R Julien

V Velay

V Vidal

Y Dahan

R Forestier

F Rézaï-Aria

Characterization and modeling of forged Ti-6Al-4V Titanium alloy with microstructural considerations during quenching process

Keywords: Behavior modeling, Microstructural evolution, Heat treatment, Forged Titanium alloy

The present investigation proposes an experimental device able to assess the thermomechanical behavior of Ti-6Al-4V Titanium alloy throughout the die-forging operation. Constitutive equations are developed to assess the influence of the process (die-forging temperature, cooling rate) and the microstructure parameters on the mechanical response of the alloy. For this purpose, a non-unified behavior model formulation is implemented, which defines two main mechanisms related to α and β phases and allows the prediction of hardening, strain rate sensitivity and temperature, combined with the phase evolution that is dependent on the cooling conditions and which can greatly affect the mechanical behavior. This identification strategy is then applied for die-forging temperatures below the β-transus temperature, which requires microstructural information provided by SEM (Scanning Electron Microscopy) observations and image analysis. Finally, the approach is extended to die-forging temperatures above the β-transus temperature.

Introduction

Titanium alloys are widely used in the aerospace industry for their well-known high mechanical strength/weight ratio [START_REF] Boyer | Materials Properties Handbook: Titanium Alloys[END_REF]. They can be used as forged semi-finished products in many industrial applications. These products are transformed into final parts by subsequent thermo-mechanical heat treatments and machining operations.

Depending on the temperature of the thermo-mechanical processing (TMP) and the heat treatments (HT), various complex microstructures can be achieved. Titanium alloys can be heat-treated above or below the β-transus temperature depending upon the specific micro-structural aspects required in terms of grain size and morphology, as well as the existence of phases and mechanical strength requested by the end-users. HT generally consists in an isothermal dwell for a certain period of time following quenching. To fulfil the microstructural characteristics and mechanical performance requirements and according to the size of the semi-finished products, an appropriate quenching environment (e.g., air, oil, water, etc.) is selected. The dimensions of the product and the quenching conditions can drastically influence the spatial quenching rates, specifically the time-temperature history, in any points in the product. In fact, quenching operations generate transient time-temperature histories in a part, moving inwards from the surface to the bulk. One of the major concerns, particularly for products with large dimensions, is to guarantee a homogeneous microstructure through all regions of the product. Another major technical concern is to avoid any excessive distortions during quenching and also hot tearing, which result in internal defect initiation, such as micro-and/or mesoscopic cracking. While the post-quenching surface cracking can be eliminated by subsequent machining, the undetectable internal cracking becomes a major parameter that can drastically disqualify a semi-finished product for reasons of nonconformity. Moreover, these operations induce important residual stresses in the part, which makes the final milling stage difficult. Predicting the internal residual strains/stresses has become mandatory in industrial practice. The relevant constitutive laws therefore need to be developed. However, one important problem is that during HT and quenching the microstructure and phases in titanium alloys evolve depending on the kinetics of the phase evolutions, in that they are time-temperature rate dependent.

Furthermore, titanium alloys have a complex mechanical behavior, exhibiting strain rate sensitivity effects [START_REF] Majorell | Mechanical behavior of Ti-6Al-4V at high and moderate temperatures -Part I: Experimental results[END_REF][START_REF] Vanderhasten | Ti-6Al-4V: Deformation map and modelisation of tensile behaviour[END_REF], which can be reproduced through viscosity laws [START_REF] Picu | Mechanical behavior of Ti-6Al-4V at high and moderate temperatures -Part II: constitutive modeling[END_REF][START_REF] Cheong | Modelling of hardening due to grain growth for a superplastic alloy[END_REF][START_REF] Semiatin | Self-consistent modeling of the flow behavior of wrought alpha/beta titanium alloys under isothermal and nonisothermal hot-working conditions[END_REF].

These effects becomes significant for temperatures greater than 500 • C [START_REF] Robert | Simulation numérique du soudage du TA6V par laser YAG impulsionnel: caractérisation expérimentale et modélisation des aspects thermomécaniques associés à ce procédé[END_REF][START_REF] Tao | Quasi-static tensile behavior of large-diameter thin-walled Ti-6Al-4V tubes at elevated temperature[END_REF]. Depending on the test temperature, this phenomenon can be combined with hardening effects induced by dislocation motions and resulting from a competition between storage and annihilation terms [START_REF] Bouaziz | Revisited storage and dynamic recovery of dislocation density evolution law: Toward a generalized kocks-mecking model of strain-hardening[END_REF][START_REF] Mecking | A universal temperature scale for plastic flow[END_REF]. At very high temperatures, other mechanisms are induced, such as grain boundary sliding [START_REF] Lin | Modelling of microstructure evolution in hot forming using unified constitutive equations[END_REF][START_REF] Lin | GA-based multiple objective optimisation for determining viscoplastic constitutive equations for superplastic alloys[END_REF]. Moreover, important microstructural changes can occur, such as grain growth, or phase fraction evolutions, which themselves greatly influence the mechanical behavior, and therefore need to be introduced in the model formulation [START_REF] Fan | Internal-state-variable based self-consistent constitutive modeling for hot working of two-phase titanium alloys coupling microstructure evolution[END_REF][START_REF] Teixeira | Simulation of coupled temperature, microstructure and internal stresses evolutions during quenching of a β-metastable titanium alloy[END_REF][START_REF] Teixeira | Influence of the β → α+β transformation on the stresses and strains evolutions during quenching of the Ti17 alloy from the β phase field[END_REF]. Indeed, these evolutions can involve strain hardening [START_REF] Cheong | Modelling of hardening due to grain growth for a superplastic alloy[END_REF][START_REF] Velay | Behavior modeling and microstructural evolutions of Ti-6Al-4V alloy under hot forming conditions[END_REF] or softening due to dynamic recrystallization [START_REF] Alabort | On the mechanisms of superplasticity in Ti-6Al-4V[END_REF] under large deformation conditions. Moreover, when the β-phase is predominant, particular phenomena, caused by the pinning-depinning effect of the dislocation, can affect the mechanical behavior (Yield point effect). This effect was investigated at the scale of the single crystal [START_REF] Laasraoui | Prediction of steel flow stresses at high temperatures and strain rates[END_REF][START_REF] Mecking | Strain hardening and dynamic recovery[END_REF] then generalized at the scale of the polycrystal [START_REF] Lin | Modelling of microstructure evolution in hot forming using unified constitutive equations[END_REF][START_REF] Momeni | Yield point phenomena in TIMETAL 125 beta Ti alloy[END_REF][START_REF] Wang | Yield-Point Phenomena of Ti-20V-4Al-1Sn at 1073 K and Its Constitutive Modelling[END_REF][START_REF] Wang | A Study of High Temperature Viscoplastic Deformation of Beta Titanium Alloy Considering Yield-point Phenomena[END_REF]. Hence, non-unified approaches can be implemented in order to define several inelastic mechanisms associated with each phase evolution [START_REF] Teixeira | Simulation of coupled temperature, microstructure and internal stresses evolutions during quenching of a β-metastable titanium alloy[END_REF][START_REF] Montheillet | Comportement rhéologique des matériaux métalliques multiphasés[END_REF][START_REF] Yoshida | A constitutive model of cyclic plasticity[END_REF][START_REF] Yoshida | A plasticity model describing yield-point phenomena of steels and its application to FE simulation of temper rolling[END_REF].

