T Jailin

Nicolas Tardif

J Desquines

Philippe Chaudet

Michel Coret

Marie-Christine Baietto

V Georgenthum

M.-C Baietto

FEMU based identification of the creep behavior of Zircaloy-4 claddings under simulated RIA thermo-mechanical conditions

Keywords: RIA, Identification, Creep, Zircaloy-4, Full-field measurements, FEMU

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

Introduction

The Reactivity Initiated Accident (RIA) is one of the design basis accidents studied for Pressurized Water Reactors. It results from a sudden insertion of reactivity within the nuclear core. Regarding the fuel rod behavior, it can be split in two consecutive phases: the Pellet Clad Mechanical Interaction (PCMI) phase, followed by the post-Departure from Nucleate Boiling (post-DNB) phase. During this latter phase, the temperature of the fuel claddings may increase up to temperatures greater than 900 °C with very fast heating rates (around 1000 °C/s), meanwhile the claddings are internally pressurized by filling gas and fission gas release [START_REF] Bessiron | Modelling of clad-to-coolant heat transfer for RIA applications[END_REF]. For safety purposes, the least penalizing transients leading to fuel rod failure must be determined, which requires a precise knowledge of the cladding constitutive law in all of these conditions. Identifying a constitutive model of the fuel cladding response in post-DNB conditions is then of great interest.

It is well-known that the thermo-mechanical properties of a material are strongly linked to its microstructure, and that the microstructure depends on the fabrication process. From this, it follows that it is sometimes necessary to work directly on structural samples, in order to take into account the effects of geometry and fabrication process of the specimen on its thermomechanical behavior. On another hand, the loading conditions may also play an important role in the material response. As an 25 example, the heating rate may affect the phase transformation of the material [START_REF] Porter | Phase transformations in metals and alloys[END_REF][START_REF] Forgeron | Experiment and Modeling of Advanced Fuel Rod Cladding Behavior Under LOCA Conditions: Alpha-Beta Phase Transformation Kinetics and EDGAR Methodology[END_REF][START_REF] Jailin | Experimental study and modelling of the phase transformation of Zircaloy-4 alloy under high thermal transients[END_REF] and thus may affect its mechanical properties [START_REF] Forgeron | Experiment and Modeling of Advanced Fuel Rod Cladding Behavior Under LOCA Conditions: Alpha-Beta Phase Transformation Kinetics and EDGAR Methodology[END_REF][START_REF] Jailin | Mechanical behavior of as-fabricated Zircaloy-4 claddings under the simulated thermo-mechanical post-DNB conditions of a Reactivity Initiated Accident (RIA)[END_REF][START_REF] Jailin | Thermo-mechanical behavior of Zircaloy-4 claddings under simulated post-DNB conditions[END_REF]. Therefore, the identification of a constitutive model must be done in conditions as close as possible to real life conditions in order to avoid extrapolations that may lead to 30 uncontrolled discrepancies. Following this idea, an innovative device was developed recently in [START_REF] Jailin | Thermo-mechanical behavior of Zircaloy-4 claddings under simulated post-DNB conditions[END_REF] to reproduce simulated post-DNB thermo-mechanical conditions directly on cladding sections made of Zircaloy-4. It is worth noting that the effects of irradiation-induced defects were not considered in this 35 work. Such effects are very complicated to study, especially for safety reasons. Even if irradiation defects tend to recover at high temperatures [START_REF] Torimaru | Changes in mechanical properties of irradiated Zircaloy-2 fuel cladding due to short term annealing[END_REF][START_REF] Ribis | Experimental study and numerical modelling of the irradiation damage recovery in zirconium alloys[END_REF], nothing indicates that this recovery is fast enough to have no impact during the post-DNB phase of a RIA. Further studies should be carried out to investigate the 40 influence of irradiation defects after high thermal transients.

In [START_REF] Jailin | Thermo-mechanical behavior of Zircaloy-4 claddings under simulated post-DNB conditions[END_REF], creep ballooning tests were performed on fuel cladding sections under conditions representative of the post-DNB phase of a RIA: the tubular sample was first internally pressurized, followed by heating until high temperature with 45 fast heating rate (≃ 1200 °C/s). The temperatures of interest ranged from 840 to 1020 °C. The ballooning deformation and the temperature at the sample surface were measured by stereo-correlation and near infrared thermography, respectively. The sample, made of Zircaloy-4 alloy, shows a (α → β) phase 50 transformation at high temperature from around 800 to 1000 °C [START_REF] Forgeron | Experiment and Modeling of Advanced Fuel Rod Cladding Behavior Under LOCA Conditions: Alpha-Beta Phase Transformation Kinetics and EDGAR Methodology[END_REF][START_REF] Banerjee | Phase Transformations: Examples from Titanium and Zirconium Alloys[END_REF][START_REF] Nguyen | Microstructure Changes during Fast β Cycles of Zirconium Alloys[END_REF][START_REF] Jailin | Experimental study and modelling of the phase transformation of Zircaloy-4 alloy under high thermal transients[END_REF]. This transformation, studied and modeled in [START_REF] Jailin | Experimental study and modelling of the phase transformation of Zircaloy-4 alloy under high thermal transients[END_REF] from equilibrium to 2000 °C/s, shows a strong dependence to the heating rate. The creep results of [START_REF] Jailin | Thermo-mechanical behavior of Zircaloy-4 claddings under simulated post-DNB conditions[END_REF] evidenced a close link between the phase transformation and the mechanical properties, as well as an important impact of the heating rate on the mechanical properties of the material. However, due to structural effect the stress state could not be accurately estimated to identify with sufficient accuracy a constitutive creep law.

To the authors knowledge no constitutive model has yet been identified in post-DNB conditions, i.e. with very fast heating rate, mainly due to experimental and identification difficulties.

Based on the finite element model updating (FEMU) method [11,[START_REF] Réthoré | Robust identification of elasto-plastic constitutive law parameters from 690 digital images using 3D kinematics[END_REF][START_REF] Mathieu | Estimation of Elastoplastic Parameters via Weighted FEMU and Integrated-DIC[END_REF][START_REF] Pottier | Contribution of heterogeneous strain field measurements and boundary conditions modelling in inverse identification of material parameters[END_REF] this paper aims to identify a creep law weakly coupled to the phase transformation, using the experimental results of [START_REF] Jailin | Experimental study and modelling of the phase transformation of Zircaloy-4 alloy under high thermal transients[END_REF][START_REF] Jailin | Thermo-mechanical behavior of Zircaloy-4 claddings under simulated post-DNB conditions[END_REF]. A single and continuous law is developed to reproduce the experimental creep results obtained in simulated post-DNB conditions from 840 to 1020 °C. The impact of the heating rate on the creep properties is also discussed using existing results from the literature. The present work also details a procedure to model the experiments, since the loading conditions were only known in a restricted region of the samples during the tests. The 3D-modeling, based on a Love-Kirchhoff assumption through the thickness, is presented as well as its sensitivity to input parameters to ensure the efficiency of the identification algorithm. This paper is decomposed as follows. First, the experimental results, the modeling of the experiments and the FEMU method are presented. The identification procedure is then validated on a virtual test case representative of the experiments. In the fourth part a creep law coupled to the phase transformation is identified based on the experimental results of [START_REF] Jailin | Thermo-mechanical behavior of Zircaloy-4 claddings under simulated post-DNB conditions[END_REF]. Finally, an extension to higher time scale is proposed in Norton's law by taking into account the grain size contribution to model the non linearity in the material response.

Experimental results, material and methods

Experimental device, specimens and results

The so-called ELLIE test facility and the experimental results are here briefly summarized. More information can be found in [START_REF] Jailin | Thermo-mechanical behavior of Zircaloy-4 claddings under simulated post-DNB conditions[END_REF] and [START_REF] Campello | Validation of a multimodal set-up for the study of zirconium alloys claddings' behaviour under simulated LOCA conditions[END_REF].

The specimens are 90 mm long fuel cladding sections made of stress relieved annealed Zircaloy-4. The tubes have an external diameter of 9.5 mm and a wall-thickness of 0.57 mm. The samples are loaded on the ELLIE device, which enables reproducing simulated post-DNB conditions. The experimental procedure can be described as follow : (i) the samples are mechanically loaded with an internal pressure and a compressive force, (ii) when the mechanical loading is stabilized the samples are heated using an induction system until a regulation temperature with a heating rate of around 1200 °C/s, (iii) the thermal and the mechanical loads are maintained to observe the creep deformation of the sample. Around twenty tests were performed in these simulated RIA conditions with temperatures of interest ranging from 840 to 1020 °C and with two relative levels of pressure of 7 and 11 bar. One additional experiment was carried out with a pressure of 9 bar to validation purposes, as presented in the following. All the experiments were performed within an enclosure where an argon flush was set up during the tests to prevent the samples from oxidation.

Figure 1 presents how the experimental results were ex-110 tracted from the tests. The thermal and the kinematics fullfields were measured by near infrared (NIR) thermography and stereo-correlation (3D-DIC), respectively. A methodology, presented in [START_REF] Jailin | Coupled NIRT / 3D-DIC for a FEMU Identification of the Thermo-Mechanical Behavior of Zr-4 Claddings Under Simulated Reactivity Initiated Accident[END_REF], enabled the two kinds of fields to be obtained using the same two CMOS cameras. Thus, the displacement 115 evolution, as well as the temperature evolution, were obtained on the whole region seen by the cameras. Using these data, the mean creep rates were computed as a function of the mean temperature at each node of the correlation mesh and for each test during the first ten seconds of the experiments, which is 120 the approximate maximum duration of the boiling crisis of a RIA. Gathering the results of all the tests in one single graph, a complex creep behavior has been evidenced. Three different creep domains have been identified from 840 to 1020 °C. The first one ranged from 840 to 910 °C. In this 125 domain, the β fraction is low (⩽ 10%), the grains are small and equiaxed, and the creep rate increases with the temperature. In the second domain, defined from 910 to 970 °C, the creep rate decreases while the temperature increases. The grain size remains small and the β fraction ranged from around 10 to 100%.

130

Because the temperature was maximal at the middle of the sample and decreased toward the jaws, two areas affected by higher creep deformation than elsewhere were obtained on the specimens located from either side of the hottest region. It resulted in the so called "butternut shape" profile on the samples at the 135 end of the experiments. The third domains corresponds to the pure β domain at 970 °C. In this domain the creep rate increases anew with the temperature. As the β grain growth is not blocked anymore by the remaining α phase, big β grains are observed in this domain. In [START_REF] Jailin | Thermo-mechanical behavior of Zircaloy-4 claddings under simulated post-DNB conditions[END_REF], a uniform stress in the two phases was 140 assumed to estimate the creep behavior of the material in the two-phase domain. However, due to structural effects that may take place within the samples during the experiments, the stress state could not be accurately estimated, at least with enough confidence to identify a constitutive law.

