Laurent Claparede

Nicolas Clavier

Adel Mesbah

Florent Tocino

Stéphanie Szenknect

Nicolas Dacheux

From sintering to chemical durability : impact of the cationic homogeneity on the (Th,U)O 2 ceramics lifecycle

In order to study the effects of the cationic distribution homogeneity on the life cycle of Th 1-x U x O 2 ceramics, including sintering and reprocessing (dissolution) steps, five different ways of preparation were set up, going from the most homogeneous solid solutions obtained through oxalic co-precipitation to a mechanical mixture of the parent oxides. Dilatometric experiments evidenced the better sintering capacity for the most homogenous compounds obtained through wet chemistry methods while dry chemistry routes led to poor density values (between 80 and 90 %TD). However, the introduction of additional mechanical grinding step prior the sintering of the powders systematically led to the homogenization of the systems.

Improved homogeneity also provide a better chemical durability associated with the congruent dissolution of thorium and uranium in solution during dissolution tests of Th 0.5 U 0.5 O 2 samples. Conversely, heterogeneous samples led to incongruent behaviors that can be smoothed by introducing a grinding step prior the preparation of sintered samples.

Since the impact of the cationic distribution homogeneity must be followed carefully during the study of the solid/solution interface evolution, in operando observations of evolving solidsolution interface by ESEM were performed during dissolution. They allowed imaging the preferential dissolution of uranium enriched zones and confirmed the important role of chemical heterogeneities present at the interface.

INTRODUCTION

Mixed actinide dioxides are currently used as fuels in Pressurized Water Reactors (PWR) and also stand as potential candidates for several Gen IV concepts including Sodiumcooled Fast Reactor (SFR) or Gas-cooled Fast Reactor (GFR) [1,2]. In this field, the reprocessing of minor actinides coming from nuclear spent fuel into mixed-oxide fuels or in UO 2 -based blankets surrounding the core is often considered [3]. In a similar way, the preparation of Pu-and U-bearing ThO 2 samples was already described in literature and proposed as advanced fuel materials [4][5][START_REF] Lung | Nuclear Science and Technology project report EUR 17771 for the European Commission[END_REF][START_REF] Gruppelaar | [END_REF]. Indeed, thoria (ThO 2 ) presents a great interest for its potential use in nuclear energy applications, as 232 Th can be used to produce 233 Th (T 1/2 = 21.83 minutes) by neutron capture then 233 Pa (T 1/2 = 27 days) and finally 233 U (fissile isotope, T 1/2 = 159200 years) by two successive   decays.

Mixed actinide dioxides were usually prepared through powder metallurgy methods (i.e. mechanical mixture of parent oxides) which could result in some heterogeneities in the cationic distribution within the material [START_REF] Buisson | Rôle de la distribution des compositions cationiques sur l'aptitude à la dissolution des combustibles MOX Caractérisation de la distribution par diffraction des rayons X sur poudre[END_REF]. As a consequence, wet chemistry methods were investigated with the aim to monitor the physico-chemical properties of the final oxides from the synthesis step. These processes are mainly based on the precipitation of crystallized precursors [START_REF] Arab-Chapelet | [END_REF] such as carbonates, nitrates, hydroxides or oxalates [10][11][12][13]. These latter remained probably the most frequently cited in the literature as they provide a rapid and quantitative precipitation of actinides from solution. Also, they led to the formation of homogenous solid solutions which ensure the homogeneity of the cation distribution at the molecular scale. This enhanced homogeneity is then thought to be one of the main factors leading to the improvement of several properties of interest of the system, including sintering capability of the mixed oxide powder obtained after thermal conversion, or chemical durability of the final ceramics [14,15].

Several authors already underlined the importance of homogeneity during the sintering of actinide mixed oxides. Indeed, the densification of the samples mainly proceeds through solid-state diffusion. Starting with heterogeneous powders generally delays the sintering process since the formation of an homogenous solid solutions is the first mechanism triggered [16][17][18]. Also, important differences in the diffusion coefficients of the cations considered can led to the formation of porosity, due to the well-known Kirkendall effect [19,20]. In this context, the initial co-precipitation of actinide cations, particularly as oxalate precursors, was found to provide homogenous oxide powders after heat treatment, which further present improved sintering capability; the temperature needed to achieve a complete densification being lowered by 100 to 200°C compared to powder metallurgy processes [15]. Nevertheless, if this effect was systematically correlated to the homogeneity of the cation distribution in the starting powder, no specific study was undertaken on this point to our knowledge.

Similarly, the multiparametric study of the dissolution of Th 1-x U x O 2 solid solutions was recently examined by varying conventional parameters such as temperature, chemical composition or leachate acidity [14,[21][22][START_REF] Heisbourg | Synthèse, caractérisation, et études cinétique et thermodynamique de la dissolution de ThO 2 et des solutions solides Th 1-x M x O 2 (M = U, Pu)[END_REF]. However, the consequences of the physicochemical modifications (including morphology) induced by the chemical way of synthesis considered on the chemical durability were only recently evocated [11,12,[START_REF] Greiling | [END_REF] while they appear as key parameters either for the understanding of the reprocessing operations or for the long-term storage of spent nuclear fuels.

In this context, this work is focused on the effects of the cationic distribution on the life cycle of Th 1-x U x O 2 ceramics, including sintering and reprocessing (dissolution) steps.

Thorium-uranium-(IV) mixed dioxides were selected as a simplified model of other tetravalent actinides oxides (such as uranium, neptunium, plutonium). Five different ways of preparation were then set up in order to prepare Th 1-x U x O 2 ceramics with different characteristics in terms of cationic distribution, going from the most homogeneous solid solutions obtained through oxalic co-precipitation to a mechanical mixture of both endmembers. . The various powders obtained were further characterized by the means of SEM and XRD to evidence the microstructural characteristics of the different samples. Their sintering capability was then evaluated through a dilatometric study combined with SEM-EDX mapping analysis. Finally, the chemical durability of the samples was determined during dissolution tests in highly acid media while in operando observations of the solid/solution interface evolution were performed by ESEM during dissolution.

II. EXPERIMENTAL SECTION

Synthesis and thermal conversion

Caution! uranium and thorium are natural radioelements, more often involving αemitters and as such are considered a health risk. Their use requires appropriate infrastructure and personnel training regarding the handling of radioactive materials.

The Following materials were used to perform the synthesis and the dissolution tests: depleted U (kindly supplied by CETAMA, France), Th(NO 3 ) 4 .4-5 H 2 O, H 2 C 2 O 4 , HCl, HNO 3 (Sigma Aldrich). The actinides solutions were prepared by dissolving metallic U or Th salt in hydrochloric acid (6M). The final concentration of cations in each solution was about 0.5M and was accurately determined by Inductively-Coupled Plasma Atomic Emission Spectroscopy (ICP-AES, Spectro Arcos) or Photon-Electron Rejecting Alpha Liquid Scintillation (PERALS, Ametek) measurements.

In order to prepare a set of Th 0.5 U 0.5 O 2 oxides exhibiting various homogeneities in the distribution between uranium and thorium, five methods were set up: I.

First, an homogenous solid solution Th 0.5 U 0. IV. In a modification of the protocol described above, both oxalate precursors were converted into ThO 2 and UO 2 by heating at 500°C, then mixed thoroughly in a mortar.

V. Lastly, protocol IV was slightly modified. In this case, both oxalate precursors were converted into ThO 2 and UO 2 by heating at 1000°C, then mixed thoroughly in a mortar.

Moreover, it is important to note that a second set of materials was prepared using the same five protocols described above but with the addition of a mechanical grinding step (30 Hz, 30 min.) within a zirconia jar prior the final treatment at high temperature. Such modification aimed to evaluate how the heterogeneities inherited from the preparation methods can be hindered by an additional grinding step performed afterwards.

Powders characterization and sintering

PXRD. The powdered samples obtained either after precipitation of oxalate precursors or after their thermal conversion into oxides were characterized by powder X-ray diffraction (PXRD) using a Bruker D8 diffractometer equipped with a Lynx-eye detector and Cu Kα 1,2 (λ = 1.54184 Å) adopting the reflection mode (parallel beam). PXRD patterns were recorded at room temperature in dedicated sealed sample holders in order to avoid radioactive contamination in the 5° ≤ 2θ ≤ 120° angular range, with a step size Δ(2θ) of 0.03° and an average total counting time of about 3 hours per sample.

The collected PXRD patterns were then refined by the Rietveld method using the Fullprof_suite program [26] in order to evaluate the average crystallite size (i.e. average length of the coherent domains). Pure silicon was used as external standard to extract the instrumental function. For each refined pattern, the following parameters were allowed to vary: zero shift, unit cell parameters, scale factors, global thermal displacement. Moreover, the modelling of the intrinsic microstructure parameters was performed for each phase by applying an anisotropic size model.

SEM.

Morphological characterization of the samples was performed using a FEI Quanta 200 environmental scanning electron microscope equipped with a backscattered electron detector (BSED) or a secondary electron detector (SE) in vacuum conditions with an acceleration voltage of 25 kV. Samples were always directly analysed without any additional preparation step such as metallization.

