D Da Fonseca

F Mompiou

T Jourdan

J.-P Crocombette

A Chartier

F Onimus

Evidence of dislocation loop preferential nucleation in irradiated aluminum under stress

Keywords: irradiation creep, dislocation loop, in-situ straining irradiation, object kinetic Monte-Carlo, Frenkel pair accumulation

To explain irradiation creep, several mechanisms have been proposed. Some are based on the effect of stress on either nucleation or growth of dislocation loops. To investigate these mechanisms in aluminum we combine in-situ transmission electron microscope irradiation under stress and two simulation approaches (object kinetic Monte-Carlo, molecular dynamics Frenkel pair accumulation). We observe the selectivity of Frank loop variants under electron irradiation and applied stress. When the stress is turned on after loop formation, there is no loop variant selectivity, suggesting the absence of preferential absorption on already formed loops. Object kinetic Monte-Carlo simulations, including the effect of stress on the diffusion of point defects, show no growth rate difference between loop variants. Frenkel pair accumulation simulations exhibit variant selectivity of nucleated loops. This shows that loop selectivity is due to preferential nucleation of well oriented loops under stress and not to differential growth of loops.

Under irradiation and applied stress, a specific deformation process known as irradiation creep arises in many materials, such as steels, nickel-based and zirconium alloys [START_REF] Hesketh | A transient irradiation creep in non-fissile metals[END_REF][START_REF] Lewthwaite | Irradiation creep in several metals and alloys at 100➦C[END_REF][START_REF] Onimus | 1.10 -irradiation creep in materials[END_REF]. Its phenomenology, well documented, is very different from thermal creep, which essentially operates at high temperature. Depending on irradiation flux, temperature and stress magnitude, irradiation creep can be responsible for deformation rates far higher than those related to thermal creep [START_REF] Mosedale | Irradiation creep in fast reactor core component materials[END_REF]. Understanding the mechanisms underlying irradiation creep is therefore of prime importance.

Several mechanisms have been proposed to explain irradiation creep [START_REF] Matthews | Irradiation creep models -An overview[END_REF]2,[START_REF] Onimus | 1.10 -irradiation creep in materials[END_REF]. Some of them are based on the effect of stress on dislocation loops and account for various observations. In some experiments, interstitial loops in planes where the normal stress is the largest have been shown to be larger [3,[START_REF] Jitsukawa | The Behavior of Irradiation-Produced Dislocation Loops under External Stress during Electron Irradiation[END_REF], resulting in a net strain. Other experiments have evidenced an increase in interstitial loop density with the normal stress on the loop habit planes, also resulting in a net strain [START_REF] Okamoto | Stress-biased loop nucleation in irradiated type 316 stainless steels[END_REF]3,5,[START_REF] Suzuki | Stress-assisted nucleation of interstitial loops in an electron-irradiated Fe-18Cr-14Ni alloy[END_REF][START_REF] Tanigawa | Numerical analysis of stress effects on frank loop evolution during irradiation in austenitic Fe-Cr-Ni alloy[END_REF][START_REF] Xu | Anisotropic dislocation loop distribution in alloy T91 during irradiation creep[END_REF]. The mechanisms which can explain such results can be classified in two main categories: the stress-induced preferred absorption (SIPA) phenomenon [START_REF] Heald | Steady-state irradiation creep[END_REF]4] and the stress-induced preferred nucleation (SIPN) of interstitial dislocation loops. SIPA is due to the anisotropic diffusion of self-interstitials and vacancies under stress [START_REF] Woo | Irradiation creep due to elastodiffusion[END_REF][START_REF] Fonseca | Numerical investigation of dislocation climb under stress and irradiation[END_REF]. It has been shown to be able to explain not only differential loop growth, but also preferential loop formation in some planes [START_REF] Wolfer | Theory of swelling and irradiation creep[END_REF]. The first version of SIPN was based on the classical nucleation theory, which proved erroneous for interstitial dislocation loops due to the absence of activation barrier for nucleation [START_REF] Garner | A reassessment of the role of stress in development of radiationinduced microstructure[END_REF]. Later, it was suggested that the reorientation of small SIA clusters under stress could explain the different loop densities on different habit planes [START_REF] Wolfer | Correlation of radiation creep theory with experimental evidence[END_REF]. However, the magnitude of this effect was found too low to match experimental results. Although this process is not a classical nucleation process, we consider it as SIPN, as opposed to the purely diffusive SIPA process.

Whether the contribution of loops to irradiation creep is due to a SIPN or a SIPA mechanism is still an open question [START_REF] Garner | Irradiation creep mechanisms: an experimental perspective[END_REF]. Results are not all consistent, with some experiments leading to opposite trends to what would be expected due to SIPN [START_REF] Tabata | Behavior of lattice defects under the constant load[END_REF] or SIPA [START_REF] Saka | Effect of external tensile stress on the growth of interstitial loops[END_REF]. In addition, internal stresses are difficult to determine and probably affect the results [START_REF] Gelles | Frank loop formation in irradiated metals in response to applied and internal stresses[END_REF][START_REF] Sato | Stress effect on formation of interstitial loops in an electron irradiated Fe-18Cr-14Ni alloy[END_REF]1]. Finally, in high dose irradiations, loops interact with the dislocation network. This interaction alters the loop size distributions and makes the sole effect of stress on loops less easy to observe, especially if irradiations are performed ex-situ [START_REF] Garnier | Irradiation creep of SA 304L and CW 316 stainless steels: Mechanical behaviour and microstructural aspects. Part I: Experimental results[END_REF][START_REF] Renault-Laborne | Evolution of microstructure after irradiation creep in several austenitic steels irradiated up to 120 dpa at 320➦C[END_REF][START_REF] Gharbi | Impact of an applied stress on c-component loops under Zr ion irradiation in recrystallized Zircaloy-4 and M5➤[END_REF]. Therefore, in-situ, low dose irradiations should be preferred to investigate stress effects on loops.

In the following we combine both experimental in-situ straining transmission electron microscopy (TEM) irradiations of aluminum thin foils and two simulation methods, namely object kinetic Monte-Carlo (OKMC) and molecular dynamics using Frenkel pair accumulation (FPA) process, to determine whether stress has an impact on the nucleation and/or on the growth of loops.

In-situ observations were conducted in a JEOL 2010HC microscope operated either at 140 kV, i.e. well below displacement threshold for setting up experiments and at 180 kV for irradiation. A pure aluminum (99.999%) annealed 0.5 mm thick foil was used for the experiments. It ensures a large grain microstructure with a strong cube texture, i.e. foil normal close to [0 0 1] and rolling direction along the [1 0 0] direction. 3 mm disc specimens were extracted from the foil plane, mechanically grinded and eventually doubled-jet electropolished to electron transparency around a central hole using a solution of perchloric acid/ethanol (5/95) at -30 • C and 30 V. Then the specimens were glued, with a cyanoacrylate glue, on a copper tensile grid and placed in a Gatan straining holder, as shown in Fig. 1.a. Samples were oriented on the copper grid so that the ⟨1 1 0⟩ direction (at 45➦ from the rolling direction) was aligned with the straining direction, with the normal of the foil close to ⟨0 0 1⟩ direction. The stress was imposed by a micrometer controlled motion of one of the sample grip while the other one stayed at rest. The method used to evaluate the applied stress is described in Supplementary Material. Irradiations were conducted at room temperature without and with external stress. At this temperature (below 0.5 T m , where T m is the melting temperature), emission of point defects from loops can be neglected and the contribution of the stress dependent emission rates to loop evolution is negligible. The beam was adjusted to obtain an electron flux of 1 × 10 5 e -/nm 2 /s at 180 kV which corresponds to 4.55 × 10 -5 dpa/s. The damage rate was computed as the electron flux times the adequate cross-section given in Oen's table [START_REF] Oen | Cross sections for atomic displacements in solids by fast electrons[END_REF]. In this condition, the irradiated zone covers a disk of 3 µm of diameter. Observations were made in the center of the irradiated area which was measured [START_REF] Mompiou | pycotem: An open source toolbox for online crystal defect characterization from TEM imaging and diffraction[END_REF] to be around 170 nm thick (see Supplementary Material). Three experiments referenced as 1, 2 and 3 (Fig. 1.b) are detailed in the following. They were all carried out in the same grain in the area where the local stress is uniaxial and parallel to the tensile axis. Fig. 2 shows weak-beam dark field (WBDF) images taken after irradiation with g = 200, using g(3g) diffraction condition, in order to have all possible Frank loops visible.

After irradiation without applied stress, the microstructure was characterized in detail. The loop nature was determined using the inside/outside method [START_REF] Maher | Neutron irradiation damage in molybdenum[END_REF][START_REF] Föll | A simple method for the analysis of dislocation loops by means of the inside-outside contrast on transmission electron micrographs[END_REF]. Only interstitial loops were observed. Furthermore, most of these loops are Frank loops with Burgers vectors b = 1/3⟨1 1 1⟩ and lying in {1 1 1} habit planes [START_REF] Yang | Electron irradiation damage in high purity aluminum[END_REF]. In the tensile configuration used, Burgers vectors 1/3[1 1 1] and 1/3[1 1 1] (denoted herein as type A loops in red in Fig. 1c-d) are both perpendicular to the straining axis, while Burgers vectors 1/3[1 1 1] and 1/3[1 1 1] (denoted herein as type B loops in green in Fig. 1c-d) are tilted 35 • away from the straining axis. Hence, the two types of loops are not equivalent with respect to the applied stress. For the type A loops, the component of the stress along the loop Burgers vector is equal to zero. Moreover, all the normals of the habit planes of the loops make a 55 • angle with the foil normal which allows the observations of the loops with the same apparent projections, but with different orientation with respect to the tensile axis: the long axis of the ellipse of the A loops is vertical, parallel to the tensile axis while the long axis of the ellipse of the B loops is horizontal, perpendicular to the tensile axis.

Experiment 1 (Fig. and 40% of type B loops. Although the proportions are not exactly equally balanced, it can be considered as a reference state. The authors want to point out that a stress-free state, with nearly balanced proportion of loops, is not systematically obtained. This is presumably due to residual stresses. Only samples with nearly stress-free state, with initially quasi-balanced proportion of loops, have thus been considered for this work.