Such approaches can translate grain-boundary strengthening caused by a lamellar microstructure [START_REF] Taleff | Pearlite in ultrahigh carbon steels: Heat treatments and mechanical properties[END_REF][START_REF] Perdrix | Influence of cooling rate on microstructure and mechanical properties of a Ti-48Al alloy[END_REF][START_REF] Perdrix | Influence of nitrogen on the microstructure and mechanical properties of Ti-48Al alloy[END_REF][START_REF] Ishii | Mechanical Properties of α + κ Twophase Lamellar Structure in Fe-Mn-Al-C Alloy[END_REF].

The present study proposes a behavior model that is able to faithfully predict the strain-stress response of the material during the quenching operation. Considering all the aspects described above, a laboratory experiment testing facility has been developed to conduct in-situ heat-treatment at temperatures beyond or lower than β-transus temperature on a cylindrical specimen [START_REF] Julien | Tensile behaviour of high temperature forged ti-6al-4v during in-situ heat treatments[END_REF]. Then, tensile tests were combined in order to assess the behavior of the alloy under transient thermo-mechanical loadings. The microstructural evolutions are greatly influenced by the cooling rate and can exhibit several phases (primary and secondary α phase and β phase), the proportions of which were assessed by SEM observations and image analysis. In such complex conditions, the mechanical behavior was characterized for several temperature levels and cooling rates. From these experiments, non-unified constitutive equations were implemented to define several mechanisms related to each phase.

They consider a rule of mixture between phases depending on the cooling conditions. This rule plays an important role on the activated mechanisms influencing the mechanical behavior. The model formulation can take account of the strain rate sensitivity and the hardening over a wide temperature range (from the die-forging temperature to the ambient temperature) and several cooling rates connected to various microstructural states. This behavior model was identified for die-forging temperatures below the β-transus temperature. Finally, the approach was successfully extended to die-forging temperatures above the β-transus temperature.

Experimental procedures

Material and device

Industrial thermo-mechanical heat treatment consists in 3 main operations, as shown in Fig. 1:

• forging at 940 • C • die-forging at 950 • C • tempering at 730 • C. t θ T β - Forging 940 • C Die-Forging 950 • C, 2 h Tempering 730 • C θ Figure 1: Thermo-mechanical industrial process
The material studied in the present work was supplied by Aubert & Duval as a billet of Ti-6Al-4V Titanium alloy after the forging operation. Cylindrical specimens were machined from this billet. At this stage, an equi-axed microstructure was observed, which included α primary nodules, decorated at grain boundaries by the β phase, as shown in Fig. 2a. Depending on the microstructure needs (equiaxed, duplex or lamellar morphology), the die-forging operation can be carried out at different temperatures that are higher or lower than the β-transus temperature (T β = 1000 • C). In the present case, the die-forging temperature considered is 950 • C (i.e. below the β-transus temperature). During the cooling/quenching from this temperature, the β phase transforms into a lamellar microstructure consisting of colonies of secondary α in the β phase (labelled β t ). At this die-forging temperature, the phase transformation is not entirely completed and a duplex microstructure is obtained, regardless of the cooling conditions (Fig. 2b at 60 • C/min). The cooling rate greatly influences the induced microstructure. Indeed, the thickness of the secondary (lamellar) α-phase diminishes as the cooling rate increases.

Tensile tests

The test conditions (see Fig. 3) were selected in order to accurately reproduce the thermo-mechanical loadings induced in the billet during the die-forging step. This analysis led to three cooling conditions being considered ( θ = {5, 60, 200} • C/min) and several strain rates (10 -4 s -1 ≤ ε ≤ 10 -2 s -1 ). Moreover, three temperature domains were investigated:

• from 950 • C to 800 • C • from 800 • C to 500 • C • from 500 • C to 20 • C
In the following section, the effects of the cooling rate, strain rate and test temperature on the stress-strain response are discussed. Interpretations are based on microstructural evolution analysis (fraction and size of α I nodules, α II lamellae or β phase).

Influence of the cooling rate

As shown by many research works [START_REF] Semiatin | Self-consistent modeling of the flow behavior of wrought alpha/beta titanium alloys under isothermal and nonisothermal hot-working conditions[END_REF][START_REF] Katzarov | Finite Element Modeling of the Morphology of β to α Phase Transformation in Ti-6Al-4V Alloy[END_REF] on the topic of microstructure evolution during cooling from temperatures above 950 • C, while the phase transformation (β ↔ α) mainly depends on the temperature, the cooling rate mainly affects the size of the primary α nodules as well as the initiation and growth of the α II phase, leading to different sizes and morphologies. Five tensile tests at different temperature levels (θ = {950, 800, 700, 500, 300, 20} • C) were conducted with a constant strain rate of 10 -2 s -1 . The present study shows that, during cooling, an important phase transformation of β into α II occurs, mainly between 950 • C and 900 • C. This result is confirmed by other research works [START_REF] Elmer | In situ observations of lattice expansion and transformation rates of α and β phases in Ti-6Al-4V[END_REF][START_REF] Lonardelli | In situ observation of texture evolution during α → β and β → α phase transformations in titanium alloys investigated by neutron diffraction[END_REF] and is illustrated in more detail in the next section (section 2.2.2). Thus, for the tensile tests carried out at and above 800 • C, it can be considered that the microstructure observed at room temperature is the one generated at the test temperature, as shown by Fig. 4 for tests conducted at 20 • C and 700 • C under various cooling conditions. For the rapid cooling rates ( θ = {60, 200} • C/min), many thin α II lamellae can be observed in the transformed β grains. At a slower cooling rate ( θ = 5 • C/min), the nucleation of α II lamellae takes a longer time to grow, leading to lamellar coarsening. Hence, the morphology of these lamellae is quite similar to that of the α I nodules. The corresponding tensile tests (see Fig. 5) show a significant hardening with the increase in the cooling rate. This feature can be related to the decrease in the α II -lamellar thickness with the cooling rate. This induces a higher number of α/β boundaries that can act as more obstacles to the dislocation movements.

5 • C/min 700 • C 60 • C/min 700 • C 200 • C/min 700 • C 5 • C/min 20 • C 60 • C/min 20 • C 200 • C/min 20 • C
This hardening is thus linked to the plasticity of the α phase (α I nodules and α II lamellae) as the evolution of the lamellar thickness can be related to the yield stress [START_REF] Katzarov | Finite Element Modeling of the Morphology of β to α Phase Transformation in Ti-6Al-4V Alloy[END_REF][START_REF] Sui | Relationship between Thickness of Lamellar α+β Phase and Mechanical Properties of Titanium Alloy[END_REF][START_REF] Gil | Formation of α-Widmanstätten structure: effects of grain size and cooling rate on the Widmanstätten morphologies and on the mechanical properties in Ti6Al4V alloy[END_REF] or to the hardness or the ductility of the material [START_REF] Perdrix | Influence of cooling rate on microstructure and mechanical properties of a Ti-48Al alloy[END_REF][START_REF] Gil | Formation of α-Widmanstätten structure: effects of grain size and cooling rate on the Widmanstätten morphologies and on the mechanical properties in Ti6Al4V alloy[END_REF]. the sequel. From these measurements, the evolution of the β fraction can be assessed (see Fig. 8). These results are very similar to those provided in the work of Elmer [START_REF] Elmer | In situ observations of lattice expansion and transformation rates of α and β phases in Ti-6Al-4V[END_REF] showing, in a Ti-6Al-4V alloy, the evolution with the temperature of the β phase amounts measured by in situ X-ray diffraction techniques. As in our present study, it seems that most of the β ↔ α phase transformation occurs at a temperature in the 800 • C -T β range. This therefore confirms that, during cooling, most of the β phase had completed its transformation into α II phase around 800 • C. Regarding the mechanical behavior, as expected, the stress-strain curves show a decrease in the flow stress with the temperature (see Fig. 9). Moreover, a significant stress relaxation occurs during the tensile dwell time for the test temperature above 500 • C, involving considerable viscous stress, whereas it is considerably reduced below this temperature.