SEM micrographs recorded at low magnification were first used to evaluate the initial specific surface area (S SA in m².g -1 ) of the pellets using the SESAM (Study of Evolving Surface Area by Microscopy) method [27,28]. With this aim, 5 images of 92 µm× 62 µm were binarized using the FiJi software to determine the surface area of the pores visible in the investigated domains. The distribution of the pore diameter was evaluated from these images using the "analyse particles" plugin implemented in the FiJi software. Then, the surface area associated to the pores was obtained assuming that the pore size distribution of the analysed domain was representative of the whole sample and that the pores were cylindrical. The determined surface area of the pores were then weighted by the fraction of open porosity deduced from apparent and pycnometric densities and the resulting surface was divided by the mass of the sample to calculate the specific surface area. For each sample, an average value of the specific surface area was deduced from the analysis of the 5 images recorded at low magnification.

Additionally, X-EDS mapping of the surface of sintered samples was performed in order to evaluate the homogeneity of the cations distribution within the pellets. Sintered samples were first polished up to a mirror grade. A selected area of about 150×100 µm was then analysed during 4 hours. These conditions ensured reliable counting statistics associated to an important number of data points (750 000) which can be considered as representative of the whole pellet surface. Also, the counting rate was kept constant for all the zones analysed, in order to avoid any analytical bias during the data analysis.

Sintering. The obtained powders were shaped by uniaxial pressing (500 MPa) in a tungsten carbide die as cylindrical green pellets of 5 mm in diameter and about 1 to 2 mm thickness. The sintering of these compacts was further monitored by dilatometric measurements using a Setaram Setsys Evolution apparatus. Heating treatments were performed under Ar flux up to 1500°C with a rate of 10°C.min -1 . An isothermal plateau was then applied during 8 hours before the sample was cooled down to room temperature at 30°C.min -1 . The resulting dense pellets were further investigated by the means of SEM (see above) and geometrical density measurements using a precision calliper.

Dissolution experiments

The leaching experiments were carried out using high-density polytetrafluroethylene (PTFE) vessels (volume of 25 mL) in order to avoid any adsorption of released elements on the containers walls. Dissolution tests were first performed under static conditions : pellets of around 200 mg were put in contact with 25 mL of 2M HNO 3 at 90°C for few hours. During this time, aliquots of 0.5 mL were regularly taken off and replaced by the same volume of fresh nitric acid solution to maintain a constant volume of solution. Depending on the dissolution conditions, i.e. on the basis of the concentration of released uranium, the aliquots were diluted with 0.2 M HNO 3 .

In order to study the impact of the nitric acid concentration, pellets were placed in contact with 25 mL of 0.1 to 4M HNO 3 at room temperature. Nitric acid solutions were prepared from analytical grade reagents (supplied by Sigma or Fisher). The leaching solution was continuously renewed using a peristaltic pump in order to avoid any saturation process.

At the outlet of the dissolution reactor, the solution passed through a 0.45 µm filter before being regularly sampled using an automatic fraction collector. The flow rate varied from 1 to 5 mL.h -1 , depending on the nitric acid concentration, in order to allow the determination of the elementary concentrations in the outflow by ICP-AES measurements. The intensity of the peaks was recorded at  = 279.394 nm, 367.007 nm and 409.014 nm for uranium and at  = 283.231 nm, 283.730 nm and 401.913 nm for thorium to avoid any interference between the studied elements.

As already described in previous works, the progress of the dissolution reaction of a solid phase is monitored through the determination of the normalized mass loss N L (i,t) (g.m -2 ) [29], which is defined as:

(

where, m i (t) (g) is the mass of element i measured in solution at time t, S (m²) is the surface area of the solid in contact with the solution and f i (g.g -1 ) is the mass ratio of the element i in the solid. The surface area of the sample is defined as:

(

where (m².g -1 ) denotes the initial specific surface area of the solid measured by the SESAM method, and m 0 (g) is the initial mass of solid introduced in the system.

The normalized dissolution rate, R L (i) (g.m -2 .d -1 ), is defined as the time-derivative of the normalized mass loss, i.e.:

(

The use of Equation ( 5) can be considered as acceptable as soon as the surface area remains constant. This assumption was found to be reasonable and led to an error lower than the experimental uncertainties for a mass loss that does not exceed 5 % of the initial mass of the solid sample [28]. The normalized dissolution rate in a stirred flow-through reactor is obtained when the steady state is reached from the following mass balance expression:

where d (L.d -1 ) is the flow rate and (M) is the elementary concentration of i in the inflow, which usually equals zero. The elementary concentrations at steady state C i (M) were determined by averaging the values measured by ICP-AES.

The dissolution was qualified as congruent when all the normalized dissolution rates were identical (i.e. when all the elements were released with the same ratios than the stoichiometry of the material or more generally when 1/3 < R L (i)/ R L (j) < 3) [30]. For a congruent dissolution, the R L (i) values determined from the release of each constitutive element in solution are then equal to the dissolution rate of the solid.

III. RESULTS AND DISCUSSION

Materials features

The various samples prepared by the protocols I to V described above, including both oxalate precursors and the final Th 0.5 U 0.5 O 2 oxides, were first characterized by SEM. Figure 1 shows selected pictures for each series. The hydrothermal synthesis of mixed oxalates (method I) led to the formation of large agglomerates of about 100 µm in length, whereas the direct precipitation of the same compounds at 60°C produced much smaller grains (about 3-5 microns), presenting the characteristic square-plate habit of the tetravalent actinides oxalates dihydrate. The thermal conversion of these powders into thorium-uranium oxide solid solutions after heating at 500°C under Ar/H 2 (4%) did not modify the starting morphology and only resulted in a slight isotropic decrease of the grains size. Once again, this observation is in very good agreement with literature concerning actinides oxalates since pseudomorphic conversion when heating was frequently described for such kinds of compounds [31].

Moreover, if a clear modification of the sample morphology can be operated depending on the precipitation mode considered (i.e. hydrothermal conditions or direct precipitation), the introduction of an additional mechanical grinding step mostly erases this effect, resulting in all cases in the formation of sub-micrometric grains as viewed in Figure 1. Nevertheless, whatever the final morphology of the oxides, X-EDS analyses confirmed the formation of homogenous solid solutions for protocols I and II with x = 0.48(2) and 0.49(2), respectively.

Powder X-Ray Diffraction

The synthesized oxalate precursors prepared following the protocols I and II were found to be pure and monophasic, their pattern being in good agreement with the monoclinic structure (C2/c space group) reported in the literature [25]. Combined with the X-EDS results, these samples can thus be attested as Th 0.5 U 0. The thermal conversion of the uranium-thorium bearing oxalate powders led systematically to the formation of pure Th 0.5 U 0.5 O 2 . From Figure 2, the introduction of a grinding step does not affect the structural properties of the final oxides prepared through protocols I and II and in both cases it leads to the formation of a solid solution as shown by the unit cell parameters reported in Table 2. Conversely, when processing through oxalate precursors mixing (method III), an additional grinding step is required to yield a single phase Th 0.5 U 0.5 O 2 solid solution, a mixture of oxides being obtained otherwise. In the case of methods IV and V, a degradation of the cationic homogeneity is systematically detected and a mixture of oxides is observed. Moreover, for the samples prepared at 500°C the crystallites size ranged from 3 to 5 nm whereas it reached 17-25 nm for those prepared at 1000°C (Table 2). However, this difference in the crystallites size and in the morphology of the powder did not affect significantly the specific surface area which ranged from 7 to 12 m 2 /g.

Sintering and characterizations of pellets

In order to evaluate the impact of the heterogeneities in the cation distribution induced by the various protocols applied during the preparation of the powders, dilatometric experiments were undertaken up to 1500°C. or in an open vessel, exhibit a two-step shrinkage already depicted in the literature for such compounds [33,34]. The first stage lying between 500 and 1100°C was then assigned to sintering and grain growth phenomena within the square-shaped agglomerates described previously in section 3.1. Conversely, the second stage, which is marked by a significant decrease of the pellet thickness, was correlated to the sintering between the agglomerates and led to the densification of the sample. This process was found to be complete as soon as from 1200°C, which appears very low in comparison to the temperatures generally reported in the literature for the sintering of actinide dioxides [START_REF] Abe | Uranium oxide and MOX production[END_REF].

The powders obtained through mixing of oxalate end-members or of oxides coming from oxalate conversion at 500°C presented a very similar behavior. In these cases, the first shrinkage stage observed between 500 and 700°C was assigned to grain growth phenomena, which are probably favored by the grinding step performed to homogenize the mixture of the powders. This process probably occurs through surface diffusion, known to present a lower activation energy in comparison to volume and grain boundaries diffusion, generally related to densification processes [START_REF] Kang | Sintering -Densification, Grain Growth & Microstructure[END_REF]. At higher temperatures, only a weak residual shrinkage was recorded, showing that the sintering capability of the samples was significantly hindered by the use of powder metallurgy processes (i.e. mixtures of large sized powders through mechanical grinding followed by shaping and high temperature heat treatment). Such conclusion was even more enlighten by the sample prepared following protocol V, since no shrinkage was noted below 1000°C for this sample. On this basis, it is likely that the heating step led to the homogenization of the distribution of the cations in the system rather than to the densification of the pellet. Indeed, Gabard et al. already evidenced the homogenization of cations due to interdiffusion processes in thoria-based materials [START_REF] Gabard | [END_REF] while the use of heterogeneous powder mixture is known to delay significantly the sintering of mixed oxides ceramics [16,38]. Beyond the shape of the dilatometric curve, which clearly evidences differences in the sintering mechanisms, the impact of the heterogeneity of cation distribution was directly evidenced by the final linear shrinkage measured after firing at 1500°C. Indeed, this latter reached about 20% for samples prepared by wet chemistry methods, while it was below 10% for protocol V. These differences were also noted on the SEM micrographs recorded at the surface of the various pellets prepared (Figure 4). Indeed, only the powders prepared through wet chemistry methods led to fully densified pellets (i.e. in the 96 -98 %TD range), which do not exhibit significant amount of open porosity at the surface. For such conditions of preparation, the initial grinding of the samples only led to a modification of the average grain size, which is lowered from about 5 -10 µm to 1 -5 µm. As attested by the dilatometric study, the use of powder metallurgy methods generally decreases the final densification state of the pellets, which was generally found between 80 and 90 %TD from geometrical measurements. The ceramics obtained following protocol III (i.e. mixture of oxalates) still present the characteristic features of sintered materials, showing the formation of grain boundaries associated to significant grain growth, but also reveal the presence of numerous open pores.