In experiment 2, the specimen was first strained at lower voltage to avoid atomic displacement until the first dislocation motion was observed. Then, the displacement of the tensile grip was slightly reduced. Irradiation started with immobile dislocations under a stress slightly below the yield stress that can be roughly estimated from dislocation curvature, i.e. of the order of 100 MPa in a thin foil (see Supplementary Material). After irradiation, the tensile grip was again slightly moved (a few microns), inducing dislocation glide, in order to check that the stress did not relax during irradiation. Fig. 2.b shows the microstructure after 41 minutes of irradiation. Contrary to experiment 1, 10% of loops are of type A and 90% of type B. This result is consistent with most of the observations in literature where the populations with the largest projection of the tensile stress vector on the normal to their habit plane are preferentially formed [START_REF] Okamoto | Stress-biased loop nucleation in irradiated type 316 stainless steels[END_REF]3,5,[START_REF] Gelles | Frank loop formation in irradiated metals in response to applied and internal stresses[END_REF][START_REF] Sato | Stress effect on formation of interstitial loops in an electron irradiated Fe-18Cr-14Ni alloy[END_REF]1,[START_REF] Suzuki | Stress-assisted nucleation of interstitial loops in an electron-irradiated Fe-18Cr-14Ni alloy[END_REF][START_REF] Xu | Anisotropic dislocation loop distribution in alloy T91 during irradiation creep[END_REF]. This strong anisotropy of loop population is a clear evidence of the influence of the applied stress, but this experiment is not self-sufficient to ascertain between preferential absorption and preferential nucleation mechanisms, as some studies explained the anisotropic population with SIPA mechanism [START_REF] Gelles | Frank loop formation in irradiated metals in response to applied and internal stresses[END_REF][START_REF] Xu | Anisotropic dislocation loop distribution in alloy T91 during irradiation creep[END_REF] based on the work of Wolfer [START_REF] Wolfer | Correlation of radiation creep theory with experimental evidence[END_REF].

A third experiment was hence conducted, in two steps, to discriminate between SIPA and SIPN. In a first step (experiment 3a), a new area was irradiated without applying an external stress, until the loops were large enough to allow their type classification. The irradiation was stopped after 16 minutes (see Fig. 2.c). As expected for an irradiation without stress, proportions are balanced with 47% of type A loops and 53% of type B loops and a measured mean diameter of 13.3 nm and 12.4 nm, respectively. This corresponds to a mean growth rate in diameter of 0.013 nm/s. Using the same protocol as in experiment 2, the sample was then stressed and irradiated for 48 minutes in the same area (experiment 3b) (see Fig. 2.d). The proportion of the loops did not significantly evolve: 40% of loops are of type A and 60% of type B. The loops have a mean diameter of 27 nm for both types.

Comparing the results of experiment 2 and 3 shows that stress has a strong impact only in early stages of irradiation, suggesting that the contribution of SIPA should be very moderate. The shrinkage of type A loops and the growth of type B loops would be expected due to SIPA but instead both types of loop grow and only the shrinkage of new small loops is observed, becoming sacrificial loops for the growth of previously formed loops. Suzuki and Sato [START_REF] Suzuki | Stress-assisted nucleation of interstitial loops in an electron-irradiated Fe-18Cr-14Ni alloy[END_REF] did an equivalent experiment to our experiment 3 in Fe-18Cr-14Ni alloy under 1 MeV electron irradiation at temperature ranging between 340➦C and 456➦C, and they drew the same conclusion. To confirm this analysis, SIPA and SIPN were evaluated separately using two different simulation methods. The results are presented in the following. The height of 200 nm in the z direction corresponds approximately to the foil thickness. Free surfaces are hence introduced in the z direction while periodic boundary conditions are considered in x and y directions. Initially, the simulation box contains no point defect. Frenkel pairs are then generated at a dose rate of 5 × 10 -5 dpa/s to mimic TEM irradiation. Point defects migrate in the simulation box and can recombine with defects of opposite type, agglomerate with each other (thus forming interstitial loops or cavities) or escape to the surfaces. In addition, immobile traps are randomly placed in the system before the irradiation starts, to mimic impurities present in the material. These impurities are assumed to strongly bind to migrating point defects. Without such traps, SIAs would all recombine with vacancies or escape to free surfaces, leaving the simulation box free of loops. A total of 10 impurities, corresponding to a concentration of 0.08 appm, is chosen to reproduce the observed loop density in experiment 3a, well below the estimated experimental concentration. This suggests that not all impurities are efficient traps for point defects. When SIAs agglomerate with an impurity to form a loop, the habit plane is chosen randomly and kept the same as the loop grows, so SIPN is purposefully discarded. More details about the parametrization are given in Supplementary Material.

The migration of point defects is affected by the internal stress generated by dislocation loops in the thin foil and the externally applied stress [START_REF] Gelles | Object kinetic monte carlo modelling of irradiation microstructures with elastic interactions[END_REF]. The elastic interaction energy between point defects and the local stress field, which biases the migration of point defects and leads to SIPA, is given by an elastic model based on the elastic dipole tensors and diaelastic polarizabilities of point defects [9]. These properties have been obtained recently in aluminum from density functional theory calculations [START_REF] Fonseca | Numerical investigation of dislocation climb under stress and irradiation[END_REF]. Image forces induced by the presence of loops near surfaces [START_REF] Wu | The elasticity of the 1/2 a 0 ⟨111⟩ and a 0 ⟨100 dislocation loop in α-Fe thin foil[END_REF] are neglected, as it has been shown that they affect dislocation loop growth only when loops are a few nanometers away from the surface [START_REF] Gelles | Object kinetic monte carlo modelling of irradiation microstructures with elastic interactions[END_REF].

Simulations were run at 300 K without or with a 100 MPa uniaxial stress applied in the y direction. Simulations were stopped after 10 minutes of irradiation. Cluster distributions are obtained from 100 independent simulations carried out with different spatial distributions of impurities.

Simulations performed without stress show an almost equal proportion of type A (49.4%) and type B (50.6%) loops with mean diameters 7.67 nm and 7.99 nm respectively, leading to an average growth rate of 0.013 nm/s. This growth rate is the same as in experiment 3a, which validates our simulation method. Typical microstructures and loop size distributions when a stress is applied are shown in Fig. 3.b and Fig. 3.e, respectively. Proportions of type A (48.5%) and type B (51.5%) loops are still balanced with mean diameters of 8.45 nm and 9.89 nm respectively. Well-oriented loops are slightly larger, indicating only a moderate effect of the stress on differential growth. This effect is however below the experimental error and hence could not be ascertained by TEM. More importantly, the simulations fail to reproduce the strong selectivity on loop variants (Fig. 3.d and Fig. 3.e). These results thus suggest that the microstructure evolution under stress cannot be explained by a SIPA mechanism. They are in line with the rather small effect of stress on absorption efficiencies of dislocations derived in a previous work [START_REF] Fonseca | Numerical investigation of dislocation climb under stress and irradiation[END_REF]. Absorption efficiencies of loops are found to be much more dependent on other factors, such as loop size and local loop environment. To further highlight the importance of SIPN, molecular dynamics simulations of defect accumulation under stress were performed. Simulation boxes oriented along [1 1 0], [1 1 0] and [0 0 1] directions were used. Their dimensions are of around 25x25x24 nm 3 and they contain 887520 atoms of aluminum whose interactions are described with an EAM potential [START_REF] Liu | Aluminium interatomic potential from density functional theory calculations with improved stacking fault energy[END_REF]. Periodic boundary conditions are applied in all three directions. Simulations are done at constant room temperature and constant stress using a Berendsen thermostat and a Parrinello-Rahman barostat, with fixed angles so that the box remains tetragonal during the simulations. TEM experiments, i.e. electron irradiations, are modelled by periodically introducing Frenkel pairs in the box, following the same FPA method as in previous works [START_REF] Limoge | Computer simulation studies of radiation induced amorphization[END_REF][START_REF] Michael | Radiation-induced amorphization of ordered intermetallic compounds CuTi, CuTi 2 , and Cu 4 Ti 3 : A molecular-dynamics study[END_REF][START_REF] Crocombette | Atomistic simulation of amorphization thermokinetics in lanthanum pyrozirconate[END_REF]7,6,8]. 200 Frenkel pairs are generated every 2 ps, which corresponds to a dose rate of 1.13 × 10 8 dpa/s, far from the experimental reality. The total simulated time is 0.5 ns, which makes long range diffusion negligible and therefore excludes any SIPA process. In addition, results cannot be directly compared with experiments at the same absolute value of dose. A snapshot of atomic configurations is taken every 2 ps, just before each insertion of point defects. The cell is then visualized with the OVITO software [START_REF] Stukowski | Visualization and analysis of atomistic simulation data with OVITO-the Open Visualization Tool[END_REF] and dislocations are identified with the DXA algorithm [START_REF] Stukowski | Automated identification and indexing of dislocations in crystal interfaces[END_REF]. Note that only closed dislocations made up of segments with the same Burgers vector are counted in the total number of Frank loops. The evolution of loops is tracked until they are too big and interact with each other thus forming a dislocation network (around ∼ 0.03 dpa).

A reference simulation was performed at zero stress. Type A loops come out at 48.6% to 51.4% for type B, thus the code reproduces correctly equiprobable loop nucleation between the two variants. In a second simulation, we set a stress of 100 MPa in [1 1 0] direction and 0 MPa in [0 0 1] and [1 1 0] directions. Under stress, we obtain the microstructure shown in Fig. 3.c and the corresponding distribution in Fig. 3.f. Among all the dislocation segments detected by OVITO, 75.8% are of Frank type (265 Frank loops are counted in Fig. 3.c). What stands out from the snapshot and the bar plot is that well-oriented loops (B type loops) are in majority (65.7% of the total). This proportion is lower than what is shown in Fig. 3.d but the result still evidences preferential orientation of Frank loops under stress (even at this relatively low applied stress for MD simulations). We found that the proportion of type B loop rises with the level of stress applied along [1 1 0] direction.

Our experiments show that the impact of stress is noticeable when it is applied from the start of the irradiation, suggesting that the anisotropic microstructure observed is due to preferential formation of well oriented loops under stress and not to the differential evolution of already existing loops. OKMC simulations confirm the small effect of SIPA and MD-FPA simulations yield an anisotropic population under stress. We thus prove that SIPN is stronger than SIPA under electron irradiation, suggesting that SIPA has only a minor contribution to irradiation creep deformation.

Measurement of sample thickness

The sample thickness was estimated from the length and direction determination of two dislocations emerging at the sample surfaces and located at the periphery of the irradiated area. The procedure is described in [7] using the pycotem software. Measurements of dislocation apparent lengths and directions were obtained at different sample inclination with respect to the main holder axis (here vertical). The apparent thickness at the observation tilt was deduced from dislocation direction and length knowing the electron beam direction. Although the foil normal has not been determined, it was assumed to be close to the beam direction at null tilt. The thickness ranges from 210±16 nm to 137±16 nm for the two dislocations, indicating that the foil has a wedge shape. For the sake of simplicity, we took an average thickness of 170 nm.