Material behavior

Phase transformation

Zircaloy-4 presents a hexagonal close-packed α phase stable under around 800 °C, and a body centered cubic β phase stable above around 1000 °C. Between these two temperatures, the 150 two phases coexist within the material. The phase transformation of the material is computed using the model characterized in [START_REF] Jailin | Experimental study and modelling of the phase transformation of Zircaloy-4 alloy under high thermal transients[END_REF], which is based on a Leblond's equation [START_REF] Leblond | A new kinetic model for anisothermal met-allurgical transformations in steels including effect of austenite grain size[END_REF]. It reads:

d f β dt =          f eq (T ) -f β τ(T ) if f β > F 500 (T ) F 500 (T ) -f β dt otherwise. ( 1
)
where T is the temperature, f β is the fraction of β, f eq (T ) is the fraction of β at equilibrium for a temperature T , F 500 (T ) is a sat-155 uration function describing the fraction of β at a temperature T during a heating where the heating rate is greater than 500 °C/s, and τ(T ) represents the characteristic time of transformation. τ(T ) is defined using an Arrhenius-type equation such as:

τ(T ) = k 0 .e - Q f R.T -1 , (2)
where k 0 is a kinetic prefactor, Q f is an activation energy and R is the perfect gas constant. The equilibrium function f eq reads:

f eq (T ) = 1 -e -[C.(T -T 0 )] n , (3)
where C, T 0 and n are three material constants characterizing the rate of transformation during heat up phase. The function F 500 is defined using the same formulation:

F 500 (T ) = 1 -e -[C 2 .(T -T 2 )] n 2 , (4)
where C 2 , T 2 and n 2 are three constants describing the fastest transformation rate observed in the material. Above 500 °C/s heating rate has no more influence on the phase transformation kinetics.

The values of the parameters of Equations ( 2), ( 3) and ( 4) are given in Table 1.

Mechanical behavior

τ(T ) ln(k 0 ) 79.62 Q f (kJ.mol -1 ) 764.2 C (°C -1 )
6.89 10 -3 f eq (T )

T 0 (°C) 750.0 n 4.14 2), ( 3) and ( 4).

C 2 (°C -1 ) 1.36.10 -2 F 500 (T ) T 2 (°C) 865.4 n 2 4.18
Since the deformation levels reached during the tests can be important, true stress and strain definitions are used in this study. The total deformation of the material can be decomposed 175 such as:

ε = ε e + ε th + ε vp , (5)
where ε, ε e , ε th , and ε vp are the total, elastic, thermal and viscoplastic deformations, respectively. The elastic behavior is described using a isotropic description with a temperature-dependent Young's Modulus E described in 180 Equation ( 6) and a Poisson's ratio of 0.342 [START_REF] Hagrman | MATPRO-Version 11: A handbook of materials properties for use in the analysis of light water reactor fuel rod behavior[END_REF].

E(T ) = 116.1 -59.10 -3 .(T + 273.15) [GPa], (6)
where the temperature T is expressed in °C.

The isotropic thermal deformation is described using a coefficient of thermal expansion (CTE) associated to each phase. The CTE of the α and β phase are set equal to 9.6.10 -6 °C-1 and 1.32.10 -5 °C-1 , respectively, according to [START_REF] Jailin | Experimental study and modelling of the phase transformation of Zircaloy-4 alloy under high thermal transients[END_REF]. The reference temperature at which the thermal strain is considered to be zero is set in each test to be the temperature obtained from the first image of the test by near infrared thermography.

Based on [START_REF] Lemaitre | Mechanics of solid materials[END_REF], the visco-plastic strain rate tensor εvp is defined such as:

εvp = 3 2 ṗ σ ′ σ vM , (7)
where σ vM is the von Mises stress, σ ′ is the deviatoric stress tensor and ṗ is the accumulated plastic strain rate, referred for convenience as the creep rate hereinafter.

The creep rate ṗi of each phase i is defined using a Norton's formulation commonly used to describe the steady stage of creep in zirconium alloys [START_REF] Campello | Identification of the steady-state creep behavior of Zircaloy-4 claddings under simulated Loss-Of-Coolant Accident conditions based on a cou-725 pled experimental/numerical approach[END_REF][START_REF] Kaddour | Experimental determination of creep properties of zirco-730 nium alloys together with phase transformation[END_REF][START_REF] Kassner | Fundamentals of Creep in Materials[END_REF][START_REF] Hayes | Mechanical and Creep Behavior of Advanced Materials[END_REF], which reads:

ṗi = A i σ n i vM exp - Q i RT , i = α; β (8)
where A i is a constant, n i is the Norton's exponent, Q i is an activation energy. The material behavior in the two-phase domain is modeled 200 assuming a serial distribution of the two phases [START_REF] Jailin | Thermo-mechanical behavior of Zircaloy-4 claddings under simulated post-DNB conditions[END_REF][START_REF] Qu | Fundamentals of micromechanics of solids[END_REF], which can be formulated such as:

Σ = σ α = σ β (9)
ṗ = ṗα .(1 -f β ) + ṗβ . f β ( 10
)
where Σ, σ α and σ β are the homogenized stress, the stress in the α phase and the stress in the β phase, respectively. f β is the β phase fraction, ṗ, ṗα and ṗβ are the homogenized 205 accumulated plastic strain rate, the accumulated plastic strain rate in the α phase and the accumulated plastic strain rate in the β phase, respectively.

It follows that 6 parameters {A i ; n i ; Q i } have to be identified 210 in the domains i = {α; β}.

It is worth precising that three assumptions were used to model the material behavior in this study. (i) It is assumed that the stress state does not influence the phase transformastress for instance. (iii) The behavior of the α and β phase are considered to follow the same law in the single and two-phase domain while their microstructure and chemical composition may be different.

The FEMU method 225

The identification problem is solved using the FEMU method, which consists in comparing experimental to finite element (FE) results in order to identify a set of parameters λ. A finite element modeling is developed to numerically reproduce the experiments, and of course a constitutive law where the set 230 of parameters λ is sought. The method also requires the definition of a cost-function, which is commonly described using a mean-square formulation on the displacements as follow:

η 2 (λ) = 1 N.γ 2 (U exp -U num (λ)) t .(U exp -U num (λ)), (11)
where U exp gathers the experimental radial displacements measured by stereo-correlation, U num (λ) gathers the numerical ra-235 dial displacements obtained with the set of parameters λ, N denotes the number of data extracted from DIC and γ is the standard deviation of noise on the measured displacements. This cost function is expected to be equal to unity if the errors are solely due to noise [START_REF] Mathieu | Estimation of Elastoplastic Parameters via Weighted FEMU and Integrated-DIC[END_REF]. Using the 3 directions of the displace-240 ments in the cost function didn't induce any differences in the results since the hoop and axial displacements are much less sensitive than the radial ones to ballooning deformations. The minimization of this cost-function can be resolved by an iterative Gauss-Newton algorithm such as:

245 δλ = H -1 J t (U exp -U num (λ)), (12)
where δλ, is the parameter increment, J is the Jacobian matrix gathering the sensitivity fields (δU num = Jδλ) and H is the approximated Hessian computed such as H = J t J.

The optimization is considered as converged when the costfunction evolution and the parameter evolution become lower 250 than 1% for an iteration. The sensitivity of the model to the parameters is naturally computed during the minimization process. It is a useful tool to ensure the relevance of the experiments performed to the constitutive model identification. This will be developed in the following.

Finite element modeling

The thermal loading was not axisymmetric during the experiments. The temperature and the displacements are only known in the region of interest (ROI) at the external surface of the sample. It follows that only this ROI was modeled to numerically 260 duplicate the experiments. The DIC surface mesh is "extruded" toward the radial direction to obtain a 3D mesh as shown in Figure 2. Two elements at the boundary of the correlation mesh are not extruded, in order to limit boundary errors and compute accurately the surface normal. For all the tests, the FE mesh of the 265 ROI represents a section of about 40°of arc length and 9 mm in height. The commercial software Abaqus is used to perform the FE analysis and to account for the geometric non-linearity. Six incompatible mode eight-node brick "C3D8I" elements [START_REF] Simo | A class of mixed assumed strain methods and the method of incompatible modes[END_REF] are used through the thickness to obtain a converged mesh.

270

The temperature obtained at the surface of the samples by NIR thermography is imposed at each time step to the FE nodes. The induction heating system led to a uniform temperature across the thickness so the temperature imposed at the surface is propagated through the sample thickness. The internal pressure measured during the test is applied on the internal surface of the tube. The outer pressure during the tests being the atmospheric one, the external surface of the cladding is then considered as a free edge. The radial, hoop and axial displacements (U r ; U θ and U z , respectively) measured by 3D-DIC are applied to the master nodes depicted in red in Figures 2 and3. These master nodes correspond to the nodes of the external surface at the ROI boundaries. It is obvious that the external section cannot neither be let free nor be constrained by propagating the displacements through the thickness, like done for the temperature. To overcome this, a Love-Kirchhoff kinematic is imposed to the external section of the ROI. The slave nodes, depicted in blue in Figure 3 The displacements obtained by 3D-DIC were noisy, with noise of around 1 µm for the in-plane displacements and about 5 µm for the out-of plane displacements. To avoid stress con-295 centration and convergence problems the displacements at the edge of the ROI are first smoothed using polynomial functions such as:

H(θ, z) = 5 ∑ i=0 i ∑ k=0 Ξ (i-k)k .θ i-k .z k (13)
where θ and z are the angular and axial coordinates of the nodes, respectively. Ξ i j are the polynomial coefficients. 3. Virtual test: performance evaluation

Description of the virtual case

To validate the identification method a virtual test case is setup. Five calculations are performed where the tubular sample is entirely modeled (cf. Figure 4). 6 "C3D8I" elements are 305 used through the tube thickness. The axial temperature is imposed using a quadratic function knowing that the temperature is maximal at the center and minimal at the jaws where it is set to 500 °C. A circumferential thermal gradient of 25 °C is imposed to better represent the experiments. The sample is then 310 internally pressurized. The visco-plastic behavior of the material is modeled using the Norton's law described in Equation ( 8 Table 2: Loading conditions of the virtual case. T max corresponds to the maximal temperature at the center of the sample and P i correspond to the internal pressure.

