J.-P Malet

A Remaître

O Maquaire

C Ancey

J Locat

Flow susceptibility of heterogeneous marly formations: implications for torrent hazard control in the Barcelonnette Basin (

Keywords: marls, earthflow, debris avalanche, debris flow, rheology, torrent hazard control

Earthflows, debris avalanches and debris flows occurring in black marls in Southeast France present differentiated morphological characteristics and velocities. The rheological response of these remoulded fine-grained flow-like landslide deposits have been investigated using several techniques (parallel-plate rheometry, coaxial rheometry, slump tests and inclined plane tests). This contribution shows that rheological characteristics can be useful in establishing the transformation process of the materials into flow-like phenomena and identifying the natural mixture of basic formations which present the weakest yield stress and viscosity. Results show that the simple shear behaviour of all these natural flows clearly distinguishes the type of movement, in particular for earthflow and debris flow. Furthermore, some proportion mixtures present the highest mobility properties, and thus the highest susceptibility for transformation into high-speed flow-like phenomena. Results may be useful for identifying potential source zones in muddy flow-like landslide prone areas.

INTRODUCTION

Fine-grained gravitational movements presenting flow characteristics are frequent on the slopes or in the channels of torrential basins comprising black marls in Southeastern France. According to eyewitness observations and on the basis of Hungr et al.'s classification (2001), three types of flow-like landslides occur in black marls: continuously slow-moving muddy earthflows (approx. 0.01 mday -1 ), fast-moving muddy debris avalanches (0.5-1 mmin -1 ) and fast-moving muddy debris flows (5 ms -1 ). Moreover, earthflows, debris avalanches and debris flows can be differentiated by the morphology of the deposits, the grain-size distribution and the geomechanical parameters. Earthflows are slower, gentler and cover short distances, while debris avalanches and debris flows are faster and cover long distances. All types of flows carry gravel and boulders up to 4 m in diameter, with more or less 60-70% mud (fraction <20 mm). The percentage of mud decreases from earthflows to debris flows.

According to [START_REF] Coussot | Recognition, classification and mechanical description of debris flows[END_REF] rheological classification, the three flows may be thought of as three members within a continuum between "muddy solid" and "muddy fluids". Earthflows exhibit a behavior at the interface of "muddy solid" and "muddy fluid" near the "fracture limit" (Fig. 1). Nevertheless the behaviour can be well represented by viscoplastic empirical models [START_REF] Ancey | Debris flows and related phenomena[END_REF]. Debris avalanches and debris flows can also be represented in this way, as they are sufficiently concentrated. They behave as "muddy fluids". Maquaire et al. (in press) have demonstrated a genetic link between these movements. On a short-term scale it is possible that the earthflows transform into debris avalanches or debris flows as they reached higher slope gradients in conjunction with having a higher proportion of sand and silt in their muddy matrix (Malet et al. in press). The transformation is governed by the progressive change in the granulometry of the materials, and by the addition of material (moraine, sandstone) other than the source material (marl). Similar transformation from fine-grained earthflows to muddy debris flows was reported by [START_REF] Colas | Glissement et coulée de La Valette dans les Alpes de Haute-Provence: présentation générale et modélisation de la coulée[END_REF], [START_REF] Johnson | Tumalt Creek debris flows and debris avalanches in the Columbia River Gorge, East of Portland, Oregon[END_REF] or Couture & Evans (in press). In the black marls outcrops of Southeastern France, earthflows, debris avalanches and debris flows involve marl, moraine and sandstone. A previous survey, grain-size distribution, consistency limits, and undrained cohesion have demonstrated that some mixtures of deposits present the highest mobility properties (Maquaire et al. in press), and thus the highest susceptibility to initiate highspeed flow-like phenomena. The direct determination of the behaviour of flow material with the help of rheometers is faced with the irretrievable problem that they generally contain particles of various sizes including big boulders [START_REF] Coussot | Recognition, classification and mechanical description of debris flows[END_REF]. Numerous studies have demonstrated that the behaviour of fine-grained flows is mainly guided by the muddy matrix, which acts as a lubricant, rather than the blocks or debris carried [START_REF] Johnson | A model for debris flow[END_REF][START_REF] Johnson | Debris flow[END_REF][START_REF] Pierson | A rheologic classification of sub-aerial sediment-water flows[END_REF][START_REF] O'brien | Laboratory analyses of mudflow properties[END_REF][START_REF] Major | Debris flow rheology: experimental analyses of fine-grained slurries[END_REF][START_REF] Coussot | Recognition, classification and mechanical description of debris flows[END_REF]. At the opposite, in the case of coarse-grained flows where rheology evolves as mixture agitation, grain concentration, and fluid-pressure change during flow initiation, transit and deposition [START_REF] Iverson | The physics of debris flow[END_REF][START_REF] Iverson | New views of granular mass flows[END_REF], simple constitutive relations (Bingham, Herschel-Bulkley) are not able to capture the complex grain-grain and water-grain interactions controlling these flows. Grain-size distribution analyses of the flows show their muddy character (more than 60% of sand, silt and clay, and more than 25% of clay and silt). In such a case of fine-grained sediments, the presence of colloidal fractions may introduce yield stress [START_REF] Hampton | Competence of fine-grained debris flows[END_REF][START_REF] Major | Debris flow rheology: experimental analyses of fine-grained slurries[END_REF]. We therefore carried out rheological measurements on the < 20 mm-fraction of the materials using parallel-plate rheometry, coaxial rheometry, slump tests and inclined plane tests.

The main objectives were to: (i) check if the differences in velocity, flowing mechanisms and other features of these muddy flows observed and measured in the field were in relation to the rheological characteristics of the muddy matrix; (ii) define how the material's grain-size distribu-tion influences the mobility, and (iii) identify the mixtures (marl, moraine, sandstone) which present the weakest yield stress and viscosity. The paper presents the results of the rheological characterisation carried out on the deposits of various flow-like landslides and on artificial mixtures of marl, moraine and sandstone. The main hypothesis is the following: little differences in the grainsize distribution of the muddy matrix can strikingly change the rheological behaviour of the flows, and thus the velocities.

DESCRIPTION OF PHENOMENA AND MATERIAL

In the Barcelonnette basin (Alpes-de-Haute-Provence, France, Fig. 2), the materials tested were taken from deposits from three muddy earthflows (Super-Sauze, EA-C1a, EA-IND; La Valette, EA-VAL; Poche, EA-POC), two muddy debris avalanches which started on the upper part of the Super-Sauze earthflow (DA-COU99a, DA-COU99-b), two debris flows with a muddy matrix in the Faucon (DF-FAU) and Riou-Bourdoux (DF-RBX) torrents, weathered marls (Mar), moraines (Mor) and weathered sandstones (San). Eyewitness observations in the morphology of the deposits (streamlines, lobes, and bulges) and field measurements (Malet et al. in press) have shown that such flow-like landslides move predominantly by distributed flow, and exhibit distributed straining. The earthflows exhibit a macro-viscous behaviour, with a wide range of velocity which varies with time. The channelized flow fossilizes a complex topography and creates accumulation lobes which may be as much as 20 meters thick, with no grading (Fig. 3a). Muddy earthflows involve the products of weathering of clay-rich rocks. Weathering produces a clayey colluvium with a consis-tency closer to the plastic than the liquid limit [START_REF] Hungr | A review of the classification of landslides of the flow type[END_REF]. Field observations and measurements show that the movement may be maintained over long distances and for long periods of time by intermittent plastic deformations accompanied by built-up of interstitial pore water pressures. There are frequent sudden accelerations in the upper part of the flows above an interstitial pressure threshold. Muddy earthflows contain a maximum of 25-30% of coarse (gravel) material, though protruding boulders of up to 4 m in diameter can be coated in the mud. The accumulation lobes present fissures a meter in depth (Fig. 3a). At the opposite, the deposits of debris avalanches and debris flows are shallow (1-4 m thick), mainly flat-topped lobes, becoming progressively more sheet like in the downward direction. The debris avalanches deposits are mainly lateral levees or small accumulation cones showing no grading (Figs 3b,d). Debris avalanches observed in marls are rapid (0.5-1 mmin -1 ), completely saturated and not channelized phenomena. The two debris avalanches observed in 1999 initiated as rotational slides but they acquired flow characteristics very quickly. The percentage of gravel is often between 30 to 35%. Most of the coarser elements are on the subsurface.