Measurement of local stress

The local stress applied during the in-situ TEM experiments was evaluated in-situ between experiment 3.a and 3.b (see letter) using the dislocation curvature of the first moving dislocation. In this area, the principal local stress is a tensile stress with an axis parallel to the sample holder as indicated in Fig. 2. First, the sample was strained, by increasing the displacement of the mobile grip, until dislocation glide was observed in the same grain where irradiation experiments were performed. Then the grip displacement was slightly reduced to stop the dislocation motion before the stress was relaxed Fig. 2.a. The glide plane, (1 1 1), was deduced from slip traces analysis knowing crystal orientation and thickness using pycotem [7]. Cross-slip was eventually observed in the (0 0 1) plane deduced from its trace analysis Fig. 2.c. The Burgers vector is then at the intersection of the two planes, i.e. along the ⟨1 1 0⟩ direction as shown in the stereographic projection in Fig. 2.b. The local shear stress was inferred from the comparison between the shape of a dislocation at a given stress computed from anisotropic elasticity, using DISDI [4] and elastic constants at 300 K from [8], and its shape on an image corrected from perspective effect (Fig. 2.d). We estimate the local shear stress to be equal to 45 ± 5 MPa (Fig. 2.d). Considering a Schmid factor of 0.47 and that the stress is locally a pure tensile stress, the applied stress is estimated to be between 85 and 106 MPa.

OKMC parametrization

In our simulations only point defects are assumed to migrate. Migration energies in the bulk, without any effect of stress, are 0.105 eV for self-interstitial atoms and 0.605 eV for vacancies [1]. Elastic properties of point defects (elastic dipoles and polarizabilities), which affect their mobility under stress, are given in Ref. [3]. Including the mobility of small point defects clusters would not change the results. The concentration of SIAs is so low that no di-interstitials form. Vacancies are assumed not to bind together, in agreement with DFT calculations [9,2]. Only SIA-impurity and vacancy-impurity complexes may form, which are assumed to be immobile. SIA-impurity binding energy is chosen equal to 1 eV in order to avoid any thermal dissociation. The binding energy of vacancy-impurity is assumed to be lower than the latter. It is chosen to equal to 0.4 eV. It is probably an upper bound for the binding energy with impurities [5]. With such a value, some cavities may form occasionally but their number is far lower than loops. This proves that cavities should not be seen, as confirmed experimentally.

SIA-impurity and vacancy-impurity clusters can grow by absorbing SIAs and vacancies, respectively. Clusters containing 2 SIAs or more are modelled as Frank loops. No thermal emission of point defects from these loops is permitted. This approximation is well justified given the low temperature at which simulations are performed. The binding energy of a vacancy with a vacancy cluster, nucleated on an impurity and modelled as a cavity, is equal to the binding energy with the impurity (0.4 eV) if the number of vacancies attached to the impurity is smaller than the number of first nearest neighbours of the impurity (assumed to be equal to 12) or if the binding energy with the impurity is higher than the binding energy with a cavity. This binding energy, which is used otherwise, is calculated with a capillary law [6] based on the binding energy of 2 vacancies (0 eV) and the formation energy of a vacancy (0.67 eV).

Vacancies and SIAs are created as Frenkel pairs, at a distance of 4a, where a is the lattice parameter. They are described as spherical defects of radius 0.16 nm, which recombine with each other if the distance between their center is lower than 2a. They agglomerate with a Frank dislocation loop if the distance to the dislocation line is lower than 2b, where b is the Burgers vector of the loop.

Figure 1 :

 1 Figure 1: (a) Straining TEM sample holder, (b) sketch of the experiments: three irradiation are conducted in the same grain close to the electropolished hole, with different stress conditions (without or with stress estimated to be around 100 MPa), the final microstructures are presented in Fig.2, (c) orientation of the two different Frank loop types with respect to the tensile axis, and (d) the corresponding stereographic projection with the associated tensile axis. Well-oriented loops are presented in green (type B) and unfavorably oriented loops are presented in red (type A). The method used to evaluate the applied stress is described in the Supplementary Material.

Figure 2 :

 2 Figure 2: Irradiations conducted in the same grain with a flux of 1 × 10 5 e -/nm 2 /s at room temperature: (a) experiment 1 without applied stress for 38 minutes, (b) experiment 2 under stress for 41 minutes, (c) experiment 3a without stress for 16 minutes, (d) experiment 3b with 48 additional minutes under stress. Red arrows point out some unfavorably oriented loops and green arrows some well-oriented loops.

Figure 3 :

 3 Figure 3: Comparison of the three methods to evaluate the main mechanism involved. On the left (a), (b), (c) are the microstructures studied with the same orientation of the applied stress showed by a white arrow. (a) Experiment 2: in-situ irradiation under stress, unfavorably oriented loops are in minority so some of them are highlighted by red ellipses, (b) four out of 100 systems simulated by OKMC method, (c) system simulated with FPA method, black lines correspond to non-Frank dislocations. For the simulations in (b) and (c), type A dislocations are in red and type B are in green. On the right (d), (e), (f) are the corresponding normalized distributions of loop diameters in nanometer.

Figure 1 :

 1 Figure 1: Two micrographs showing two dislocations of interest d1 and d2 for thickness determination: (a) left part of the irradiation area, where the sample is thicker because it is further away from the electropolished hole and (b) right part of the irradiation area, closer to the hole. Scale bars correspond to 500 nm.

Figure 2 :

 2 Figure 2: (a) Micrograph showing a curved gliding dislocation d. The slip traces left at sample surfaces are visible on the dislocation wake after its motion from left to right. (b) Stereographic projection of the crystal as observed. (c) Micrograph showing the cross-slip of dislocation d. Primary and cross-slip traces are highlighted in red and blue respectively. Their apparent widths are also indicated perpendicular to the traces. They match the plane shown in the stereographic projection. (d) Image of the dislocation corrected from perspective effect and the best theoretical shape. The fit was chosen from a series of theoretical shapes when varying the applied shear stress from 20 to 100 MPa with 10 MPa step (right). The local tensile axis is reported on the stereographic projection.

Acknowledgments

The authors are grateful to Estelle Meslin, Benoit Arnal and Guilhem Sagnes for their help with sample preparation and TEM operation.

Supplementary Material for: Evidence of dislocation loop preferential nucleation in irradiated aluminum under stress D. Da Fonseca


Fabien Onimus
email: fabien.onimus@cea.fr.

Jean-Luc Béchade

Claude Prioul

Philippe Pilvin

Isabelle Monnet

Sylvie Doriot

Bénédicte Verhaeghe

Didier Gilbon

Laurence Robert

Laurent Legras

Jean- Paul Mardon

B Kammenzind

Guest Editors. P Rudling

Keywords: dislocation channeling, irradiation, loop, internal stress, zirconium alloy

TEM investigations have been performed on irradiated samples after deformation covering various testing conditions for different recrystallized Zr alloys. It is shown that for transverse tensile tests and internal pressure tests performed at 350°C, only basal channels are observed for strain levels up to uniform elongation, whereas only prismatic and pyramidal channels are observed for axial tensile test. Then, focusing on internal pressure testing, irradiation hardening and strain hardening behavior have been analyzed in detail and compared to TEM observations. It is proposed that dislocation channeling phenomenon leads to high strain incompatibility between channels and surrounding grains and therefore to high microscopic internal stresses. These high microscopic internal stresses are believed to induce strong kinematic hardening at the macroscopic scale, as suggested by the mechanical behavior analysis. Finally, a micro-mechanical model based on microscopic deformation mechanisms and using homogenization techniques is proposed, taking into account the observed channeling phenomenon.

Introduction

Zirconium alloys, mainly used as fuel cladding tubes in light water nuclear reactor, undergo fast neutron irradiation leading to strong irradiation damage. Since the cladding is the first confinement barrier of the radioactive material, it is essential to keep its mechanical integrity throughout its lifetime, especially under service conditions. Consequently, an important part of the research undertaken on these materials aims at increasing the knowledge of irradiation effects on mechanical properties and microstructure in order to develop a predictive model for irradiated Zr alloys. Lately a micro-mechanical model, based on homogenization theory and deformation mechanisms, has been developed [START_REF] Geyer | Modeling of the Anisotropic Viscoplastic Behavior of Fully Annealed Zircaloy-4 Tubes by a Polycrystalline Approach[END_REF][START_REF] Brenner | Thermal Creep of Zr-Nb1%-O Alloys: Experimental Analysis and Micromechanical Modeling[END_REF] for non-irradiated material. Nevertheless, in order to extend this type of approach to irradiated zirconium alloys, the deformation mechanisms have to be clearly characterized and understood.

It is known that neutron irradiation leads to a significant increase in strength and a strong reduction in uniform elongation with an associated highly localized necking as reviewed in [START_REF] Douglass | The Metallurgy of Zirconium[END_REF]. From a microscopic point of view, it has been established that in zirconium alloys irradiation damage at low fluence [START_REF] Northwood | Characterization of Neutron Irradiation Damage in Zirconium Alloys -An International Round-Robin Experiment[END_REF] consists mainly in the formation of interstitial and vacancy <a> type prismatic loops, with prismatic habit planes. As reviewed by Bement [START_REF] Bement | Fundamental Materials' Problems in Nuclear Reactors[END_REF] and Hirsch [START_REF] Hirsch | Point Defect Cluster Hardening[END_REF], the increase in strength observed in irradiated materials is usually attributed to the presence of the high density of small irradiation-induced loops which can act as obstacles against dislocation glide. However, these obstacles can be overcome by dislocations when a sufficient stress is applied; then the loops can be dragged by dislocations. This process of removal of irradiation loops produces a cleared zone free of defects inside the grain. These obstacle-free channels will therefore constitute preferred areas for further dislocation gliding, leading to plastic strain localization at the grain scale with regions of very high plastic strain surrounded by regions of almost zero plastic strain. This mechanism, called dislocation channeling, has been observed in many different irradiated materials as reviewed by Wechsler [START_REF] Wechsler | Dislocation Channeling in Irradiated and Quenched Metals, The Inhomogeneity of Plastic Deformation[END_REF]. In the case of zirconium alloys, several authors have reported defect free channels [START_REF] Coleman | Deformation Parameters of Neutron Irradiated Zy-4 at 300°C[END_REF][START_REF] Onchi | The Inhomogeneous Deformation Behavior of Neutron Irradiated Zircaloy-2[END_REF][START_REF] Williams | Effects of Boiling Water Reactor Irradiation on Tensile Properties of Zircaloy[END_REF][START_REF] Pettersson | Evidence for Basal or Near-Basal Slip in Irradiated Zircaloy[END_REF][START_REF] Adamson | Effects of Neutron Irradiation and Oxygen Content on the Microstructure and Mechanical Properties of Zircaloy, Microstructure and Mechanical behavior of materials[END_REF][START_REF] Fregonese | Failure Mechanisms of Irradiated Zr Alloys Related to PCI: Activated Slip Systems, Localized Strains, and Iodine-Induced Stress Corrosion Cracking[END_REF][START_REF] Régnard | Activated Slip Systems and Localized Straining of Irradiated Alloys in Circumferential Loadings[END_REF], however, there is yet no agreement on the type of activated slip systems. Indeed, for transverse tensile test, Adamson [START_REF] Adamson | Effects of Neutron Irradiation and Oxygen Content on the Microstructure and Mechanical Properties of Zircaloy, Microstructure and Mechanical behavior of materials[END_REF] reported mainly prismatic and pyramidal channels for an irradiated high oxygen ([O] = 1400 ppm) Zr alloy, whereas Régnard [START_REF] Régnard | Activated Slip Systems and Localized Straining of Irradiated Alloys in Circumferential Loadings[END_REF] observed mainly basal channels and, to a lesser extent, pyramidal and prismatic channels, for irradiated recrystallized Zy-4. As a conclusion, it is worth noticing that there is no agreement between the different authors and that too few observations have been performed for other types of loading directions and testing conditions. It also has to be highlighted that the link between macroscopic mechanical behavior and dislocation channeling mechanism has rarely been investigated in detail [START_REF] Odette | Modeling of Multiscale Mechanics of Flow Localization-Ductility Loss in Irradiation Damaged bcc Alloys[END_REF]. In order to develop a predictive model based on deformation mechanisms, a more statistical characterization is necessary. It is also important to have a better understanding of the effect of test conditions, such as loading direction, strain rate, test temperature, and strain level, on deformation mechanisms.