Conversely, all the samples prepared from the mixture of oxide powders led to an incomplete sintering process. More particularly, the micrographs clearly show that the densification only proceeded within the agglomerates, leading to dense zones separated by large pores. Moreover, it is important to note that an additional mechanical treatment of the powders prepared through protocols III to V was not associated to a significant increase of their sintering capability, the final microstructure of the pellets being strongly comparable to their counterparts obtained from raw powders. SEM micrographs recorded at the surface of the pellets prepared by using the five protocols after pelletizing then heating for 8 hours at 1500°C under Ar atmosphere

Raw powder

Finally, the surface of the various sintered pellets produced was investigated by the means of X-EDS mapping in order to evaluate the distribution of uranium and thorium. In this aim, the relative deviation regarding to the average chemical composition (illustrated by the uranium incorporation rate in mixed oxide, x) was plotted statistically from the 750000 data points considered (Figure 5). For raw powders series (i.e. without any additional grinding step), two types of distribution were clearly evidenced. The first one was associated to wet chemistry methods of preparation (i.e. protocols I and II) and corresponds to an highly homogeneous composition close to the calculated values. In this case, the FWHM value (Full Width at Half Maximum) was found to be close to 0.11, meaning that most of the data points analyzed led to chemical compositions ranging from Th 0.55 U 0.45 O 2 to Th 0.45 U 0.55 O 2 .

Nevertheless, one must note that for protocol II, a small shoulder corresponding to a limited fraction of uranium-enriched zones was observed. On the other hand, and as expected from the results already compiled in the literature [18,39,40], the dry chemistry routes led to heterogeneous samples, illustrated by the dramatically wide aspect of the distribution curve.

Indeed, even if a small peak was noticed for compositions closed to Th 0.85 U 0.15 O 2 , the incorporation rate of uranium appeared to be very scattered up to pure UO 2 spots. Also, the absence of data points corresponding to pure ThO 2 allows to get first insights on the homogenization processes, which seem to proceed mainly through the diffusion of uranium in thorium-enriched zones, in good agreement with the values of diffusion coefficients reported in the literature [41,42]. Besides, the introduction of an additional mechanical grinding step prior the calcination and sintering of the powders systematically led to the homogenization of the systems. In these conditions, the FWHM values were all in the 0.04 -0.05 range for protocols I to IV, but remained higher than 0.13 for the sample obtained from the mixture of oxide powders initially fired at 1000°C. Indeed, such a high conversion temperature probably led to low specific surface areas associated with large crystallites, which hinder the reactivity of the powders.

Behavior of the pellets during dissolution tests : macroscopic description

In order to underline the impact of the cationic distribution on the chemical durability of thorium-uranium(IV) mixed oxides, dissolution experiments were performed on the Th 0.5 U 0.5 O 2 sintered samples prepared according to the five different protocols described previously. In order to avoid any impact from other parameters on the dissolution of the materials, preliminary calcination step, shaping of the powders and sintering (T = 1500°C, t = 8 hours, Ar atmosphere) remained unchanged for all the samples prepared.

Dissolution tests were undertaken in 2 M HNO 3 at 90°C. The evolution of the normalized mass losses are plotted in Figure 6 and in Figure 7, while the corresponding normalized dissolution rates are gathered in Table 3. When no milling step was performed before the sintering of the pellets, the normalized dissolution rates calculated from the release of uranium were found to increase by around one order of magnitude when shifting from wet chemistry methods (protocols I and II) to powder metallurgy processes (III -V). This observation appears in very good agreement with the results obtained from the X-EDS mappings (see Figure 5), which suggests a much improved cationic distribution when starting from co-precipitated oxalic precursors. Moreover, the dissolution of the oxide ceramics was found to be congruent for these two preparation methods (I and II) since the congruency ratio, r = R L,0 (U)/ R L,0 (Th), was close to 1.

However, even if the R L,0 (Th) appeared to be only slightly modified when considering dry chemistry routes of preparation, the congruency ratio r appeared to increase significantly (r > 3) for synthesis protocols III, IV and V, which underlines the preferential dissolution of uranium-enriched zones in the material. Indeed, we recently reported the dissolution of powdered samples of Th

Normalized dissolution rates R L,0 (U) and R L,0 (Th) (in g.m -2 .d -1 ) and congruency ratio r = R L,0 (U)/R L,0 (Th) determined during the dissolution, in 2M HNO 3 and at T = 90°C, of Th 0.5 U 0.5 O 2 sintered samples prepared following the five synthesis protocols, and with the presence or not of a grinding step prior the sintering step. Furthermore, when a grinding step was performed prior the sintering step, the normalized dissolution rates determined from the thorium and uranium releases in solution were found to be very similar (typically between 7.6 and 12.5 g.m -2 .d -1 ) whatever the way of synthesis considered. This result corresponds to the significant decrease of the uranium normalized dissolution rates for the samples prepared following protocols III, IV and V (Table 4). As instance, a decrease by a factor of 6 of the normalized dissolution rate was observed for the sample prepared through protocol IV. Once again, these results demonstrate that the application of a preliminary grinding step before the shaping and sintering of the powders enhances the homogeneity of the final solids by limiting the presence of uraniumenriched zones. However, as normalized dissolution rates remained almost unchanged for samples prepared by wet chemistry methods, such a grinding step could be avoided for this king of protocol, limiting the risks of contamination during the handling of radioactive powders.

Grinding

Table 4.

Ratio between the normalized dissolution rates obtained for raw samples R L,0 (i) NB and milled samples R L,0 (i) B determined during the dissolution in 2M HNO 3 at 90°C of sintered samples of Th 0.5 U 0.5 O 2 prepared following the 5 protocols. 5 and in Table 6 for samples prepared by protocol II and V, respectively. The variations of the congruency ratio, r = R L,0 (U)/ R L,0 (Th), versus the acidity of the solution are plotted in Figure 8 for the samples prepared following the five protocols. The results obtained for the heterogeneous compound show that the congruency ratio is strongly dependent on the solution acidity. Indeed, this ratio increases from 1.6 (in 10 -1 M HNO 3 ) to 3.6 (in 4M HNO 3 ), thus suggesting that the dissolution is incongruent ( Table 6). This result can be explained by the oxidation of uranium (IV) to uranium (VI), which is much more pronounced when the nitrate concentration of the medium increases [43].

Protocol

Thus, uranium-enriched zones are more quickly dissolved when the nitric acid concentration is higher than 0.5M. Conversely, uranium-depleted areas retain their refractory character, leading to a clearer difference between uranium-enriched and uranium-depleted zones within From the data plotted on Figure 9, the normalized dissolution rates were found to 6. congruency ratio R L (U) / R L (Th) of 3.8 ± 0.1. Based on this observation, one could conclude on the incongruency in the elemental releases, as previously reported. Apparently, this behavior could result from the presence of fluctuations in terms of cationic distribution within the pellet submitted to dissolution. In order to confirm this assessment, the reactive surface was then evaluated by ESEM at different dissolution stages. With this objective, ESEM pictures were taken on the same selected zone for dissolution times of 0, 3 and 5 hours. The associated BSE micrographs were binarized using the Fiji software to determine the surface area of the altered zones. The results are given in Figure 10 whereas the associated extracted data are gathered in Table 7.

(A) (A') (B') (C') (B) (C) Th U
From the analyses of the hyper-maps, evolution of the reactive surface is observed, especially in terms of topography and chemical composition. Indeed, uranium-enriched zones are dissolved preferentially, thus underlining the significant impact of the cationic heterogeneity on the chemical durability of the pellet. Moreover, from the data reported in table 7, it is also possible to evaluate the chemical composition of the altered zones as follows:

From this calculation, it appeared that the composition of the dissolved zones equals x U = 0.78 which agrees with the ratio between cation releases in solution (r(t) = C U(t) / C Th(t) = 4.3). Indeed, making the assumption that both elements are released congruently from dissolved uranium enriched zones, the chemical composition of such zones would be Th 0.21 U 0.79 O 2 solid solution.

In order to take in account then correct this bias, one must consider the chemical composition of the dissolved zones. Consequently, the parameter f i (g.g -1 ) (mass ratio) was changed to 0. As it was expected, the normalized dissolution rates of both elements become very closed (R L,0 (U) = 14.1 ± 0.1 g.m -2 .d -1 and R L,0 (Th) = 13.5 ± 0.3 g.m -2 .d -1 ), with a congruency ratio, R L,0 (U)/R L,0 (Th), of 1.04 ± 0.03 in agreement with the congruent dissolution within the less durable zones. It is thus important to note that such kinds of normalization, required to obtain comparable results in terms of composition and reactivity, leads to satisfactory results for homogeneous compounds. On the contrary, its application can be limited for materials showing chemical heterogeneities in terms of cationic distribution. Thus, it can sometimes induce bias leading to misinterpretation of the data. This point becomes very important to consider in case of materials showing strong heterogeneity that is the case for spent nuclear fuel [45][46][START_REF] Carrott | The chemistry of (U,Pu)O 2 dissolution in nitric acid[END_REF], hence the uranium enriched areas could be dissolved quickly the other zones owing to its oxidation.