The displacements and the temperature are extracted from a 320 region of 40°of arc length and 9 mm in height at the center of the sample to simulate the experimental results U exp . To be representative of the experiments, white noise of 1 µm and 5 µm is added on the in-plane and out-of-plane displacements, respectively, to challenge the convergence stability of the algorithm.

325

The ROI is then modeled following the procedure described in section 2.4. The set of initializing parameters of the creep law is finally strongly perturbed leading to the set λ 0 with ln(A)= 4, n = 1.5 and Q = 120 kJ.mol -1 .

Sensitivity of the method 330

Since all the external boundaries of the ROI are fully constrained, one may ask if the calculations are sensitive to a parameter perturbation. To study the sensitivity, the mean sensitivity of a node S mean is defined such as:

S mean = 1 3 3 ∑ i=1 U num (λ) -U num (λ + δλ i ) U num (λ) , (14)
where λ = {A, n, Q} and δλ i is a small perturbation applied to the parameter i. U num (λ) gathers the numerical radial displacements obtained with the set of parameters λ.

It is then possible to obtain the spatial distribution of sensitivity as well as its temporal evolution at a given node. Examples of sensitivity results are given in Figure 5 for the Test 4 and for a parameter perturbation of 0.2%.

The sensitivity of the test is important with a maximal value of around 0.8% at t = 10s at the center of the ROI. The sensitivity at the boundaries is null, because all the degree of freedom are constrained, and increases toward the center. Having 345 an important sensitivity is important to minimize the number of iterations needed and limit the risk of finding a local minimum.

Results of the virtual test case

The FEMU method is applied using the set λ 0 to initialize the algorithm. The results are given in Figure 6.

The mean error on the radial displacement is around 0.5% and the parameters are found with maximal errors of 0.8% compared to their reference values (cf. The virtual test case shows excellent robustness of the identification algorithm and fast convergence. Only 8 iterations are needed to obtain the parameters at 99% of their reference values, while they were initially strongly perturbed. Furthermore, this test case enables validating both the use of a Love-Kirchhoff kinematic across the external section and the smoothing of the displacements at the boundaries of the extracted mesh by polynomial surfaces.

Application to the experiments

The method is then applied to the experiments performed in the ELLIE bench simulating post-DNB conditions presented in [START_REF] Jailin | Thermo-mechanical behavior of Zircaloy-4 claddings under simulated post-DNB conditions[END_REF].

Initialization of the algorithm

The identification process needs (N e × (N λ + 1)) calculations per iterations, where N e is the number of experiments modeled and N λ is the number of sought parameters. To have a maximum test conditions, the 24 experiments of [START_REF] Jailin | Thermo-mechanical behavior of Zircaloy-4 claddings under simulated post-DNB conditions[END_REF] are used in the 375 identification process. It leads to a number of 168 calculations per iterations. In order to minimize the number of iterations it is useful to find the best initial set of parameters λ 0 . To do so, two assumptions are formulated to be able to resolve analytically the mechanical problem. First, the stress state during the tests 380 is approximated assuming that the sample can be represented as an open thin-walled cylinder (since a force was applied during the tests to compensate the end caps effect). follows:

σ vM = P i .(R i +U r ) e.( 1
+ ε rr ) , ( 15
)
where P i is the internal pressure, R i is the initial inner radius, U r is the radial displacement, e is the initial thickness and ε rr is the 385 radial deformation obtained using the displacements measured by 3D-DIC and the assumption of volume conservation in the plastic domain.

Second, using the radial displacement measured by stereo-390 correlation, the strain rate ṗ is approximated such as:

ṗ = Ur R e (t) , ( 16
)
where Ur is the radial displacement rate, and R e is the external radius of the sample. The set of parameters λ 0 is then identified using the Equations (1), [START_REF] Nguyen | Microstructure Changes during Fast β Cycles of Zirconium Alloys[END_REF], [START_REF] Campello | Validation of a multimodal set-up for the study of zirconium alloys claddings' behaviour under simulated LOCA conditions[END_REF] and [START_REF] Jailin | Coupled NIRT / 3D-DIC for a FEMU Identification of the Thermo-Mechanical Behavior of Zr-4 Claddings Under Simulated Reactivity Initiated Accident[END_REF]. All the radial displacement values 395 obtained in the tests are used to perform the initialization. As a reminder, only the first ten seconds of the tests are studied for the moment.

Results of the identification

The final values of the parameters obtained after the FEMU 400 procedure are given in Table 4. The residual evolution is presented in Figure 7. The algorithm converged after only 4 iterations. The parameters do not evolve significantly while the residual decreases by 27%. It denotes a high sensitivity of the tests to a parameter perturbation. The final residual is equal to 405

times the noise level.

A new scalar is introduced to quantify the error between the model and the experiments: ξ. It represents the mean error on the radial displacement rates of a test, which reads:

ξ(%) = 1 N N ∑ i=1 Uexp r -Unum r Uexp r × 100, ( 17
)
where N is the number of nodes on the correlation mesh of a Figure 8 compares the experimental radial displacement rates to the computed ones. The mean approximated strain rate during the first 10 seconds is plotted as a function of the temperature for all the tests and at each node of the correlation mesh. The mean error of each test, described by the Equation [START_REF] Leblond | A new kinetic model for anisothermal met-allurgical transformations in steels including effect of austenite grain size[END_REF], is also given in the upper graph. The evolution of the phase fraction is also given at equilibrium and for a heating rate representative of the experiments (≃ 1200 °C/s), computed using Equation [START_REF] Bessiron | Modelling of clad-to-coolant heat transfer for RIA applications[END_REF]. The calculations reproduce the experimental results with good agreements. The mean error ξ on the radial displacement rates is equal to 10.5%, which is considered as very good for these kinds of experiments. The error is more important at the lower temperatures, where the signal/noise ratio is low. A slightly more important error is observed around 970 °C, when the material switches from the two-phase domain to the β domain. A high sensitivity of the metallurgical state to the temperature is expected around this transition [START_REF] Jailin | Experimental study and modelling of the phase transformation of Zircaloy-4 alloy under high thermal transients[END_REF][START_REF] Jailin | Thermo-mechanical behavior of Zircaloy-4 claddings under simulated post-DNB conditions[END_REF].

ln(A α ) n α Q α (kJ.mol -1 ) ln(A β ) n β Q β (kJ.

Validation on an additional experiment

Once identified, the model is validated using an additional experiment, which was not used during the identification process. The internal pressure during the test was 9 bar and the loading conditions are given in Table 5. The results of the calculation are presented in Figure 8.

The residual for this test is 2.47 and the mean error ξ is 10.7%. It corresponds to the mean value obtained for all the tests modeled. Figure 9 presents the displacements and the fraction of β extracted from an axial line at the center of the ROI at t = 10s. The temperature distribution is also given. First, it is worth noticing that two more deformed areas are present and are not located at the hottest point which is at the middle. In Table 5: Experimental conditions of the additionally experiment performed. The internal pressure was 9 bar. T ROI represents the temperature computed at the center of the ROI by NIR thermography. dT /dt is the mean value of the thermal transient measured by the pyrometer. OS + is the difference between the maximal temperature reaches at the end of the transient and the regulation temperature [START_REF] Jailin | Thermo-mechanical behavior of Zircaloy-4 claddings under simulated post-DNB conditions[END_REF]. ∆T gives the mean thermal distribution within the ROI at t = 10s.

this domain of temperature, the creep rate decreases while the temperature increases, leading to the "butternut shape" profile 445 on the sample already mentioned in Section 2 (it is the second domain of creep identified in [START_REF] Jailin | Thermo-mechanical behavior of Zircaloy-4 claddings under simulated post-DNB conditions[END_REF]). Higher strains are observed at lower temperatures, where the β fraction is lower. It can be observed that this "butternut shape" profile is well reproduced by the model. The mean error on the displacements between 450 the calculation and the experiment is 4.8%. The model is thus considered as valid within the range of loading conditions investigated.

Discussion

This work succeeded in reproducing the creep response of 455 Zircaloy-4 in the whole domain of loading conditions investigated in [START_REF] Jailin | Thermo-mechanical behavior of Zircaloy-4 claddings under simulated post-DNB conditions[END_REF] using a single and continuous equation. However some points and assumptions need to be discussed. A Norton's creep law is used to model the creep behavior of the two phases while the steady stage of creep is not reached 460 yet. However, as the test duration analyzed is short (10 seconds) and as the level of deformation remains low at 10 seconds the The assumption was made that the phases are present in a serial distribution in the two phase domain. It led to good results, and succeeded to reproduce the "butternut" profile on the sample in the two-phase domain. However additional data in the 470 pure α domain at equilibrium, i.e. at lower temperatures, are still needed to validate this serial assumption.

A stress exponent around unity is identified for the α phase, which is characteristic of a diffusion creep mechanism. On the other hand, a dislocation creep mechanism is observed in the β 475 phase, evidenced by a stress exponent of around 4.5. These observations are in good agreement with the literature for this low level of applied stress [START_REF] Kaddour | Experimental determination of creep properties of zirco-730 nium alloys together with phase transformation[END_REF][START_REF] Campello | Identification of the steady-state creep behavior of Zircaloy-4 claddings under simulated Loss-Of-Coolant Accident conditions based on a cou-725 pled experimental/numerical approach[END_REF][START_REF] Donaldson | A transition stress in the creep of an alpha phase zirconium alloy at high temperature[END_REF]. Since the results of the literature were obtained in isothermal conditions with low heating rate, it may be concluded that there is no impact of the heating 480 rate on the activated creep mechanism up to 1200 °C/s. It is also worth noticing that in [START_REF] Jailin | Thermo-mechanical behavior of Zircaloy-4 claddings under simulated post-DNB conditions[END_REF][START_REF] Jailin | Experimental study and modelling of the phase transformation of Zircaloy-4 alloy under high thermal transients[END_REF] a small grain size was evidenced in the α phase, and much bigger grains were observed in the pure Figure 8: Comparison between the numerical creep rates, computed using the final set of parameters λ f and the equations given in Section 2 (serial assumption), and the experimental results of [START_REF] Jailin | Thermo-mechanical behavior of Zircaloy-4 claddings under simulated post-DNB conditions[END_REF]. The evolution of the phase fraction against temperature is also given at equilibrium and under a heating rate of 1200 °C/s. While the stress exponents identified are in the expected order of magnitude under the tested conditions, the activation energies identified following a very fast heating are much higher 490 in the two phases than the ones identified in the literature in isothermal conditions. The activation energy in the α phase for a diffusion mechanism is usually estimated around 150kJ/mol [START_REF] Donaldson | A transition stress in the creep of an alpha phase zirconium alloy at high temperature[END_REF][START_REF] Kaddour | Experimental determination of creep properties of zirco-730 nium alloys together with phase transformation[END_REF][START_REF] Campello | Identification of the steady-state creep behavior of Zircaloy-4 claddings under simulated Loss-Of-Coolant Accident conditions based on a cou-725 pled experimental/numerical approach[END_REF] for isothermal tests, and the one in the β phase for a dislocation mechanism is about 140kJ/mol [START_REF] Hunt | High temperature strain behavior of Zircaloy-4 and Zr-2.5 Nb fuel sheaths[END_REF][START_REF] Rosinger | Steady-state creep of zircaloy-4 fuel cladding from 940 to 1873 k[END_REF][START_REF] Sills | Predicting high-temperature transient deformation from microstructural models[END_REF][START_REF] Kaddour | Experimental determination of creep properties of zirco-730 nium alloys together with phase transformation[END_REF].