The first part of this paper describes a Transmission Electron Microscopy (TEM) investigation of plastic deformation mechanisms in neutron irradiated recrystallized Zr alloys tested with different loading conditions. The second part of this paper analyzes the mechanical behavior for internal pressure test loading (at 350°C) in detail and compares the analysis to the TEM observations. Finally, a micro-mechanical model based on the observed deformation mechanism is proposed for the internal pressure test at 350°C.

Materials and Experimental Procedure

Materials and Mechanical Testing

In order to have a better insight into the mechanisms involved in plastic deformation of irradiated Zr alloys, numerous mechanical tests and TEM investigations have been performed on various recrystallized zirconium alloys specimens. The four materials (Table 1) used for this study are all recrystallized Zr alloys: Zy-4 (two different chemical compositions for sheet and cladding), M5 alloy, and a previous experimental grade alloy called here Zr-1%Nb-O and referred to as M5-0 in [START_REF] Gilbon | Irradiation Creep and Growth Behavior, and Microstructural Evolution of Advanced Zr-Base Slloys[END_REF]. These materials have been irradiated in different reactors (MRT and PWR), at various irradiation temperatures and up to various fluences (Table 2). This variability in irradiation conditions is assumed to be a second order effect in considering the deformation mechanisms. After irradiation, different types of mechanical tests have been performed on these four materials: transverse tensile tests on rolled sheet, ring tensile tests ( ) on rings machined out of tubes in a double dog-bone shape after irradiation [START_REF] Desquines | Mechanical Properties of Zircaloy-4 PWR Fuel Cladding with Burnup 60-64 MWd/kgU and Implications for RIA Behavior[END_REF], internal pressure tests ( zz = ) on tubes, and axial tensile tests ( zz ) on tubes. All internal pressure tests have been performed using a dynamically controlled pressurization device. This is particularly important for the constant strain rate tests. The ring tests have been performed using the procedure described in [START_REF] Desquines | Mechanical Properties of Zircaloy-4 PWR Fuel Cladding with Burnup 60-64 MWd/kgU and Implications for RIA Behavior[END_REF]. In addition to the classical strain hardening tests, stress-relaxation tests (during 72 h) and creep test (during 500 h) were conducted using internal pressure loading. Stress-strain curves and straintime curves are given in Figs. 6a andb. The irradiation and testing conditions of the eleven specimens studied here are given in Table 2. In addition to these data, it has to be pointed out that the textures of these materials are typical of recrystallized Zr alloy cladding or rolled sheet. It also has to be noticed that the A specimen was cut out from a 1-mm thick sheet in the TD-LD plane, whereas the B specimen was cut from a 10-mm thick sheet in the TD-ND plane. This last specimen is particularly interesting because it allows the investigation of the channel 3D structure.

TEM Procedure

After mechanical testing, all the thin foils were prepared in hot cell as described in [START_REF] Gilbon | Effect of Irradiation on the Microstructure of Zircaloy-4[END_REF]. In addition, on some thin foils a notch was left in order to know the orientation of the sample.

First, in order to confirm previous observations [START_REF] Coleman | Deformation Parameters of Neutron Irradiated Zy-4 at 300°C[END_REF][START_REF] Onchi | The Inhomogeneous Deformation Behavior of Neutron Irradiated Zircaloy-2[END_REF][START_REF] Williams | Effects of Boiling Water Reactor Irradiation on Tensile Properties of Zircaloy[END_REF][START_REF] Pettersson | Evidence for Basal or Near-Basal Slip in Irradiated Zircaloy[END_REF][START_REF] Adamson | Effects of Neutron Irradiation and Oxygen Content on the Microstructure and Mechanical Properties of Zircaloy, Microstructure and Mechanical behavior of materials[END_REF][START_REF] Fregonese | Failure Mechanisms of Irradiated Zr Alloys Related to PCI: Activated Slip Systems, Localized Strains, and Iodine-Induced Stress Corrosion Cracking[END_REF][START_REF] Régnard | Activated Slip Systems and Localized Straining of Irradiated Alloys in Circumferential Loadings[END_REF] and to identify precisely the activated slip systems according to the loading direction (transverse, internal pressure, and axial), a statistical TEM investigation was carried out on the five specimens listed in Table 2 (A-E, tested at 350°C). For each thin foil taken from these specimens, many grains were studied successively in arbitrary areas. The local orientation was determined by means of conventional diffraction pattern indexing in order to draw the stereographic projection for each grain. Examinations were carried out at relatively high magnification in order to determine whether grains contain channels, by tilting the specimen successively in the entire accessible angular domain (±45°, ±30°) of the double tilt sample holder. Then, in order to identify the channeling slip system, the traces of the channels were systematically analyzed on the stereographic projection. Moreover, the tilt conditions were recorded for the maximum contrast between channels and surrounding material. These conditions correspond to the situation where the channeling plane nearly contains the electron beam. These two complementary methods allow the channeling slip system to be determined without ambiguity especially in the case of basal channeling. Finally, the mean number of channels per grain and the mean channel width have been roughly characterized. Extended experimental details and results are given in [START_REF] Onimus | A Statistical TEM Investigation of Dislocation Channeling Mechanism in Neutron Irradiated Zirconium Alloys[END_REF].

In a second step, using similar procedures, the six other specimens listed in Table 2 (F-K) were studied qualitatively in order to extend the understanding of the deformation mechanisms to other testing conditions (strain rate, creep, temperature, strain level). TEM observations are shown schematically on Fig. 5a and Figs. 6a andb.

Results of TEM Investigations

Statistical TEM Analysis

Experimental Observations-On the two transverse tensile specimens (A and B), 81 grains have been studied. It has been observed that 38 grains over 81 contained basal channels (Fig. 1a). No other type of channel has been observed. The other 43 grains did not exhibit any sign of plastic deformation. On the C and D specimens tested in internal pressure loading, 80 grains have been studied, and 43 grains over 80 contained basal channels (Fig. 1b). As in the previous case, no other type of channel was observed. It has also to be underlined that for all of the A, C, and D specimens, connecting channels were very seldom observed (only two grains over 64 grains with basal channels observed), whereas on the B specimen many basal channels were connected from grain to grain (17 grains with connecting channels over 21 grains with basal channels). For all these four specimens, the width of the channels was always less than 100 nm, and the mean width was between 40 and 80 nm. It also has been estimated that there are on average 3-5 channels per grain. On specimen E, tested under uni-axial loading (axial tensile test), 41 grains have been studied, and 20 of these 41 grains contained prismatic and pyramidal channels, no basal channel was observed. In two cases, it has not been possible to distinguish between prismatic and pyramidal channels, however it has been determined that 14 grains contained prismatic channels, and 4 grains contained pyramidal channels as shown on Figs. 2a andb. In addition to that, it has to be noticed that the channels were more difficult to observe in this case and did not seem fully cleared of defects. The mean width of channels has been roughly estimated at 40 nm, the maximum width is 50 nm, and there are, on average, five channels per grain. All of these results are reported in Table 3. Discussions-The TEM observations of the A, B, C, and D specimens lead to the conclusion that only basal channeling occurs at 350°C for transverse tensile test and internal pressure test and for plastic strain up to uniform elongation. This result has to be compared with the deformation mechanisms in non-irradiated material. Indeed, it has been established, as reviewed by Douglass [START_REF] Douglass | The Metallurgy of Zirconium[END_REF] and Tenckhoff [START_REF] Tenckhoff | Deformation Mechanisms, Texture, and Anisotropy in Zirconium and Zircaloy[END_REF], that for low plastic strain, plastic deformation occurs essentially by <a> dislocations gliding on prismatic planes for axial, transverse, or closed end pressure loading. This is due to the fact that the Critical Resolved Shear Stress (CRSS) of the prismatic slip system is lower than that of the other slip systems. The important change in slip system activation with irradiation can be attributed to a modification in the hierarchy of the CRSS. This can be explained by the difference in the interactions between loops and dislocations, which implies that dislocation channeling is easier in the basal plane than in the prismatic plane. Indeed, as it has been proposed by Fregonese [START_REF] Fregonese | Failure Mechanisms of Irradiated Zr Alloys Related to PCI: Activated Slip Systems, Localized Strains, and Iodine-Induced Stress Corrosion Cracking[END_REF] and Régnard [START_REF] Régnard | Activated Slip Systems and Localized Straining of Irradiated Alloys in Circumferential Loadings[END_REF], the fact that in the hexagonal closed packed structure (h.c.p.) all the <a> Burgers vectors lie in the basal plane implies that the junctions created between a <a> loop and a <a> dislocation gliding in the basal plane are always glissile. Therefore, the loops can be easily swept up by basal dislocations, which would lead to an easy dislocation channeling and to a low increase in the critical shear stress. On the contrary, in the case of a <a> dislocation gliding on the prismatic plane interacting with a <a> loop, the junction created is sessile in two out of three cases, which leads to a lower ability for dislocation channeling and therefore to a higher CRSS. This explanation also applies for pyramidal slip system.