Conclusion

Five different protocols of synthesis were set up in order to prepare Th 0.5 U 0.5 O 2 ceramics with different characteristics in terms of cationic distribution, going from the most homogeneous solid solutions obtained through oxalic precipitation to the least, prepared by powder metallurgy from mixture of thorium dioxide and uranium dioxide.

Clear differences were observed when performing the sintering of the powders depending on their origin. A classic two-step shrinkage composed by grain growth phenomena within the square-shaped agglomerates followed by sintering between theses agglomerates was observed for samples prepared by grouped or separate oxalate precipitation.

Conversely, using powder mixtures only led to a weak residual shrinkage at high temperature,

showing that the sintering capability of the samples was decreased by the use of powder metallurgy processes. Such conclusion was even more enlighten by the sample prepared following the protocol with a mixture of oxides heated at 1000°C, since no shrinkage was noted below 1000°C in this case.

SEM micrographs show that the use of wet chemistry routes helped the elaboration of fully densified pellets (i.e. in the 96 -98 % TD range) with only few quantity of open porosity at the surface. Conversely, the use of powder metallurgy methods decreases the final densification state of the pellets, with values ranging between 80 and 90% TD f. The surface of the sintered pellets prepared was also investigated through X-EDS mapping showing improved homogeneity and expected chemical composition for the samples produced by protocols I and II. On the contrary, heterogeneities in the cation distribution were observed for samples prepared by dry chemistry routes. Nevertheless, the introduction of an additional mechanical grinding step prior sintering can partially correct this bias.

The impact of cationic distribution on the chemical durability of Th 0.5 U 0.5 O 2 solid solutions was further examined by performing dissolution tests in acidic media. The normalized dissolution rates calculated from the release of uranium were found to increase by around one order of magnitude when shifting from wet chemistry methods (protocols I and II)

to powder metallurgy processes (III -V). Moreover, dissolution tests on homogeneous compounds suggest congruent dissolution, which was not the case for heterogeneous samples.

However, dissolution became congruent for all the samples when using a grinding step of the powders prior sintering.

In operando monitoring of the solid/liquid interface evolution during dissolution was finally performed by ESEM on heterogeneous Th 0.5 U 0.5 O 2 sample. For this kind of materials, the use of normalization induces an important bias leading to the misinterpretation of the dissolution data. Indeed, if the macroscopic analysis of the dissolution concluded on incongruent dissolution, the microscopic examinations of the samples during dissolution through operando experiments associated this apparent incongruent dissolution to the preferential dissolution of uranium enriched zones (in which uranium and thorium were dissolved congruently).

To conclude, these results showed that potential chemical heterogeneity in terms of cationic distribution have to be considered very carefully is one wants to better understand the behavior of actinide oxide base ceramics during reprocessing step or long-term storage in the field of underground repository. Especially, this aspect becomes crucial when making association between the strongly refractory ThO 2 (or CeO 2 ) with the less durable UO 2 .

Figure 1 .

 1 Figure 1. Selected SEM micrographs recorded for raw series of a) I-Th 0.5 U 0.5 (C 2 O 4 ) 2 .2H 2 O; b) I-Th 0.5 U 0.5 O 2 ; c) II-Th 0.5 U 0.5 (C 2 O 4 ) 2 .2H 2 O; d) II-Th 0.5 U 0.5 O 2 and ground series of e) II-Th 0.5 U 0.5 (C 2 O 4 ) 2 .2H 2 O and f) II-Th 0.5 U 0.5 O 2 .

Figure 2 .

 2 Figure 2. PXRD patterns collected for Th 0.5 U 0.5 O 2 raw (a) and ground (b) samples (*: XRD peaks of the sample holder).

Figure 3

 3 evidences drastic differences in the sintering process of the powders depending on their origin. First, the solid solutions obtained from the initial precipitation of Th 0.5 U 0.5 (C 2 O 4 ) 2 .2H 2 O, either under hydrothermal conditions

Figure 3 .

 3 Figure 3. Dilatometric curves recorded during the sintering of raw powders using the five ways of preparation: I-II (light blue), III (red), IV (pink) and V (dark blue).

  Figure 4.SEM micrographs recorded at the surface of the pellets prepared by using the five protocols after pelletizing then heating for 8 hours at 1500°C under Ar atmosphere

Figure 5 .

 5 Figure 5.Statistical analysis of the distribution of the cations within Th 0.5 U 0.5 O 2 sintered samples prepared through the different protocols studied.

Figure 6 .

 6 Figure 6. Evolutions of normalized weight losses N L (U) observed during the dissolution in HNO 3 2M at 90°C of Th 0.5 U 0.5 O 2 sintered samples prepared following protocols I (■), II (♦), III (▼), IV (•) and V (▲) without (A) or with (B) preliminary grinding step.

Figure 7 .

 7 Figure 7. Evolutions of normalized weight losses N L (Th) observed during the dissolution in HNO 3 2M at 90°C of Th 0.5 U 0.5 O 2 sintered samples prepared following protocols I (■), II (♦), III (▼), IV (•) and V (▲) without (A) or with (B) preliminary grinding step.

  1-x U x O 2 solid solutions by varying several parameters such as temperature, chemical composition, leachate acidity, … [43]. Conversely to what observed for Th 1-x Ce x O 2 solid solutions [44], the chemical composition induces strong modifications of the chemical durability of the leached samples. The chemical durability of Th 1-x U x O 2 solid solutions was significantly affected by the uranium mole loading in the samples due to oxidation of tetravalent uranium into uranyl during the dissolution process.

  order to underline the behaviour of Th 1-x U x O 2 mixed oxides with different distributions of cations, several Th 0.25 U 0.75 O 2 pellets were dissolved at room temperature in various nitric acid solutions (0.05 M to 4 M HNO 3 ) and using dynamic conditions in order to avoid any saturation process. The initial normalized dissolution rates R L,0 (Th) and R L,0 (U) are gathered in Table

1 Table 6 . 1 Figure 8 .

 1618 Figure 8.Variation of the congruency ratio, r = R L,0 (U) / R L,0 (Th), versus the acid concentration, obtained during the dissolution of homogenous (■) and heterogeneous (•) sintered samples of Th 0.25 U 0.75 O 2 et room temperature

3 . 4 . 2

 342 Consequently, the release of uranium in solution is much more important, suggesting the existence of an incongruent dissolution from a macroscopic point of view. Behavior of the pellets during dissolution tests: evolution of solid/liquid interface during dissolution test of a heterogeneous Th 0.5 U 0.5 O In order to better analyze the incongruent character of the dissolution of heterogeneous Th 0.5 U 0.5 O 2 samples, the solid/liquid interface evolution was monitored by in operando ESEM observations during the dissolution of a Th 0.5 U 0.5 O 2 pellet prepared following the protocol (IV), at room temperature and in 2M HNO 3 (Figure 6). Indeed, such samples can present local heterogeneities in terms of composition which can alter the determination of the normalized dissolution rates.

Figure 9 .

 9 Figure 9. Evolution of normalized weight losses N L (U) (•) and N L (Th) (■) observed during the dissolution of heterogeneous Th 0.5 U 0.5 O 2 sintered samples in HNO 3 2M at room temperature.

  1 ± 0.1 g.m -2 .d -1 and 23.4 ± 0.1 g.m -2 .d -1 for thorium and uranium, respectively, leading to a

Figure 10 .

 10 Figure 10. X-EDS hyper-maps showing the evolution of the solid/liquid interface of the pellet during the dissolution experiments at different times (0h (A), 3h (B) and 5h (C)) and the associated binarized BSE micrographs (A', B' and C').

Figure 11 .

 11 Figure 11. Evolutions of normalized weight losses N L (U) (•) and N L (Th) (■) observed during the dissolution of heterogeneous Th 0.5 U 0.5 O 2 sintered samples in HNO 3 2M at room temperature after being corrected by the real mass ratio of the element i in the solid.

Table 1 .

 1 Refined unit cell parameters for the synthesized Th 1-x U x (C 2 O 4 ) 2 .2(H 2 O) compounds.

	5 (C 2 O 4 ) 2 .2H 2 O solid solutions. The refined unit

Table 2 .

 2 Refined unit cell parameters and average crystalline size obtained for the mixed oxides Th 1-x U x O 2 (raw and ground series)

	Method	x (calc)	x (EDS)	a (Å)	Volume (Å 3 )	Average crystallites size (Å)
	Raw series
	I	0.50		5.5268(1)	168.82(1)	48 ± 4
	II	0.50	0.49(2)	5.5222(3)	168.40(1)	33 ± 6
		0	0	5.6102(4)	176.58(2)	38 ± 2
	IV	1	1	5.4405(3)	161.03(1)	56 ± 14
		0	0	5.6003(1)	176.64(1)	139 ± 29
	V	1	1	5.4712(1)	163.77(1)	238 ± 54
	Ground series
	I	0.50	0.48(1)	5.5343(3)	169.5(2)	28 ± 5
	II	0.50	0.49(2)	5.5264(5)	168.78(2)	38 ± 8
	III	0.5	0.48(2)	5.5349(4)	169.56(2)	33 ± 3
		0	0	5.6078(3)	176.36(2)	44 ± 4
	IV	1	1	5.4434(3)	161.29(1)	51 ± 4
		0	0	5.5974(1)	175.38(1)	177 ± 38
	V	1	1	5.4679(1)	163.48(1)	253 ± 15

Table 5 .