The debris flows are faster (5 ms -1 ) than the debris avalanches and concentrate in torrential channels; two characteristics of debris flows are a deposit with inverse grading and lateral levees (Figs 3c,e). These lobes are flat, with convex sides. Lateral sorting is poor, whereas vertical rough sorting is high [START_REF] Major | Pebble orientation on large, experimental debris flow deposits[END_REF]. The coarser clasts and boulders are concentrated at the top of the flow surface. The vertical section (Figs 3c,e) shows several successive beddings of varying granulometry: (i) an indurated stratum corresponding to the flow hardpan; this may be covered by a thin stratum of fine material deposited by a subsequent stream flow, (ii) an alluvial stratum or hyperconcentrated flow stratum, consisting of fine loose material with a high lateral sorting, and (iii) a debris flow stratum with coarser material and debris set in a fine matrix. The percentage of gravel is often comprised between 35 to 40%.

The grain-size distribution of the material and the plasticity domain are the factors which influence the differentiated response of materials to the movement. The comparison of geomechanical and rheological characteristics of the natural mixtures (earthflows, debris avalanches, debris flows) with those of the artificial mixtures comprising the three superficial formations in the Barcelonette Basin allows to: (i) identify the influence of each superficial formation in the behaviour of each flow-like landslide type, (ii) define the mixtures the most susceptible to mobility (weathered black marl, weathered sandstone, moraine). Two groups of materials were tested (Table 1): a combination of marl and moraine mainly found on the north-facing slope (EA-C1a, EA-IND, DA-COU99a, b, EA-POC, UM artificial mixtures) (Fig. 2) and the second combining marl, moraine and sandstone found only on the south-facing slope (EA-VAL, DF-RB, DF-FAU, AM artificial mixtures). The ponderal proportions are set out in Figure 4. All the materials are relatively homogeneous from a mineralogical standpoint, mainly comprising illites, chlorites and kaolinites. There is no swelling clay in these formations [START_REF] Malet | Caractérisation rhéologique des coulées de débris et des laves torrentielles du bassin marneux de Barcelonnette (Alpes-de-Haute-Provence, France)[END_REF]. The materials are characterized by their high percentage of gravel, increasing from the earthflow deposits (25-30%) to the debris flow deposits (35-40%). The gravel fraction is larger than the sandy fraction. Moreover, after some forty analyses on the fraction passing the 20-mm sieve, the three types of flow are clearly distinguished even though the granulometric characteristics of the samples are close. Muddy earthflow deposits contain an abundant fine-grained matrix with an essentially silty-clay texture, as muddy debris avalanche deposits have a silty-sand texture and debris flows deposits a sandy-silt texture (Table 1, Fig. 4). The earthflow deposits are rich in silt and clay (EA-C1a, EA-POC) as they consist mainly of weathering products of marl; the materials in the marl/moraine or marl/sandstone mixtures (EA-IND, EA-VAL) have more significant sandy fractions. In addition these values should be considered alongside the age of the earthflows; the older the earthflow the higher the percentage of clay and silt due to the weathering of the marls. The grain-size distribution of the debris avalanche is at the interface between earthflow and debris flow. Finally the DF-FAU and DF-RBX deposits are of the classic sandy-silt texture of debris flows with a muddy matrix.

The composition of the matrix allows distinction of the types of movement. This demonstrates a clear connection between the grain-size distribution composition and the type of flow and this result agrees with the flow classification based on the texture of the matrix suggested by [START_REF] Hungr | A review of the classification of landslides of the flow type[END_REF]. The debris flows contain less than 30% of silts and clays. These can therefore be distinguished from earthflows which are much more cohesive. The debris avalanches present an intermediate texture -this is a remarkable feature and should be noted. The artificial mixtures tested (marl, moraine and sandstone) represent the granulometric spectrum of the various phenomena (Fig. 4, Table 1). The percentage of fine material increases with the amount of marl in the mixture. The Atterberg limits, and in particular the plasticity index (I p ), show a clear distinction between earthflow and debris avalanche (I p =12-18) and debris flow (I p =7-8), which exhibits very little plasticity. The liquidity limits, which are between 26% (DF-FAU) and 37% (EA-POC), guided our choice of total solid volume fraction for the rheological tests. It therefore emerges from these geomechanical characteristics that the more muddy and plastic the material the slower the flow velocities appear to be (for a similar slope gradient). The Atterberg limits for the artificial mixtures have low plasticity indices (9<I p <13). The plasticity index varies positively with the increase of the proportion of marl, the least plastic mixtures being UM4, UM3, AM1, AM5 and AM6.

METHODOLOGY AND THE VALIDITY OF RESULTS

The rheological characterisation was conducted alongside rheometric tests (0-400 µm fraction), slump tests and inclined plane tests (0-20 mm fraction) in order to define the behaviour over a wide granulometric spectrum, taking account of the wide variety of total solid volume concentrations observed (Fig. 1).

Rheometrical tests

The fluids subjected to rheometric tests were suspensions of nine materials. The tests were carried out on six total solid volume fractions . A given volume of solid material was added to water and then mixed with a laboratory mixer to 600 t.min -1 for one minute and then to 400 t.min -1 for 30 minutes. The fluids were tested with a Rotovisco RV20 (Haake) rheometer equipped with rough parallel-plates (diameter, 50 mm; roughness, 250 m; thickness of the sheared sample, 2.8 mm) and a Rotovisco RV12 (Haake) equipped with coaxial cylinders (MV-I sensor with a large gap: 0.96 mm). The experimental process, by imposed speed stages, and the precautions taken to limit the effects of disturbances (wall slip, edge effect, fracturing, particle migration, sedimentation) are described in detail in [START_REF] Magnin | Cone-and-plate rheometry of yield stress fluids. Study of an aqueous gel[END_REF]. After a 30-minutes homogenisation period the fluid was introduced into the rheometer and sheared for two minutes at a shear rate of 1000 s -1 , then for 28 minutes at a rate of 100 s -1 . We used the experimental protocols described by [START_REF] Coussot | On the behavior of fine mud suspensions[END_REF] for the parallel-plate geometry and by Locat & Demers (1988) for the coaxial cylinder geometry. The use of two geometries enabled us to investigate an extended range of shear rates. The protocols involve a succession of speed ramps (from 1.87 s -1 to 18700 s -1 for the parallel-plate rheometry and from 0.02 s -1 to 1200 s -1 for the coaxial rheometry) where each shear rate is maintained for a period of 15 seconds. The range of shear rates used in this study is 2 to 3 times higher than that met with this type of flow in the field [START_REF] O'brien | Physical processes, rheology and modelling of mudflows[END_REF]. Three viscoplastic empirical models (Bingham, bi-linear -Locat 1997 -and Herschel-Bulkley) were adjusted to the experimental data [START_REF] Malet | Caractérisation rhéologique des coulées de débris et des laves torrentielles du bassin marneux de Barcelonnette (Alpes-de-Haute-Provence, France)[END_REF] for each material and each total solid volume fraction, taking account of the depth of the hollow observed in most of the experiment with the parallel-plate geometry [START_REF] Coussot | Determination of yield stress fluid behaviour from inclined plane test[END_REF]. The fluids are not thixotropic and the tests were carried out at sufficiently low shearing rates, so the yield stress c should be well defined [START_REF] Coussot | On the behavior of fine mud suspensions[END_REF].

