schematic illustration of welding heat cycle test (Satoh test)

Figure 4 . 12 :

 412 Figure 4.12: Experimental stress evolution and comparison with both TRIP and β-scale transition rule simulations to thermal shrinkage. The cooling causes a volume decrease and therefore a tensile stress state in the specimen. If the cooling is fast enough, austenite will transform to martensite. The effect of this metallurgical transformation on the overall stress occurs obviously in the temperature range (400 • C -300 • C).In fact, since martensite has bigger volume and higher yield strength, the tensile stress is reduced due to the expansion caused by the martensite that results in a compressive stress.

Figure 4 . 13 :

 413 Figure 4.13: Overall view of the Disk-Spot test

Figure 4 . 14 :

 414 Figure 4.14: View of the ceramic pins and displacement sensors locations

Figure 4 . 15 :

 415 Figure 4.15: Schematic showing the displacement sensors locations

Figure 4 . 16 :

 416 Figure 4.16: Temperature fields at the end of heating, respectively at the end of cooling

Figure 4 . 17 :

 417 Figure 4.17: Metallurgical state at the end of heating, respectively at the end of cooling

  FIG. 4.23 to FIG. 4.25 show the comparison between experimental and simulated tangential residual stresses σ θθ respectively at the mid-section of the disk and at the top and bottom surfaces. If the simulations are very close to the experiments in the disk mid-section, important differences are observed at both the top and the bottom surfaces. Finally FIG. 4.26 allow to compare ex-

Figure 4 . 18 :

 418 Figure 4.18: Von-Mises stress fields at the end of heating, respectively at the end of cooling

Figure 4 . 19 :

 419 Figure 4.19: Deflection at the end of heating, respectively at the end of cooling

Figure 4 . 20 :

 420 Figure 4.20: Predicted and measured radial stresses at the mid-section of the disk

Figure 4 . 21 :

 421 Figure 4.21: Predicted and measured radial stresses at the top surface

Figure 4 . 22 :

 422 Figure 4.22: Predicted and measured radial stresses at the bottom surface

Figure 4 . 23 :

 423 Figure 4.23: Predicted and measured tangential stresses at the mid-section of the disk

Figure 4 . 24 :

 424 Figure 4.24: Predicted and measured tangential stresses at the top surface

Figure 4 . 25 :

 425 Figure 4.25: Predicted and measured tangential stresses at the bottom surface

Figure 4 . 26 :

 426 Figure 4.26: Predicted and measured normal stresses at the mid-section of the disk

Figure 4 . 27 :

 427 Figure 4.27: Comparison of the normal predicted stresses after deposing the first welding pass for the two different simulations, β model on the right

Figure 4 . 28 :

 428 Figure 4.28: Comparison of the normal predicted stresses after deposing the ninth welding pass for the two different simulations, β model on the right

Figure 4 . 29 :

 429 Figure 4.29: Comparison of the normal predicted residual stresses after deposing at the end of the welding operation for the two different simulations, β model on the right Two different simulations were carried out: the first without considering the tempering model

Figure 4 . 30 :

 430 Figure 4.30: Comparison of the Von-Mises predicted stresses after deposing the first welding pass for the two different simulations, β model on the right

Figure 4 . 31 :

 431 Figure 4.31: Comparison of the Von-Mises predicted stresses after deposing the ninth welding pass for the two different simulations, β model on the right

Figure 4 . 32 :

 432 Figure 4.32: Comparison of the Von-Mises predicted residual stresses after deposing at the end of the welding operation for the two different simulations, β model on the right

  B.1. The calculations were then performed using a constant heat convection/radiation exchange coefficient h of 20 and 252 W/(m 2 K). In fact, these last two values correspond approximately to the values of the curve h = h (T) for the two respective temperatures 200 • C and 1400 • C. The temperature fields obtained at different moments during the first and second welding passes for different h values are shown below.