 5 Normalized dissolution rates R L,0 (U) and R L,0 (Th) (expressed in g.m -2 .d -1 ) determined during the dissolution of homogenous Th 0.25 U 0.75 O 2 pellets (prepared following protocol II) at room-temperature and in various nitric acid solutions.

	Acid concentration	4 M	2 M	1 M
	R L (U)	(2.6 ± 0.1) × 10 -1	(1.2 ± 0.1) × 10 -1	(2.5 ± 0.1) × 10 -2
	R L (Th)	(2.7 ± 0.1) × 10 -1	(1.1 ± 0.1) × 10 -1	(2.4 ± 0.1) × 10 -2

Table 7 .

 7 Data extracted from the analysis of X-EDS hyper-maps obtained by the X-EDS hyper-maps treatment

	Time (hours)	Th (mol.%)	U (mol.%)	x U	x Th	Dissolved zones %	x U in dissolved zones
	0	19.84	13.50	0.40	0.60	1.9
	3	21.25	12.08	0.36	0.64	7.4	0.77
	5	22.12	11.21	0.34	0.66	10.6	0.79

Acknowledgements

The authors would like to thank Johann Ravaux and Renaud Podor (ICSM/L2ME) for performing the SEM experiments. This work also benefited from financial support of the French National Research Agency (ANR project # ANR-08-BLAN-0216) and from the CNRS Interdisciplinary Research Program MaProSu (Matériaux et Procédés de Remplacement/Substitution).


Pierre-Olivier Barrioz

J Ér Émy Hure

Benoît Tanguy

On void shapes and lattices effects on coalescence criterion

Void coalescence is known to be the last microscopic event of ductile fracture in metal alloys, and corresponds to the localization of plastic flow in between voids. Limit-analysis framework has been used to provide coalescence criteria that have been subsequently recasted into effective macroscopic yield criteria, leading to homogenized models for porous materials valid for high porosities. Such coalescence models remain up to now restricted to cubic or hexagonal lattices of spheroidal voids. Based on limit-analysis kinematic approach, a methodology is first proposed to get upper-bound estimates of coalescence stress for arbitrary void shapes and lattices. Semi-analytical coalescence criteria are derived for elliptic cylinder voids in elliptic cylinder unit-cells for isotropic matrix material, and validated through comparisons to numerical limit-analysis simulations. The practical relevance of these criteria for realistic void shapes and lattices is finally assessed numerically.

Introduction

Ductile fracture of metal alloys is mainly related to the nucleation, growth and coalescence of voids [START_REF] Puttick | Ductile fracture in metals[END_REF]. Experimental observations have provided guidance into the development of homogenized models of porous materials accounting for the presence of voids with additional state variables. Homogenized models can then be used to simulate crack growth in ductile materials and to predict fracture toughness [START_REF] Pardoen | An extended model for void growth and coalescence[END_REF], e.g., in structural analysis. The reader is referred to the recent reviews on ductile fracture mechanisms, modeling and computational aspects [START_REF] Benzerga | Ductile fracture by void growth to coalescence[END_REF][START_REF] Pineau | Failure of metals I: Brittle and ductile fracture[END_REF][START_REF] Besson | Continuum models of ductile fracture: a review[END_REF]. One of the key ingredient of these models is the yield criterion describing the effective or macroscopic plastic behavior of porous materials. Growth regime, i.e., when voids do not interact with each others, is by far the most widely studied part of void growth to coalescence ductile fracture. Following seminal contributions [START_REF] Gurson | Continuum theory of ductile rupture by void nucleation and growth: Part I -Yield criteria and flow rules for porous ductile media[END_REF][START_REF] Rousselier | Three-dimensional constitutive equations of damage and fracture[END_REF][START_REF] Ponte Castañeda | The effective mechanical properties of nonlinear isotropic composites[END_REF] based respectively on limit-analysis, thermodynamics and variational approach, yield criteria for porous materials have been proposed accounting for void shapes [START_REF] Gologanu | Continuum Micromechanics[END_REF][START_REF] Madou | A Gurson-type criterion for porous ductile solids containing arbitrary ellipsoidal voids. I. Limit-analysis of some representative cell[END_REF][START_REF] Madou | A Gurson-type criterion for porous ductile solids containing arbitrary ellipsoidal voids. II. Determination of yield criterion parameters[END_REF], anisotropy [START_REF] Benzerga | Plastic potentials for anisotropic porous solids[END_REF][START_REF] Han | A yield function for single crystals containing voids[END_REF][START_REF] Paux | An approximate yield criterion for porous single crystals[END_REF] or both [START_REF] Morin | A Gurson-type criterion for plastically anisotropic solids containing arbitrary ellipsoidal voids[END_REF][START_REF] Mbiakop | An analytical model for porous single crystals with ellipsoidal voids[END_REF][START_REF] Song | A finitestrain homogenization model for viscoplastic porous single crystals: I -Theory[END_REF], to name but a few. Coalescence regime, i.e., when voids strongly interact with each others through localized plastic flow in between adjacent voids, has been far less studied than growth. Thomason [START_REF] Thomason | A theory for ductile fracture by internal necking of cavities[END_REF][START_REF] Thomason | Three-dimensional models for the plastic limit-loads at incipient failure of the intervoid matrix in ductile porous solids[END_REF] provided coalescence stress assuming internal necking of voids embedded in an isotropic perfectly plastic matrix. Yield criterion was proposed in [START_REF] Benzerga | Anisotropic ductile failure. part I: experiments[END_REF] based on the coalescence stress, and was subsequently used in combination with growth yield criterion [START_REF] Benzerga | Micromechanics of coalescence in ductile fracture[END_REF] to provide a complete physically-based homogenized modeling of ductile fracture. Thomason's coalescence criterion has been shown to be in good agreement with experimental data [START_REF] Weck | Experimental investigation of void coalescence in metallic sheets containing laser drilled holes[END_REF][START_REF] Weck | Visualization by X-ray tomography of void growth and coalescence leading to fracture in model materials[END_REF], and was used in its original form or with phenomenological modifications to account for strain hardening [START_REF] Pardoen | An extended model for void growth and coalescence[END_REF][START_REF] Scheyvaerts | The growth and coalescence of ellipsoidal voids in plane strain under combined shear and tension[END_REF], secondary voids [START_REF] Fabrègue | A constitutive model for elastoplastic solids containing primary and secondary voids[END_REF] or penny-shaped cracks [START_REF] Benzerga | Micromechanics of coalescence in ductile fracture[END_REF], and for the presence of shear loading conditions [START_REF] Tekoglu | A criterion for the onset of void coalescence under combined tension and shear[END_REF]. Recently, significant efforts have been devoted to reassess and/or extend Thomason approximate coalescence criterion which is limited in practice to spheroidal voids in isotropic plastic material (obeying von Mises plasticity) under axisymmetric loading conditions. Benzerga and Leblond [START_REF] Benzerga | Effective yield-criterion accounting for microvoid coalescence[END_REF] and Morin et al. [START_REF] Morin | Coalescence of voids by internal necking: Theoretical estimates and numerical results[END_REF] proposed analytical upper-bound estimates of the coalescence stress for spheroidal voids in isotropic plastic material under axisymmetric loading conditions, subsequently extended by Torki et al. [START_REF] Torki | On void coalescence under combined tension and shear[END_REF][START_REF] Torki | Theoretical and numerical analysis of void coalescence in porous ductile solids under arbitrary loadings[END_REF] to account for combined tension and shear loading conditions. Upper-bound estimate of coalescence stress for penny-shaped cracks under arbitrary loadings in isotropic plastic material has been proposed in [START_REF] Hure | Theoretical estimates for flat voids coalescence by internal necking[END_REF]. Anisotropic materials (obeying Hill's plastic criterion) have also been considered in [START_REF] Keralavarma | A criterion for void coalescence in anisotropic ductile materials[END_REF][START_REF] Morin | Influence des effets de forme et de taille des cavits, et de l'anisotropie plastique sur la rupture ductile[END_REF]. Anisotropic coalescence criterion has also been proposed considering interfacial effects for anisotropic materials under arbitrary loading conditions [START_REF] Gallican | Anisotropic coalescence criterion for nanoporous materials[END_REF].

One of the key parameter of void coalescence is the intervoid distance, which comes from voids lattice in the coalescence plane. Thomason [START_REF] Thomason | Three-dimensional models for the plastic limit-loads at incipient failure of the intervoid matrix in ductile porous solids[END_REF] considered a cubic lattice of voids, while models developed later consider hexagonal lattices (through the approximation of cylindrical unit-cell).