495

Activation energies of about 600kJ/mol were identified in the two phases when the material was heated with a very fast heating rate. Additional microstructure analysis should be carried out to better understand the underlying mechanisms behind this change in the activation energies. It is also possible that the 500 activation energy accommodates in the minimization process to fit the experimental data, because of the assumptions made to model the material behavior (i.e. a serial distribution of the two phases, no impact of the stress state on the phase transformation, and no change in the creep law in the α and β phases 505 between the single and two-phase domains). More experimen-tal data (especially results in the pure alpha domain) is needed to discriminate whether these energy values are true or come from one (or several) of the bias(es) mentioned above.

From a numerical point of view one may notice that the residual obtained with the initial set of parameters is close to the converged residual. It has been demonstrated that this is not due to a low sensitivity of the calculations to the parameter values. The low evolution of the residual can be explained by the high quality of the initialization, which was carried out with a large amount of data, and with assumptions that appear to be not far from the truth. In some case it may be useful to spend some time in finding a good starting set, in order to save an important calculation time afterwards and possibly avoid diverging problems.

In [START_REF] Jailin | Thermo-mechanical behavior of Zircaloy-4 claddings under simulated post-DNB conditions[END_REF] only the first ten seconds of the experiments were analyzed since it is the characteristic time of a reactivity initiated accident. The relatively small strain levels reached at 10 seconds enables the Norton's law to reproduce well the experimental data, but it cannot be extended to longer test times. The following section proposed to extend the identification up to the first 100 seconds of eight tests located in the two-phase domain. In order to take into account the secondary stage of creep the impact of the grain size evolution on the creep properties is modeled. Furthermore, since the deformations are higher at 100 s the structural effects have to be taken into account with a finite element based identification method, which enables reproducing the 3D stress state in the material.

Extension to 100s

In order to be more representative of other accidents, like the loss of coolant accident, the identification procedure was also performed on eight experiments carried out in the two-phase domain during the first 100 seconds of the tests.

The primary stage of creep is taken into account by adding the grain size evolution in the Norton's creep law [START_REF] Lee | Effect of Grain Size on Creep Properties of Type 316LN Stainless Steel[END_REF][START_REF] Kassner | Fundamentals of Creep in Materials[END_REF]. The creep rate now reads :

Ė = ∑ i A i D 0i D i g i σ n i vM exp - Q i RT f i , with i = {α; β}, (18)
where the subscript i defined the phase, D 0i is the initial grain size of the phase i, D i is the grain size of the phase i.

The grain size evolution is modeled using the following equation [START_REF] Kassner | Fundamentals of Creep in Materials[END_REF][START_REF] Douglass | The Metallurgy of Zirconium[END_REF] :

dD i dt = K i exp - Q di RT t τ 1-m i m i , with i = {α; β}. ( 19
)
where K i is a constant, Q di is an activation energy, m is the temporal exponent and τ is a characteristic time. In this application τ is set to 10s since it is the characteristic time of a reactivity initiated accident. 16 parameters have finally to be identified in Equations ( 18) and ( 19).

An initial set of parameters is identified using Equations ( 15) and ( 16) and an assumption of a serial distribution of the phases. Not enough data on the microstructure evolution are available 555 in the literature. It follows that the uniqueness of the solution is not ensured. To obtain physically acceptable results the initial grain size was constraint to be inferior than 10 µm and the maximal grain size at 100s was bounded to be inferior than 50µm. These boundaries were set following the microstructure data of 560 [START_REF] Sills | Predicting high-temperature transient deformation from microstructural models[END_REF] for Zircaloy-4 and of [START_REF] Trégo | Influence of grain size on the high-temperature creep behaviour of M5 Framatome zirconium alloy under vacuum[END_REF] for M5 alloy). The idea here is to show that a grain growth contribution in the creep law enables well reproducing the hardening of the material. Additional data on the microstructure evolution would be necessary to obtain the true evolution of the grain size during the experiments.

565

The evolution of the objective function is given in Figure 10. In this case the initial error is high because the ballooning deformation of the sample generates a complex stress state within the sample, which therefore cannot be well approximated by the Equation ( 15). The residual decreases quickly in four iterations 570 up to a converged value equal to 2.7 times the noise level. The values of the parameters of Equations ( 18) and ( 19) are given in Table 6. The evolution of the grain size for the α and β phase is given in Figure 11 for a test performed at 930 °C. The α grain size 575 does not evolve significantly with time while the β grains grow from very small grain size (nucleation) to several tens of microns. These results are consistent with the results of the literature [START_REF] Sills | Predicting high-temperature transient deformation from microstructural models[END_REF][START_REF] Trégo | Influence of grain size on the high-temperature creep behaviour of M5 Framatome zirconium alloy under vacuum[END_REF]. Figure 12 compares the radial displacement normalized by the initial inner radius obtained experimentally and 580 numerically at the center of the ROI. The grain size contribution enables the secondary stage of creep to be well reproduced with mean error on the radial displacement rates of 6.6%. Adding a grain growth contribution in the creep law seems then to be a good way of reproducing the hardening of the material in the 585 two-phase domain. Other sources could also be responsible of the non-linear hardening observed during the tests, such as a microstructure evolution of the material or a strain hardening mechanism. This may be investigated in further studies. 18) and ( 19) with i = {α; β}.

Figure 11: Evolution of the α and the β grain sizes obtained for a temperature of 930 °C using the Equation ( 19) and the set of parameters given in Table 6.

Summary and Conclusion

In this work, a creep model weakly coupled to the phase transformation of the material has been identified for Zircaloy-4 fuel claddings in simulated post-DNB conditions, based on the results of [START_REF] Jailin | Thermo-mechanical behavior of Zircaloy-4 claddings under simulated post-DNB conditions[END_REF]. This model, being the very first identified in such conditions, is expected to be a very good start in the study of the creep response of zirconium alloys under very high heating rate. The temperatures of interest in this work range from 840 to 1020 °C, the heating rate is about 1200 °C/s, and the stress level ranges from around 5 to 8 MPa. The identified model reproduces the first 10 seconds of the experiments with a mean error on the radial displacement rates of around 10%. It was shown that the heating rate seems to have no impact on the activated creep mechanisms. A diffusion creep is identified in the α phase and a dislocation creep is identified in the β phase. These creep mechanisms are consistent with the grain sizes characterized in [START_REF] Jailin | Thermo-mechanical behavior of Zircaloy-4 claddings under simulated post-DNB conditions[END_REF] on the post-mortem samples. On the contrary, the activation energies identified in simulated post-DNB conditions are much higher than the usual values found in LOCA conditions. Additional microstructure analysis should be carried out to better understand the underlying mechanisms of such a difference. The model of this work is currently being implemented in the SCANAIR software developed by the French Institute for Radiological Protection and Nuclear Safety (IRSN), which simulates a reactivity initiated accident [START_REF] Moal | SCANAIR: A transient fuel performance code. Part One: General modelling description[END_REF][START_REF] Georgenthum | SCANAIR a transient fuel performance code. Part two: Assessment of modelling capabilities[END_REF].

An extension of the creep law to the first 100 seconds has also been proposed for 8 experiments performed in the two-phase domain to reproduce the non linearity of the creep response at higher time scale. This extension is based on adding the grain growth contribution in the Norton's law. The identification led to a very good agreement between experimental and numeri-620 cal results with mean error on the radial displacement rates of around 7%. Although there is a lack of data about the grain size evolution to ensure the uniqueness of the parameter values, these results are expected to be of great interest for the zirconium community in the nuclear industry.

625

From a numerical point of view, an identification procedure has been proposed to identify a constitutive model based on experiments carried out on tubular samples, where the loading conditions are solely known in the region seen by the cameras. Only the ROI was modeled to reproduce the experiments. The 630 boundary conditions are imposed using the displacements measured by stereo-correlation, and a Love-Kirchhoff kinematic was imposed to constraint the external section of the ROI. The method was first validated on a virtual test case representative of the real experiments before being applied to the true one.

635

The sensitivity of the calculations to the creep parameters was also presented to ensure a fast convergence of the algorithm and minimize the risk of finding a local solution.

  145

Figure 1 :

 1 Figure 1: Procedure used in [6] to extract the experimental results from the creep tests.

  255

Figure 2 :

 2 Figure 2: Extrusion of the FE mesh from the correlation mesh.

Figure 3 :

 3 Figure 3: (a) U r , U θ and U z are imposed to the master nodes, the slave nodes are constrained to stay on the normal ⃗ n ; (b) computation of the normal ⃗ n using the correlation nodes.

  300

Figure 4 :

 4 Figure 4: FE modeling of the virtual case.

  ). The parameters are set such as ln(A) = 11, n = 4 and Q = 250 kJ.mol -1 . These 3 parameters composed the reference set λ re f .315Five simulations are carried out with different loading conditions as presented in Table2. The computations are interrupted at a test time of 10 seconds, being the characteristic time of a reactivity initiated accident.

  Uexp r gathers the radial displacement rates of the test and Unum r gathers the radial displacement rates obtained by calculations with the set λ f .

Figure 5 :

 5 Figure 5: (a) Sensitivity surface extracted at t = 10s for the Test 4 ; (b) temporal evolution of the sensitivity at the center of the ROI (red point in Figure (a)).

T

  ROI ( °C) dT /dt ( °C/s) OS + ( °C) ∆T ( °C)

Figure 6 :

 6 Figure 6: Results of the virtual case: (a) evolution of the residual η ; (b) evolution of ln(A)/ln(A ref ) ; (c) evolution of n/n ref ; (d) evolution of Q/Q ref .

Figure 7 :

 7 Figure 7: Evolution of the residual η for the identification of a creep law reproducing the first ten seconds of all the experiments performed in simulated post-DNB conditions (serial assumption).