However, we have seen that for an axial tensile test (E specimen), prismatic channeling and pyramidal channeling occur. This is due to the fact that since the material is characterized by a strong texture with the <c> axis tilted about 30° around the normal (or radial) direction in the TD-ND plane (r-plane), a large majority of the grains present a very low resolved shear stress on the basal plane for an axial tensile test. Therefore, in order to accommodate the plastic deformation, other slip systems, such as prismatic or pyramidal, must be activated for axial tensile test loading conditions. This can be achieved by the sweeping up of only one type of <a> loop or by another mechanism such as the one proposed by Carpenter [START_REF] Carpenter | Dislocation Channeling by Prism Slip in HCP Metals[END_REF]. Nevertheless, it is seen that the sweeping up of loops is limited for these slip systems since the channels do not seem to be fully cleared of defects.

The 3D structure of the basal channels has also been investigated thanks to the thin foils taken in the B specimen. Indeed, on the B specimen, propagation of channels from grain to grain has been observed for 11 grains out of 17 with basal channels, whereas in all other specimens (A, C, and D) this feature has only been observed for 2 grains out of 64 with basal channels. This can be explained by the fact that the propagation of channels from grain to grain occurs in the shearing direction. In this case, since the material exhibits a strong texture (<c> axis in the ND-TD or r-plane as shown on Fig. 3a), for internal pressure test and transverse tensile test, the best orientation for the shearing direction is contained in the ND-TD plane (or r-plane) and is tilted 45° from the normal direction. Therefore, in order to observe the propagation of grains, it is necessary to take the thin foil in the ND-TD plane (Fig. 3b), as proved by TEM investigations of the B specimen. On the contrary, if the thin foil is taken in the TD-LD plane (or z-plane), as in the case of A, C, and D specimens, the propagation of channels will be seldom observed in agreement with the TEM observations. The propagation of channels in surrounding grains is probably due to the high stress concentration on the grain boundary due to the strong plastic strain localization in the channel.

FIG. 3 Schematic drawing describing the propagation of basal channels for a transverse tensile test: (a) Main grains orientation, (b) Propagation of channels in the specimen frame.

All of these observations obtained for materials tested at 350°C, and for a strain rate of 3 10 -4 s -1 show a good coherency and lead to a better understanding of the dislocation channeling mechanism in irradiated zirconium alloys. Nevertheless, in order to extend these conclusions to broader test conditions (test temperature, strain rate, plastic strain level, and creep), six other specimens have been studied.

Extending the Analysis to Different Loading Conditions

Effect of Stress Relaxation-On the H specimen, many basal channels have been observed as in the case of the C and D specimen as shown on Fig. 4c. This proves that the basal channels created during the strain hardening phase have not been affected by the stress relaxation phase performed during 72 h.

Effect of Strain Rate-Basal channels have also been observed in the I specimen tested under internal pressurization at 350°C and at a strain rate of 3 10 -6 s -1 (Fig. 4d). These channels are similar to the channels observed for tests performed at 3 10 -4 s -1 (C, D, H specimens). This proves that within this range of strain rate, the plastic deformation mechanism is not affected by the strain rate and occurs by basal channeling (before uniform elongation).

Effect of Creep-On the K specimen, a start of recovery of the irradiation microstructure has been noticed, after a creep test performed at 350°C and during 500 hours. A small decrease in <a> loops density seems to have occurred, probably assisted by stress, leading to homogeneously deformed areas, similar to non-irradiated materials. However, basal channels could still be observed in this specimen, as shown in Fig. 4f.

Effect of Test Temperature-On the J specimen, it has been observed that the irradiation microstructure has been partially annealed, showing larger loops than in the previous specimens. In this case, the plastic deformation seems to take place homogeneously since no channels but many dislocation lines are observed as shown on Fig. 4e. It is believed that these dislocations glide in the prismatic plane as in the case of non-irradiated material.

Effect of Plastic Strain Level-Thin foils have been taken in the gauge length of the F and G ring specimens, the test being interrupted after the load drop beyond the uniform strain limit, before failure, as shown on Fig. 1 reported in [START_REF] Régnard | Activated Slip Systems and Localized Straining of Irradiated Alloys in Circumferential Loadings[END_REF]. On these thin foils, many channels have been observed. However opposite to the observations on the A and B specimens studied in the first part, in this case, the channels correspond to the basal as well as to the pyramidal and the prismatic slip system as shown on Figd. 4a and b. This suggests that before uniform elongation, only basal channeling occurs, whereas for strain level higher than uniform elongation, other types of channels start to appear, probably due to the need to accommodate the plastic deformation in the necking zone.

The TEM results are summarized in Table 4, and the analysis is reported in Table 5. It has also to be underlined that only six specimens have been chosen here, but all of these observations have been confirmed on other specimens deformed in the same conditions.

Mechanical Behavior Analysis: Irradiation Induced Hardening and Strain Hardening Behavior

In order to understand the effect of dislocation channeling mechanism on the mechanical behavior of irradiated Zr alloys, the tests performed in internal pressure loading on the C, D, and H specimens, respectively, recrystallized Zy-4, Zr-1%Nb-O alloy, and M5 alloy (tests performed at 350°C, with a strain rate of 3 10 -4 s -1 ), presented above, have been analyzed in details. The analysis has been mainly focused on irradiation induced hardening and on the strain hardening behavior of irradiated material, also considering the TEM observations performed on the C, D, and H specimens. The cylindrical geometry of the internal pressure specimen allows very precise extensometry during the mechanical testing, therefore the mechanical behavior for this loading condition is known very accurately. Stresses and strains are computed for internal pressure tests by using the usual thin wall approximation (e.g., [START_REF] Ferrer | The Effect of Small Concentrations of Sulfur on the Plasticity of Zirconium Alloys at Intermediate Temperatures[END_REF]). For strain levels higher than uniform elongation, pyramidal and prismatic channels start to appear for transverse test (ring).

Results

Dimensional Measurements-Very accurate dimensional measurements, using laser imaging techniques, have also been performed on the D specimen in order to investigate the distribution of the plastic strain at the specimen scale. The profiles of the external diameter of the D specimen, obtained by the laser imaging technique, before and after testing show that the deformation is homogeneous in the zone where the thin foils have been taken, although at the grain scale the deformation is localized inside the channels.

Yield and Flow Stress-The stress-strain curves of the three specimens are reported on Fig. 5a and compared to non-irradiated materials for the same testing conditions. Thanks to the good sensitivity of the extensometers, the flow stress has also been plotted as a function of the plastic strain. This type of representation allows a much better insight into the stress-plastic strain behavior and therefore to the strain hardening behavior, as shown in Fig. 5b.

The TEM investigation has shown that channeling already occurs before 0.2 % plastic strain, therefore the flow stress at 0.2 % plastic strain does not correspond to the irradiation induced hardening due to the loops since these loops are already cleared up inside the channels. In order to estimate the hardening due to loops, it has been chosen to consider the flow stress at a very low plastic strain level (yield stress), say 0.005 % plastic strain. However, this stress is not known accurately (+20 MPa), since the elastic slop is not known precisely. The flow stress at 0.2 % plastic strain ( 0.2 % ) can thus be associated with the plastic flow occurring inside the channels. These data are reported in Table 6 and plotted as a function of the fluence in Fig. 7.

As reported by many authors [START_REF] Douglass | The Metallurgy of Zirconium[END_REF][START_REF] Higgy | Effect of Neutron Irradiation on the Tensile Properties of Zy-2 and Zy-4[END_REF][START_REF] Baroch | Effect of Irradiation at 130, 650, and 775°F on Tensile Properties of Zy-4 at 72, 650, and 775[END_REF][START_REF] Yasuda | Deformation and Fracture Properties of Neutron-Irradiated Recrystallized Zircaloy-2 Cladding under Uniaxial Tension[END_REF], it is shown that irradiation leads to a strong increase in flow stress. It can be noticed on Fig. 7 that the flow stress as well as the yield stress saturates as the fluence increases, considering that all the different materials are equivalent to a first order. It can also be noticed that for a non-irradiated material, the difference between yield and flow stress is low (of the order of 50 MPa), whereas once the material is irradiated, the difference between flow stress and yield stress is very high (of the order of 190 MPa). This shows that the strain hardening is higher for irradiated material than for non-irradiated material. Strain Hardening Rate-In order to investigate more accurately the strain hardening behavior of irradiated materials, the strain hardening rate has been computed for the mechanical test performed on the D specimen and compared to non-irradiated material. The strain hardening rate is plotted as a function of the stress as reported on Fig. 8. It is shown that the strain hardening rate is two orders of magnitude higher in irradiated material (d /d p 1000 GPa) than in nonirradiated material (d /d p 10 GPa) at the onset of the plastic flow. However, for irradiated material, the strain hardening rate decreases rapidly to values of the same order of magnitude as in non-irradiated material (d /d p 10 GPa), whereas for non-irradiated material, the hardening rate decreases slowly.

FIG. 8-Strain hardening rate versus hoop stress for the internal pressure test performed on irradiated Zy-4 (D specimen) compared to non-irradiated material (2 specimen). Discussion on Mechanical Behavior Considering TEM Observations

It has been shown that irradiation leads to a strong hardening, as illustrated on Fig. 5 for yield stress ( 0.005 % ) and flow stress at 0.2 % ( 0.2 % ). The increase of the yield stress ( 0.005 % ) with irradiation can be directly related to the high loop density induced by irradiation. Indeed, according to the classical dispersed barrier hardening model reviewed by Refs. [START_REF] Bement | Fundamental Materials' Problems in Nuclear Reactors[END_REF][START_REF] Hirsch | Point Defect Cluster Hardening[END_REF][START_REF] Brown | Strengthening Methods in Crystals[END_REF], dislocation loops act as obstacles against dislocation glide. However, when a sufficient shear stress is applied, dislocations can overcome or sweep up all the obstacles encountered in the slip planes and macroscopic plastic deformation occurs. This stress level defines the macroscopic yield stress, which has been chosen in our case as the flow stress at very low plastic strain (0.005 % plastic strain).