It has been proposed in [START_REF] Torki | On void coalescence under combined tension and shear[END_REF] to consider equivalent porosity in the coalescence band to go from hexagonal lattices to cubic lattices. While such procedure lead to predictions in reasonable agreement with numerical simulations, it clearly calls for a theoretical improvement. Moreover, void aspect ratios are also influential parameters in void coalescence, but only spheroidal voids -having two axis length equal and thus only one aspect ratio -have been considered so far. As illustrated for example in [START_REF] Cao | A model for ductile damage prediction at low stress triaxialities incrorporating void shape change and void rotation[END_REF] with unit-cells simulations under non-axisymmetric loading conditions showing complex evolutions of void aspect ratios prior to coalescence, or in [START_REF] Navas | Void growth and coalescence in a three-dimensional npnperiodic void cluster[END_REF] through numerical simulations with non-periodic void clusters, including general void shapes and lattices in coalescence criterion is definitely required. Based on limit-analysis kinematic approach, a methodology is proposed in Section 2 to obtain formally trial velocity fields for arbitrary unit-cells. A semi-analytical coalescence criterion is then derived for the particular case of an elliptic cylinder void in elliptic cylinder unit-cell, under the assumption of isotropic matrix material and loading axes aligned with the principal axes of unitcell and void. Comparisons between the predictions and numerical results obtained through numerical limit-analysis are detailed in Section 3. The practical relevance of these criteria for realistic void shapes and lattices is finally discussed based on numerical results in Section 4, as well as potential extensions to more general void lattices.

Theoretical estimates of coalescence stress

Underline A and bold A symbols refer to vectors and second-order tensors, respectively. A cartesian orthonormal basis {e 1 , e 2 , e 3 } is used and position vectors are denoted X = {X,Y, Z} or x = {x, y, z}. Only isotropic materials are considered in the following.

Void coalescence deformation mode is defined from a general point of view as localized plastic flow in a layer Ω coa (of normal e 3 ) linking adjacent voids ω associated with an almost rigid motion (through elastic unloading) outside the coalescence layer Ω\Ω coa [START_REF] Koplik | Void growth and coalescence in porous plastic solids[END_REF], leading to macroscopic uniaxial straining conditions1 D D D = D 33 e 3 ⊗ e 3 at the scale of some (periodic) unit-cell Ω. For practical reasons, approximations of periodic unit-cells are classically used, as shown on Fig. 1, for which the boundary conditions for any velocity field v due to plastic flow in order to assess coalescence are:

v (x ∈ S lat ) .n S lat = 0 v x ∈ Ω\Ω ± coa = ±D 33 He 3 (1)
where S lat is the lateral surface, H the half-height of the unitcell (Fig. 1), and n stands as the normal vector. The relevance of the unit-cell approximation will be discussed in Section 4. The isotropic matrix material around voids is assumed to obey von Mises perfect plasticity with associated plastic flow.

Analytical limit-analysis

In order to evaluate the macroscopic stress Σ 33 at which coalescence can occur for a given void shape and lattice, homogenisation along with limit analysis is used (see, e.g., [START_REF] Benzerga | Ductile fracture by void growth to coalescence[END_REF] for details). For periodic boundary conditions, macroscopic stress Σ Σ Σ and strain rate D tensors are related to their microscopic counterparts by volume averaging:

Σ Σ Σ = 1 volΩ Ω σ σ σ dΩ D = 1 volΩ Ω d dΩ (2)
with σ σ σ the Cauchy stress and d the microscopic strain rate tensor. Hill-Mandel lemma reads:

1 volΩ Ω σ σ σ : d dΩ = Σ Σ Σ : D (3)
Upper-bound theorem of limit analysis enables to assess the limit-load of the unit-cell Ω containing voids ω, and is, for a perfectly plastic material obeying von Mises' criterion:

Σ Σ Σ : D = Π(D) ≤ Π + (D) (4)
with

Π + (D) = σ 0 d eq Ω-ω = 1 volΩ Ω-ω σ 0 d eq dΩ (5)
where σ 0 is the yield stress, d eq = [2/3]d : d the equivalent strain rate (d = [ t t t ∇ ∇ ∇v + ∇ ∇ ∇v]/2), and v a velocity field kinematically admissible with D and verifying the property of incompressibility tr(d) = 0. Π + (D) = Π(D) when v is the velocity field solution. Π(D) will be referred to as the macroscopic plastic dissipation and superscript + will be omitted in the following for clarity. In presence of a velocity field having a purely tangential discontinuity along an interface S d , the plastic dissipation related to the discontinuity is:

Π sur f (D) = 1 volΩ Sd σ 0 √ 3 ||v t ||dS (6)
where ||v t || is the absolute value of the velocity jump. The macroscopic limit stress or yield locus is obtained from Eq. 4 by the equation:

Σ Σ Σ = ∂Π(D) ∂D (7)
Analytical expression for macroscopic stress according to Eq. 7 that will stand as coalescence load requires the choice of trial velocity fields that should be (1) kinematically admissible with coalescence boundary conditions and (2) incompressible. Few of such trial velocity fields have been provided in previous studies [START_REF] Thomason | Three-dimensional models for the plastic limit-loads at incipient failure of the intervoid matrix in ductile porous solids[END_REF][START_REF] Benzerga | Effective yield-criterion accounting for microvoid coalescence[END_REF][START_REF] Morin | Coalescence of voids by internal necking: Theoretical estimates and numerical results[END_REF][START_REF] Hure | Theoretical estimates for flat voids coalescence by internal necking[END_REF][START_REF] Keralavarma | A criterion for void coalescence in anisotropic ductile materials[END_REF]. However, finding trial velocity fields for arbitrary unit-cells is not an easy task, which has limited the development of general void coalescence criterion.

Towards trial velocity fields for arbitrary unit-cells

The starting point of the methodology proposed to get a trial velocity field for arbitrary unit-cell is a trial velocity field V defined for a reference unit-cell (of coordinates X) satisfying the conditions described in Section 2.1:

∇ X .V (X) = 0 (8) V X ∈ S re f lat .n S re f lat = 0 V X ∈ Ω re f \Ω re f coa = ±D 33 He 3 (9)
where Eq. 8 corresponds to the incompressibility condition and Eq. 9 to the coalescence boundary conditions. Considering an arbitrary unit-cell of coordinates x defined such that:

x = φ(X) (10)
preserving the boundary surface. A trial velocity field v such that:

v(x) = ψ(V (X)) (11)
will satisfy the property of incompressibility and coalescence boundary conditions if:

∇ x .v = ∇ x {ψ[V (φ -1 (x))]} = 0 (12) ψ V [φ -1 (x ∈ S lat )] .n S lat = 0 ψ V [φ -1 (x ∈ Ω\Ω coa )] = ±D 33 He 3 (13)
Finding solutions to Eqs. 12,13 with respect to the functions φ and ψ leads to a trial velocity field satisfying the property of incompressibility and compatible with coalescence boundary conditions on the unit-cell Ω. While this provides an effective -albeit not trivial -procedure to get such velocity field and thus to upper-bound estimates of the coalescence stress through Eqs. 5,7, it should be noted here that nothing ensures that such trial velocity field is actually a good one, and thus leading to an accurate estimate of the coalescence stress, which should ultimately be checked through comparisons to numerical simulations.

A solution to Eqs. 12 and 13 can be found for elliptic cylinder unit-cell, and is described in the next section.

Trial velocity field for elliptic cylinder unit-cells

Elliptic cylindrical unit-cell Ω of half-height H and semi-axes L 1 and L 2 containing a coaxial void ω (of semiaxes R 1 and R 2 and half-height h) is now considered (Fig. 2). Four dimensionless ratios can be defined:

W 1 = h R 1 W 2 = h R 2 χ 1 = R 1 L 1 χ 2 = R 2 L 2 (14)
where W i are the out-of-plane (with respect to the coalescence plane) aspect ratios of the void, χ i the in-plane dimensionless length of the inter-void ligament. An additional dimensionless parameter c = h/H can be defined, but does not play any role in coalescence criterion derived hereafter, as long as the phenomenon considered is coalescence in layers, and not coalescence in columns [START_REF] Benzerga | Ductile fracture by void growth to coalescence[END_REF].

Fig. 2: Elliptic cylinder unit-cell with coaxial elliptic cylinder void considered in this study. The principal axes of the mechanical loading are assumed to be the same of the ones of the unit-cell (and void).

In order to apply the methodology described in Section 2.2 to the elliptic cylinder unit-cell, the reference trial velocity field chosen is the one proposed in [START_REF] Benzerga | Effective yield-criterion accounting for microvoid coalescence[END_REF] for cylindrical unit-cell of radius L 1 :

V X (X,Y ) = D 33 H 2h L 2 1 X 2 +Y 2 -1 X V Y (X,Y ) = D 33 H 2h L 2 1 X 2 +Y 2 -1 Y V Z (Z) = D 33 H h Z (15)
for |Z| ≤ h. The following function φ allows to map the cylindrical unit-cell to the elliptic cylinder unit-cell of semi-axis L 1 and L 2 :

x = φ(X) =          X Y L 2 L 1 Z (16)
Eqs. 12 and 13 can be satisfied by defining the function ψ such as:

v = ψ(V ) =          V X V Y L 2 L 1 V Z (17)
Finally, the trial velocity field v can be written as:

                 v x (x, y, z) = D 33 H 2h L 2 1 x 2 + α 2 y 2 -1 x v y (x, y, z) = D 33 H 2h L 2 1 x 2 + α 2 y 2 -1 y v z (x, y, z) = D 33 H h z (18)
for |z| ≤ h, and where the dimensionless parameter α is defined such that:

α = L 1 /L 2 (19)

Coalescence criterion for elliptic cylinder unit-cell 2.4.1 General case

The equivalent plastic strain rate d eq can be computed with the trial velocity field defined in Eq. 18, as well as the tangential discontinuities at the top/bottom surfaces ||v t ||. We make use of the change of coordinates systems, from cartesian to elliptic: x = r cost and αy = r sint:

             d eq = D 33 H h 1 + L 4 1 3r 4 1 + α 2 -1 2α sin 2t 2 ||v t || = r D 33 H 2αh L 2 1 r 2 -1 1 + (α 2 -1) cos 2 t (20)
inside in the coalescence layer Ω coa , while d eq = 0 outside. Macroscopic plastic dissipation (Eq. 5) can now be computed Π = Π 1 + Π 2 , where Π 1 corresponds to the volumic plastic dissipation, while Π 2 is related to the velocity field discontinuity. Only half of the unit-cell is considered due to symmetry.