The shape and the range of the flow curves are similar in the range of shearing rates common to both rheometers. The Herschel-Bulkley parameters estimated from the two geometries are identical, in particular for the yield stress [START_REF] Malet | Caractérisation rhéologique des coulées de débris et des laves torrentielles du bassin marneux de Barcelonnette (Alpes-de-Haute-Provence, France)[END_REF]). The differences are in the margin of error specified by Coussot et al. (1996) which indicate that the difference in the shear stress measurements between one geometry and another is between 5 to 15 Pa.

Slump tests

The test comprises filling a cylinder (diameter, 42 mm; height, 82.5 mm) with the fluid (0-20 mm), raising the cylinder rapidly and allowing the material to sink under its own weight. The difference between the initial and the final height of the material and the thickness of the unsheared zone enables the estimation of the yield stress c using the formula devised by [START_REF] Pashias | A fifty cent rheometer for yield stress measurement[END_REF]. In addition to reproducibility, this test enables the characterization of material at higher solid fractions. Making allowances for uncertainties in rheometrical and slump test measurements the agreement between rheometrical data and slump tests results is very good. The "relative error" ( c / c ) fluctuates around a mean value which is close to 0.15, whatever the total solid volume fraction (Fig. 5a).

Inclined plane tests

We used a 4 m-long rectangular channel as an inclined plane; its width reached 0.25 m and its slope could be varied between 4 and 40°. The bottom surface was plywood. During the experiments we noticed that wall slip was negligible. The channel was equipped with a damming system upstream. The system was filled with 0.002 m 3 of material (fraction 0-20 mm) and then a bottom gate was opened quickly so that the fluid could flow downstream. In general the fluid appeared to flow slowly during a period of between 10 and 30 min. The asymptotic depth was measured far enough from the channel tips downstream and upstream for the fluid depth to correspond to the uniform depth. Yield stress was determined using the relation proposed by Coussot et al. (1996), taking into account the asymptotic flow depth and the shape of the lateral levee. Total solid volumes ranging between 0.35 and 0.50 were tested for each material.

The inclined plane tests results were quite reproducible: differences as low as 5% were recorded from one test to another. As for the slump tests, the "relative error" in the inclined plane tests results fluctuates around a mean value close to 0.15 (Fig. 5b).

RESULTS

Behaviour of flow-like landslide deposits (natural mixtures)

All the materials present the same type of shear-thinning behaviour toward high shearing rates. Towards low shearing rates (typically between 0.05 and 1 s -1 ), regardless of the total solid volume fraction, an inflection of the curve is observed for all materials from landslides, while material which has not moved (Mar, Mor, San) shows no inflection. The thixotropy of materials is low [START_REF] Malet | Caractérisation rhéologique des coulées de débris et des laves torrentielles du bassin marneux de Barcelonnette (Alpes-de-Haute-Provence, France)[END_REF].

All the material presented a marked viscoplastic behaviour over the range of shear rates under consideration (Fig. 6a) and this is well represented by a Herschel-Bulkley model (determination coefficients greater than 0.85). All the materials indicate Herschel-Bulkley parameters ( c , ) which increase with the solid volume concentration, with an exponent n between 0.25 and 0.40. The yield stress and the plastic viscosity varied respectively from 1 to 480 Pa and from 0.1 to 150 Pa. We observed a significant differentiation between the materials in the range of total solid volume concentrations from 0.35 to 0.50 while the materials present the same rheological parameters for high concentrations of water.

The master curves of the various materials were calculated in order to determine the rheological characteristics from simple field experiments (Fig. 6b). The shearing rates and the shear stresses were reduced by c . In this way, in a graph ( / c , / c ), all the flow curves of a given deposit are adjusted on a master curve which tends towards a straight slope at high shearing rates and 1 at low shearing rates. In spite of the differences in grain-size distribution it is possible to represent the flow behaviour by a single master curve; the parameters of which depend only on the shear stress and the total solid volume concentrations. The master curve equations are given in Figure 6b. The results of the rheological analyses reflect the variations in velocity observed in the field. It is interesting to note the similarity of EA-IND and DA-COU99a master curves, which show that the EA-IND material corresponds to the debris avalanche's source zone DA-COU99a. Only the superficial formations (weathered marl, sandstone, moraine) present a more marked bilinear behaviour, which can be explained by the fact that these formations have no history of flows.

Behaviour of artificial mixtures and comparisons with natural materials

The behaviour of all the artificial mixtures is also very well represented by a Herschel-Bulkley model (for the range of shear rates tested). The evolution of the rheological characteristics (yield stress, viscosity) as a function of the total solid volume fraction have been placed on a graph in order to compare the behaviour of the natural materials with that of the artificial mixtures (Fig. 7).

It should be recalled that two types of mixture were tested, the first being a group of mixtures (marl and moraine) associated with localised movements on the north-facing slope (EA-POC, EA-IND, EA-C1a, DA-COU99a, b) and the second (marl, moraine and sandstone) associated with movements on the south-facing slope (EA-VAL, DF-RBX, DF-FAU). The natural and artificial mixtures both present the same type of relationship. For total solid volume fractions between 0.30 and 0.60 the yield stress may vary by as much as three times, while the viscosity varies only by twice as much. For both groups the yield stress and the viscosity increase with the percentage of black marl in the mixture and a flow threshold can be observed as soon as the proportion of black marls in the mixture is equal to at least 50%. There are no significant differences, however, in the behavior of artificial mixtures comprising moraine or sandstone. Moraine and sandstone present similar yield stresses for the same total solid fraction, while viscosity increases more rapidly with the solid fraction for moraine.

In the marl-moraine-sandstone mixtures the hypothetical mixtures with the greatest susceptibility to flow are all characterised by a moderate proportion of black marl in the mixtures (<50%). For the marl-moraine mixtures the UM4 mixture (40% moraine, 60% marl) presents the greatest susceptibility to flow. The behaviour of this mixture is very close to the behaviour of the EA-IND material (material from the DA-COU99a debris avalanche (Malet et al. in press).

The change in behaviour of the mixtures, depending on the constituents of the materials, relates particularly to the viscosity. A distinction in the mixtures can be observed, depending on the volume concentration. This is the case, for example, in the AM5 mixture which present a very low viscosity for = 0.35, but it is much greater (higher than for the other mixtures) for total solid volume fractions over 0.45.

CONCLUSION

This study has demonstrated that materials from flow-like landslides which develop into earthflows, debris avalanches and debris flows triggered in black marl present a rheological behaviour in simple shear which can be described by a Herschel-Bulkley model where the geomechanical parameters increase with the solid volume concentration. The results obtained by various methods (rheometrical tests, slump tests, inclined plane tests) are satisfactory, because the differences noted for the yield stress and the viscosity were small (< 20 %). The similarities in the behaviour of materials enabled us to propose normalized master curves. The differences in the plasticity index for each material appear less obvious than the rheological parameters which distinguish two groups of materials according to the percentage of marl and moraine and/or sandstone. Nevertheless a classification of materials according to the flow mode using viscosity as parameter may be observed. The material which flowed fastest presented the lowest viscosity, regardless of the volume concentration; the materials in situ are the most viscous (Mar). A mixture of 40 % of moraine (or sandstone) and 60 % of marls presented the weakest yield stress and viscosity for field-measured solid concentrations. The presence of sandstone or moraine associated with marl did not change the behaviour of the mixture as a whole.