Lastly, at 950 • C, a yield point phenomenon is observed. It is probably due to a pinning-depinning process of the dislocations in a Cottrell atmosphere [START_REF] Cottrell | Dislocation Theory of Yielding and Strain Ageing of Iron[END_REF][START_REF] Hahn | A model for yielding with special reference to the yield-point phenomena of iron and related BBC metals[END_REF], as observed in BCC metals. Indeed, in BCC metals (as the β phase), the dislocations can be pinned by interstitials, in which case a higher force is required in order to leave such dislocations away from their Cottrel atmosphere. Thus, during the first loading at 950 • C, this higher force is responsible for the upper yield point (stress peak). After unpinning, the dislocations can move easily at a lower stress leading to a slight stress softening. During the dwell time, the initial Cottrel atmosphere is recovered (static recovery) involving a new stress peak after the second loading.

This phenomenon is predominant at 950 • C where the mechanisms related to the β phase play an important role, but vanishes at lower temperatures where the plastic deformation is mainly governed by the α phase.

Behavior modeling

The previous analysis leads to the definition of 3 mechanisms acting on the mechanical behavior. The first one is related to the β phase, whereas the two others are related to the α phase through the nodular part (α I ) and lamellar part (α II ).

Non-unified Constitutive Equations

A homogeneous deformation (Eq. ( 2)) is assumed in each phase [START_REF] Teixeira | Simulation of coupled temperature, microstructure and internal stresses evolutions during quenching of a β-metastable titanium alloy[END_REF] and a strain partition of the total strain into elastic and plastic parts is considered (Eq. ( 3)).

ε t = ε t α I = ε t α II = ε t β ( 2
)
ε t φ = ε e φ + ε p φ ∀φ = α I , α II , β (3)
Hooke's law is given by Eq. 4 for each phase. And each strain component can be related to a phase ratio Z Φ (Eq. 5).

σ φ = C φ (ε t φ -ε p φ ) ∀φ (4
)

ε e = φ Z φ ε e φ ; ε p = φ Z φ ε p φ with: φ Z φ = 1 ∀φ (5)
A von Mises yield surface is assumed for each phase, as shown by Eq. 6. Its evolution is defined through an isotropic hardening variable R φ .

f φ = σ eq φ -R φ -σ 0 φ = 0 ∀φ (6)
σ eq φ and σ 0 φ , are respectively the equivalent stress and the elasticity limit related to the phase φ.

This approach is in agreement with the thermodynamics of the irreversible process defined by two potentials, the Helmotz free energy ψ and the dissipation potential Ω.

The free energy can be partitioned into elastic and inelastic parts ψ = ψ e + ψ in and, in the present study, formulated for each phase (Eq. 7)

.

̺ψ e = 1 2 φ Z 2 φ C φ ε e φ : ε e φ ; ̺ψ in = 1 2 φ Z 2 φ b φ Q φ r 2 φ ∀φ (7)
The state laws giving the Cauchy stress and the macroscopic isotropic hardening variable derive from this potential (Eq. 8).

σ = ̺ ∂ψ e ∂ε e = φ Z φ σ φ ; R = ̺ ∂ψ in ∂r = φ Z φ R φ = φ Z φ b φ Q φ r φ (8)
with r φ the internal variable associated to the isotropic hardening. b φ and Q φ are temperature-dependent coefficients.

The dissipation potential Ω allows definition of the evolution of the internal variables (Eq. 9). It includes, first, a static recovery part (Eq. 10) and a classical viscoplastic potential formulated in the form of a power law. However, its expression differs from one phase to another. Indeed, a similar form is used to describe the primary and secondary alpha phase (Eq. 11), whereas a particular form is considered for the β phase (Eq. 12) so as to reproduce the yield point phenomenon. The viscoplastic flow derives from this potential (Eq. 13).

Ω = φ Z 2 φ Ω p φ + Ω r φ ∀φ (9)
Ω r φ = a φ R 2 φ 2 b φ Q φ ∀φ (10)
Ω p α = K α n α + 1 f α K α nα+1 ∀α = α I , α II (11)
Ω p β = b ρ ρ m M D n β + 1 f β D n β +1 (12)
εp = ∂Ω/∂σ = φ Z φ 3 2 S φ σ eq φ ṗφ = φ Z φ εp φ ∀φ (13)
where S φ is the deviatoric part of σ φ .

The cumulative plastic strain for each phase is given by Eq. 14 for the α phase and Eq. 15 for the β phase.

ṗα = Ω ′ α (f α ) = f α K α nα ∀α = α I , α II (14)
ṗβ = Ω ′ β (f β ) = b ρ ρ m M f β D n β (15)
Lastly, the evolution equation related to the isotropic hardening for each phase is determined from Eq. 16.

ṙ = -∂Ω/∂R = φ Z φ ṙφ with: ṙφ = ṗφ (1 -b φ r φ ) -a φ r φ ∀φ (16)
The positivity of intrinsic dissipation D ensures good agreement of the model formulation with thermodynamic principles. It can be expressed by Eq. 17.

D = φ σ φ : εp φ - φ R φ ṙφ (17)
The positivity of D can be proved by Eq. 18.

D = φ f φ + R φ + R 2 φ Q φ ṗφ + φ a φ b φ R φ Q φ 2 ≥ 0 (18)

Introduction of the microstructural parameters

The viscoplastic flows (Eq. 14 and 15) require the identification of material parameters for each phase, which were determined by using SEM observations and image analysis. Thus, the influence of the microstructural evolutions related to the cooling rate during the quenching stage can be introduced into the model formulation.

α I phase

The proposed model acts on the coefficient K α I and establishes a relationship between this parameter and the average size of the primary α nodules d α I , as shown by Eq. 19 following the Hall-Petch law.

K α I = K 1 d -n d α I (19)
with K 1 a temperature-dependent material parameter and n d the Hall-Petch coefficient, n d = 0.5.

α II phase

Similarly, the proposed law introduces a relationship, given by Eq. 20, between K α II and the thickness of the α II lamellae L.

K α II = K 2 L -n L (20)
K 2 and n L are material parameters. L depends on the cooling rate θ (see Eq.

1).

β phase

The mobile dislocations are at the root of the yield point phenomenon [START_REF] Yoshida | A constitutive model of cyclic plasticity[END_REF] and the density of these dislocations ρ m is a part f m of the density of the total dislocations ρ t (Equation 21). Moreover, this part evolves between a starting value f m0 and an asymptotic value f ma [START_REF] Yoshida | Constitutive modeling of large-strain cyclic plasticity for anisotropic metals[END_REF]. Finally, an empirical law is used to define the relation between the densities of the total dislocations and the cumulative plastic strain.