Figure B. 1 :

 1 Figure B.1: Heat exchange convection/radiation coefficient evolution

Figure B. 3 :Figure B. 4 :

 34 Figure B.3: Fisrt pass, temperature field at t = 13 s and t = 50 s for h = 252 W/(m 2 K)

Figure C. 1 :

 1 Figure C.1: Double ellipsoid heat source configuration

  used a multiplicative correction term d 0 d g γ in the expression of ṗ (See equation (D.2)).

Figure D. 1 :

 1 Figure D.1: Austenite grain size function of austenitizing conditions [1] Austenitized at 1050 • C with a heating rate of 10 • C/s (a) and 50 • C/s (b)

Figure D. 2 :

 2 Figure D.2: Example of Ashby and Easterling prediction diagram of grain size in HAZ [2]

  ing equation: ḋg = b (T ) d g lim (T )d g (D.11)Where b (T ) and d g lim (T ) (the saturation value of the grain size at a temperature T ) are parameters that may depend on temperature as follows:b (t) = b 0 exp Q R T (t) (D.12)And an Arrhenius law for d g lim (T ) : d g lim (T ) = d g 0 exp -

Figure D. 3 :

 3 Figure D.3: Austenitic heat treatment

Figure D. 4 :Figure D. 5 :√ 2 * 93 ≈

 45293 Figure D.4: ThermoGravimetric Analysis (TGA) machine optical microscope equipped with a digital camera

Figure D. 6 :

 6 Figure D.6: Heat treatments at 1200 • C (20 h) and 1300 • C (20 min) respectively

  presented earlier in this chapter. In what follows, a quick recap of equations.ḋg = b (T ) d g lim (T )d g b (t) = b 0 exp Q R T (t) d g lim (T ) = d g 0 exp -W R T

Figure D. 7 :Table D. 2 :

 72 Figure D.7: Grain size estimation using the counting method

Figure D. 8 :

 8 Figure D.8: Grain growth evolution respect to holding time

Figure D. 9 :

 9 Figure D.9: Maximum grain size function of holding temperature

  FIG. F.1 to FIG. F.5 show the comparison between experimental and simulated curves during monotonic tensile tests on austenitic phase at 500 • C, 600 • C, 700 • C, 800 • C and 900 • C respectively.

Figure F. 1 :Figure F. 2 :Figure F. 3 :Figure F. 4 :Figure F. 5 :

 12345 Figure F.1: Comparison between experimental and simulated curve during a monotonic tensile test on austenitic phase at 500 • C

Figure F. 6 :

 6 Figure F.6: Comparison between experimental and simulated curve during a cyclic test on quenched martensite x T = 0 at 20 • C

Figure F. 7 :Figure F. 8 :Figure F. 9 :Figure F. 10 :

 78910 Figure F.7: Comparison between experimental and simulated curve during a cyclic test on quenched martensite x T = 0 at 200 • C

Figure F. 11 :Figure F. 12 :

 1112 Figure F.11: Comparison between experimental and simulated curve during a monotonic tensile test on quenched martensite x T = 0 at 200 • C

Figure F. 13 :

 13 Figure F.13: Comparison between experimental and simulated curve during a cyclic test on partially tempered martensite x T = 0.4 at 20 • C

Figure F. 14 :Figure F. 16 :

 1416 Figure F.14: Comparison between experimental and simulated curve during a cyclic test on partially tempered martensite x T = 0.4 at 200 • C

Figure F. 17 :Figure F. 18 :Figure F. 20 :

 171820 Figure F.17: Comparison between experimental and simulated curve during a cyclic test on partially tempered martensite x T = 0.4 at 500 • C

Figure F. 21 :Figure F. 22 :Figure F. 23 :Figure F. 24 :

 21222324 Figure F.21: Comparison between experimental and simulated curve during a cyclic test on partially tempered martensite x T = 0.6 at 200 • C

Figure F. 31 :Figure F. 32 :

 3132 Figure F.31: Comparison between experimental and simulated curve during a cyclic test on fully tempered martensite x T = 1 at 500 • C