Moreover the key difference between muddy earthflows, muddy debris avalanches and muddy debris flows appears to be controlled by the fluid phase of the flow. The fluid phase consists of air, water, clay, silt and sand. The fluid phase in the debris avalanche and debris flow apparently has lower strength and viscosity in comparison to that of the earthflow. Nevertheless all flows may be described as macro-viscous flows.

These relations between yield-stress, viscosity and the textural properties of the material may be useful for identifying potential source zones in flow-like landslides areas. It may be useful in torrent hazard control by mapping flow susceptibility in terms of the grain-size distribution of the surficial formations.

Figure 1 .

 1 Figure 1. Position of the flow-like fine-grained landslides occurring in black marls in the conceptual rheological classification (function of fine content and total solid volume fraction) proposed by Coussot & Meunier (1996).

Figure 2 .

 2 Figure 2. Morphological sketch of the Barcelonnette basin showing locations of the landslides deposits and extension of the landslide source material.

Figure 3 .

 3 Figure 3. Morphological characteristics of the deposits. Schematic vertical stratigraphic sections of: (a) the EA-C1a earthflow deposit, (b) DA-COU99a debris avalanche deposit, and (c) DF-FAU debris flow deposit. Photographs of the stratigraphy at (d) the toe of the DA-COU99a debris avalanche and (e) DF-FAU debris flow.

Figure 4 .

 4 Figure 4. Textural composition of (a) landslide deposits and (b) artificial mixtures. Dotted lines represent the classification proposed by Hungr et al. (2001).

Figure 5 .

 5 Figure 5. Error on yield stress determination using slump tests (a) and inclined plane test (b) as a function of solid fraction for each landslide deposits. The yield stress determined by the parallel-plate rheometry ( c ref ) is taken as reference.

Figure 6 .

 6 Figure 6. Fitting curve for the (a) COU99 material and (b) master curves of the landslides and in situ deposits.

Figure 7 .

 7 Figure 7. Variation of the rheological properties (yield stress and viscosity) as a function of the total solid volume, using the rheometrical tests, the slump tests and the inclined plane tests. Figure 7a represents the landslide material and artificial mixtures involving marl and moraine, Figure 7b represents the landslide material and artificial mixtures composed of marl, moraine and sandstone.

Table 1 .

 1 Geomechanical characteristics of flow-like landslides and artificial mixtures.

		Grain-size						Consistency	Density
		Sand Silt Clay Gravel Q 25	Q 50	Q 75	W P W L I P	sat
		%	%	%	%	mm	mm	mm	kg.m -3
	Landslides or in situ deposits
	EA-C1a	25	22	15	38	0.0170 0.600 10.0	16 32 16	2140
	EA-IND	31	29	10	30	0.0023 0.400 2.50	17 33 16	1790
	EA-VAL	32	20	11	37	0.0016 0.975 4.00	19 31 12	1730
	EA-POC	20	42	9	29	0.0009 0.019 1.91	20 37 17	1830
	DA-COU99a	35	24	9	32	0.0018 0.422 2.50	15 30 15	1700
	DA-COU99b	27	30	10	33	0.0021 0.436 3.75	16 29 13	1670
	DF-FAU	30	20	6	44	0.0190 0.800 8.30	19 26 7	1630
	DF-RBX	36	16	10	38	0.0190 0.455 4.30	21 29 8	1650
	Mor	38	12	9	41	0.0205 0.435 4.40	19 30 11	1590
	San	37	11	5	47	0.0315 0.550 6.75	17 25 8	/
	Mar	37	14	12	37	0.0220 0.400 4.20	21 29 8	1710
	Artificial mixtures
	UM2	29	35	10	25	0.0015 0.270 2.45	18 31 13	/
	UM3	30	29	9	32	0.0126 0.312 2.87	17 28 11	/
	UM4	31	26	8	35	0.0208 0.421 3.19	16 26 10	/
	AM1	31	17	7	45	0.0218 0.765 7.55	18 27 9	/
	AM3	38	18	9	35	0.0156 0.603 6.14	21 33 12	/
	AM5	40	12	7	41	0.0297 0.715 7.16	16 27 11	/
	UM6	38	11	8	43	0.0302 0.726 7.08	18 28 10	/

This work was supported by grants from the French Ministry of Research under the ACI CatNat contract MOTE (MOdélisation, Transformation, Ecoulement des coulées boueuses dans les marnes) and from the CNRS under the INSU-PNRN contract ECLAT (Ecoulement, Contribution de LAves Torrentielles dans les basins versants marneux). Contribution INSU N° 326. Contribution EOST N° 2002.19-UMR 7516. Special thanks are due to M. Nelson for the reviewing of the English version of the paper. The authors wish to thank Prof. A.M. Johnson (Purdue University) for his valuable comments. ACKNOWLEDGEMENTS 11


Keywords: rst2023-rennes:486659 plastiques, fleuves éphémères, typologie et distribution des déchets, suivi et quantification de la pollution, OSPAR

est destinée au dépôt

Si la pollution plastique en domaine marin est avérée et suivie depuis plus de 50 ans, le principal vecteur de transfert des déchets se trouve être les cours d'eau. La Fiumara d'Amendolea, située au sein du parc national de l'Aspromonte en Calabre (Italie), est une rivière à régime intermittent et éphémère (IRES -Intermittent Rivers and Ephemeral Streams ). La particularité de ce régime hydrodynamique a nécessité la mise en place d'un nouveau protocole de quantification et de suivi de la pollution plastique à l'échelle du profil de dépôt. Afin d'intégrer le premier réservoir (Sink ) du domaine marin, un suivi a donc été réalisé sur la plage de Condofuri, en périphérie de l'embouchure de la Fiumara. Le travail présenté ici concerne la partie amont du système, en complément de l'étude d'Heckmann et al.

Les paramètres analysés sont la végétation, la topographie, les structures morpho-sédimentaires, la quantité/typologie de déchets comptabilisés le long du profil de la Fiumara.

Les premiers résultats montrent que la source des déchets est directement liée aux dépôts individuels le long des routes et des ponts, particulièrement à proximité des zones urbaines, ainsi qu'aux décharges sauvages dans la Fiumara elle-même. La distribution des plastiques se caractérise par une diminution rapide des quantités de macro-plastiques dans le lit de la rivière à partir des zones sources principales. Au sein de la Fiumara, les éléments plastiques sont majoritairement piégés dans les barres axiales, les barres latérales et les chenaux inactifs, avec une distribution naturellement également fonction de l'évolution du profil topographique. La végétation ne semble jouer un rôle significatif dans le piégeage des plastiques qu'au niveau des barres latérales. Notre étude souligne l'impact des pratiques anthropiques sur l'environnement. Inspiré des récents développements autour du protocole OSPAR, ce travail fait un premier état des lieux (T0). La répétition de la nouvelle méthodologie proposée permettra de mettre en place des indicateurs pertinents pour le suivi statistique de la pollution plastique. A terme ce travail fournira une quantification de la contribution d'une rivière à régime intermittent et éphémère dans l'export de plastique jusqu'au domaine marin.