Figure 9 :

 9 Figure 9: Comparison between the experimental and the numerical displacements obtained along an axial line at the middle of the ROI for the 9 bar test.

Figure 10 :

 10 Figure 10: Evolution of the residual η for the identification of a creep law with a grain size contribution reproducing the first 100 seconds of eight experiments performed in the two-phase domain.

  ln(A i ) n i Q i (kJ.mol -1 ) g i K i Q di (kJ.mol -1 ) m i D 0i (µm)

Table 1 :

 1 Parameters of Equations (

Table 3 )

 3 .

		Converged parameters Relative error
	ln(A)	10.91	0.8%
	n	3.98	0.6%
	Q (kJ.mol -1 )	248.9	0.4%
	Residual η(λ)	1.09

Table 3 :

 3 Converged parameters and relative errors compared to the reference set λ re f .

  mol -1 )

	λ 0	48.7	1.11	544.2	43.4	4.53	603.9
	λ f	48.6	1.11	545.0	43.2	4.45	607.6

Table 4 :

 4 Values of the initial and final set of parameters.

Table 6 :

 6 Values of the parameters of Equations (

tion. Phase transformation studies under loading should be carried out to validate this assumption. (ii) Because of the very fast recrystallisation at high temperature[START_REF] Dunlop | Modelling 740 isothermal and non-isothermal recrystallisation kinetics: Application to Zircaloy-4[END_REF], an isotropic behavior was considered. Anisotropy could be taken into account for lower temperatures, using a Hill definition of the equivalent

Acknowledgments

The authors acknowledge with thanks financial support of 640 this work from EdF and Institut Carnot Ingénierie@Lyon. The authors want also to express their thanks to J. Réthoré for the useful discussions about inverse problems and to ATYS Consulting Group for helping on the induction problematic.

The three parameters are found at 95% of the reference parameters at the fifth iteration and at 99% at the eighth iteration. From the eighth iteration the residual evolution and the parameter evolution become lower than 0.5%, the algorithm is then considered as converged. The final residual is equal to 1.09.


T Jailin

N Tardif

J Desquines

P Chaudet

M Coret

M.-C Baietto

V Georgenthum

Thermo-mechanical behavior of Zircaloy-4 claddings under simulated post-DNB conditions

Keywords: Fast thermal transients, Full-field measurements, Creep, Zircaloy-4, RIA

The thermo-mechanical behavior of Zircaloy-4 claddings under simulated post-DNB RIA conditions was investigated. Around twenty experiments were performed in simulated post-DNB conditions, i.e. creep ballooning tests with heating rates greater than 1000 • C/s. Two different levels of pressure of 7 and 11 bar were tested for temperatures of interest ranging from 840 • C to 1020 • C. A complex creep behavior was highlighted in this range of temperature. It appears very well correlated to the phase content present within the material during fast thermal transients. Tests with low thermal transients were also performed and evidence a strong impact of the heating rate on the thermo-mechanical properties of the claddings.

Introduction

During the operating life of Pressurized Water Reactors (PWR) the nuclear reaction is controlled within the nuclear core using control rod made of neutron poison material. The Reactivity Initiated Accident (RIA) is one of the design basis accidents studied for PWR. It supposes an inadvertent insertion of reactivity in the nuclear power reactor. It may result from several possible scenarios, the most studied being the control rod ejection [START_REF] Rudling | Nuclear Fuel Behaviour under RIA Conditions[END_REF]. The RIA accident can be decomposed in two consecutive phases: the Pellet Clad Mechanical Interaction (PCMI) phase, followed by the post-Departure from Nucleate Boiling (post-DNB) phase. This second phase of the accident starts at the boiling crisis of the coolant, which leads to a brutal decreasing of the thermal transfer between the fuel claddings and the water. Subsequently, a high increase of the fuel claddings temperature occurs with thermal transients that can be greater than 1000 • C/s up to temperatures exceeding 900 • C [START_REF] Bessiron | Modelling of clad-to-coolant heat transfer for RIA applications[END_REF]. In the same time, an internal pressure can appear within the claddings due to fission gas release. All these conditions may lead to the ballooning of the fuel rod until its potential burst. The aim of this work is to characterize the thermo-mechanical behavior of Zircaloy-4 fuel rods under such conditions.

Zircaloy-4 is widely used as fuel claddings material in PWR. It presents an allotropic phase transformation (α → β) between 800 • C to 1000 • C upon heating at equilibrium. It switches from an hexagonal closed packed structure at low temperature, to a body centered cubic structure at high temperature [START_REF] Banerjee | Phase Transformations: Examples from Titanium and Zirconium Alloys[END_REF]. A dual phase domain (α + β) is defined between the two single phase domains, where the two phases are present within the material.

It is well-known that the phase transition is highly dependant on 30 the thermal history [START_REF] Forgeron | Experiment and Modeling of Advanced Fuel Rod Cladding Behavior Under LOCA Conditions: Alpha-Beta Phase Transformation Kinetics and EDGAR Methodology[END_REF][START_REF] Holt | The beta to alpha phase transformation in Zircaloy-4[END_REF][START_REF] Mittemeijer | Analysis of the kinetics of phase transformations[END_REF]. Moreover, the metallurgical state of the claddings has a strong impact on its thermo-mechanical properties [START_REF] Forgeron | Experiment and Modeling of Advanced Fuel Rod Cladding Behavior Under LOCA Conditions: Alpha-Beta Phase Transformation Kinetics and EDGAR Methodology[END_REF][START_REF] Kearns | Effect of alpha/beta phase constitution on superplasticity and strength of Zircaloy-4[END_REF][START_REF] Sell | Effects of alpha-beta transformation on high temperature (LOCA) creep behavior of Zr-alloys[END_REF]. Garde for instance [START_REF] Garde | Micrograin superplasticity in Zircaloy at 850 • C[END_REF], pointed out a superplastic behavior in the dual phase domain. Massih [START_REF] Massih | High-temperature creep and superplasticity in zirconium alloys[END_REF] proposed to model it by adding a grain boundary sliding contri-35 bution on the deformation, using a modified Ashby's model [START_REF] Ashby | Diffusion-accommodated flow and superplasticity[END_REF]. Some authors noticed a complex creep behavior in the dual phase domain that does not follow a classical power creep law, usually used to describe the creep behavior in Zirconium alloys [START_REF] Fréchinet | Transformations et comportement du Zircaloy-4 en conditions anisothermes[END_REF][START_REF] Kaddour | Experimental determination of creep properties of zirconium alloys together with phase transformation[END_REF][START_REF] Trégo | Comportement en fluage à haute température dans le domaine biphasé (α + β) de l'alliage M5[END_REF].

40

A lot of experiments were performed in the last decades to investigate the thermo-mechanical behavior of fuel rods in the three domains {α, (α + β), β}. A special attention has been payed to accidental situations where the loading conditions can be very demanding. Due to the difficulty of reproducing well-45 controlled fast thermal transients, most of the experimental devices were developed to reproduce simulated LOCA (Loss Of Coolant Accident) conditions [START_REF] Kaddour | Experimental determination of creep properties of zirconium alloys together with phase transformation[END_REF][START_REF] Réocreux | A study of fuel behavior in PWR design basis accident: an analysis of results from the PHEBUS and EDGAR experiments[END_REF][START_REF] Dominguez | Effect of pre-oxide and hydrogen on creep of Zircaloy-4 at 1123K[END_REF], or PCMI conditions [START_REF] Cazalis | The PROMETRA program: Fuel cladding mechanical behavior under high strain rate[END_REF][START_REF] Abe | Development of advanced expansion due to compression (A-EDC) test method for safety evaluation of degraded nuclear fuel cladding materials[END_REF]. These experiments are characterized by isothermal tests or tests with low thermal transients. Extrapolating the 50 data to post-DNB conditions is therefore somewhat adventurous... Chaieb et al. [START_REF] Chaieb | Anisothermal behaviour of unirradiated CWSR zircaloy-4 fuel clads under RIA conditions[END_REF], investigated the plastic properties of Zircaloy-4 under thermal transients up to 560 • C/s. They performed uniaxial and biaxial tensile tests with constant displacement rate, on claddings section and cruciform samples, respec-55 tively. They highlighted an impact of the heating rate on the mechanical properties for tests carried out with a constant strain rate of 0.5s -1 .

Most of the experiments carried out to simulate post-DNB conditions were performed in experimental nuclear reactors 60 [START_REF] Papin | Summary and Interpretation of the CABRI REP-Na Program[END_REF][START_REF] Ishikawa | A study of fuel behavior under reactivity initiated accident conditions[END_REF]. In this kind of tests the thermo-mechanical conditions cannot be well-controlled. Furthermore the many multi-physics couplings that take place within the nuclear core make the data analysis difficult. It follows that the creep behavior of the fuel rods remains poorly known after a fast thermal transient, especially in the dual phase domain where the metallurgical structure of the material may be far from the equilibrium state.

In this work, the experimental device ELLIE, developed by Campello and Tardif [START_REF] Campello | Identification of the steady-state creep behavior of Zircaloy-4 claddings under simulated Loss-Of-Coolant Accident conditions based on a coupled experimental/numerical approach[END_REF][START_REF] Campello | Validation of a multimodal set-up for the study of zirconium alloys claddings' behaviour under simulated LOCA conditions[END_REF], was updated to reproduce simulated thermo-mechanical post-DNB conditions on fuel cladding sections [START_REF] Jailin | Mechanical Behavior of As-Fabricated Zircaloy-4 Claddings Under the Simulated Thermo-Mechanical Post-DNB Conditions of a Reactivity Initiated Accident (RIA)[END_REF][START_REF] Jailin | Multimodal setups for the study of fresh Zircaloy-4 claddings under simulated thermal-mechanical RIA conditions[END_REF]. Creep ballooning tests were performed with well-controlled thermal transients up to 1500 • C/s using an induction heating system. Around twenty tests were carried out in simulated post-DNB conditions at two different levels of pressure for temperatures of interest ranging from 840 • C to 75 1020 • C. The use of full-field measurements methods enabled obtaining a large amount of data from each experiment by taking profit of the tests heterogeneities. These experiments highlighted a complex creep behavior in the domain of temperature investigated. It appears very well correlated to the phase content present within the material during fast thermal transients (� 1200 • C/s). Tests with low thermal transients were also performed and showed a strong impact of the heating rate on the thermo-mechanical properties of the claddings.

The first section of this paper deals with the description of 85 the test bench ELLIE. The second section presents two representative experiments performed in the α and (α + β) domains.

The third section details the tests performed in "Simulated RIA" conditions and highlights a complex behavior in the dual phase domain. The fourth section presents the impact of the heating 90 rate on the material creep behavior.