  Austenite fraction vs. temperature: experimental results vs. models predictions coupled nucleation-and growth JMA type model with temperature dependence of the growth law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.32 Austenite fraction vs. temperature: experimental results vs. models predictions coupled nucleation-and growth Leblond type model . . . . . . . . . . . . . . . . 2.33 Complex dilatometry tests : isothermal loadings after heating at 10 • C/s up to 20 or 50% of austenite transformation, and variable heating rate, viz. shifting from 50 to 10 • C/s at 20% of austenite transformation . . . . . . . . . . . . . . . . . . 2.34 Complex dilatometry tests : austenitization at 50 • C/s after partial austenitization at 50 • C/s up to 20 or 50% of austenite transformation and quenching . . . . . . 2.35 Austenite fraction vs. time: experimental results vs. models predictions: isothermal loadings after heating at 10 • C/s up to 20 or 50% of austenite transformation 2.36 Austenite fraction vs. time: experimental results vs. models predictions: variable heating rate, shifting from 50 to 10 • C/s at 20% of austenite transformation . . . 2.37 Austenite fraction vs. time: experimental results vs. models predictions: complete austenitization at 50 • C/s heating rate subsequent to 50% austenitization at constant 50 • C/s heating rate and cooling down to 500 • C . . . . . . . . . . . . . 2.38 Result of dilatometry tests on more or less tempered martensite . . . . . . . . . . 2.39 Austenite Start Temperature A S function of the tempering factor x T . . . . . . . . 2.40 A Sat parameter influence on Austenite Start Temperature A S . . . . . . . . . . . . 2.41 A Sat vs tempering factor x t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.42 Brachet type model: Austenite fraction vs. temperature: experimental results vs. model predictions for x t = 0, 0.3, 0.6 and 1 . . . . . . . . . . . . . . . . . . . . 2.43 K JMA and m JMA function of x t : experimental results vs. models . . . . . . . . . . 2.44 JMA type model: Austenite proportion vs. temperature: experimental results vs.

	List of Figures	vii
	2.31

model predictions for x t = 0, 0.3, 0.6 and 1 . . . . . . . . . . . . . . . . . . . .

  Comparison between experimental and simulated curve during a monotonic tensile test on fully tempered martensite x T = 1 at 500 • C . . . . . . . . . . . . . . . . . 172

	List of Figures	xi
	F.34

• C . . . . . . . . . . . . . . . . . F.33 Comparison between experimental and simulated curve during a monotonic tensile test on fully tempered martensite x T = 1 at 300 • C . . . . . . . . . . . . . . . . . A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016 xii List of Figures A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016

  1.1 Chemical composition of Grade 91 steel [10] . . . . . . . . . . . . . . . . . . . 4 1.2 Main physical properties of T91 [13] . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 The heat capacity parameter evolutions . . . . . . . . . . . . . . . . . . . . . . . 18 1.4 Evolution of the thermal conductivity and convection/radiation coefficient . . . . 19 1.5 Welding machine efficiency for different passes . . . . . . . . . . . . . . . . . . 20 1.6 Evolution of the thermo-mechanical parameters . . . . . . . . . . . . . . . . . . 23 2.1 Values of the Kirkaldy temperature function parameters . . . . . . . . . . . . . . 35 2.2 Identified Koistinen-Marburger model parameters . . . . . . . . . . . . . . . . . 44 2.3 Design matrix of tempering heat treatments . . . . . . . . . . . . . . . . . . . . 46 2.4 Identified tempering model parameters . . . . . . . . . . . . . . . . . . . . . . . 49 2.5 Identified Hardness model parameters . . . . . . . . . . . . . . . . . . . . . . . 51 2.6 Brachet type model: Identified material parameters . . . . . . . . . . . . . . . . 60 2.7 JMA type model without temperature dependence of the growth law: Identified Identified Austenite Start model parameters . . . . . . . . . . . . . . . . . . . . 64 2.11 Identified A Sat model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.12 Identified K JMA and m JMA model parameters . . . . . . . . . . . . . . . . . . . . 68 3.1 Identified Thermo-mechanical model parameters for the austenitic phase . . . . . 78 3.2 Mechanical isothermal test campaign on matensite and tempered martensite material 79 3.3 Identified thermo-mechanical model parameters for martensite . . . . . . . . . . 82 4.1 Identified parameters of the TRIP models . . . . . . . . . . . . . . . . . . . . . 89 4.2 Identified parameters of the β-model . . . . . . . . . . . . . . . . . . . . . . . . 92 D.1 Autenite grain size for different holding times and temperatures . . . . . . . . . 146 D.2 Identified Vereecke grain growth model parameters . . . . . . . . . . . . . . . . 147

material parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.8 JMA type model with temperature dependence of the growth law: Identified material parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.9 JMA type model with temperature dependence of the growth law: Identified material parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.10