Π 1 = 1 volΩ Ω σ 0 d eq dΩ (21)
With the change of variables considered, dΩ = dx dy dz = r dr dt dz/α, and volΩ = (πHL 2 1 )/α:

Π 1 = D 33 σ 0 πL 2 1 L 1 R(t) 2π 0 1 + L 4 1 3r 4 1 + α 2 -1 2α sin 2t 2 rdrdt (22)
where R(t) can be written as:

R(t) = R 1 cos(t) 2 + (1 -cos(t) 2 )[R 2 1 /(αR 2 ) 2 ] (23)
The 2D integral can not be computed analytically, and can only be reduced to a 1D integral (once the integration over the variable r is done, similarly to [START_REF] Morin | Coalescence of voids by internal necking: Theoretical estimates and numerical results[END_REF]).

The plastic dissipation Π 2 related to the velocity field tangential discontinuity is computed according to Eq. 6:

Π 2 = 1 volΩ S top σ 0 √ 3 ||v t ||dS (24)
Again, with the change of variables considered, dS = dx dy = r dr dt/α:

Π 2 = D 33 σ 0 2 √ 3πL 2 1 hα L 1 R(t) 2π 0 (L 2 1 -r 2 )dr 1 + (α 2 -1) cos 2 tdt (25
) Finally, according to Eq. 7 that reduces to Σ 33 = Π/D 33 and normalizing length by L 1 , the coalescence stress can be written as:

Σ 33 ≤ σ 0 π 1 R(t)/L 1 2π 0 1 + 1 3r 4 1 + α 2 -1 2α sin 2t 2 rdrdt + σ 0 2 √ 3πW 1 χ 1 α 1 R(t)/L 1 2π 0 (1 -r 2 )dr 1 + (α 2 -1) cos 2 tdt (26)
Eq. 26 gives an upper-bound of the coalescence stress for an elliptic cylinder unit-cell containing a coaxial elliptic cylinder void, for an isotropic material and in absence of shear stresses with respect to the principal axes defined by the unitcell. It extends previous studies aiming at predicting coalescence stress for spheroidal voids. Some progress could be made to compute partially the integrals (or using Cauchy-Schwartz inequality to get an upper-bound). One should note however that the computations of these integrals can easily been done numerically. Some simplifications can be made considering homothetic void and unit-cell, as detailed in the next section.

Homothetic voids and unit-cells

The particular case of homothetic void and unit-cell, that corresponds to R 1 = αR 2 or χ 1 = χ 2 = χ, allows to simplify the coalescence stress (Eq. 26). In order to get analytical coalescence estimate, Cauchy-Schwartz inequality is used to get an upper-bound of the volumic plastic dissipation:

Π 1 ≤ D 33 σ 0 πL 2 1 L 1 R 1 rdr 2π 2π 0 1 + L 4 1 3r 4 1 + α 2 -1 2α sin 2t 2 dt ≤ 2D 33 σ 0 L 2 1 L 1 R 1 1 + β 4 L 4 1 r 4 rdr (27)
where a new dimensionless ratio is defined:

β 4 = α 4 + 6α 2 + 1 24α 2 (28)
Upon integration over r, the upper-bound of the volumic plastic dissipation is:

Π 1 ≤ D 33 σ 0 -β 4 + χ 4 + β 4 + 1 - β 2 2 log ( β 4 + 1 + β 2 )( β 4 + χ 4 -β 2 ) ( β 4 + 1 -β 2 )( β 4 + χ 4 + β 2 ) (29)
Eq. 29 depends only on two geometrical ratios χ and β and verifies the property Π 1 (α) = Π 1 (α -1 ) which corresponds to a permutation of the axes e 1 and e 2 . An analytical upperbound of the plastic dissipation Π 2 can also be obtained with Cauchy-Schwartz inequality:

Π 2 ≤ D 33 σ 0 √ 6W 1 χ √ α 2 + 1 α χ 3 -3χ + 2 3 ( 30
)
The analytical expression for the upper-bound of the surfacic plastic dissipation (Eq. 30) can alternately be written in a symmetric form (with respect to W 1 and W 2 ):

Π 2 ≤ D 33 σ 0 √ 6χ W 2 1 +W 2 2 W 1 W 2 χ 3 -3χ + 2 3 (31)
Finally, the upper-bound estimate of the coalescence stress can be written (using Eq. 7 that reduces to Σ 33 = Π/D 33 ):

Σ 33 σ 0 ≤ -β 4 + χ 4 + β 4 + 1 - β 2 2 log ( β 4 + 1 + β 2 )( β 4 + χ 4 -β 2 ) ( β 4 + 1 -β 2 )( β 4 + χ 4 + β 2 ) + 1 √ 6χ W 2 1 +W 2 2 W 1 W 2 χ 3 -3χ + 2 3 ( 32
)
For α = 1 (and thus W 1 = W 2 = W ), i.e., for cylindrical unitcells and voids, Eq. 32 reduces to the expression given in [START_REF] Benzerga | Effective yield-criterion accounting for microvoid coalescence[END_REF][START_REF] Morin | Coalescence of voids by internal necking: Theoretical estimates and numerical results[END_REF]:

Σ 33 σ 0 ≤ χ 3 -3χ + 2 3 √ 3W χ + 1 √ 3 2 -1 + 3χ 4 + log 1 + 1 + 3χ 4 3χ 2 (33)
Eq. 32 corresponds to an upper-bound of the coalescence stress for elliptic cylindrical voids in homothetic unit-cells, for isotropic material and in absence of shear stresses.

Eqs. 26,32,33 are upper-bounds of the coalescence stress. In particular, and due to the some limitations of the trial velocity field used, Eq. 33 overestimates numerical results by an approximately constant multiplicative factor 2 from results 2 Refined calibration has been proposed in [START_REF] Torki | On void coalescence under combined tension and shear[END_REF] which is close to Eq. 34 for W ∈ [0.5 : 3] and χ ∈ [0.3 : 0.7] (and equivalent for W 1).

presented in [START_REF] Morin | Coalescence of voids by internal necking: Theoretical estimates and numerical results[END_REF]:

Σ 33 σ 0 ≈ 0.9 Σ 33 σ 0 upper-bound [START_REF] Gallican | Anisotropic coalescence criterion for nanoporous materials[END_REF] where [Σ 33 /σ 0 ] upper-bound is taken as Eq. 33 (or Eqs. 26,32). Therefore, in the following, Eq. 34 will be compared to numerical results, keeping in mind that the theoretical derivation gives only the upper-bound used in Eq. 34. The ability of the proposed analytical expression to predict coalescence stress for ellipsoidal voids will be assessed in Section 3.

3 Assessment of theoretical coalescence criterion

Numerical limit-analysis

In order to assess coalescence stress derived through limit-analysis by choosing a trial velocity field (Eq. 7), exact coalescence stresses are computed through numerical simulations known as numerical limit-analysis (see, e.g., [START_REF] Madou | A Gurson-type criterion for porous ductile solids containing arbitrary ellipsoidal voids. II. Determination of yield criterion parameters[END_REF][START_REF] Tekoglu | Void coalescence in ductile solids containing two populations of voids[END_REF]): the problem defined in Fig. 1 is solved (classically with finite-element method (FEM)) under the small perturbation hypothesis, with elastic-perfectly plastic material (obeying von Mises criterion). Macroscopic stresses are computed through volume averaging (Eqs. 2). A loading parameter is increased until saturation of the macroscopic stresses that correspond to exact coalescence stresses, up to numerical errors. As an alternative to FEM simulations, Fast Fourier Transform (FFT-) based solver [START_REF] Moulinec | A numerical method for computing the overall response of nonlinear composites with complex microstructure[END_REF] has been used in this study, as in [START_REF] Paux | Plastic yield criterion and hardening of porous single crsytals[END_REF]. FFT simulations rely on periodic unitcell discretized in voxels. Different constitutive equations can be applied to subsets of voxels in order to model heterogeneous unit-cells. Material voxels constitutive equations correspond to elasto-plasticity with von Mises criterion (of Young's modulus Y , Poisson's ratio ν and yield stress σ 0 ), while void voxels are pure elasticity with zero rigidity. Loading parameters are either average strains E E E or stresses Σ Σ Σ: to assess coalescence corresponding to uniaxial straining (in absence of shear stresses), E E E = E 33 e 3 ⊗ e 3 is applied, where E 33 is the scalar loading parameter. In order to be able to simulate non-periodic unit-cells as the ones considered in Section 2.2 (Fig. 2), a fictive orthotropic elastic material is added around the elliptical unit-cell, with particular elastic moduli:

ν 12 = ν 23 = ν 13 = 0 Y 1 = Y 2 = G 12 Y Y 3 = G 13 = G 23 Y (35)
In the limit defined by Eqs. 35 for the fictive elastic material around the elliptic-cylinder unit-cell, the boundary conditions used in the theoretical approach can be recovered, which is checked in Section 3.2. An example of the typical discretized unit-cell used is shown in Fig. 3. AMITEX FFTP [41] software was used for all simulations performed in this study, along with MFront software [START_REF] Helfer | Introducing the opensource mfront code generator: Application to mechanical behaviors and materials knowledge management within the PLEIADES fuel element modelling platform[END_REF] for generating constitutive models, with σ 0 /Y = 10 -4 . A convergence study w.r.t the number of voxels was performed for all numerical results shown hereafter.