ρ m = f m ρ t ; ρ t = ρ 0 + C ρ p β (21)
where C ρ and a ρ are material parameters.

The strain rate had a significant influence on the yield point phenomenon, therefore, the following time evolution of f m is assumed (Equation 22).

ḟ m = -λ ṗκ β (f m -f ma ) (22)
with: f m (t = 0) = f m0 and λ and κ are material parameters.

This equation differs from the literature [START_REF] Teixeira | Simulation of coupled temperature, microstructure and internal stresses evolutions during quenching of a β-metastable titanium alloy[END_REF]. Indeed, in the present study, the yield point phenomenon increases with the strain rate, which is an effect that has not been observed on steels [START_REF] Yoshida | A constitutive model of cyclic plasticity[END_REF][START_REF] Yoshida | A plasticity model describing yield-point phenomena of steels and its application to FE simulation of temper rolling[END_REF] or on metastable β-titanium alloys [START_REF] Wang | Yield-Point Phenomena of Ti-20V-4Al-1Sn at 1073 K and Its Constitutive Modelling[END_REF][START_REF] Wang | A Study of High Temperature Viscoplastic Deformation of Beta Titanium Alloy Considering Yield-point Phenomena[END_REF].

As shown previously, the tensile dwell time induces a stress relaxation which can be reproduced by introducing a static recovery term into the isotropic hardening component. This phenomenon involves a dislocations rearrangement with a decrease in the dislocation density [START_REF] Hull | Introduction to Dislocations[END_REF][START_REF] Kerisit | EBSD coupled to SEM in situ annealing for assessing recrystallization and grain growth mechanisms in pure tantalum[END_REF][START_REF] Smallman | Modern Physical Metallurgy[END_REF]. Moreover, during this dwell time at high temperature, some interstitial atoms can diffuse back around the dislocations, leading to the re-pinning of dislocations in the Cottrel atmosphere. Therefore, the yield point phenomenon is again observed during the second loading. In order to account for this effect in the model formulation, Eq. 22 is modified by Eq. 23 and a static recovery term is added, describing the decrease in the density of the mobile dislocations during dwell time.

ḟ m = -λ ṗκ β (f m -f ma ) -µf δ m ( 23
)
where µ and δ are material parameters.

All the constitutive equations are given in Appendix A (table A.1).

Results

Identification Strategy

Young's Modulus

The evolution of Young's modulus with the temperature is obtained by using the relationship given by Eq. 24.

E = (Z α I + Z α II )E α + Z β E β ( 24
)
The same values are assumed for α I nodules and α II lamellae. A tensile test at 1030 • C was performed to determine the modulus of the β-treated alloy. A literature review [START_REF] Teixeira | Influence of the β → α+β transformation on the stresses and strains evolutions during quenching of the Ti17 alloy from the β phase field[END_REF][START_REF] Joshi | Titanium Alloys : An Atlas of Structures and Fracture Features[END_REF][START_REF] Brandes | The effect of oxygen and stress state on the yield behavior of commercially pure titanium[END_REF][START_REF] Vidal | Experimental Study of the Deformation Mechanisms in Textured Alpha-titanium Alloy Sheets[END_REF][START_REF] Fréour | Application of inverse models and XRD analysis to the determination of Ti-17 β-phase coefficients of thermal expansion[END_REF] allows Young's modulus evolution to be determined at lower temperatures. Then, knowing the phase fraction Z φ , the Young's modulus values of the α phase are deduced from Eq. 24. The results obtained are in a good agreement with the values found in the literature, as shown in Fig. 10.

Time-dependent parameters

The tensile tests performed with a cooling rate of 60 • C/min were used to identify the viscous parameters K φ and n φ . The stress relaxation curves σ-σ i = f (time)

were plotted in a bi-logarithmic diagram to determine these parameters for temperature level, where σ i is the non-viscous stress corresponding to the stabilized stress value at the end of the relaxation time. The static recovery term a φ of the isotropic hardening variable allows a better description of the relaxation curve, as shown in Fig. 11. This term a = a φ is assumed to be equal for each phase and is obtained by an optimization procedure for each temperature level.

The curve gives the value of n = n φ , which is assumed to be the same for each phase. The K φ parameter depends on the phase, as shown in Eq. 19 and 20. Assuming K α I = K α II given by the bi-logarithmic curve, L (provided by Eq. come from the work of Wang for a titanium alloy [START_REF] Wang | Yield-Point Phenomena of Ti-20V-4Al-1Sn at 1073 K and Its Constitutive Modelling[END_REF]. Moreover, the parameters related to the dislocation densities ρ 0 , C ρ and f m0 are also found in the literature [START_REF] Wang | A Study of High Temperature Viscoplastic Deformation of Beta Titanium Alloy Considering Yield-point Phenomena[END_REF]. Lastly, the parameter a ρ is identified from the tests performed at 950 • C and 900 • C, its value is usually between 0.7 and 1.5 [START_REF] Yoshida | A constitutive model of cyclic plasticity[END_REF][START_REF] Hahn | A model for yielding with special reference to the yield-point phenomena of iron and related BBC metals[END_REF][START_REF] Hull | Introduction to Dislocations[END_REF]. The parameters λ, δ, κ and µ related to the volume fraction of mobile dislocation are also identified at 950 • C and 900 • C where the yield point phenomenon is observed, they are selected to fit the stress-strain curve and are determined by an optimization procedure. The volume fraction of mobile dislocations f m evolves between an initial value f m0 and an asymptotic one f ma [START_REF] Yoshida | A constitutive model of cyclic plasticity[END_REF][START_REF] Yoshida | A plasticity model describing yield-point phenomena of steels and its application to FE simulation of temper rolling[END_REF]. f m0 is constant with the temperature, whereas f ma decreases in order to take into account the decrease in the mobile dislocations density at lower temperatures. This evolution will only be activated at 950 • C and 900 • C in order to consider the yield point phenomenon. For the other temperature levels, this phenomenon vanishes by taking f ma = f m0 .

Hardening parameters

The parameters Q = Q φ and b = b φ of the isotropic hardening variable and the elasticity limit σ 0 = σ 0 φ are assumed to be equal for each phase. An optimization procedure is used for a cooling rate of 60 • C/min considering the whole database. Lastly, the model predictions for several cooling rates are illustrated in Fig. 14 at θ = 500 • C (a) and at θ = 20 • C (b).

All the values of the model parameters are given in Appendix B (Tables B.2-B.6).

Model extension on β-treated Ti-6Al-4V

In this section, the model is extended to a β-treated alloy. For this purpose, an experimental test campaign similar to the previous one was performed. It used the same starting samples obtained after the Forging operation (see Fig. 1) but the experimental procedure was changed by considering a solution annealing at between the model response and the experiment were less than 15% for the other test conditions. This is is quite acceptable since the only parameters that have to be changed concern microstructural features of the β-heat treated alloy.