Table 1 . 1 :

 11 Chemical composition of Grade 91 steel

	C	Si	Mn	P	S	Cr	Mo	Ni
	wt% 0.099 0.216 0.405 0.007 0.002 8.305 0.951 0.130
	Nb	V	Al	N	Cu	Sn	As	Ti
	wt% 0.075 0.201 0.011 0.034 0.054 0.006 0.003 0.004

Table 1 . 2

 12

	Temperature ( • C)	20	100 200 300 400 500 600
	Modulus of elasticity (GPa)	218 213 207 199 190 181 168
	Thermal conductivity (W/(m K))	26	27	28	28	29	30	30
	Coefficient of linear expansion (MK -1 )	0	10.9 11.3 11.7 12.0 12.3 12.6
	Specific heat capacity (J/(kg K))	440 480 510 550 600 660 770
	Weight per volume (Mg/m -3 )	7.77

: Main physical properties of T91

[START_REF] Vallourec | The T91/P91 Book[END_REF]

design of the steel, the chemical composition was balanced according to a formulation modified from the work of Newhouse et al., as reported by Swindeman et al.

[START_REF] Swindeman | Issues in replacing cr-mo steels and stainless steels with 9cr-1mo-v steel[END_REF]

, on the Chromium-Nickel Balance (CNB) given in equation

(1.1)

Table 1 . 4 :

 14 Evolution of the thermal conductivity and convection/radiation coefficient

	2 )

Table 1 .

 1

	•	1	2	3	4	5	6	7	8	9	10	11
	Efficiency	0.91 0.94 0.97 0.91 0.97 0.69 0.74 0.74 0.71 0.7 0.64

5: Welding machine efficiency for different passes

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016

Table 1 .

 1

			)	N	σ y (MPa)	b	R ∞ (MPa)
	20	186.8	128	6.2	422	21.4	287
	400	178.4	245	6.2	254	85.4	190
	500	157.3	470	5.9	164	155.0	172
	700	132.7	600	4.8	20	325.8	80
	900	84.6	225	5.64	0	121.7	13
	1000	12.6	125	5.64	0	0	0
	1100	8.5	86	5.64	0	0	0
	1200	2.0	54	5.64	0	0	0

1.6 and illustrated in FIG. 1.24 to FIG. 1.26. T ( • C) E (GPa) K (MPa6: Evolution of the thermo-mechanical parameters

Table 2 . 1 :

 21 Values of the Kirkaldy temperature function parameters question (ferritic, pearlitic or bainitic respectively). The factor n can take into account the carbon content of steel. Q is the activation energy of newly formed phases. TAB. 2.1 gives the values of these two factors: In case of ferritic or pearlitic transformation, the newly formed phase proportion function F y i is written as:

	Transformation n Q (kJ/K)
	ferritic	3	23
	pearlitic	1	27.5
	bainitic	2	27

RT

(2.9) Each phase transformation (ferritic, pearlitic, bainitic, martensitic, austenitic) is proportional to cooling rate (T CR -T ) where T CR is equal to Ae 3 , Ae 1 , or B S depending on the transformation in

  Where k = 7.10 5 mm 2 /s is a growth constant, Q = 241.2 kJ/mol the activation energy and R = 8.314 kJ/(mol K). The chemical composition of the steel is taken into account by the function F C which is the basis of this metallurgical model. The various expressions of F C with the weight percentage composition are:

							solid-solid phase transformation
	∂d ∂t	=	1 2d	k exp -	Q RT	(2.14)
	For ferritic : F C =	1 59.6Mn + 1.45Ni + 67.7Cr + 244Mo	(2.15)
	For pearlitic : F C =	1 1.79 + 5.42(Cr + Mo + 4Ni)	(2.16)
	For bainitic : F C =	10 4 2.34 + 10.1C + 3.8Cr + 19Mo	(2.17)

2

(2.