The mechanical analysis also shows that the flow stress at 0.2 % plastic strain is increased by irradiation. However, in this case, TEM investigations of the D specimen have shown that for 0.2 % plastic strain, basal channels are observed. Since inside the channels irradiation loops have been cleared up by gliding dislocations, leaving the following dislocations with a defect free zone, it is surprising that irradiation hardening is still observed for this plastic strain level. In addition to that, it is seen on Fig. 5b that for the C specimen the flow stress increases between 0.2 and 0.5 % of plastic strain, and many channels can be observed for 0.5 % plastic strain. Assuming that for the C specimen channels were already present in the material at 0.2 % plastic strain as for the D specimen, it is also surprising that the flow stress increases since dislocation channeling corresponds to irradiation defects clearing, and therefore to local softening inside the channels. This phenomenon can be explained by considering the polycrystalline aspect of the material. Indeed, the strong localization of the plastic strain inside a channel leads to high strain incompatibility between the channel and the surrounding grains, which should induce elastic stress in return or microscopic internal stress. In addition, since only basal channeling occurs for the internal pressure test (only two independent deformation modes), compatibility conditions between grains cannot be satisfied. This phenomenon should also induce microscopic internal stress. These microscopic internal stresses are associated, at the macroscopic scale, to the kinematic hardening component of the flow stress or back stress. Therefore, in the case of irradiated material one would expect a strong kinematic hardening to occur because of the strong plastic strain localization inside the channels and the occurrence of only basal channeling. This high kinematical hardening could then balance the local softening inside the channel, leading to a hardening behavior at the macroscopic scale as observed for the C specimen. This would therefore explain that the plastic deformation at the specimen scale is still homogeneous, according to laser measurements, although at the grain scale, the plastic deformation occurs in a localized manner through dislocation channeling.

The high strain hardening rate measured at the onset of plastic flow in the case of irradiated material can also be interpreted as being a consequence of the localization of the plastic strain at the grain scale. Indeed, if a channel embedded into an elastic media is considered, for a small amount of macroscopic plastic strain, a high amount of plastic strain inside the channels is needed since it is assumed that all the plastic strain takes place inside the channels. Therefore, this high amount of local plastic strain will lead to a high increase of internal stress, thus leading to a high macroscopic strain hardening rate. However, it is shown that the strain hardening rate decreases very rapidly in the case of irradiated material (Fig. 8) compared to non-irradiated material. The decrease of the strain hardening rate can be mainly attributed to the fact that the channels propagate from grain to grain due to the strong stress concentration at the grain boundaries, at the edge of the channels. This very low macroscopic strain hardening rate, due to channel propagation, would then induce strain localization at the specimen scale, which therefore can explain the strong decrease in uniform elongation with irradiation reported by many authors [START_REF] Douglass | The Metallurgy of Zirconium[END_REF][START_REF] Higgy | Effect of Neutron Irradiation on the Tensile Properties of Zy-2 and Zy-4[END_REF][START_REF] Rieger | Strength and Ductility of Neutron Irradiated and Textured Zy-2[END_REF].

An extended analysis conducted on a larger data base (analysis of specimen unloading behavior, strain rate sensitivity, and stress relaxation), which will not be presented here, has also given some more clues that internal stress is actually higher for irradiated material than for non-irradiated material. By using only the internal pressurization tests, it has not been possible to measure the internal stress level. Additional tensile-compression tests in the transverse direction are necessary to measure internal stresses for the transverse tensile direction.

Micromechanical Modeling for Internal Pressure Test Loading

On the basis of TEM investigations and mechanical behavior analysis, a micromechanical model has been developed for irradiated zirconium alloys. Starting from the dislocation scale, the dislocation-loop interaction has been modeled in a simple manner. Then the channel microstructure has been schematically described at the grain scale in order to model the macroscopic behavior of irradiated polycrystalline Zr alloys by means of homogenization techniques. Only internal pressure tests performed at 350°C have been modeled. Since only basal channeling has been observed in this case, only basal glide is taken into account. A table of mathematical symbols used is given in Table 7.

Modeling at the Dislocation Scale

Hardening-It is known that the high density of <a> loops act as obstacles against dislocation glide, in the same way as the forest dislocations. According to the classical dispersed barrier hardening model [START_REF] Bement | Fundamental Materials' Problems in Nuclear Reactors[END_REF][START_REF] Hirsch | Point Defect Cluster Hardening[END_REF][START_REF] Brown | Strengthening Methods in Crystals[END_REF], the hardening phenomenon can be taken into account by adding a term, proportional to the square root of the loop density (N), to the critical resolved shear stress leading to the expression (Eq 1) for the basal critical shear stress, where i is the obstacle force, the shear modulus, b the Burgers vector, d the mean loop diameter and 0 B the critical resolved shear stress for the basal plane when the material is not irradiated.

Nd b i B 0

(1) Softening-It has been shown that when a dislocation gliding on the basal plane interacts with <a> loops, glissile junctions are created, and the loop can be swept up on its cylinder, leading to clearing up of irradiation loops inside a band of the grain and therefore to local softening. The clearing up process can be modeled, as proposed by Rodney [START_REF] Rodney | Irradiation Hardening by Interstitial Loops: Atomistic Study and Micromechanical Model[END_REF] and Pokor [START_REF] Pokor | Effect of Irradiation Defects on the Work Hardening Behavior[END_REF], by considering that all the loops lying within a distance H/2 to the dislocation plane (basal plane in this case) are captured and cleared up by the dislocation. Taking into account the dislocation density inside the channel gliding at the velocity v , it is shown that during a time dt, the number of cleared loops inside the channel is NH vdt. Since the plastic strain rate inside the channel (microscopic strain rate) is related to the dislocation density inside the channel and to the . Therefore, the equation governing the evolution of the loop density can be deduced, written here (Eq 2) in a more general way by adding the contribution of all the dislocations gliding on the three basal slip systems ( ). Thermally Activated Flow Law-The plastic strain rate inside the channel is given by the constitutive flow law, which is related to the thermally activated motion of dislocations. According to Refs. [START_REF] Friedel | Dislocations[END_REF][START_REF] Cagnon | Thermodynamique de la déformation et essais mécaniques -Théorie de l'activation thermique[END_REF], the plastic strain rate can be expressed, if reverse motion is taken into account, as Eq 3, where s is the applied resolved shear stress on the basal system s ( s B ) inside the channel, is the basal critical (athermal) resolved shear stress which takes into account the hardening due to irradiation induced loops, V B is the basal activation volume, and is a coefficient which includes mobile dislocation density and lattice vibration characteristic frequency. The applied resolved shear stress inside the channel is here computed as Eq 4, where s is the two order orientation tensor of the considered slip system defined as given in Eq 5, where l s and n s are unitary vectors corresponding respectively to the shearing direction and to the normal of the slip plane.

B s B s ch s N b H N (2)

Modeling at the Grain Scale

Shape, Orientation, and Channel Volume Fraction-At the grain scale the specific channel microstructure has been schematically modeled as an oblate spheroid embedded into a spherical grain (Fig. 9). The thickness of this penny shaped ellipsoid corresponds to the width of the channel d ch . Since only basal channels are taken into account, the short dimension d ch of the flat ellipsoid corresponds to the <c> axis of the h.c.p. crystallographic structure and the plane of the flat ellipsoid corresponds to the basal plane. Homogeneous isotropic elasticity is also assumed. Considering a spherical shaped grain, the diameter of the channel corresponds to the diameter of the grain d g . This very simple geometry allows Eshelby calculations for an ellipsoidal inclusion with uniform eigenstrain to be applied. Indeed, it has been shown by [START_REF] Eshelby | The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problem[END_REF][START_REF] Mura | Micromechanics of Defects in Solids[END_REF] that the stress inside the inclusion for an oblate spheroid embedded into an infinite elastic matrix, with macroscopic applied stress and eigenstrain inside the channels p ch , is given in Eq 6 where C is the four order elastic modulus tensor, S ch is the Eshelby four order tensor for an oblate spheroid [START_REF] Eshelby | The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problem[END_REF][START_REF] Mura | Micromechanics of Defects in Solids[END_REF], I is the identity four order tensor, and p ch is the plastic strain inside the channel.

p ch ch ch S I C : ) ( : (6)
It has to be underlined that homogeneous and isotropic elasticity are also assumed for this modeling. Equation 6 is referred here as the concentration rule. Since it is assumed that all the plastic deformation of the grain is localized into the channels, the plastic strain of the grain, , is given by Eq 7, where f ch corresponds to the volume fraction of channels per grain and p ch to the mean plastic deformation inside the channels. p ch ch p g f [START_REF] Wechsler | Dislocation Channeling in Irradiated and Quenched Metals, The Inhomogeneity of Plastic Deformation[END_REF] This volume fraction can be very simply defined as Eq 8, assuming roughly a rectangular shaped channel and grain. In this expression, n ch is the mean number of channels per grain, d ch is the mean width of channels, and d g corresponds to the size of the equiaxed grain. V g and V ch account for the volume of the grain and for the volume of the channels, respectively. It can be noticed that the concentration rule (Eq 6) actually describes the microscopic internal stress created by the localization of plastic strain inside the channels. Indeed, for a low plastic strain inside the grain ( p g ), the plastic strain inside the channel ( p ch ) is large, since the volume fraction of channels is of the order of 1-2 %. Therefore, although inside the channel the flow stress is low because of the clearing of loops, at the macroscopic scale, the flow stress is high due to the second term of Eq 6.

Knowing the shear strain rate on each basal slip system ( ), the plastic strain rate inside the channels can be computed as Eq 9, taking into account the orientation of the grain with respect to the macroscopic orientation of the specimen. The plastic strain rate of the grain is then deduced by differentiating Eq 7. Homogenization techniques do not aim at describing all the individual grains inside the representative volume of the polycrystalline material. With the homogenization techniques, only the grains having different crystallographic orientation are treated independently. Therefore, it is here considered that the sphere actually represents all the grains with the same crystallographic orientation, given by Euler's angles ( 1 , , 2 ), and characterized by the volume fraction of grains with the same crystallographic orientation (f g ) obtained by X-ray diffraction texture analysis. The oblate spheroid inclusion inside the sphere therefore represents all the channels of all the grains with the same crystallographic orientation, characterized by the mean volume fraction of channels per grain with the same orientation (f ch ).

Channels Volume Fraction Evolution-The creation and multiplication of channels with plastic deformation is closely linked to the increase of the internal stresses with local plastic deformation. Indeed, it can be considered that when a critical level of internal stress for one channel is reached, the channel will stop to deform, and another channel will be created inside the grain. For an isolated channel inside an infinite elastic matrix, the internal stress increases linearly with the local plastic strain, as shown in Eq 6. Therefore, for a critical value of plastic deformation inside the channel, let's say c , the channel will stop to deform, and another channel will be created. As a consequence, the total plastic strain ch due to n ch channels, can be written as c ch ch n . Then, according to its definition, the evolution of the volume fraction of channels (f ch ) can be expressed as a function of the total plastic strain in all the channels of the grains with the same crystallographic orientation as in Eq 10, where the contributions of the three basal slip systems are added.