Discussion

The quenching of industrial parts generates transient temperature variations that induce plastic straining due to thermal self-constraining. The degree of selfconstraining depends on the dimensions of a part and on the heat transfer mechanisms between the part and the quenching environment (for example oil or water quenching ...) that control the mean global cooling rate [START_REF] Bourouga | Roles of heat transfer modes on transient cooling by quenching process[END_REF]. In general the residual stresses are investigated at room temperature, and can be measured through X-Ray Diffraction or hole drilling methods. In-situ measurements of the residual stresses are not possible and the only rational manner is to use advanced thermo-mechanical modeling and numerical simulations analysis. Therefore relevant and reliable constitutive laws have to be developed. However, alloys such as Ti-6Al-4V alloy are very much prone to microstructural evolution at high temperature or during transient temperature-time conditions. In many approaches, the reliability of the constitutive laws is examined by laboratory testing of heat-treated alloys. The high temperature assessments are thus run on specimens heated up again to a prescribed temperature for mechanical testing. In the present investigation, the main objective was to assess and to model the thermo-mechanical behavior of Ti-6Al-4V alloy, through stepwisetemperature mechanical testing by first conducting an in-situ heat-treatment and then by quenching with a controlled rate to a prescribed temperature and finally by conducting the mechanical testing at this temperature. Such combining of timetemperature-mechanical scenarios are not commonly reported in the literature and often absent in the open literature. Although, many investigations deal with the mechanical behavior of Titanium alloys especially Ti-6Al-4V alloy, most of them consider the thermodynamic equilibrium conditions. However, some comparisons can be made with the present study considering the influence of the microstructure on the mechanical response of the material.

First, the phase analysis (Fig. 8) illustrates a drastic decrease of the β phase from 950 to 800 • C with a β phase around 78% at 950 • C. Similar results are found in the literature at this temperature [START_REF] Elmer | In situ observations of lattice expansion and transformation rates of α and β phases in Ti-6Al-4V[END_REF][START_REF] Lonardelli | In situ observation of texture evolution during α → β and β → α phase transformations in titanium alloys investigated by neutron diffraction[END_REF]. This means that the role of the β phase cannot be neglected and the plasticity induced has to be linked to this phase. The yield point phenomenon illustrated in Fig. 12 is observed on many Body-Centered Cubic (BCC) materials [START_REF] Hull | Introduction to Dislocations[END_REF][START_REF] Smallman | Modern Physical Metallurgy[END_REF] such as the β phase for titanium alloys. It can be related to the dislocations locked by the solute atoms then broken away from the pinning points at a high stress level. It is also associated with discontinuous yielding with the increase of new mobile dislocations generated from the grain boundary [START_REF] Momeni | Yield point phenomena in TIMETAL 125 beta Ti alloy[END_REF][START_REF] Wang | Yield-Point Phenomena of Ti-20V-4Al-1Sn at 1073 K and Its Constitutive Modelling[END_REF][START_REF] Duan | Flow behavior and microstructure evolution of TB8 alloy during hot deformation process[END_REF][START_REF] Li | Flow stress behavior and deformation characteristics of Ti-3Al-5V-5Mo compressed at elevated temperatures[END_REF]. The behavior model implemented in the present study is based on a rule of mixture between two mechanisms, one related to the α phase (divided in primary and secondary phases), the second to the β phase. At high temperatures, the constitutive equations associated with the β phase are predominant, and the formulation is based on the works performed by Yoshida et al on BCC materials [START_REF] Wang | Yield-Point Phenomena of Ti-20V-4Al-1Sn at 1073 K and Its Constitutive Modelling[END_REF][START_REF] Wang | A Study of High Temperature Viscoplastic Deformation of Beta Titanium Alloy Considering Yield-point Phenomena[END_REF][START_REF] Yoshida | A constitutive model of cyclic plasticity[END_REF][START_REF] Yoshida | A plasticity model describing yield-point phenomena of steels and its application to FE simulation of temper rolling[END_REF]. As discussed previously, these equations are modified to take into account some particular effects observed, such as the increase of the yield point with the strain rate and the dislocations rearrangement with a decrease in the dislocation density during dwell times (see Eq. 23).

Secondly, at lower temperatures, the effect of the α phase increases while that of the β phase decreases. Moreover, Fig. 4 shows a decrease of the α II -lamellar thickness with the cooling rate involving an increase of the flow stress (Fig. 5).

This result is induced by an increase of the α/β boundaries acting as obstacles to the dislocation movements. These results are also shown in several investigations [START_REF] Perdrix | Influence of nitrogen on the microstructure and mechanical properties of Ti-48Al alloy[END_REF][START_REF] Gil | Formation of α-Widmanstätten structure: effects of grain size and cooling rate on the Widmanstätten morphologies and on the mechanical properties in Ti6Al4V alloy[END_REF][START_REF] Semiatin | The effect of alpha platelet thickness on plastic flow during hot working of Ti-6Al-4V with a transformed microstructure[END_REF]. This effect is described in the constitutive equations through the viscous flow where the K Φ (Φ = {α I , α II }) parameter evolves with the thickness of the α II lamellae L (see Eq. 20), itself related to the cooling rate, or with the average size of the primary α nodules d α I (Eq. [START_REF] Mecking | Strain hardening and dynamic recovery[END_REF].

Finally, the strain rate sensitivity is reduced for temperatures inferior to 500 • C

[8] involving a significant hardening effect which is assumed similar for each phase.

On the other hand, Fig. 12 and 13 show an important strain rate effect for the temperatures exceeding 600 • C as mentioned in [START_REF] Vanderhasten | Ti-6Al-4V: Deformation map and modelisation of tensile behaviour[END_REF] compared to the hardening effect.

Conclusions

In the present work, an experimental device was developed to reproduce the mechanical behavior of Ti-6Al-4V throughout the die-forging operation. A nonunified behavior model was implemented and the following conclusions can be drawn.

• For a (α + β) dual-phase alloy, the phase transformation is greatly influenced by the cooling rate conditions, which themselves play an important role in the strain-stress response of the material.

• The non-unified behavior model is able to predict the mechanical behavior, assuming an initial phase proportion (α I , α II or β).

• Depending on the test temperature, the model gives a good prediction of the strain rate and hardening effects. Moreover, based on a modified Yoshida model formulation, it can describe the yield point phenomenon observed at high temperature and the static recovery effect exhibited during the dwell times.

• Lastly, the model was successfully extended to the behavior prediction of a β-treated Ti-6Al-4V alloy.

Appendix B. Model coefficients for α + β treated Ti-6Al-4V

Figure 2 :

 2 Figure 2: Starting microstructure of Ti-6Al-4V : (a) after Forging, (b) after time-temperature heat treatment corresponding to the Die-Forging (cooling rate: 60 • C/min) [Kroll Reagent | SEM | × 2000]

Figure 3 :

 3 Figure 3: Test procedure (a) In-situ heat treatment; (b) Isothermal loading path

Figure 4 :

 4 Figure 4: SEM observations for tensile tests performed at 700 • C and 20 • C for different cooling rates: 5 • C/min (left) , 60 • C/min (center) , 200 • C/min (right) [Kroll Reagent | SEM | × 2000]

  Image analysis was conducted to determine the α II lamellar thickness L ( θ = {60, 200} • C/min) and the α I nodule size ( θ = 5 • C/min). Fig. 6 illustrates the evolution of α II -lamellae thickness L for different cooling rates. As shown, this trend can be aligned to a power law. By plotting the curve given by Eq. 1 in a bi-logarithmic diagram, parameter B can be determined from the value of the slope. Regarding the α I nodules, the observations do not exhibit a significant evolution, regardless of the test conditions.The average size considered next is thus 15µm.