[START_REF] Vallourec | The T91/P91 Book[END_REF]

Where the austenite grain size (µm) is obtained using Easterling and Ashby [ASHB 82] growth model given in equation

(2.13)

. For more information on austenite grain growth please refer to ANNEX D.

  22) solid-solid phase transformation By combining equations (2.21) and (2.22) we get:

  7Mn -16.9Ni + 29.1Si + 16.9Cr + 6.

	A f ( • C) = 910 -203	38W + 290As C -15.2Ni + 44.7Si + 104V + 31.5Mo + 13.1W √	(2.37) (2.38)

Table 2 . 3

 23

	450 • C 1			13	19 25
	500 • C 2			14	20 26
	550 • C 3	7	10	15	21 27 31 33
	650 • C 4	8	11 16		22 28
	750 • C 5		17		23 29 32	34
	800 • C 6	9	18 12		24 30

: Design matrix of tempering heat treatments

  • C/s] was derived from the database illustrated in FIG. 2.24. The calculated austenite formation kinetics and phase fractions are plotted as a function of temperature (see FIG. 2.26 -corresponding to 50

Table 2 . 6 :

 26 Brachet type model: Identified material parametersA eq = 802 • C K JMA = 19.68 10 -38 T 0 = 0.0748 • C

	m JMA = 6.092	t 0 = 3.23 10 -8	n JMA = 0.309
	Table 2.7: JMA type model without temperature dependence of the growth law: Identified mate-
			rial parameters
	A eq = 802 • C K JMA = 19.68 10 -38 T 0 = 0.0748 • C
	m JMA = 8.514	Q = 327 kJ/mol	t ∞ = 108 10 3 s n JMA = 0.317

Table 2 . 8 :

 28 JMA type model with temperature dependence of the growth law: Identified material parametersA eq = 802 • C K eq = 0.056 • C -1 m eq = 1.151 τ L1 = 1.711 s τ L3 = -0.032 s A L = 0.588 s B L = 0.0129 • C -1 s

Table 2 . 9 :

 29 JMA type model with temperature dependence of the growth law: Identified material parameters The results of this identification process are illustrated by (FIG. 2.29, FIG. 2.31, FIG. 2.30 and FIG. 2.32) where the responses of the models are compared to the experimental dilatometry curves at constant heating rates.

  • C/s up to complete austenitization (see FIG. 2.34). The responses of the models for this test are compared to the experimental plots in FIG. 2.37. Leblond type model seems to be very simple to respond to such large temperature rate ranges.

Table 2 .

 2 10: Identified Austenite Start model parameters

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016

Table 2 . 11 :

 211 Identified A Sat model parameters

Table 2 .

 2 1.8610 -38 2.1210 -36 0.127 9.34 10.45 0.887 12: Identified K JMA and m JMA model parameters

  Thermo-mechanical behaviour of each phase tified parameters are given in TAB. 3.1. Note that it is assumed that for the austenitic phase, the -8 1.15 10 -7 1.89 10 -7 1.45 10 -7 7.17 10 -8

	T ( • C)	500	600	700	800	900
	E (GPa)	149	130	106	74	55
	σ y (MPa)	109	83	44	14	0
	R ∞ (MPa)	282	232	167	0.5	0
	b	12	16	31	204	112
	K	2.7	4.2	4.5	7.8	6.8
	ε∞	6.3 10

Table 3 . 1 :

 31 Identified Thermo-mechanical model parameters for the austenitic phase

Table 3 . 3 :

 33 Identified thermo-mechanical model parameters for martensite

Table 4 . 1 :

 41 Identified parameters of the TRIP models which seems crucial for a predictive characterization of residual stresses. The transformation plasticity induced during austenitization can be considered negligible, because at these temperatures the material is almost purely viscous.