Modeling at the Polycrystal Scale

In order to deduce the macroscopic behavior from the knowledge of the microscopic behavior, a polycrystalline model, developed by Pilvin [START_REF] Pilvin | The Contribution of Micromechanical Approaches to the Modelling of the Inelastic Behaviour of Polycrystals[END_REF] and Geyer [START_REF] Geyer | Modeling of the Anisotropic Viscoplastic Behavior of Fully Annealed Zircaloy-4 Tubes by a Polycrystalline Approach[END_REF], has been adapted to irradiated material. According to this approach, the calculation occurs in four successive stages for each time step. First, the macroscopic stress ( is computed from the applied loading. Then, the stress inside the channel ( ch ) is calculated by using the concentration rule. The plastic strain rate inside the channel ( p ch ) is then computed thanks to the constitutive laws, and finally the macroscopic plastic strain rate ( p E ) is deduced from the computed microscopic plastic strain rate. However, in order to take into account the morphology of the channels in the case of irradiated material, the self-consistent computation described in [START_REF] Geyer | Modeling of the Anisotropic Viscoplastic Behavior of Fully Annealed Zircaloy-4 Tubes by a Polycrystalline Approach[END_REF][START_REF] Pilvin | The Contribution of Micromechanical Approaches to the Modelling of the Inelastic Behaviour of Polycrystals[END_REF] cannot be performed. This involves the use of a simplified approach as described below.

Loading Computation-From the macroscopic loading given in a mixed manner (stress and strain), the macroscopic stress is computed by using Hooke's law, assuming homogeneous isotropic elasticity (Eq 11) with the isotropic elastic modulus tensor defined as Eq 12, the behavior of the material being elasto-visco-plastic.

p E E C (11) 1 1 2 1 2 I C ( 12
)
Stress Concentration Rule-Thanks to the concentration rule, the stress inside the channels can be computed, knowing the macroscopic applied stress. However, since the channels can propagate from grain to grain, the assumption that the channels can be considered as an ellipsoid embedded into an infinite elastic matrix cannot hold anymore. In order to take into account the channel propagation, the plastic strain inside the channels is replaced by a local effective plastic strain called ch , as shown in Eq 13. This semi-empirical effective plastic strain ch evolves non linearly with the plastic strain ( p ch ), given by Eq 14, taking into account, in a simple manner, the plastic strain accommodation of the surrounding grains due to channel propagation from grain to grain. Indeed, as the plastic strain inside the channel ( p ch ) increases, the effective strain ( ch ) saturates, leading to the saturation of the microscopic internal stresses which takes into account that the surrounding material does not behave elastically anymore. This effective strain ch has no physical reality, the plastic strain in the channel is In addition to this, another term, called image stress by Mura [START_REF] Mura | Micromechanics of Defects in Solids[END_REF], is added to stress concentration rule (Eq 13) in order to satisfy that the macroscopic stress equals the mean value of the microscopic stresses (Eq 15). g G g g f [START_REF] Odette | Modeling of Multiscale Mechanics of Flow Localization-Ductility Loss in Irradiation Damaged bcc Alloys[END_REF] However, in order to satisfy this last condition, several assumptions have to be made. First, it is necessary to know the value of the stress in the grain outside the channels. Here, a simple assumption is made that outside the channels, the stress is of the order of the macroscopic applied stress (Eq 16).

outside [START_REF] Gilbon | Irradiation Creep and Growth Behavior, and Microstructural Evolution of Advanced Zr-Base Slloys[END_REF] Then, in order to satisfy the condition on the average of the stress (Eq 15), the two order tensor B (in Eq 13) is defined as Eq 17. This simplified approach, although interesting, would require further improvements by using a self-consistent scheme such as the one developed in [START_REF] Geyer | Modeling of the Anisotropic Viscoplastic Behavior of Fully Annealed Zircaloy-4 Tubes by a Polycrystalline Approach[END_REF][START_REF] Brenner | Thermal Creep of Zr-Nb1%-O Alloys: Experimental Analysis and Micromechanical Modeling[END_REF][START_REF] Pilvin | The Contribution of Micromechanical Approaches to the Modelling of the Inelastic Behaviour of Polycrystals[END_REF][START_REF] Lebenshon | A Self-Consistent Anisotropic Approach for the Simulation of Plastic Deformation and Texture Development of Polycrystals: Application to Zirconium Alloys[END_REF][START_REF] Turner | Modeling the Mechanical Response of Rolled Zircaloy-2[END_REF][START_REF] Berveiller | An Extension of the Self-Consistent Scheme to Plastically-Flowing Polycrystals[END_REF] in order to avoid the difficult introduction of the image stress as pointed out in [START_REF] Mura | Micromechanics of Defects in Solids[END_REF]. Nevertheless, this is only possible if the morphology of the channels is not taken into account.

Macroscopic Plastic Strain Computation-Since the average of the microscopic strain is equal to the macroscopic strain (Eq 18), and assuming homogeneous elasticity, it can be shown that the macroscopic plastic strain is equal to the average of the microscopic plastic strain (Eq 19).

G g g g f E (18) p g G g g p f E (19)
Assuming that no plastic strain occurs outside the channels ( 0 p outside ), it is shown that the macroscopic plastic strain can be expressed as Eq 20.

ONIMUS ET AL. ON IRRADIATED ZIRCONIUM ALLOYS 19

p ch G g ch g p f f E (20)
For each time step, all these formulas are computed for each grain (all grains with the same orientation), leading to the calculation of the macroscopic mechanical behavior by integrating constitutive laws at the grain scale. A file containing 240 orientations ( 1 , , 2 , f g ), representative of the texture of the zirconium alloys of the cladding is used. The slip system geometry and the orientation of the ellipsoids are explicitly taken into account in the modeling.

Results

The modeling includes several coefficients and parameters that have to be adjusted based either on TEM observations, microscopic data, or macroscopic results. First, a fitting of the nonirradiated modeling has been performed on non-irradiated M5 according to the approach described in [START_REF] Geyer | Modeling of the Anisotropic Viscoplastic Behavior of Fully Annealed Zircaloy-4 Tubes by a Polycrystalline Approach[END_REF][START_REF] Pilvin | The Contribution of Micromechanical Approaches to the Modelling of the Inelastic Behaviour of Polycrystals[END_REF], but this is not presented here. For non-irradiated material, coefficients for prismatic, basal, and first order pyramidal (<a> and <c+a>) slip systems are fitted. The coefficients obtained for non-irradiated material such as elastic coefficients (µ = 28.6 GPa, = 0.4), reference strain rate ( s 10 0

10

-1 ), basal critical shear stress ( B = 40 MPa), and basal activation volume (V B = 60b 3 ), are given in Table 7. Only the coefficient for basal slip system are given here since for irradiated material tested in internal pressure test only basal systems are activated. These coefficients are in agreement with the data given in the literature [START_REF] Mills | The Plastic Deformation of Zirconium-Oxygen Alloy Single Crystals in the Range 77-950 K[END_REF][START_REF] Conrad | Effect of Interstitial Solutes on the Strength and Ductility of Titanium[END_REF][START_REF] Akhtar | Basal Slip in Zirconium[END_REF].

Then, several coefficients have been chosen according to TEM examinations, such as the initial loop density before straining (N 0 = 5 10 22 m -3 ), the mean loop diameter (d = 8 nm), the mean grain size (d g = 8 m), and the mean width of the channels (d ch = 210 nm). The value chosen here for the mean width of the channels is larger than the actual observed width so as to have a better description of the strain hardening at the onset of plastic flow. The rate of channel increase, given by c , is adjusted in order to obtain a good order of magnitude fit for the channel volume fraction (f ch 2 %). It is also considered that the capture distance is equal to the loop diameter (H = d = 8 nm). The obstacle force is also chosen according to literature data. In this case, since the created junctions are glissile in the basal plane, the obstacle force is relatively weak. The value chosen here is i = 0.1.

Finally, the fitting for irradiated material has been performed on an extended data base taking into account various strain rates (3 10 -4 s -1 to 3 10 -6 s -1 ) and stress relaxation tests but only for internal pressure tests performed at 350°C. The values for the coefficients chosen and the three fitting parameters obtained are given in Table 8. Only three empirical fitting parameters are used here (D, M, m). These three empirical parameters can be related to physical phenomenon. The D parameter is related to the plastic accommodation of the surrounding grains, and the M and m parameters are related to the visco-plastic accommodation of the surrounding grains, leading to static recovery of internal stresses. The apparent channels mean width has been previously adjusted in order to obtain a good description of the strain hardening rate at the onset of plastic flow. This coefficient indeed has a significant effect on the strain hardening rate at the onset of plastic flow. The agreement between simulation and experimental results is illustrated in Fig. 10. It has to be underlined that contrary to the non-irradiated approach, no self consistent calculations are made in this case, the empirical fitting parameters D, M, and m being adjusted on the macroscopic behavior.

Discussion

Taking explicitly into account the strain localization in the channels, the model is able to reproduce the high strain hardening rate observed at the beginning of the plastic flow due to the strong internal stress, the saturation of the strain hardening rate, and also the high stress relaxation. However, three empirical fitting parameters (D, M, and m) are used for this modeling leading to a limited prediction capability. It also has to be underlined that by taking the shape and the orientation of the channels into account in a homogenization model, several assumptions and simplified hypotheses have to be made. Moreover, this approach does not use a self-consistent scheme, from the homogenization theory point of view, contrary to the modeling developed for non-irradiated material [START_REF] Geyer | Modeling of the Anisotropic Viscoplastic Behavior of Fully Annealed Zircaloy-4 Tubes by a Polycrystalline Approach[END_REF][START_REF] Brenner | Thermal Creep of Zr-Nb1%-O Alloys: Experimental Analysis and Micromechanical Modeling[END_REF][START_REF] Pilvin | The Contribution of Micromechanical Approaches to the Modelling of the Inelastic Behaviour of Polycrystals[END_REF][START_REF] Lebenshon | A Self-Consistent Anisotropic Approach for the Simulation of Plastic Deformation and Texture Development of Polycrystals: Application to Zirconium Alloys[END_REF][START_REF] Turner | Modeling the Mechanical Response of Rolled Zircaloy-2[END_REF][START_REF] Berveiller | An Extension of the Self-Consistent Scheme to Plastically-Flowing Polycrystals[END_REF]. Other types of modeling are currently being studied in order to overcome these limitations.