Experimental setup and procedure

Samples preparation

The specimens were cut from as-fabricated stress relieved annealed (SRA) Zircaloy-4 claddings (see the chemical com-95 position in Table 1). The initial microstructure of the material is composed of thin and elongated grains and presents a texture having the [0002] direction at +/-30 • from the radial direction of the tube [START_REF] Li | The effect of hydride morphol-710 ogy on the failure strain of stress-relieved Zircaloy-4 cladding with an outer surface pre-crack under biaxial stress states[END_REF]. The samples were 90mm long with an outer diameter of 9.5mm and a wall-thickness of 0.57mm.

100

Their eccentricity was small and was estimated between 1 and 3% [START_REF] Campello | Identification of the steady-state creep behavior of Zircaloy-4 claddings under simulated Loss-Of-Coolant Accident conditions based on a coupled experimental/numerical approach[END_REF]. In order to perform stereo-correlation measurements the specimens were painted with a black undercoating and a white speckle pattern. Three thermocouples (type K, wires of 79µm diameter) were spot-welded on the sample outer surface in the region of interest (ROI) seen by the cameras (Figure 1.a).

Sn

Fe O Cr C Zr 1.29 0.21 0.13 0.11 0.016 bal.

The test bench ELLIE

The experimental device ELLIE was presented in detail in [START_REF] Campello | Validation of a multimodal set-up for the study of zirconium alloys claddings' behaviour under simulated LOCA conditions[END_REF][START_REF] Jailin | Multimodal setups for the study of fresh Zircaloy-4 claddings under simulated thermal-mechanical RIA conditions[END_REF]. Some reminders are given hereinafter. The test bench is composed of a 10kN servo-hydraulic tensile machine. Fig- 110 ure 1 presents a picture and a scheme of a sample mounted on the device. The sample was connected onto the test machine using custom grips [START_REF] Tardif | Experimental study of the frac-715 ture kinetics of a tubular 16MnNiMo5 steel specimen under biaxial loading at 900 and 1000 • C . Application to the rupture of a vessel bottom head during a core meltdown accident in a pressurized water reactor[END_REF] reducing the distance between the jaws to 58mm. An internal pressure up to 70 bar can be imposed and controlled within the claddings during the whole tests using ar-115 gon gas. A 30kW induction generator (ThermoInduzione TI3-30KW CT-110) was used to heat the samples with heating rates up to 1500 • C/s. The induction generator frequency (92kHz) was chosen so that the thermal loading was uniform across the wall-thickness of the sample, even during fast thermal tran-120 sients. The PID regulation was optimized to get low overshoots and the induction coil was designed to obtain the hottest point at the center of the ROI. The temperature regulation was achieved using a pyrometer (CTLM-2HCF4-C3, wavelength of 1.6µm). All the experiments were performed within an enclosure where 125 an argon flush was set up during the tests to prevent the samples from oxidation.

Tests processing

Two kinds of tests were performed to investigate the impact of the thermal transient on the material thermo-mechanical 130 properties. Experiments, called "Simulated RIA", were performed with high heating rates to reproduce post-DNB conditions. The test procedure is depicted in Figure 2 Experiments with low heating rates, called "Simulated 140 LOCA", were also carried out following the procedure used in [START_REF] Campello | Identification of the steady-state creep behavior of Zircaloy-4 claddings under simulated Loss-Of-Coolant Accident conditions based on a coupled experimental/numerical approach[END_REF]. The experimental procedure of these tests is presented in Figure 2.b. The thermal loading was first applied with a heating rate of 3 • C/s. The mechanical loading was applied once the temperature was stabilized at the target value.

145

In all tests, a compressive force was applied simultaneously to the internal pressure to compensate the end caps effect. This force was calculated using the initial inner radius R i of the claddings and the internal pressure P i applied:

F z = P i .π.R 2 i . (1)
For the two kinds of experiments the test time was set to zero 150 once the thermal and mechanical loading were stabilized. This time offset is referred to t 0 in Figure 2.

Optical means of measurement

Full-field optical techniques enable obtaining a large amount of data in one single experiment. The kinematic field can be ob-155 tained on the sample outer surface using digital images correlation (DIC) for instance. The thermal field can be either obtained using infrared (IR) thermography or near-infrared thermography (NIRT) techniques. The richness of these optical methods enables heterogeneous tests to be performed with complex 160 loading conditions. Under very high thermal transients a tiny eccentricity between the specimen and the induction coil leads to a loss of axisymmetry of the thermal loading. Therefore it was chosen to force this eccentricity so that the hottest point on the sample

Stereo-correlation (3D-DIC)

The 3D-surface displacements field was obtained using the 185 global stereo-correlation software UFreckles developed by J. Réthoré [START_REF] Réthoré | Robust identification of elasto-plastic constitutive law parameters from digital images using 3D kinematics[END_REF], [29]. A finite elements basis of quadrilateral elements with linear shape functions was used to perform the images analysis with an element size of 35x35px 2 (1 px � 9µm). An example of a correlation mesh is given in Figure 4.b.

190

In order to avoid chromatic aberrations at high temperatures the calibration of the optical system was performed at low tem- perature by lighting the sample in the near infrared domain using a LED ring at the wavelength λ = 860nm.

It is worth noting that because of the device geometry a "standard" calibration using calibration grids couldn't be performed. The calibration step was then achieved using the well-known geometry of the sample. The noise was finally estimated to be lower than 1µm for the in-plane displacements and about 5µm for the out-of-plane displacements. The NIRT method was used to compute the thermal field at each time step. This method links the digital level intensity I of an image (i.e. the gray level of each pixel) to the temperature T using a radiometric model (Eq. 2) based on Planck's law:

205 T = K 1 ln � K 2 I + 1 � . ( 2
)
Two constants K 1 and K 2 have to be identified in this model [START_REF] Campello | Identification of the steady-state creep behavior of Zircaloy-4 claddings under simulated Loss-Of-Coolant Accident conditions based on a coupled experimental/numerical approach[END_REF].

Thermocouples are subjected to an inertia effect due to their intrusive nature, and the speckle pattern disturbs the optical thermal measurements. Therefore, an identification procedure that links the thermocouples, the pyrometer and the NIRT was developed to identify the radiometric model in a robust way. Furthermore, since the same images were used to compute both the thermal and the kinematics fields, a weak coupling between the two fields was developed. It follows that the thermal field was obtained at the exact same time and location that the kinematic field. No interpolation or smoothing operation were finally necessary on the thermal field. The accuracy on the temperature was estimated around 0.7% of the measured temperature [START_REF] Campello | Validation of a multimodal set-up for the study of zirconium alloys claddings' behaviour under simulated LOCA conditions[END_REF]. More details on the procedure followed can be found 220 in [START_REF] Jailin | Coupled NIRT / 3D-DIC for a FEMU identification of the thermo-mechanical behavior of Zr-4 claddings under simulated Reactivity Initiated Accident[END_REF].

Results from two representative simulated RIA experiments

Phase fraction evolution

Before everything else the evolution of the β phase fraction 225 with the temperature is given in Figure 5 for equilibrium condition and upon a heating rate of 1200 • C/s. The equilibrium data can be found in [START_REF] Forgeron | Experiment and Modeling of Advanced Fuel Rod Cladding Behavior Under LOCA Conditions: Alpha-Beta Phase Transformation Kinetics and EDGAR Methodology[END_REF]. The 1200 • C/s curve was obtained by a stress-free dilatometry experiment performed at an imposed heating rate of 1200 • C/s on a GLEEBLE-3500 device [START_REF] Jailin | Experimental study and modelling of the phase transformation of Zircaloy-4 alloy under high thermal transients[END_REF]. The

Experimental procedures of the two tests

Results from two tests that followed the experimental protocol presented in Figure 2.a are now used to describe a typical experiment analysis. The procedures of these two tests are depicted in Figure 6. One test, referred for convenience to the 245 α-test in the following, was carried out with a regulation temperature of 850 • C and an internal pressure of 20 bar. The other test was run in the dual-phase domain (α + β) with a regulation temperature of 940 • C and an internal pressure of 11 bar. The heating rates obtained for the α and (α + β) experiments 250 were 1216 • C/s and 1260 • C/s, respectively. Tiny overshoots (of +4 • C and +8 • C, respectively) can be observed. The times offsets of the two tests are represented by t 0 in Figure 6. In both case the temperature stabilized at +/-1 • C around the target value about 3.5s after t 0 .

255

The heating rate is plotted during the transient for the two tests in Figure 7. Another test, carried out in the β domain, is also included in Figure 7, in order to see the evolution of the heating rate during the whole phase transition. A good repeatability of the thermal slope is obtained between the tests.

260

A slight decrease of the heating rate is observed during each experiment between 650 and 700 • C. It may be due to the intermetallic Zr(Fe, Cr) 2 dissolution. It can also be observed in the (α + β) and the β experiments that the heating rate decreases during the phase transition from around 890 • C. It reaches a Images were recorded at a frequency of 55Hz during the first fifteen seconds of the tests and then at a frequency of 1Hz until the end. Kinematic and associated thermal fields were computed on the outer surface of the specimen by 3D-DIC and NIRT, respectively, following the method described in section 2. Two very different creep regimes were evidenced. In the αexperiment the strain profile obtained may be seen as "standard", i.e. the more deformed area was localized at the hottest region. On the contrary, in the (α + β)-experiment two more pronounced ballooning areas are visible, located at the upper and lower side of the ROI. These areas of larger strains were not localized at the hottest region like in the α-test. Figure 10 presents the thermal and the kinematics data extracted from an Figure 11 shows for both tests the evolution of the radial displacement of all the nodes of the correlation meshes. The temperature of each point was computed by NIRT at each time step. Two main results can be highlighted. First, it can be observed

Analysis of the two tests

305

It is shown in the following that this "non-standard" creep behavior occurring in the dual phase domain led at the end to the so-called "butternut shape" profile on the specimens. This profile is characterized by the presence of two ballooning areas on the samples while a single hottest region was localized at the 310 center of the ROI.

Secondly, only a stationary creep is observed in the αexperiment while a primary creep occurred in the (α + β)-test. This primary creep is characterized by a decreasing of the strain rate during the first 150 seconds. The creep rate reached then a 315 constant value, characteristic of the secondary stage of creep. sponse recorded by the pyrometer is given in Figure 12. Two overshoot values, OS + and OS -, are here defined. OS + is the difference between the maximal temperature reached at the end of the transient (at t A ) and the regulation temperature. OS -is the difference between the lowest value reached after t A and the regulation temperature. In all tests OS -is about -30 • C and the test time from which the temperature reaches +/-1 • C around the regulation temperature is lower than 4 seconds (t B -t A in Figure 12). The mean temperature obtained in the ROI, the mean thermal heterogeneity within the ROI at t = 10s, the heating rates obtained during the thermal transient, the OS + overshoots and the internal pressure of the tests are given in Table 2.