Table D . 1 :

 D1 Autenite grain size for different holding times and temperatures

				• C T = 1200 • C
	t = 20 min	35	45	67
	t = 1 h		97
	t = 10 h	74	141	239
	t = 15 h		145
	t = 20 h			256

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016

Thermo-mechanical behaviour of each phase Figure 3.20: Evolution of the thermo-mechanical response with tempering for the monotonic test at 500 • C for a respective tempering of X T = 0 and X T = 0.6

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016

Chapter 4

Modelling of the multiphasic behaviour

The phases obtainable during multipass welding of X10CrMoVNb9-1 steel are morphologically very different. Consequently, their mechanical properties are, of course, different. For example, austenite -which appears at high temperature -is by nature much more viscous than martensite. Thus, studying the macroscopic behaviour of a representation volume element becomes a homogenization problem which is extremely difficult to solve without making numerous assumptions.

In this chapter, two different approaches to model multi-phasic behaviour are presented. The first approach is base on Voigt assumption with the addition of extra strain on softer phase during metallurgical transformation in order to insure an equal total strain in all phases. The second approach is based on a local/global homogenization method known as the β-scale transition rule. Both approaches were simulated numerically for simple and more complex welding cases. The two approaches were compared in term of prediction of global deformation (e.g. Satoh test) or residual stress (the multi-pass welding test). • mm for the first three passes;

• mm for the third, fourth, fifth and sixth passes;

• mm for the seventh, eighth and ninth passes;

• and 13 mm for the rest of the passes.

The filling of the groove had required 16 passes of welding.

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016

Sensitivity analysis of the heat exchange convection/radiation coefficient

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016

Sensitivity analysis of the heat exchange convection/radiation coefficient Appendix C

Analytical model and equivalent heat source

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016

Analytical model and equivalent heat source

The aim of this annex is to recapitulate the available empirical models for equivalent heat sources for muti-pass welding simulation. The models presented in this annex are 3D models. Several thermal models are herein presented with a distinction made between the models used for analytical calculation and models used for numerical calculation:

C.1 3D modified Rosenthal model: (analytical model)

This model can be used as a solution of a conduction problem in steady state conditions of infinite plate subjected to a punctual heat source moving at constant speed on one side. This model is described by the following mathematical equation:

Where

And where T denotes the temperature ( • C), T 0 the initial temperature ( • C), Q the effective power (W), λ the thermal conductivity (W/(m K)), ν the torch speed (m/s), h the plate thickness, k the thermal diffusivity (m 2 /s) and t the time. The disadvantage of this model is that it reduces the source to a point which tends to overestimate the temperature field near the source.

C.2 Rosenthal-Rykalin model: (analytical model)

Because of low computing time and its great simplicity, this model is more widespread than the one before. The equation varies with the plate thickness of the plate. For a thick plate:

For a thin plate :

Where r = x 2 + y 2 denotes the distance (m), s the plate thickness (m), ρ the density (kg/m 3 ) and c the heat capacity (J/(kg K)). Although we gained in computation time and simplicity, the problem of ponctual source remains present.

C.3 V groove model: (numerical model)

This model represents a process of GTA narrow groove welding on tube. The temperature field is given by solving the equation of heat conduction in three dimensions. The heat source is a semicircle constant flow applied on the muzzle walls. The disadvantage of this model is that it does not take into account the phenomena of redistribution by convection.

Austenite grain growth

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016 equation:

with a single constant K depending on temperature and d g 0 an integration constant de,oting the initial average grain size before growth. Note that the model of Hillert is a special case of the Beck model of (with the assumption that d g d g 0 ). Hillert has also studied the influence of second phase particles (eg carbides) which tend to slow grain boundaries motilities' ( [START_REF] Hillert | On the theory of normal and abnormal grain growth[END_REF], [START_REF] Humphreys | A unified theory of recovery recrystallization and grain growth based on the stability and growth of cellular microstructures. part ii: The effect of second-phase particles[END_REF], [START_REF] Stasko | Effect of nitrogen and vanadium on austenite grain growth kinetics of low alloy steel[END_REF], [START_REF] Fernandez | Effect of microalloying elements on the austenitic grain growth in a low carbon hsla steel[END_REF]). This phenomenon can be introduced into the equation (D.5) as:

Where r denotes the average radius of secondary particules and f their proportions. In this model, 3 f 4 r α represents the limit value at saturation of the grain size.