  Distribution des plastiques le long du continuum terre-mer : exemple de la Fiumara d'Amandolea (Calabre, Italie). Secteur Amont : du bassin versant à la confluence principale Adrien Embs-Schneider, Victoria Heckmann, Max Rousselet, Cyril Doulin, Constance De-Ghaisne-De-Bourmont, Julien Bailleul, Pascal Barrier, Sébastien Potel, Sebastien Rohais To cite this version: Adrien Embs-Schneider, Victoria Heckmann, Max Rousselet, Cyril Doulin, Constance De-Ghaisne-De-Bourmont, et al.. Distribution des plastiques le long du continuum terre-mer : exemple de la Fiumara d'Amandolea (Calabre, Italie

hal-04323763 Distribution des plastiques le long du continuum terre-mer : exemple de la Fiumara d'Amandolea (Calabre, Italie). Secteur Amont : du bassin versant à la confluence principale.

  Adrien Embs-Schneider * 1 , Victoria Heckmann 1 , Max Rousselet 1 , Cyril Doulin 1 , Constance De Ghaisne De Bourmont 1 , Julien Bailleul 1 , Pascal Barrier 1 , Sébastien Potel 1 , and Sébastien Rohais 2 1 U2R 7511, Basins-Reservoirs-Resources (B2R), UniLaSalle -University of Picardie Jules Verne, 60026, Beauvais, France. -Institut Polytechnique UniLasalle -France 2 IFP Energies nouvelles -IFP Energies Nouvelles, IFP Energies Nouvelles -France

). Secteur Amont : du bassin versant à la confluence principale. 28e Réunion des Sciences de la Terre, Géosciences Rennes; SGF, Oct 2023, Rennes (FR), France.


Antonio Fernández

GSyC Anta Ladyr

Miguel A Mosteiro
email: mosteiro@cs.rutgers.edu

Christopher Thraves
email: christopher.thraves-caro@irisa.fr

An Early-stopping Protocol for Computing Aggregate Functions in Sensor Networks

Keywords: Sensor networks, Aggregate computation, Earlystopping algorithm, Failure model, Average computing

In this paper, we study algebraic aggregate computations in Sensor Networks. The main contribution is the presentation of an early-stopping protocol that computes the average function under a harsh model of the conditions under which sensor nodes operate. This protocol is shown to be time-optimal in presence of unfrequent failures. The approach followed saves time and energy by relying the computation on a small network of delegate nodes that can be rebuilt fast in case of node failures and communicate using a collisionfree schedule. Delegate nodes run simultaneously two protocols, namely, a collection/dissemination tree-based algorithm, which is shown to be optimal, and a mass-distribution algorithm. Both algorithms are analyzed under a model where the frequency of failures is a parameter. Other aggregate computation algorithms can be easily derived from this protocol. To the best of our knowledge, this is the first optimal early-stopping algorithm for aggregate computations in Sensor Networks.

I. INTRODUCTION

A Sensor Network is a simplified abstraction of a large monitoring infrastructure, formed of sensor nodes (or sensors) that create a radio communication network from scratch. Each sensor node is equipped with communication, processing, and sensing capabilities. Nodes can collaborate to process the sensed data but, due to unreliability, a monitoring strategy can not rely on individual sensors data. Instead, the network should use aggregated information from groups of sensor nodes [START_REF] Boyd | Randomized gossip algorithms[END_REF], [START_REF] Chen | Robust computation of aggregates in wireless sensor networks: distributed randomized algorithms and analysis[END_REF], [START_REF] Kempe | Gossip-based computation of aggregate information[END_REF]. Popular examples of relevant aggregate functions are the computation of the maximum or the average of some variable sensed by the nodes in some area. Nevertheless, any algebraic aggregate function of the sensed input-values is also of interest.

Typically, in Sensor Networks, the aggregated information is collected by a small number of distinguished nodes This research was supported in part by Comunidad de Madrid grant S-0505/TIC/0285; MICINN TIN2008-06735-C02-01 and MEC PR2008-0015; EU Marie Curie European Reintegration Grant IRG 210021; NSF grants CCF0621425, CCF 05414009, CCF 0632838; and French ANR Masse de Donnèes project ALPAGE.

The first author worked partially while on leave at Bell labs.

called sinks. Given that the information has to be collected to be of any use, a sink node is generally assumed to be failure-free, and to have access to more resources than a regular sensor node. For some applications, it might be useful to compute aggregations restricted to specific areas of the network, and to route the result of those computations to the sink nodes. However, lack of position information and limitations on storage space prevents area delimitation and routing. Hence, for the most restrictive and general scenario, only aggregation among all nodes is feasible. Additionally, the result must be propagated to all nodes in the network to guarantee that sink nodes receive it. Algebraic aggregate functions are well defined. However, the implementation of such computations in practice, and specially in the harsh Sensor Networksetting, has to deal with various issues that make even the definition of the problem difficult. First, the input-values at each node might change over time. Therefore, it is necessary to fix to which time step those input-values correspond. This fact implies that any protocol has to achieve some form of global synchronization. Second, the multi-hop nature of Sensor Networks makes impossible to completely aggregate these values in one single time step. Hence, arbitrary node failures make the design of protocols challenging. Furthermore, it has been shown [START_REF] Bawa | Estimating aggregates on a peer-to-peer network[END_REF] that the problem of computing an aggregate function among all nodes in a network where some nodes join and leave the network arbitrarily in time is intractable. The only limit on adversarial failures that is customarily used in the Sensor Networks literature is a guarantee on connectivity among active nodes in each time step. An active node at time t is a node that is up and running at time t. However, for any Sensor Network, it is easy to give a node-failure schedule that maintains such connectivity but partitions the network.

The topic of this paper is the efficient computation of aggregate functions on a Sensor Network. The efficiency is measured here in two dimensions: time and energy. The energy efficiency is evaluated in terms of number of transmissions, as customary in the Sensor Networks literature. These efficiency metrics are strongly influenced by colli-sions, especially because no collision detection mechanisms are available in this setting. The response of the algorithm to sensor failures is also an important characteristic of any protocol. Some algorithms have to restart in presence of failures, while others simply compute an aggregated value that may be only an approximation.

Hierarchical aggregate computations where the few compute for the many have been studied. The most frequent hierarchical approach is to construct a tree that spans all nodes in the network [START_REF] Kollios | Robust aggregation in sensor networks[END_REF], [START_REF] Madden | Tag: a tiny aggregation service for ad-hoc sensor networks[END_REF]. The spanning tree is used to collect and gradually aggregate the input-values at each level of the tree, relying the partial results to the root. Then, the root computes the overall aggregate result and distributes it down the tree. Due to memory size limitations, it might not be possible to implement these techniques unless the degree of each node in the tree is bounded. Another drawback of this approach comes from its rigid structure. If an internal node of the tree fails during the computation, the tree is partitioned, and the result, if computed, may not consider the input-values of an unbounded number of nodes. Furthermore, these nodes may never obtain the result.

Non-hierarchical computations have also been studied [START_REF] Boyd | Randomized gossip algorithms[END_REF], [START_REF] Chen | Robust computation of aggregates in wireless sensor networks: distributed randomized algorithms and analysis[END_REF], [START_REF] Kempe | Gossip-based computation of aggregate information[END_REF]. The approach of choice is to aggregate the information at every node of the network in a mass-distribution fashion as in load balancing algorithms [START_REF] Ghosh | Dynamic load balancing by random matchings[END_REF], [START_REF] Rabani | Local divergence of markov chains and the analysis of iterative load-balancing schemes[END_REF]. In this manner, all nodes arrive at the final result concurrently. A potential shortcoming is the energy consumption overhead of having all nodes transmitting and computing. Furthermore, the fact that all nodes communicate with other nodes during all the algorithm greatly increase collisions with the consequent time and energy cost. On the other hand, nonhierarchical approaches are more resilient to failures.