Conclusions

The statistical TEM investigation has shown, thanks to numerous observations, that for transverse tensile tests and internal pressure tests performed at 350°C and up to the uniform elongation limit, only basal channeling occurs, whereas for non-irradiated material tested in the same conditions, only prismatic systems are activated. This change in the principal slip system with irradiation is believed to be due to the fact that the junctions created between dislocations and loops are always glissile for dislocations gliding in the basal plane, whereas it is sessile (in two cases over three) for dislocations gliding in the prismatic plane. However, it has been shown that for an axial tensile test, since basal planes are not well orientated with respect to the loading direction. Considering the strong texture of the material, other slip systems are activated in order to accommodate the plastic strain. Thus, prismatic and pyramidal channeling occurs by means of other mechanisms for this loading direction, and no basal channels can be seen. It has also been shown that for transverse tensile tests, basal channels propagate from grain to grain in the shearing direction in order to accommodate the strong stress concentration created on grain boundaries.

The extension to other test conditions using a qualitative TEM analysis has shown that for lower strain rate (3 10 -6 s -1 ) deformation still occurs via basal channeling. However, for creep tests performed at 350°C during 500 h at a stress of 300 MPa, the irradiation microstructure starts to recover, leading to homogeneous like deformation zones. Nevertheless, basal channels can still be observed. Finally, it has been shown that for test performed at 400°C (during 72 h), the irradiation microstructure has been partially annealed, leading to homogeneous deformation at the grain scale probably occurring by prismatic glide. It has also been shown that for strain levels higher than the uniform elongation limit measured in transverse tensile loading, prismatic and pyramidal channels start to appear (in addition to the basal channels already present in the material) in order to accommodate the plastic strain in the necking zone.

The role of the dislocation channeling mechanism on the macroscopic mechanical behavior has been investigated by comparing the strain hardening behavior in internal pressure test at 350°C to TEM observations. It is proposed that the strong localization of the plastic deformation at the grain scale, in channels, as well as the occurrence of only basal slip, induces high internal stresses which compensate the local softening in the channels. As a consequence, it is proposed that an important part of the macroscopic flow stress is due to the kinematic hardening associated with the development of the internal stresses. It is also proposed that the high hardening rate at the onset of plastic flow is due to the low volume fraction of channels that deforms plastically. The saturation of the flow stress is assumed to be linked to the propagation of channels to surrounding grains due to high local stress concentration.

A micro-mechanical modeling has been developed to take into account the dislocation channeling process. The loop-dislocation interaction has been simply modeled at the dislocation scale, taking into account the dispersed barrier hardening due to the high density of <a> loops and the softening due to loop sweeping up inside the channels. Then, at the grain scale the channels' microstructures have been schematically described, in order to use homogenization techniques, taking explicitly into account the shape, the orientation, and the volume fraction of the channel inside the grains. Finally, a polycrystalline model has been adapted to irradiated material, using simple assumptions and simple approaches, in order to model the behavior of the material at the macroscopic scale. This model is able to reproduce the strong strain hardening, the saturation of the strain hardening, as well as the stress relaxation. Nevertheless, further developments are needed in order to avoid the introduction of the image stress [START_REF] Mura | Micromechanics of Defects in Solids[END_REF] and the use of fitting parameters.

  (a) Propagating basal channels in B specimen and (b) basal channels in C specimen. (a) (b) FIG. 2 (a) Prismatic and (b) pyramidal channels observed in E specimen.

  FIG. 4 (a) Basal channels in the F specimen, (b) prismatic channels in the G specimen, (c) basal channels in the H specimen, (d) basal channel in the I specimen, (e) annealed irradiation microstructure, <a> dislocations probably gliding in prismatic planes in the J specimen, (f) partially annealed irradiation microstructure, basal channel in the K specimen.

FIG. 5

 5 FIG. 5 Internal pressure tests on C, D, and H specimens and compared to non-irradiated materials (1, 2, 3): (a) stress-strain curves, (b) stress-plastic strain curves. (Only the strain hardening phase is represented in the case of the H specimen on Fig. 5b).

FIG. 6

 6 FIG. 6 (a) Ring tests performed on the F and G specimens; (b) creep test performed on the K specimen.

FIG. 7

 7 FIG. 7 Evolution of yield stress and flow stress with irradiation fluence (n/m 2 E>1 MeV).

  dislocation mean velocity by the Orowan equation ( v b ), the number of loops cleared up inside the channel during the time dt can be expressed as dt N b H

FIG. 9

 9 FIG.9Schematic modeling of the channeling process. Basal channels are described as flat ellipsoid.

  p ch . The third term added in Eq 14 takes also into account the static recovery of this effective plastic strain ch especially useful for stress relaxation tests.

  -(a) and (b) test simulation performed with the modeling compared to the mechanical test carried out on the H specimen.

TABLE 1 -

 1 Alloys chemical composition (weight %).

	Alloy	Sn	Nb	Fe	Cr	O	Zr
	Zy-4 (sheet)	1.40	...	0.20	0.105	0.125	bal.
	Zy-4 (cladding)	1.30	...	0.210	0.100	0.125	bal.
	Zr-1 %Nb-O (cladding)	...	1.0	0.02	...	0.125	bal.
	M5 (cladding)	...	1.0	0.035	...	0.130	bal.

TABLE 2 -

 2 Materials, irradiation conditions, and test conditions of the specimens.

	Specimen Material	Reactor Fluence, n/m 2	Irradiation	Mechanical Test	Test	Strain
				(E>1 MeV)	Temp.		Temp.	Rate, s -1
	A	Zy-4 sheet	Siloé	0.6×10 25	280°C	Transverse tensile	350°C 1.6×10 -4
	B	Zy-4 sheet	Siloé	0.6×10 25	280°C	Transverse tensile	350°C 1.6×10 -4
	C	Zr-1%Nb-O cladding	PWR	12×10 25	350°C	Internal pressure	350°C 3×10 -4
	D	Zy-4 cladding Osiris 0.4×10 25	350°C	Internal pressure	350°C 3×10 -4
	E	Zr-1%Nb-O cladding	PWR	12×10 25	350°C	Axial tensile	350°C 3×10 -4
	F	Zy-4 cladding Siloé	3.1×10 25	350°C	Ring tensile (interrupted)	350°C 1.6×10 -4
	G	M5 cladding Siloé	2.4×10 25	380°C	Ring tensile (interrupted)	350°C 1.6×10 -4
	H	M5 cladding PWR	3.5×10 25	350°C	Internal pressure (stress relaxation at E=0.8 %, 72 h)	350°C 3×10 -4
	I	M5 cladding PWR	3.5×10 25	350°C	Internal pressure (stress relaxation at E=0.8 %, 72 h)	350°C 3×10 -6
	J	M5 cladding PWR	3.5×10 25	350°C	Internal pressure (stress relaxation at E=0.8 %, 72 h)	400°C 3×10 -6
	K	M5 cladding PWR	8×10 25	350°C	Internal pressure (creep at 300 MPa during 500 h)	350°C	-

TABLE 3 -

 3 Results of the statistical TEM observations of the five different specimens.

	Specimen	Mechanical Test	E p	Studied Grains	Grains with Channels Channeling Plane
	A	Transverse tensile	0.4 %	52	21	B
	B	Transverse tensile	0.36 %	29	17	B
	C	Internal pressure	0.5 %	47	28	B
	D	Internal pressure	0.2 %	33	15	B
	E	Axial tensile	1.6 %	41	20	P and 1

TABLE 4 -

 4 Results of the qualitative TEM observations of the different specimens.

	Specimen Mechanical Test	Test	Strain Rate,	E p	Deformation	Activated Slip
			Temperature	s -1	%	Mechanism	System
	F G H	Ring tensile Ring tensile Internal pressure	350°C 350°C 350°C	1.6×10 -4 1.6×10 -4 3×10 -4	6.7 7.4 0.25 channeling channeling channeling	B, 1 , P B, 1 , P B
		(stress relaxation at E=0.8 %)
	I	Internal pressure	350°C	3×10 -6	0.3	channeling	B
		(stress relaxation at E=0.8 %)
	J	Internal pressure	400°C	3×10 -6	0.33 homogeneous B or 1
		(stress relaxation at E=0.8 %)
	K	Internal pressure (creep at	350°C	…	3.2	homogeneous	B
		300 MPa during 500 h)				and channels

TABLE 5 -

 5 Discussion on TEM observations obtained for various testing conditions.

	Effect	Specimens vs. Specimens	Conclusions
	Strain rate	I vs. H, C, D	No effect of strain rate between 3 10 -6 s -1 and 3 10 -4 s -1 for internal
	(350°C)		pressure test at 350°C.
	Creep (350°C)	K vs. I, H, C, D	Start of recovery of irradiation microstructure. Basal channels and
			homogeneous deformation.
	Temperature	J (400°C) vs. I (350°C) Partial annealing of irradiation microstructure. Homogeneous
			deformation for test performed at 400°C, no channel observed.
	Strain level	F, G vs. A, B
	(350°C)

TABLE 6 -

 6 Flow stress for 0.005 % and 0.2 % of plastic strain for different materials and test conditions (irradiated materials compared to non-irradiated materials).

	Specimen	Material	Mechanical Test	Test	Strain Rate;
				Temperature	s -1	MPa	MPa
	1	Zr-1%Nb-O Internal pressure	350°C	3×10 -4	150	210
	2	Zy-4	Internal pressure	350°C	3×10 -4	150	200
	3	M5	Internal pressure	350°C	3×10 -4	170	200
	C	Zr-1%Nb-O Internal pressure	350°C	3×10 -4	280	575
	D	Zy-4	Internal pressure	350°C	3×10 -4	270	540
	H	M5	Internal pressure	350°C	3×10 -4	300	590
			(stress relaxation at E=0.8 %)

TABLE 7 -

 7 Symbols used in the description of the model.

	s		Resolved shear stress on the system s
		0	Basal system critical resolved shear stress for non-irradiated material
	µ		Effective athermal critical shear stress for irradiated material
	N		Loop density
	d		Loop diameter
	i		Obstacle force for irradiation induced loops
			Shear modulus
	b		Burgers vector
			Dislocation density
	H		Annihilation distance
			Shear strain rate
		0	Reference shear strain rate
	V B	Basal activation volume
	k		Boltzmann constant
		s	Orientation two order tensor
			Macroscopic stress
			Gliding direction
	n	s	Normal direction of slip plane
	C		Four order elastic modulus tensor
	S ch	Eshelby four order tensor
	I		Identity four order tensor
	f ch	Volume fraction of channels per grain
	n ch	Number of channels per grain
	d ch	Channel mean width
	f g		Volume fraction of grain
	d g		Grain size

ch Stress inside the channel ch Strain inside the channel p ch Plastic strain inside the channel outside Stress outside the channel, inside the grain g Stress inside the grain g Strain inside the grain p g Plastic strain inside the grain s l c Critical value of shear for multiplication of channels ch Empirical effective plastic strain inside the channel

TABLE 8 -

 8 Coefficients and parameters of the modeling.