In the following it is chosen to limit the study to the first ten seconds after the reference time t 0 (Figure 2.a) of the experiments since it is the approximate maximum duration of the boiling crisis of a RIA.

Experimental results

Figure 13 gathered the mean radial displacements rates normalized by the initial radius obtained during the first ten sec-Figure 12: Pyrometer response and regulation temperature obtained for the test referred as RIA 7 in table 2. OS + is the difference between the maximal temperature reached at the end of the transient (at t A ) and the regulation temperature. OS -is the difference between the lower value reached after t A and the regulation temperature. t B is defined as the time from which the temperature reaches +/-1 • C around the regulation temperature. P i = 7 bar dT /dt is the mean value of the thermal transient measured by the pyrometer. OS + is the difference between the maximal temperature reaches at the end of the transient (at t 0 ) and the regulation temperature (Figure 12). ΔT gives the mean thermal distribution within the ROI at t = 10s.

P i = 11 bar Exp. no T ROI ( • C) dT /dt ( • C/s) OS + ( • C) ΔT ( • C) Exp. no T ROI ( • C) dT /dt ( • C/s) OS + ( • C) ΔT ( • C
onds of the tests. These points were computed by linearizing the radial displacement evolution of each node of the correlation meshes of all tests. The associated temperatures are the mean ues were computed whereas the stationary creep stage was not reached yet. General tendencies can nevertheless be extracted.

As expected the tests performed with an internal pressure of 11 bar led to higher creep rates than the one performed with an internal pressure of 7 bar. A similar trend of the creep rate evolution with the temperature is observed between the two levels of pressure.

Three different domains of creep behavior can be identified in Figure 13. The first one is defined below � 910 • C, i.e. in the α and (α, α + β) domains, where f 1200 • C/s β � 10%. In this first domain the creep rate increases with the temperature. It follows that only one central ballooning area was observed on the samples, like in the α-experiment detailed in the previous section. Label 1 in Figure 14 shows the post-mortem shape of a sample loaded at 870 • C. Only one balloon is present at the location of the hot spot induced by the induction heating device. The sample referred to label 2 in Figure 14 was issued from an experiment performed in the upper temperature range of this first creep domain (906 • C). A single large deformed region was obtained on the specimens. This large deformed region is characteristic of the transition in the creep behavior between the first and the second domain identified. In this temperature range (around 910 • C), the creep rate does not strongly depend on the temperature.

The second domain is defined from 910 • C to 970 • C, i.e. be- The presence of these two eccentric ballooned areas can be explained by the second domain of creep defined between 910 • C to 970 • C where the creep rates are higher at low temperatures. Figure 13 gathers the creep rates measured during this study.

405

It can be observed that in the two phase region the creep rate decreases with increasing temperature. Conversely, it increases with temperature in the α and β domains. This two phase creep regime induces the double ballooning profile, as observed in Figure 14.

Metallurgical analysis

Following the tests, metallurgical analysis were performed on the samples using an optical microscope. Axial and radial sections were cut from the specimens in the region facing the Figure 15 shows four micrographs obtained after an axial cut performed at the center of the RIA 25 sample. This test lasted 135 seconds with a regulation temperature of 968 • C and an internal pressure of 7 bar. The maximal temperatures associated to the micrographs were computed by NIRT. The maximal temperature of the micrograph (a) is not known accurately because it was outside the ROI. The maximal temperature was then estimated knowing that the temperature at the jaws is around 500 • C. A strong evolution in the metallurgical structure depending on the maximal temperature reached is observed. A fully recrystallized micostructure, with small and equiaxed grains, is visible in Figure 15.a. In this area the temperature was lower than the (α + β) → β transition. In Figure 15

Discussion

In this work creep ballooning tests were performed between 840 • C and 1020 • C. Figure 13 gathers the mean radial displacements rate of the tests, assuming secondary creep regime, during the first ten seconds. The thermo-mechanical behavior ob-450 served in the temperature range investigated seems to be very well related to the phase fraction present within the material obtained at 1200 • C/s. This complex behavior is characterized in the dual phase domain by a decreasing of the strain rate as the temperature increases. Similar observations has already been 455 seen by some authors in the literature.

Frechinet [START_REF] Fréchinet | Transformations et comportement du Zircaloy-4 en conditions anisothermes[END_REF] and Kaddour [START_REF] Kaddour | Experimental determination of creep properties of zirconium alloys together with phase transformation[END_REF] noticed a decreasing of the strain rate in the (α + β) domain with increasing temperature, for tensile tests performed on Zircaloy-4 and Zr-1%NbO tubular samples. For tests performed on sheet samples in M5 alloy, 460 Trego [START_REF] Trégo | Comportement en fluage à haute température dans le domaine biphasé (α + β) de l'alliage M5[END_REF] observed a comparable behavior, but only in the upper range of the (α + β) domain. He also highlighted a steep transition in the creep behavior and in the grains size at the pure β domain transition. He explained this transition by the vanishing of the remaining α phase that was present at the β grains 465 boundaries. This remaining α phase would prevent the β grains growth. Its complete transformation into β phase would then release the β grains growth and thus lead to the third domain of creep behavior identified in Figure 13. Though the testing conditions are different from this work these conclusions are 470 in good agreements with (i) the β phase evolution obtained by the dilatometry experiment, (ii) the strong transition in the creep behavior observed around 970 • C, and (iii) the metallurgical observations performed on the samples post-mortem.

Structural effects may take place within the samples during the tests due to the thermal heterogeneity on the specimens. Finite element simulations and metallurgical studies will be carried out to identify more accurately the deformation mechanisms of the material.

The decrease of the creep rate in the second domain may result from a serial distribution of the two phases within the material. Assuming this, it follows:

Σ = σ α = σ β (3)
Ė = εα .(1 -f β ) + εβ . f β (4)
Where Σ, σ α and σ β are the homogenized stress, the stress in the α phase and the stress in the β phase, respectively. f β is the β phase fraction. And Ė, εα and εβ are the homogenized strain rate, the strain rate in the α phase and the strain rate in the β phase, respectively. The strain rates can be substituted by the displacement rate so that:

Ur = Uα r .(1 -f β ) + Uβ r . f β (5)
Where Ur , Uα r and Uβ r are the homogenized radial displacement rate, the radial displacement rate in the α phase and the radial displacement rate in the β phase, respectively.

Assuming that the creep behavior of individual phases follow a thermally activated Norton's creep law it can be written that under constant applied stress:

Ui r = A i . exp � B i T � with i = {α; β} (6)
Where T is the temperature and {A i ; B i } represent four constants associated to the pure monophasic domains α and β.

Assuming small strains and using the Equations 5 and 6 with the phase fraction evolution f 1200 • C/s β obtained by dilatometry at 1200 • C/s it becomes possible to obtain an homogenized evolution of the radial displacement rate from 840 • C to 1020 • C. Figure 13 depicts the results for the experiments performed at 7 and 11 bar. This simple homogenized behavior, based on a linear mixing rule on the strains [START_REF] Qu | Fundamentals of micromechanics of solids[END_REF], enables reproducing quite well the experimental data obtained in the dual phase domain. The authors point out that the assumptions postulated will have to be supported by finite elements simulations. These calculations will enable to account for structural effects inherent of ballooning tests.

Impact of the thermal transient

Tests performed in "Simulated LOCA" conditions

Some experiments were performed in "simulated LOCA" conditions to investigate the impact of the thermal transient on the material creep properties. The experimental procedure of these tests is described in Figure 2.b and the nominal loading conditions applied are given in Table 3

Comparison between the two testing conditions

The mean radial displacements rates obtained during the first ten seconds of the "Simulated LOCA" tests are presented in Figure 16. It can be observed that the creep rates obtained subsequently to a slow heating transient are significantly smaller 520 than the one obtained in "simulated RIA" conditions. For the same nominal loading conditions of temperature and pressure differences in the creep rates exceeding a decade are obtained depending on the thermal transient.

Butternut shape profiles were also obtained for the exper-525 iments carried out in the dual phase domain in "Simulated LOCA" conditions. The decreasing of the creep rate while the temperature increases appears clearly in Figure 16 for the tests referred to "LOCA 06", "LOCA 07" and "LOCA 09" in Table 3. Furthermore, similarly to the experiments performed in 530 "Simulated RIA" conditions, a high increase of the strain rate is also observed between the tests "LOCA 09" and "LOCA 14", located at the (α + β) → β transition. These results showed a strong impact of the thermal history on the material mechanical properties. The allotropic phase 535 transformation is a non-instantaneous phenomenon highly dependant on the heating rate. Moreover, other metallurgical phenomena like grains growth, variants selection, transformation induced plasticity or dynamic recrystallisation for instance, are time dependant and linked to the crystallographic structure of 540 the material. Two samples initially identical may be in a completely different metallurgical state at high temperature depending on their thermal history. Further investigations should be conducted in order to better identify the mechanisms responsible of the differences observed between "Simulated LOCA" 545 and "Simulated RIA" experiments.

Summary and conclusions

A creep ballooning device was set-up to reproduce simulated RIA post-DNB conditions on fuel rods sections. More than twenty ballooning tests were performed at regulation tempera-550 tures ranging from 840 • C to 1020 • C with heating rates around 1200 • C/s. Two different levels of pressure of 7 and 11 bar were investigated. The use of full-field optical methods enabled heterogeneous tests to be performed from which a large amount of data could be recorded and further processed. These tests evidenced three domains of behavior that appear to be very-well correlated with the phase fraction present within the material obtained at such heating rates:

• The first domain is defined from 840 • C to 910 • C ( f 1200 • C/s β � 10%). In this temperature range the behavior may be seen as "standard" with a creep rate increasing with the temperature.

• From 910 • C ( f

1200 • C/s β � 10%) to 970 • C ( f 1200 • C/s β � 100%
) the trend reverses and the creep rate decreases while the temperature increases. The experiments performed in this domain led to the so-called "butternut shape" profile on the samples.

• Above 970 • C, i.e. at the pure β domain transition, the creep behavior changes steeply and the creep rate increases anew with the temperature. In this domain the grains size observed post-mortem is much more important and contained needle shaped precipitates, characteristic of a fully ex-β structure.