These arguments indicate that both pure approaches, hierarchical and non-hierarchical, may have advantages and shortcomings. The algorithm presented in this paper benefits from the good properties of both approaches interleaving them. If failures are not too frequent, the tree-based algorithm provides a result with low time and energy complexity. If the frequency of failures prevents the tree-based computation from finishing, the mass-distribution algorithm will compute an approximation of the result. Hence, the combined algorithm is early stopping.

In order to reduce collisions and energy consumption, a two-level hierarchy of nodes is used. The actual computation is done by a small set of nodes, called delegate nodes, that collect the sensed input-values from the non-computing nodes, called slug nodes. Model. Sensor nodes are expected to be deployed at random in large quantities over an area of interest. Hence, we model the connectivity of nodes with the Geometric Graph Model, noted as G n,r , where n nodes are deployed at random in R2 in a unit area, and an edge between two nodes exists if and only if they are located at an Euclidean distance of at most a parameter r. We further characterize the area of deployment assuming that, if expanded in all directions by a distance of r, the new area would not be asymptotically bigger. As customary in the Sensor Networks literature, we assume that nodes are deployed densely enough to ensure network connectivity and sensing coverage even under failures. Nevertheless, we do not restrict ourselves to an specific spatial distribution. Given that we will use a radius of transmission reduced by a constant factor in some algorithms, we further assume that such density is adjusted accordingly by a constant factor to still accomplish connectivity and coverage using the reduced radius. This assumption does not change the asymptotic cost. A straightforward application of the bound in [START_REF] Gupta | Critical power for asymptotic connectivity in wireless networks[END_REF] for uniform nodedistribution gives a bound of r ∈ Ω( log n/n) to achieve connectivity with high probability (w.h.p.) 1 , even with nonuniform node-distribution, since the radius cannot be smaller if some areas have smaller density of nodes.

Regarding models of sensor node constraints, we use the following model, a relaxation (regarding failures) of the Weak Sensor Model [START_REF] Farach-Colton | Bootstrapping a hop-optimal network in the weak sensor model[END_REF]. The communication among neighboring nodes is through broadcast on a shared channel. Time is assumed to be slotted, and each transmission occurs in a given slot (or step). The length of a slot is the time to transmit one message. All nodes have the same clock frequency, but no global synchronizing mechanism is assumed. A node receives a message in a slot if and only if exactly one of its neighbors transmits in the slot. There is no collision detection mechanism available and the channel is assumed to have only two states: single transmission and silence/collision. A sensor node can not receive and transmit in the same slot. Nodes are woken up by an adversary, perhaps at different times. Sensor nodes may store only a constant number of O(log n) bit words 2 . We assume that sensor nodes can adjust their power of transmission to only a constant number of levels. Nodes are assumed to have limited life cycle. Other restrictions include: short transmission range (r << 1), only one shared channel of communication, and lack of position information.

As pointed out before, the problem can not be solved under arbitrary adversarial failures. We consider a scenario where, upon starting the algorithm, some nodes fail due to lack of power supply or any other event such that, as a consequence, the node stops participating in the algorithm and all the information stored in its memory is lost. A node may recover from a failure later (for instance, after replenishing its battery) but no information was kept in its memory. Thus, we assume that a node that recovers after a failure has to start the protocol from scratch. The rate or time at which failures occur is modeled as follows. Given two parameters f ≥ 0 and T > 0, it is assumed that the number of failures is bounded by f and the time between any pair of consecutive failures is at least T time steps.

Regarding the assignment of input-values, it is assumed to be adversarial, i.e., we do not assume any specific distribution of input-values among nodes but just a worstcase scenario. Without loss of generality, we assume those values to be positive. We also assume that nodes are assigned a unique ID of O(log n) bits also adversarially and they start "knowing" only the total number of nodes n. However, the deployment of nodes is not an uncontrolled experiment. So, information about the resultant topology can be introduced at a sink node after deployment. The presence of one distinguished node called sink is assumed. The sink does not fail and "knows" tight bounds on the maximum degree ∆ and on the maximum diameter during the whole execution of the algorithm D. Related Work. There is a large body of literature on aggregate computations in sensor networks that includes both, theoretical and experimental work. Many of these results are obtained under models that do not include important restrictions such as, limited memory size [START_REF] Kollios | Robust aggregation in sensor networks[END_REF], lack of position information [START_REF] Gupta | Scalable fault-tolerant aggregation in large process groups[END_REF] and limited range of transmission [START_REF] Kempe | Gossip-based computation of aggregate information[END_REF].

A hierarchical approach to aggregate information is presented in [START_REF] Gupta | Scalable fault-tolerant aggregation in large process groups[END_REF]. The solution proposed defines a tree structure that requires node location information to carry out the aggregation at a cost of O(n log 2 n) messages in O(log 2 n) rounds. Contention resolution and other communication issues are assumed to be resolved by an underlying protocol.

Non-hierarchical gossip-based protocols for average computations in arbitrary networks were studied [START_REF] Boyd | Randomized gossip algorithms[END_REF], [START_REF] Kempe | Gossip-based computation of aggregate information[END_REF]. All gossip-based algorithms are characterized in [START_REF] Boyd | Randomized gossip algorithms[END_REF] with a matrix that models how the algorithm evolves sharing values in pairs iteratively. It is shown there that, given a value ǫ > 0, and an arbitrary network of n nodes, where each node i holds a value ν i and all nodes start synchronously; then, with probability at least 1 -ǫ, in O(log n + log(n/ǫ)/(1λ max (( I + P )1/2))) rounds, each node i running a gossipbased algorithm characterized by the matrix P , computes a value

ν ′ i such that i (ν ′ i -ν) 2 / i ν 2 i ≤ ǫ 2
, where ν is the average i ν i /n and λ max (•) is the second largest eigenvalue. Additionally, an algorithm that takes advantage of the broadcast nature of radio networks is included in [START_REF] Kempe | Gossip-based computation of aggregate information[END_REF] giving similar bounds. In these papers no details about collision resolution are included, and the algorithms presented require ω(1) memory size.

Another mass-distribution algorithm was presented in [START_REF] Chen | Robust computation of aggregates in wireless sensor networks: distributed randomized algorithms and analysis[END_REF], although relying on a different randomly chosen local leader in each round to perform such distribution. It is shown that, given a value ǫ > 0, and a Sensor Networkof n nodes with underlying graph G with algebraic connectivity a(G) 3 , where each node i holds a value ν i 3 A characterization of the deployment topology by the second smallest eigenvalue of the Laplacian matrix of G. and all nodes start synchronously, with probability at least

1 -ǫ 2 / i (ν i -ν) 2 , in O(∆ 3 log( i (ν i -ν) 2 /ǫ 2 )/a(G))
rounds, each node i running the algorithm computes a value ν ′ i such that |ν ′ i -ν| ≤ ǫ ∀i, where ν is the average i ν i /n. If the deployment topology is known in advance, a parameter probability p can be tuned to improve that bound to O(∆ log( i (ν i -ν) 2 /ǫ 2 )/pa(G)) rounds. The expected number of transmissions is bounded by O(n∆ 2 log(( i (ν iν) 2 )/ǫ 2 )/a(G)), again, aside from communication and synchronization overhead. This result was extended recently [START_REF] Chen | Analysis of distributed random grouping for aggregate computation on wireless sensor networks with randomly changing graphs[END_REF] to networks with a time-varying connection graph.