Figure 5 :

 5 Figure 5: Stress-Strain response for different cooling rates at 20 • C(a); and 700 • C(b) and a constant strain rate of 10 -2 s -1

Figure 6 :

 6 Figure 6: Evolution of the α II -lamellae thickness with the cooling rate

Figure 7 :

 7 Figure 7: SEM observations for tensile tests performed at 950 • C(a); and 800 • C(b) for a cooling rate of 60 • C/min[Kroll Reagent | SEM | × 2000]

Figure 8 :

 8 Figure 8: Evolution of the β-phase fraction with the temperature

Figure 9 :

 9 Figure 9: Stress-Strain response for constant strain (10 -2 s -1 ) and cooling (60 • C/min) rates at different temperatures θ ≤ 600 • C(a); θ ≥ 700 • C (b)

K

  α and n α are temperature-dependent parameters of the phase α = (α I , α II ). a φ defines the static recovery term in the hardening variables of each phase. Especially for the β-phase, D is a material parameter, b ρ is the Burgers vector, ρ m the density of mobile dislocations and M the Taylor factor.

Figure 10 :

 10 Figure 10: Temperature evolution of Young's modulus for α and β phases: (a) literature (b) values used in this study

1 )Figure 11 :

 111 Figure 11: Comparison between the relaxation range provided by the model and the experiment for a cooling rate of 60 • C/min and a temperature of 700 • C

4. 2 .

 2 Fig. 12 illustrates a comparison between simulation and experiment at 950 • C.At this temperature, the mechanism related to the β phase is predominant and the

Figure 12 :Figure 13 :

 1213 Figure 12: Yield point prediction at 950 • C and several strain rates (line: simulation, marker : experiment

Figure 14 :Figure 15 :

 1415 Figure 14: Computed Strain-Stress data (line) compared to Experimental results (marker ) for several cooling rates and at θ = 500 • C (a); θ = 20 • C (b).

Figure 16 :

 16 Figure 16: Computed Strain-Stress data (line) compared to Experimental results (marker ) for several cooling rates and at θ = 500 • C (a); θ = 20 • C (b).

Fig. 15 and

 15 Fig.[START_REF] Teixeira | Influence of the β → α+β transformation on the stresses and strains evolutions during quenching of the Ti17 alloy from the β phase field[END_REF] and Fig.16illustrate the comparison between the computed Strain-Stress data and experimental results in the case to the β-heat treated Ti-6Al-4V, at 950 • C and for several strain rates (Fig.15a), at intermediate temperatures (T={20, 500, 700} • C) and for several cooling rates (Fig.15b and 16a-b). Aside from the temperature of 950 • C where the stress levels are very low, discrepancies

Table B .

 B 2 illustrates the phase-independent model parameters, tables B.3 and B.4 the α-phase dependent model parameters and tables B.5 and B.6 β-phase dependent model parameters -1 ] 5 10 -1 2.5 10 -1 8. 10 -2 2 10 -2 8 10 -3 3 10 -3 2 10 -4 1.4 10 -4 4 10 -5

	Table B.2: Phase independent and temperature dependent model parameters
	θ [ • C]	950	900	800	700	600	500	400	300	20
	σ 0 [M P a]	1	1	3	10	34	145	327	386	650
	Q [M P a]	1	2	55	85	92	97	100	101	103
	b	1	5	190	317	400	425	433	434	435
	a [s n	3.3	3.4	3.6	4.9	7.8	9.8	11.2	11.7	12

Table B .

 B 3: α-phase and temperature dependent model parameters (1) Table B.5: β-phase and temperature dependent model parameters (1) Table B.6: β-phase dependent model parameters (2)b ρ [cm] M ρ 0 [cm -2 ] C ρ [cm 2 ]

	θ [ • C]	950 900	800 700	600 500	400	300	20
	E α [GP a]	35	47		60.7	70	77.4	86	85.1 84.1 109.5
	Z αII		0	0.22				0.44
	K 1	23.7	24		62.8 77.3 73.5 51.5 20.8 20.2 32.3
	K 2		95	96		251 309	294 208	83	81	131
		Table B.4: α-phase dependent model parameters (2)
			Z αI d αI [mm] n d	n L	B
			0.22	15.10 -3	0.5 0.105 66.7 10 -3
	θ [ • C]	950		900		800		700	600 500 400 300 20
	E β [GP a]	39.3		48.2		60.7		69	74	81	79	78	90
	D	181		182		370		472	493 372 160 148 290
	Z β	0.78		0.56					0.18
	f ma	5. 10 -3 1. 10 -3 5.5 10 -4 4.5 10 -4			4.10 -4
								f m0		λ	κ	µ	δ
	2.5 10 -8 2.76	6.4 10 10

2.1 10 11 4 10 -4 18 10 3 2 1.25 2

Acknowledgement

The authors very much acknowledge the financial support received through an FUI grant in the framework of the collaborative project TiMaS (Titanium Machining and Simulation), led by Airbus. The authors also gratefully acknowledge Figeac Aéro for the machining of samples.

Multi-axial formulation

Uni-axial formulation Yield criterion

Appendix C. Model coefficients updated for β treated Ti-6Al-4V


V Velay

G Bernhart

Z Zhang

L Penazzi

CYCLIC BEHAVIOUR MODELLING OF MARTENSITIC HOT WORK TOOL STEELS

Keywords: Tempered martensitic steels, stress-strain modelling, constitutive model identification, numerical simulation

In this paper, an unified elasto-viscoplastic behaviour model based on internal states variables, is investigated in order to describe the thermo-mechanical stress-strain fatigue response of tempered martensitic steels. This model includes an isotropic part (drag stress) describing the cyclic softening of martensitic steels. The memory effect is introduced to take into account the influence of the plastic strain range on the amount of the cyclic softening. The kinematic part (back stress) of the model allows the description of complex load conditions to which tool steels are subjected. Thus, strain recovery (Baushinger effect) and time recovery terms (cyclic behaviour including tensile and compressive dwell time and ratcheting) are considered. Moreover, the identification methodology of the model parameters from only two sets of experimental data is presented; the coefficients of the kinematic and isotropic parts are determined successively. The main difficulty consists in reaching a good description both of the cyclic behaviour for different strain rates and the ratcheting effect.

INTRODUCTION

Martensitic tool steels are used in forming processes like forging or extrusion for their good mechanical strength at high temperature combined with sufficient ductility. They undergo thermo-mechanical cyclic loads which are very hard to evaluate from an experimental point of view and whose levels strongly depend on the location on the structure. So, numerical simulation seems to be a significant way to reach this information in order to optimize the tools design and improve their lifetime.

Thus, a good understanding of the martensitic steel behaviour is necessary. The heat treatment to which they are subjected consists in annealing, austenitizing followed by a quenching and one or two tempering operations. This treatment leads to a complex microstructure [1]. The quench stage changes the austenite into martensite and the tempering (560°C) gives more ductility to the material [START_REF] Delagnes | Comportement et tenue en fatigue isotherme d'aciers à outils Z38CDV5 autour de la transition fatigue oligocyclique-endurance[END_REF]. At the end, the microstructure is composed of thin lathes where the dislocation density generated during the quench associated with carbides precipitation occurring during the tempering makes the microstructural investigation difficult.