Such a behavior was already exhibited by Trego on M5 material in simulated LOCA conditions. He postulated that the changes in the creep mechanism observed at the (α + β) → β transition can be explained by the vanishing of the remaining α phase that was located at the β grains boundaries. Small α grains are expected to behave like precipitates, anchoring β grain boundaries and behaving as well as obstacles for dislocation motion. The complete vanishing of the α grains would released the β grains growth that was before inhibited and thereby lead to a high increasing of the creep rate.

A simple homogenization law, based on the assumption that the two phases are in a serial distribution, has been proposed to model the creep behavior in the (α + β) domain. Using the phase fraction, obtained by a stress-free dilatometry experiment performed at very high heating rate, combined with a linear mixing rule, an homogenized behavior could be obtained from 840 • C to 1020 • C. It enables reproducing quite well the creep rate evolution in all the temperature range addressed in the study. Further analysis using finite element simulations will be conducted in order to take into account the potential structural effects occurring within the samples. Thanks to the fullfield measurements a constitutive creep law could be identified 595 accurately by a finite element method updating (FEMU) algorithm.

Experiments were also performed with slow heating rates to investigate the impact of the thermal history on the creep material properties. The results showed significantly higher creep 600 rates for the experiments carried out with a high heating rate. Two initially identical samples may be in a completely different metallurgical state at high temperature since metallurgical phenomena -phase transition, grains growth...-are time dependant. Additional metallurgical studies should be performed to 605 identify and model the mechanisms responsible of the differences observed.

  .a. In these experiments the mechanical loading was first applied. Once it was stabilized the sample was heated up in open loop with heat-135 ing rate around 1200 • C/s. The thermal regulation switched in closed loop when the temperature measured by the pyrometer reached a value 20 • C lower than the target temperature (red circle in Figure 2.a).

2

 2

Figure 1 :Figure 2 :

 12 Figure 1: (a) Picture of the sample mounted on the device; (b) Scheme of the sample and the loading onto the device.

165

  was localized in front of the cameras. The non-axisymmetry of the thermal loading obtained induced heterogeneous creep strains. 3D optical full field measurements methods were used to record these loading heterogeneities.A scheme of a top view of the experimental set-up is depicted in Figure 3. Two CMOS cameras 12MPx (VC-12MC-M65E0-FM) recorded images of the ROI during the tests with a frequency up to 55Hz. The length of the ROI is characterized by the distance between the upper and the lower induction coils and was about 12mm. An example of an image of the ROI is 175 given in Figure 4. The two cameras were equipped of 200mm macro optics (NIK AF MICRO-NIKKOR 200MM F/4 D IF-ED). Near infrared band-pass filters (wavelength λ = 865nm +/-10nm) were mounted on the optics. The pictures recorded by the cameras were used to compute both the kinematic field 180 and the thermal field by stereo-correlation and NIRT methods, respectively. The two cameras were separated by an angle of 18 • to perform a 3D surface reconstruction.

Figure 3 :

 3 Figure 3: Scheme of a top view of the experimental set-up ELLIE.

Figure 4 :

 4 Figure 4: (a) Scheme of the sample mounted on the device; (b) Example of an image with the correlation mesh.

200 2 . 6 .

 26 Near InfraRed Thermography (NIRT)

  230phase fraction accuracy is estimated to be 5% of the phase fraction between 10 and 90% of β content. Due to the nature of the dilatometry experiments the phase fraction cannot be welltrusted at the boundaries of the dual phase domain (i.e. between 0 and 10% of β and between 90 and 100% of β). To avoid the in-235 ertia effect of the thermocouples the temperature was measured using a pyrometer leading to a thermal uncertainty of 7 • C.In the following the phase fraction obtained at equilibrium and upon a constant heating rate of 1200 • C/s are referred to f eq β and f 1200 • C/s β , respectively.

  240

Figure 5 :

 5 Figure5: β phase fraction obtained on SRA-Zircaloy-4 samples at equilibrium conditions[START_REF] Forgeron | Experiment and Modeling of Advanced Fuel Rod Cladding Behavior Under LOCA Conditions: Alpha-Beta Phase Transformation Kinetics and EDGAR Methodology[END_REF] and at a constant heating rate of 1200 • C/s[START_REF] Jailin | Experimental study and modelling of the phase transformation of Zircaloy-4 alloy under high thermal transients[END_REF]

Figure 6 :

 6 Figure 6: Experimental procedure of two tests with the setting values and the measurements obtained. (a) A test performed in the α domain with a regulation temperature of 850 • C and an internal pressure of 20 bar; (b) A test performed in the (α + β) domain with a regulation temperature of 940 • C and an internal pressure of 11 bar. Kinematic and thermal fields are presented in Figure 8 and 9 at times t 1 , t 2 , and t 3 .

Figure 8

 8 Figure8presents the kinematic field and the associated ther-

Figure 7 :

 7 Figure 7: Evolution of the heating rate for the α test, the (α + β) test and a β test.

300 in Figure 11

 11 .a that in the α-experiment the points of maximal radial displacements correspond to the hottest one. Conversely, in the (α + β)-experiment the Figure 11.b shows that the points of maximal radial displacement are the coolest one. Theses observations are in good agreement with the Figures 8, 9 and 10.

4 .

 4 Experiments performed in "Simulated RIA" conditions4.1. Tests performedMore than twenty tests were performed to study the thermomechanical behavior of Zircaloy-4 claddings in the dual phase 320 domain under simulated post-DNB conditions. Heating rates around 1250 • C/s were obtained in these tests. Two different levels of internal pressure, of 7 and 11 bar, were investigated. All the experiments followed the experimental procedure described in Figure2.a.

Figure 8 :

 8 Figure8: Kinematics and thermal field computed respectively by 3D-DIC and NIRT. These field were obtained at {t 1 = 50s; t 2 = 150s; t 3 = 350s} (see in Figure6) for a test performed in the α domain with a regulation temperature of 850 • C and an internal pressure of 20 bar. Data extracted from the red axial line are presented in more details in the Figure10.

Figure 9 :

 9 Figure9: Kinematics and thermal field computed respectively by 3D-DIC and NIRT. These field were obtained at {t 1 = 50s; t 2 = 150s; t 3 = 350s} (see in Figure6) for a test performed in the (α + β) domain with a regulation temperature of 940 • C and an internal pressure of 11 bar. Data extracted from the red axial line are presented in more details in the Figure10.

Figure 10 :Figure 11 :

 1011 Figure 10: Thermal and kinematics data extracted from the red axial lines for (a) the α experiment Figure 8, and (b) the (α + β) experiment Figure 9.

380 tween f 1200 •

 1200 C/s β � 10% and f 1200 • C/s β � 100%. In this second domain the creep rate evolution reverses and decreases while the temperature increases. For the experiments performed in this domain two well-defined ballooning areas were obtained on the samples. This double balloons profile led to the so-385 called "butternut shape" on the specimens, already mentioned for the (α + β)-experiment in section 3. The so-called "butternut shape" is observed on the sample referred to label 3 in Figure14. This specimen was issued from an experiment carried out with a regulation temperature of 954 • C and with an internal 390 pressure of 7 bar. The authors recall that in all tests a single hot spot was localized at the center of the samples as presented in Figure10.b. The temperature is then decreasing toward the jaws as illustrated in Figure14.The third domain is defined above 970 • C. It begins at the 395 (α + β) → β transition. Above this transition the creep rate increases strongly with the temperature. Three ballooning areas were obtained on the specimens experiments performed in this domain, as shown for instance in the sample labelled 4 in Figure 14. One localized at the center of the sample where the 400 temperature was maximal and two from either side of the ROI.

  410

Figure 13 :

 13 Figure 13: Mean radial displacement rates normalized by the initial radius obtained during the first 10 seconds of the tests. The fraction of β was obtained by a stress-free dilatometry experiment carried out with a constant heating rate of 1200 • C/s. Samples issued of tests labelled 1 to 4 are presented Figure 14.

Figure 14 :

 14 Figure 14: Example of 4 samples that have been tested. For each case the temperature was maximal at the center of the sample and was decreasing towards the jaws. t f is the final test time of the experiments.

  .b the grains are slightly bigger and few oxygen enriched set of parallel thin nee-dles are observed. Large grains, containing coarse needles, are 435 observed in Figure 15.c and in Figure 15.d, where the maximal temperatures were 969 • C and 976 • C, respectively. This is characteristic of a fully ex-β structure. In all the samples, the areas that exceeded 970 • C present a microstructure with large ex-β grains containing needle shaped 440 precipitates comparable to the one illustrated in Figures 15.c.d.Structures with small and equiaxed grains were obtained when the temperature was lower than 970 • C. These observations are consistent with the (α + β) → β transition obtained in the dilatometry experiments (Figure5).

  445

Figure 15 :

 15 Figure 15: Metallurgical analysis obtained from an axial cut of the RIA 25 sample along the red dashed line. The test lasted 135 seconds. Four micrographs are shown with their corresponding maximal temperature reached. The central image shows the thermal field computed by NIRT at the end of the test.

Figure 16 :

 16 Figure 16: Mean radial displacement rates normalized by the initial radius obtained during the first 10 seconds of the tests. The results of the mixing rule modeling are presented for each level of internal pressure. The fraction of β was obtained by a stress-free dilatometry experiment carried out with a constant heating rate of 1200 • C/s.

Table 1 :

 1 Stress relieved annealed Zircaloy-4 composition (in wt%).

Table 2 :

 2 Experimental conditions of the "Simulated RIA" experiments performed. T ROI represents the temperature computed at the center of the ROI by NIRT.

	)

  .

	Exp. no	T ROI ( • C) P i (bar)	ΔT ( • C)
	LOCA 10	863	11	848 -870
	LOCA 08	901	11	884 -905
	LOCA 04	931	7	922 -934
	LOCA 06	973	7	965 -976
	LOCA 11	987	11	969 -999
	LOCA 09	989	7	968 -998
	LOCA 13	1010	7	1003 -1015

Table 3 :

 3 Experimental conditions of the "Simulated LOCA" experiments performed. T ROI is the temperature computed at the center of the ROI by NIRT.

Acknowledgments

The authors acknowledge with thanks financial support of this work from EdF and Institut Carnot Ingénierie@Lyon. The authors want also to express their thanks to J. Réthoré for allowing access to the UFreckles software and ATYS Consulting 615 Group for helping on the induction problematic.

Data availability

The raw and processed data of the current study are available from the corresponding author on request. 610

value of each node and were computed by NIRT. The evolution of the β phase fraction with the temperature is also plotted in Figure 13 for the equilibrium condition and for a heating rate of 1200 • C/s.

The authors point out that these mean displacement rates val-