D.1.2.3 Grey and Higgins model

Other models, still based on knowledge of the grain joints migration velocities, include the effect of particles of second phase ( [207] [208]). One of the most famous is that of Grey and Higgins ( [START_REF] Grey | Solute limited grain boundary migration: a rationalisation of grain growth[END_REF]) :

Where λ and d g lim are material parameters. The disadvantage of this model is the differential equation, which, to be resolved, requires knowledge of the initial value of the grain size, a priori unknown.

D.1.2.4 Ashby and Easterling model

The Ashby and Easterling model ( [2]) is a diffusion controlled model applicable for anisothermal loading cases, unlike the models presented previously. According to Ashby and Easterling, grain growth is assumed to be controlled by the diffusion and surface energy and without prior nucleation. The variation in grain size is written as:

Where f (d g ) is an adjustable function, A and Q material parameters of the Arrhenius type law. Using this kinetic model of grain growth, identified from experimental measurements on several grades of steel and using available information concerning the carbides dissolution, the authors have constructed diagrams (FIG.D.2) of prediction of grain size as a function of the maximum temperature reached at a point in the HAZ and its characteristic cooling rates δt 500-800 for each steel.

D.1.2.5 Leblond and Devaux model

The kinetic model of grain size proposed by Leblond and Devaux ([115]) is a special case of the Ashby and Easterling model:

10)

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016

Extension of Leblond type model

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016

Extension of Leblond type model

In this annex we present the extension of Leblond phase transformation model for case of n phases. Leblond developed for the FEM codes Framatome and Sysweld, a model based KJMA evolution law, involving a proportion of transformed phase at equilibrium and a constant time ( [115] [116]) as already seen in Chapter 2:

Where : y γL is the new phase proportion y eq (T ) is at equilibrium phase proportion τ(T ) is a time constant.

Leblond chooses to bring it's the evolution law in differential form, he shows on simple examples that the addition rule and that the principle of isokinetic transformations may be in default if the proportion of the new phase in equilibrium is different from unity. He does not take into account an incubation time, the constant time is used to wedge the start and finish transformation times of the experimental curves. The advantage of equation (E.1) is its simplicity, is actually the derivative of the equation of Johnson-Mehl-Avrami for n = 1 and T constant in isothermal condition: ẏγL = n JMA k JMA 1/n JMA y eq (T )y γL ln y eq (T ) y eq (T )y γL

In anisothermal conditions, equation (E.2) is not beter than its simplified form (E.1) since the terms in dT /dt are neglected. The goal is not to multiply the coefficients to obtain a physical description, Leblond keeps to the expression (E.1). The model says to be able to simulate all phase transformations that can occur in a steel, equation (2.66) is balanced in order to simulate the inverse transformation. Let's consider two phases 1 and 2 and the transformation 1 to 2. The evolution equation (2.66) can be written:

If we set: p 1 = 1y γL , p 2 = y γL and

This allows to describe the 1 to 2 transformation as well as the 2 to 1 and 2 transformation by noting respectively k 12 (T ) , l 12 (T ) the coefficients for the first transformation and k 21 (T ) , l 21 (T ) the coefficients for the second one. The extension to a n-phase system can be obtained by replacing 1 by i and 2 by j by where {i, j} ∈ [1, ..., n].

Leblond identifies his model from a CCT diagram and obtain good results for ferritic and pearlitic transformations. The model however cannot reproduce the bainitic transformation which force Leblond to introduce a temperature rate dependence Ṫ to the coefficients:

With this writing the identification of bainite becomes possible, Leblond justifies this change by the difference in structure of bainite obtained by rapid or slow cooling. The coefficients K i j (T )

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016

Appendix F

Mechanical tests and identification results

A thermo-metallurgical-mechanical model for the numerical simulation of multipass GTA welding of martensitic X10CrMoVNb9-1 steel -Farah HANNA 2016