Besides the microstructural aspect, the different kinds of loads induced by the forming process itself have an influence on the behaviour. Thus, in this work, a cyclic constitutive behaviour model adapted to martensitic steels and industrial forming conditions is investigated in the framework of the thermodynamics of irreversible processes.

EXPERIMENTAL BEHAVIOUR AND MODELLING

Fatigue experimental behaviour of martensitic steels

The 42 HRC (Rockwell hardness) 55NiCrMoV7 martensitic steel is investigated in isothermal fatigue conditions for a temperature range between 20 and 500° C. The continuous softening from the first cycle until rupture is typical of such materials. If the semi-stress amplitude is represented versus the number of cycles, this softening can be divided in three successive stages (see Figure 1). Indeed, the strong softening stage occurring during the first hundred cycles is followed by a pseudo-stability one (weak softening) during the largest part of the tool life. At last, crack propagation occurs, defined by a fast decrease of the stress amplitude before the rupture [START_REF] Delagnes | Comportement et tenue en fatigue isotherme d'aciers à outils Z38CDV5 autour de la transition fatigue oligocyclique-endurance[END_REF].

Constitutive model selection

Selection of a constitutive model has to take into account the specificity of the tempered martensitic steel behaviour as well as the load paths during the forming process.

First, the fast evolutions during each inelastic transient can be described through kinematic variables (back stresses); the slow evolutions during the successive cycles corresponding to the cyclic softening of the material may be reproduced through an isotropic hardening variable (drag stress) [START_REF] Nouailhas | Modélisation de l'écrouissage et de la restauration en viscoplasticité cyclique[END_REF]. Second, model has to reproduce material response under fatigue-relaxation loads and structure effects, which induce non-symmetrical stress-controled loads leading to ratcheting effects. This requires in general a more complex formulation of the kinematic hardening.

During the last years, several models have been formulated for such goals. The Armstrong & Frederick modelling introduces an evanescent memory term called dynamic recovery or strain recovery, which corresponds to a time independent recovery effect of the structure and simply describes an instantaneous recall effect. To take into account the dwell time and ratcheting effect, several methods have been followed: one of them implies a modification of the dynamic recovery term introducing a threshold stress [START_REF] Chaboche | Modelling of the cyclic response and ratcheting effects on Inconel 718 alloy[END_REF], another one defines two stress states leading to two strain mechanisms and creating interactions between the different hardening variables [START_REF] Cailletaud | Study of plastic/viscoplastic models with various inelastic mechanisms[END_REF].

In this work, a static recovery term is added in the kinematic hardening variable [START_REF] Malinin | Theory of creep with anisotropic hardening[END_REF]. It is completely time-dependent and describes a slow recovery of the steel crystalline structure which occurs at high temperature by annihilation of dislocations and relaxation of internal stresses. Moreover, as the ratcheting strain is over-estimated, a third kinematic variable is introduced [START_REF] Chaboche | Cyclic plasticity modelling and ratcheting effects[END_REF]. At last, all the strain levels reached during the different tests may be described by the three kinematic variables.

Slow cyclic softening is described through a drag stress including two terms, which describe the two different softening rates of the material [1]. At last, the influence of the plastic strain level on this softening range is taken into account through an asymptotic term of the isotropic variable [START_REF] Chaboche | Constitutive Equations for Cyclic Plasticity and Cyclic viscoplasticity[END_REF].

Model formulation

Model formulation is based on the thermodynamics of irreversible processes; it is defined from two potentials, the free energy Ψ and the dissipation potential Ω .

Free energy formulation Ψ defining the associated variables. It is divided into an elastic and viscoplastic part 2 .

A A el el el vp el ; : 2 1 ; ε ε ρ ρ ρ ρ = Ψ Ψ + Ψ = Ψ
is the elastic stiffness tensor

2 2 2 2 1 1 3 1 2 1 2 1 : 3 1 r bQ r Q C i i i i vp + + = Ψ ∑ = α α ρ
Thus, state laws can be written:

) 1 ( ; ; ; 3 , 2 , 1 ; 3 2 ; 2 2 2 1 1 1 r bQ R r Q R i C X A i i i el = = = = = α ε σ
σ defines the stress state and vp ε the viscoplastic strain.

2 Note: T the fourth order tensors and T the second order tensors. Dissipation potential Ω defining the internal variables. It is divided in two components.

∑ = = 3
vp Ω is the viscoplastic potential, defined as a function of the yield surface f and R Ω is the recovery potential.

) (u uH u =
with H the Heavyside function.

) 2 ( ; ; 1 ) ( ; ) ( 0 2 1 3 1 1 R R R X J f K f n K f f i i n vp R vp - - -       - = + = Ω Ω + Ω = Ω ∑ = + σ with: ( ) σ σ σ σ ′ ′ ′ = ; : 2 3 J
is the deviatoric part ofσ .

( )

∑ = + + = Ω 2 1 1 1 i m i i i i R i M X J m M 0 R defines the initial elastic limit of the material, i m n K ,
, and i M are material and temperature-dependent parameters.

As a consequence, evolution equations of the internal variables can be written:

( ) ( ) ( ) 1 ; r ; ) 3 ( ; 2 , 1 ; 2 3 2 3 ; 2 3 ; 2 2 1 3 br p r p i M X J X J X X C D p n p K f p vp m i i i i i i i vp i vp n i - = = = = - - = = = ε α ε α ε
where n is the unit normal to the yield surface 0 = f , which defines the elastic domain ( )

0 < f
and p the cumulated plastic strain.

We have:

1 ; = ∂ ∂ ∂ ∂ = n f f n σ σ .
p & is the cumulated plastic strain rate, which can be written:

vp vp p ε ε & & & : 3 2 =
The memory effect is written as usual; evolution rules for q and ξ are:

( ) ) 4 ( ; : ) ( ) 1 ( 2 3 ; : ) ( ; * * * p n n n F H p n n F H q q η ξ η ξ ε - = = - - =
where * n is the unit normal to the memory surface 0 = F .

So, we have:

1 ; * * = ∂ ∂ ∂ ∂ = n F F n p p ε ε
The parameter η induces a progressive memorisation.

The influence of this effect on the material softening is translated through the asymptotic value of the isotropic hardening variable:

q e Q Q q Q µ 2 0 2 ) ( - ∞ + =
Note that the positivity of the intrinsic dissipation D verifies the thermodynamic principles:

i i i vp I A D & & : : ∑ - = ε σ i A and i
I represent the associated and internal variables.

Finally, we get:

( ) ( ) ( ) 0 . 2 1 2 2 2 2 0 ≥                   + + + + = ∑ = i i m i i i i i X J M X J X J C D Q R R p p f D i & &
Thus, the complete formulation requires the identification of 19 coefficients.

One-dimensional model formulation. Parameters of the three-dimensional model developed previously have been determined from push-pull tests, so it is necessary to write the formulation in that case.