Results. An early-stopping protocol that computes the average function in Sensor Networks under a harsh model of sensor restrictions is presented in this paper. Although this protocol builds incrementally over known techniques, the careful combination of them in the restricted Sensor Networksetting is not trivial. It is shown here that, in presence of f non-frequent failures, w.h.p., the protocol returns a value and terminates in O(∆ + D + f log 2 n) steps which, given that D and ∆ cannot be both asymptotically smaller than a polynomial, is optimal if f ∈ o(n c ) for any c ∈ O(1); and that the overall number of transmissions is in O(n((f + 1) log n + ∆/ log n + log ∆)) in expectation. In presence of frequent failures, nodes running the protocol still converge to some result, whose accuracy with respect to the average depends on the failure model and the distribution of input-values. The aim in this case is to obtain a result that does not diverge significantly from one node to another. More precisely, it is shown that, w.h.p., the protocol takes O(∆ + D + (f + log(1/ε) + log(ν max /ν min ))/Φ 2 min ) time steps and O(n(log n + ∆/ log n + log ∆ + (f + log(1/ε) + log(ν max /ν min )))/Φ 2 min log n) expected transmissions, where ε > 0 is the maximum relative error (i.e., the maximum relative difference between two results), Φ min is the minimum conductance [START_REF] Jerrum | Conductance and the rapid mixing property for markov chains: the approximation of permanent resolved[END_REF] of the network underlying the Markov chain characterizing the algorithm, and ν max and ν min are the maximum and minimum input-values respectively.

A time-optimal protocol to compute the maximum function can be easily derived from the average protocol by flooding the delegates network with the maximum inputvalue seen so far. Other aggregate functions such as the sum, quantiles, or count, can be computed using a protocol for average without extra cost as described in [START_REF] Chen | Robust computation of aggregates in wireless sensor networks: distributed randomized algorithms and analysis[END_REF], [START_REF] Kempe | Gossip-based computation of aggregate information[END_REF].

To the best of our knowledge, this is the first optimal early-stopping algorithm for aggregate computations in Sensor Network.

II. LOWER BOUND

A lower bound on the time steps needed to compute an aggregate function in a Sensor Networkis established in the theorem below. For the sake of brevity, the details of the proof that uses the adversarial assignment of input-values and the topology, is left to the full version of this paper [START_REF] Fernández Anta | An early-stopping protocol for computing aggregate functions in sensor networks[END_REF]. Let F : R n → R, n ∈ N be an algebraic aggregate function over n real numbers. We say that F is one-node sensitive if, for any choice of values ν 1 ∈ R n , there exists another choice of values ν 2 ∈ R n such that ν 1 and ν 2 differ only in one value, and F( ν 1 ) = F( ν 2 ). Given a Sensor Networkof n nodes, where each node is assigned an input-value, we say that a protocol to compute an aggregate function over these values is assignment oblivious if it is independent of the specific assignment of input-values.

Theorem 1. Given a Sensor Networkof n nodes, where D is the diameter of the network and ∆ the maximum degree, under the model described in Section I, and independently of randomization and failures, Ω(D + ∆) time steps are needed in order to compute a one-node-sensitive algebraic aggregate function using an assignment-oblivious protocol.

III. UPPER BOUNDS

The computation of aggregate functions is carried out by a protocol following a template called Aggregate Computation Scheme. A key factor of our approach is the inclusion of a preprocessing phase that defines a delegate-slug hierarchy and a schedule of transmissions to avoid collisions. Such preprocessing is asynchronous and uses time slots that are not used in the Aggregate Computation Scheme. Hence, it is also used as a maintenance algorithm in case of node failures because nodes running it do not collide with nodes running the main part. For the sake of clarity, both parts, preprocessing and the main procedure, are described separately omitting these details.

A. Preprocessing and Maintenance

The preprocessing/maintenance algorithm includes two phases. Due to the memory size limitations, in the first phase delegate nodes are defined so that each delegate node is within range of Θ(1) delegates. Additionally, a second phase establishes schedules of transmissions so that each group delegate-slugs can communicate without colliding with neighboring groups. The first and second preprocessing phases can be implemented as in [START_REF] Farach-Colton | Sensor network gossiping or how to break the broadcast lower bound[END_REF]. Further details can be found in the full version of this paper [START_REF] Fernández Anta | An early-stopping protocol for computing aggregate functions in sensor networks[END_REF]. The following upper bound on the number of delegate nodes can be proved using that the hexagonal lattice is the densest of all possible plane packings [START_REF] Fejes-Tóth | über einen geometrischen satz[END_REF], that the radius lower bound to achieve connectivity w.h.p. under uniform distribution of nodes is r ∈ Ω( log n/n) [START_REF] Gupta | Critical power for asymptotic connectivity in wireless networks[END_REF], the assumption of complete coverage, and the assumption regarding the area of deployment.

Remark 2. Given a Sensor Networkof n nodes deployed at random to ensure connectivity and coverage over a unit area such that, if expanded in all directions by r, the expanded area is still in O(1), after running the first phase of preprocessing as described, there are O(n/ log n) delegate nodes.

The following lemma establishes formally the efficiency of these phases. Further details can be found in [START_REF] Farach-Colton | Bootstrapping a hop-optimal network in the weak sensor model[END_REF]- [START_REF] Farach-Colton | Sensor network gossiping or how to break the broadcast lower bound[END_REF]

B. The Aggregate Computation Scheme

After preprocessing, local synchronism, collision detection among slugs and their delegates, and non-colliding transmission schedules among delegates are available, because nodes use only reserved time slots. We omit in the analysis this constant factor overhead for clarity. In the Aggregate Computation Scheme, a slug node uses a radius of transmission αr, whereas a delegate node uses βr. Also for clarity, the scheme is described assuming that nodes do not fail and later this assumption is removed. In order to obtain worst-case bounds, we assume that all nodes are active.

Let the set of delegate nodes and the set of slug nodes defined in preprocessing be M and S respectively. For each slug node i, denote the set of its delegates as M (i). For each delegate node j, denote the subset of delegate nodes located at one-hop of j as N (j). Each node j ∈ M keeps track of its delegate-neighborhood N (j). Furthermore, node j updates N (j) online by keeping track of the beacon messages of its delegate-neighbors. This bookkeeping can be done by storing the IDs of the neighboring delegates, because |N (j)| ∈ Θ(1). For each node k in the network, denote the input-value as ν k .

The Aggregate Computation Scheme includes the following four phases. TRIGGER: the sink node broadcasts (τ 1 , D, ∆), where τ 1 is the time slot to measure the inputvalues, D is the diameter of the network and ∆ the maximum degree, thus, synchronizing the computation; COL-LECTION: delegate nodes aggregate slugs input; COMPUTA-TION: delegate nodes compute the aggregate function; and DISSEMINATION: delegate nodes distribute the result. The details of the implementation of each of these phases follow. Trigger Phase. The TRIGGER phase can be implemented as follows. Upon receiving the message, delegates flood the network of delegates with the message using only reserved slots. Each delegate node forwards the message broadcasted, including the ID of the node from which it has received the message first. In this manner, a BFS spanning tree among the delegate nodes is obtained at the same time that the trigger signal is disseminated in preparation for our treebased algorithm. Due to the broadcast nature of a Sensor Network, while passing the message among delegates, slug nodes receive also τ 1 . Since only reserved slots are used, the total time taken by this phase is in O(D) and using the Remark 2 the total number of transmissions in this phase is in O(n/ log n). Hence, τ 1 is tuned to ensure that active nodes receive this message on time to start the COLLECTION phase. For nodes becoming active late, upon becoming active, nodes run the preprocessing phase, which includes an initial waiting period. Nodes in this period that hear that the computation has already started do not join the computation, although they do complete the preprocessing phase in preparation for future queries. The following lemma establishes formally the bounds of this phase.