Comparisons to numerical results

Semi-analytical coalescence stress derived in Section 2 (Eq. 34) are compared to numerical results. For homothetic void and unit-cell, results are given in Fig. 4a,b,e. In the particular case of cylindrical void and unit-cell (corresponding to α = 1), numerical results obtained with FFT simulations are equal to the ones obtained in previous studies with FEM [START_REF] Morin | Coalescence of voids by internal necking: Theoretical estimates and numerical results[END_REF][START_REF] Hure | Theoretical estimates for flat voids coalescence by internal necking[END_REF], validating the use of a fictive elastic material described in Section 3.1 to impose a given macroscopic strain to a non-periodic unit-cell. For α = 1, numerical results are in good agreement with the analytical predictions, capturing the increase of coalescence stress for both α > 1 and α < 1. The agreement is particularly good for large values of the parameters W 1 and χ 1 , but deviations appear for W 1 = 0.5 and χ 1 = 0.4. This was somehow expected as the reference trial velocity field as been already shown to lead to predictions in less good agreement with numerical results in such situations [START_REF] Torki | On void coalescence under combined tension and shear[END_REF]. Analytical equivalent strain rate field (derived from the trial velocity field) and equivalent strain field of the simulation (Fig. 4f) taken at the height z = h share some common points, explaining the good agreement between numerical results and theoretical predictions. One should finally note that, whatever the value of the parameter α, the porosity in the coalescence layer is constant, and therefore the strategy proposed in [START_REF] Torki | On void coalescence under combined tension and shear[END_REF] to use a coalescence criterion derived for a given unit-cell to another one (based on equivalent porosity) would not be able to capture the results shown here.

Two other situations are assessed in Fig. 4. The first one (Fig. 4c) corresponds to the case of an elliptical cylinder unit-cell with a cylindrical void. For such situations, it was not possible to derive (simple enough) analytical coalescence stress, and therefore the integral equation (Eq. 26 along with Eq. 34) for the coalescence stress is used to compare to numerical results. A good agreement is also observed in this situation, for different values of void aspect ratio W 1 , which can again be explained by the fact the trial velocity field used captures some aspects of the real deformation mode, as shown on Fig. 4g. Similar conclusions also hold for the case of cylindrical unit-cells with elliptic cylinder voids, as shown on Fig. 4d,h. Comparisons of the proposed coalescence criteria to numerical results validate the use of Eqs. 26,32,34 to describe coalescence stress accounting for the effect of unit-cells and voids shapes, under the assumption that the principal axes of mechanical loading are the same as the ones of the unitcell and void. On the contrary, i.e. in presence of additional shear stresses, the derivation proposed in [START_REF] Torki | On void coalescence under combined tension and shear[END_REF] could be used, which remains to be done and validated against numerical results. More importantly, the unit-cells considered so far in this study are only approximations of some space-filling unit-cells with more realistic void shapes, which is detailed in the following section.

Discussion

Hexagonal-type lattices of ellipsoidal voids and associated unit-cells (Fig. 5) are considered as a more realistic description of void lattices and void shapes in a coalescence layer. Another classical choice would have been to choose cubictype lattices -which are as realistic than hexagonal ones -but are not considered in the following as being poorly described by elliptic unit-cells 3 .

Fig. 5: Hexagonal-type lattices of ellipsoidal voids. Unitcells in red solid lines are used to perform numerical simulations, and results are compared to predictions from the elliptic cells inscribed in the Voronoi cells.

FFT coalescence simulations have thus been performed on periodic unit-cells shown in solid red lines on Fig. 5, corresponding to deformed hexagonal lattices. Numerical results are compared to theoretical predictions (Eq. 34) assuming:

(1) that the corresponding unit-cell is the elliptic cylinder inscribed in the Voronoi cell of voids (red dashed line and black solid line in Fig. 5, respectively), ( 2) an effective value of the intervoid ligament χ is chosen to account for its variation along the height for ellipsoidal voids. For the latter, different approach can be considered: equivalent porosity in the coalescence layer (that would lead to χ m i = 2/3χ i ) as in [START_REF] Torki | On void coalescence under combined tension and shear[END_REF], or average value leading to χ m i = [π/4]χ i . Hereafter, the effective value has been calibrated with numerical results on hexagonal lattices (α = 1) and spheroidal voids χ m i = 0.85χ i , and used for other situations. Numerical results are compared to theoretical predictions in Fig. 6a,c,d, showing the ability of the predictions to capture all the trends due to void shapes and lattices. However, the agreement is less quantitative than for previous comparisons based on similar geometry between theoretical analysis and numerical simulations. Discrepancies appear mainly for large (or low) values of the parameter α, of low values of χ i . Both were somehow expected: the former can be understood as the assumption of representing an elliptic cylinder to represent the cell around each voids fails as α 1 or α 1, as shown on Fig. 6b, where the Voronoi cell tends to become of rectangular shape. The latter comes from the fact that for spheroidal voids, the effective intervoid ligament is lower than its maximal value, where the reference trial velocity field is known to become less accurate, even for cylindrical unit-cell for cylindrical voids. Both inaccuracies could in principle be handled by refining the theoretical analysis with refined reference trial velocity field and cubic-type unit-cells.

Conclusions and Perspectives

Void coalescence deformation mode is strongly sensitive to both void shapes and intervoid distances. As a result, void lattices play a key role, as shown recently in [START_REF] Navas | Void growth and coalescence in a three-dimensional npnperiodic void cluster[END_REF]. However, up to now, coalescence criteria have been derived assuming idealized hexagonal or cubic lattices of spheroidal voids, while criteria derived for arbitrary void shapes and lattices will be ultimately required. As a step towards this goal, coalescence criteria have been derived for elliptic cylinder unit-cells with elliptic cylinder voids, based on limit-analysis framework and on a methodology allowing getting trial velocity fields from known reference trial velocity fields. Coalescence stress predictions have been shown to be in good agreement with numerical results performed with the same geometry, and also in relatively good agreement for spacefilling arrangements of voids well approximated by elliptic unit-cells, i.e., for large values on the intervoids ligaments χ i 0.5 and in-plane cell aspect ratio up to a factor 2. Various extensions of this study could be considered. A first one corresponds to the case of a coalescence layer composed of a random arrangement of voids: the Voronoi cell around each void could be used or approximated as the unit-cell on which limit-analysis framework can be done by solving Eqs. 12,13.

A second extension is to develop evolution laws for the parameters W i and χ i that will be required for implementing the coalescence criterion as a yield criterion in constitutive equations for porous materials. In particular, the evolution of χ i can not be inferred from mass conservation as usually done when χ 1 = χ 2 . Last, the combined influence of void shapes / lattices and shear stresses (defined as macroscopic stresses not in the principal axes of voids) should be studied.

Fig. 1 :

 1 Fig. 1: Reference cylindrical unit-cell Ω re f and deformed unit-cell Ω. Coalescence deformation mode corresponds to localized plastic flow in a coalescence layer Ω coa associated with (almost) rigid body motion of the outer parts Ω\Ω coa , thus to uniaxial straining conditions D D D = D 33 e 3 ⊗ e 3 .

Fig. 3 :

 3 Fig. 3: One-eighth of the typical periodic unit-cell with cylindrical void used for FFT simulations to assess numerically coalescence stress. Three different constitutive equations are used for the blue, white and red voxels: zero rigidity, elasto-plastic von Mises plasticity and fictive elastic material, respectively. Macroscopic strain is imposed: E = E 33 e 3 ⊗ e 3 . Macroscopic coalescence stress Σ 33 is computed through volume averaging over the white and blue regions only.

Fig. 4 :

 4 Fig. 4: (a,b,c,d,e) Coalescence stress for elliptic cylinder unit-cells with elliptic cylinder voids, as a function of the parameter α, for various values of W 1 and χ 1 . Solid lines correspond to Eq. 34, squares to numerical results. (f,g,h) Comparisons of the analytical and numerical strain-(rate) fields (Arbitrary units). Numerical results are taken at an height z = h.

Fig. 6 :

 6 Fig.6: (a,c,d) Coalescence stress for hexagonal-type lattices of ellipsoidal voids as a function of the parameter α, for various values of W 1 and χ 1 . Squares correspond to numerical results, solid lines to Eq. 34 considering the elliptic unit-cell inscribed in the Voronoi cell of the void. (b) Evolution of the equivalent strain field taken at z = h obtained with numerical simulations as the parameter α decreases. Arbitrary units.

In absence of shear stresses with respect to the coalescence layer, which are not considered in this study

An obvious solution to deal with cubic-type lattices is to start from a reference trial velocity field defined for cubic unit-cell, as one of those proposed by Thomason[START_REF] Thomason | Three-dimensional models for the plastic limit-loads at incipient failure of the intervoid matrix in ductile porous solids[END_REF], which is left for a future study