The expression of the yield surface is then given by (2) :

) 5 (

0 1 2 3 1 R R R X f i i - - - - = ∑ = σ
The equations (1), giving the expression of the state laws, can be written in the onedimensional case:

) 6 ( ; ; ; 3 , 2 , 1 ; 2 2 2 1 1 1 r bQ R r Q R i C X E i i i el = = = = = α ε σ E represents the Young modulus.
Moreover, the equations (3), providing the internal variables evolution, are given by: ); 1 ( ; ;

) 7 ( ; 2 , 1 ; ; 2 2 1 3 1 3 1 br p r p r i X M X p X C D p X sign p vp i m i m i i i i vp i i i vp vp i i - = = = = - - =       - = = - = ε α ε α σ ε ε
The expression of the asymptotic value of the isotropic hardening is always given by: ( )

q e Q Q q Q µ 2 0 2 - ∞ + =
At last, the memory effect, given by ( 4), is expressed by: ( )

( ) ( ) ( ) ξ ε ξ ε σ η ξ ξ ε σ η ξ ε -         -       - - =         -       - = - - = ∑ ∑ = = vp vp i i vp i vp sign X sign p X sign p q q F 3 1 3 1 1 ) 8 (

EXPERIMENTAL TEST DECRIPTION

Two types of fatigue tests are performed on a 55NiCrMoV7 tool steel (42 HRC hardness) for the temperatures 20, 300, 400 and 500° C.

The first one (see Figure 2 and Figure 3) may be divided in two different steps:

-the first step is a symmetrical total strain controled push-pull low cycle fatigue test, with a fixed strain amplitude ∆ε t =1.6 %, itself divided in three substeps.

-Substep 1 consists in a number of fatigue cycles so as to reach a cumulated plastic strain of 4 at a constant strain rate of 10 -2 s -1 . -During the substep 2, strain rate is varied from 10 -2 to 10 -3 and 10 -4 s -1 ; and three cycles are performed at each strain rate. -At last, fatigue-relaxation cycles are included with a strain rate of 10 -2 s -1 and where relaxation-time is varied from 30 s (3 cycles) to 90 s and 600 s (2 cycles for each one). -the second step consists in a non-symmetrical stress controled fatigue test at a constant stress rate of 100 MPa.s -1 . Ten cycles are performed.

The second test (Figure 4) consists in a symmetrical total strain fatigue test at constant strain rate 10 -2 s -1 , in which strain amplitude is varied from ± 0.6, to ± 0.7, ± 0.8 and ± 0.9 before coming back to ± 0.7 %. The number of cycle for each strain amplitude is selected so as to reach a cumulated plastic strain of 1 during the number of cycles performed at each strain amplitude.

Semi-stress amplitude

Number of cycles

Part of test considered to identify the isotropic component

Part of test considered to identify the kinematic component : strain-controled loads ε = ±0.8%

-3 cycles at 10 -2 s -1 -3 cycles at 10 -3 s -1 -3 cycles at 10 -4 s -1

-3 cycles at 10 -2 s -1 ; t d = 30 s -2 cycles at 10 -2 s -1 ; t d = 90 s -2 cycles at 10 -2 s -1 ; t d = 600 s 10 stress-controled cycles:

σ mean = 225 Mpa σ& = 100 Mpa.s -1 N = 26 cycles ε ε & , ∆ non constants N ≈ 1000 cycles ε ε & , ∆ fixed 780

MODEL PARAMETER IDENTIFICATION

The parameter identification of the behaviour model is presently performed with the SiDoLo software [START_REF] Pilvin | SiDoLo version 2.4. Notice d'utilisation[END_REF]. The yield surface [START_REF] Chaboche | Modelling of the cyclic response and ratcheting effects on Inconel 718 alloy[END_REF], the state laws [START_REF] Cailletaud | Study of plastic/viscoplastic models with various inelastic mechanisms[END_REF], the evolution equations [START_REF] Malinin | Theory of creep with anisotropic hardening[END_REF] and the memory effect [START_REF] Chaboche | Cyclic plasticity modelling and ratcheting effects[END_REF] are written on a one-dimensional form for the identification stage.

Identification is performed in two successive stages:

-kinematic and viscous parameter identification is completely performed with results of the first type of test (excluding substep 1). Knowing that during optimisation processes many local minima may be reached, a progressive process was implemented. First, results of the substep 2 are used to get an approximation of the viscous parameters. Then, substep 3 results are included to get a first level of the kinematic parameters (strain and time recovery); at this time, viscous parameters are constrained. At last, stress controled test are included to get the final set of values especially those for the second and the third kinematic strain recovery variables, which have a significant impact on ratcheting strain. At this point, the elasticity limit is an apparent value corresponding to the stabilised material.

-isotropic parameter identification is performed assuming all kinematic and viscous parameters determined previously fixed. Results of substep 1 of test 1 and test 2 are taken into account. The usefulness of test 2 is related to the strain memory parameters indentification which requires softening information for various plastic strain amplitudes.

T = 500° C

Thus, two different tests are sufficient (one sample per test) to determine completely all the coefficients defined in the model formulation. At the end of this process, one set of parameters is determined for each temperature.

As shown in the following figures for a temperature of 500° C, such a progressive process allows to determine a set of parameters which gives a good fitting of experimental results. Figures 5, 6 and 7 simulations take mainly into account the viscous and kinematic parameters; whereas figure 8 shows that the isotropic variable allows a good agreement between simulated and experimental softening. Similar results were obtained for the other temperatures investigated.

CONCLUSION

An elasto-viscoplastic cyclic model was formulated to describe the behaviour of tempered martensitic tool steels subjected to typical loads induced by industrial processes. For the temperature range between 20 and 500° C, this model is able to describe both the cyclic softening of the 55NiCrMoV7 steel and ratcheting phenomenon induced by structure effects inside the tool. Moreover, some aspects linked with the forming process like strain rate variation or the description of the dwell times, are equally considered. Besides, note that important microstructure evolutions occur when the temperature level exceed 560° C (tempering temperature), and of course, in that case, the limits of the model are reached, indeed, the previous formulation is unable to describe such evolutions. The following work will be focused on model validation. The different parameters, even if only are identified from one-axial tests, are adequate to get a good description of the three dimensional behaviour of the material [START_REF] Cheikh | Elastoviscoplasticité à température ambiante[END_REF]. After implementation of the three dimensional model in ABAQUS TM software, axisymmetric samples with various stress concentrations will be simulated and results compared to experimental ones.

Fig. 1 :

 1 Fig.1: Softening stages of the 55NiCrMoV7 martensitic steel[START_REF] Bernhart | High temperature low cycle fatigue behaviour of martensitic forging tool steel[END_REF]

X

  defines the kinematic stress of the hardening and i R the isotropic one. They are respectively the associated variables to the internal ones i α and i r .

  b are material and temperature-dependent parameters.

Fig. 2 :Fig. 3 :

 23 Fig. 2 : Experimental continuous softening (step 1, substep 1)

Fig. 4 :

 4 Fig. 4 : Experimental softening with different strain amplitude levels

Fig. 5 :Fig. 6 :Fig. 7 :Fig. 8 :

 5678 Fig.5 : Strain-controled fatigue test

  [1] Z.Zhang, D.Delagnes, G.Bernhart (2002) Anisothermal Cyclic Plasticity Modelling of Martensitic Steels. International journal of fatigue. Under press.

Corresponding author Tel.: +33-5-63-49-31-68; Fax: +33-5-63-49-32-42; Email: vincent.velay@enstimac.fr

& & & & & & & & & & & &

& & & &

& & & & & & & & & & & & &

& & & &