Lemma 4. After the sink node starts disseminating the trigger message, all delegate nodes have received the message within O(D) steps and the overall number of transmissions is O(n/ log n).

Collection Phase. At time τ 1 , nodes start running the COLLECTION phase using the input-values at that time step. Slug nodes communicate in this phase using the following procedure. In each round, slug nodes choose uniformly at random a slot within a window of slots to transmit their messages. Starting with a window of size c 1 ∆, the window size is repeatedly halved in each round down to c 2 log n, where c 1 > 0 and c 2 > 0 are constants chosen appropriately. After that, a final round of c 3 log 2 n steps where nodes repeatedly transmit with probability c 4 / log n is included. Again, c 3 > 0 and c 4 > 0 are constants chosen appropriately [START_REF] Farach-Colton | Sensor network gossiping or how to break the broadcast lower bound[END_REF]. We refer to this protocol as the windowed protocol. Further details can be found in the full version of this paper [START_REF] Fernández Anta | An early-stopping protocol for computing aggregate functions in sensor networks[END_REF].

Each slug node i ∈ S begins the COLLECTION phase choosing one of its delegates to pass its input-value, to ensure that each input-value is used exactly once in the computation. Using the windowed protocol, each slug node transmits a message to the delegate chosen. The message transmitted contains ν i and the ID of the delegate chosen. Given the availability of delegate acknowledgements, a delegate receives exactly one input-value per slug node.

Each delegate node j ∈ M running the COLLECTION phase maintains two magnitudes that we call sum σ j and weight ω j . Each delegate node j initializes the sum σ j = ν j and the weight ω j = 1. Upon receptions, delegates update these values appropriately. Sum and weight values are polynomially upper bounded so memory restrictions are not violated. The following lemma establishes formally the correctness and efficiency of the COLLECTION phase. The proof uses well-known techniques and the details are left to the full version of this paper [START_REF] Fernández Anta | An early-stopping protocol for computing aggregate functions in sensor networks[END_REF] for brevity. Lemma 5. Let V be the set of n nodes in a Sensor Network, ν i be the input-value of node i ∈ V , and let M be the set of delegate nodes. There exists a τ 2 ∈ O(∆ + log 2 n) such that, after running the COLLECTION phase with that τ 2 , the following holds. (i) V has been partitioned in |M | disjoint subsets {V 1 , V 2 , . . . , V |M | } and each node j ∈ M holds two values σ j and ω j such that, ∀k ∈ {1, . . . ,

|M |}; ∀j ∈ M : j ∈ V k ⇒ (σ j = i∈V k ν i ∧ ω j = |V k |), w.h.

p. (ii) The time taken by the algorithm is in

O(∆ + log 2 n). (iii)
The number of transmissions of delegate nodes during this phase is in O(n(∆/ log n + log n)), and the expected number of transmissions of slug nodes during this phase is in O(n(log n + log ∆)).

Computation and Dissemination Phases. Upon completion of the COLLECTION phase, slug nodes standby waiting for the delegates to compute in the COMPUTATION phase and send back to them the result in the DISSEMINATION phase. In the following sections, the two approaches used -treebased and mass-distribution -are described separately for clarity, although they are run simultaneously in two different slots reserved to communicate among delegates. If the result of the tree-based computation is obtained, the massdistribution-based computation is just stopped. Otherwise, the mass-distribution algorithm continues until some result is returned.

Before moving to the details of the analysis of both algorithms, recall that thanks to the delegate/slug hierarchy, a failure of a slug node after passing its input-value does not impact the protocol neither in time nor in correctness. On the other hand, if a unique copy of an input-value is lost before being passed to other nodes that value is inevitably lost. Furthermore, given the shared nature of the channel, no algorithm can guarantee that all input-values are passed to some other node in less than ∆ time steps under adversarial failures. Hence, we consider from now on slug failures only during the COLLECTION phase.

If a delegate node fails early enough before the time slot in which input-values are obtained, the lack of its beacon triggers the execution of the preprocessing phase by its slug nodes and this failure does not impact the protocol. On the other hand, if the failure occurs at a time slot such that its slug nodes do not have enough time to elect new delegates and receive the synchronization message of the TRIGGER phase, its slug nodes do nothing. Given that most of the slug nodes have more than one delegate, this failure does not impact the protocol except in some marginal cases (boundary nodes or multiple neighboring delegates failure). Given that a delegate failure during the COLLECTION phase has the same impact as a failure of a delegate during the first round of the COMPUTATION phase. Thus, we consider from now on only delegate failures during the COMPUTATION phase. Tree-based Algorithm. For the sake of clarity, we describe first the algorithm assuming that nodes are activated early enough to receive the trigger, stay active long enough to receive the result of the computation, and do not fail. The slug and delegate failures described in Section III-B are considered afterwards. The tree-based algorithm is well known and simple to describe. Once a rooted tree is built, it includes three steps: the root broadcasts a query to all nodes in the tree, then nodes convergecast the aggregated input-values to the root and finally the root computes the function and broadcasts back the result to all nodes in the tree. The details follow.

While broadcasting the time slot τ 1 of the input-values that have to be used in the computation, in the TRIGGER phase, a BFS rooted tree of constant degree is built among delegate nodes by making each delegate node keep track of its tree neighbors. The root of such a tree is either the sink node (if delegate) or a delegate node at one hop of the sink node (if slug). Without loss of generality we assume it is the sink node. At τ 1 , all nodes run the COLLECTION phase using the windowed protocol as described. Then, at time τ 1 + τ 2 , the COMPUTATION phase starts. In this phase, each delegate node i aggregates the input-values by passing to its parent in the tree the average and weight of the subtree rooted at i. Thus, the root of the tree receives the average and weight of the subtrees rooted at its children and computes the total average. Finally, in the DISSEMINATION phase, the root node floods the network of delegates with the result which in turn is disseminated to the slug nodes by each delegate node upon receiving it.

In order to handle failures, the tree-based algorithm is enhanced as follows. Upon defining the tree, each delegate node broadcasts to its slugs its view of the tree topology, i.e., its parents and children, and the slugs store that information. Since the tree has constant degree, such a bookkeeping is feasible. Slug nodes detect the failure of their chosen delegate due to the lack of beacon. In presence of such a failure, slug nodes compete to replace the missing delegate running the preprocessing phase at a O(log 2 n) cost (Lemma 3). Given the assumption of coverage even under failures, there must exist enough slug nodes to replace the failed delegate. Due to the geometry, more than one of them may become delegate but only a constant number. Upon becoming delegates and using the view of the tree broadcasted by the failed delegate, the new delegates repair the structure locally at a O(1) cost and continue with the computation appropriately. The details are omitted for brevity. Then, if the time between failures is large enough, this procedure repairs the structure successfully. The following theorem shows the overall efficiency of the Aggregate Computation Scheme when the tree-based algorithm is used. Theorem 6. Given a Sensor Networkwith a set of nodes V running the Aggregate Computation Scheme as described, where ∆ is a tight upper bound on the maximum number of neighbors of any node, D is a tight upper bound of the diameter of the network during all the execution of the algorithm, and τ 1 is the time slot at which the input-values are assigned. Under the model described in Section I, if the number of node failures after τ 1 is bounded by f ≥ 0, and T is the minimum time between any pair of consecutive failures. There exist positive constants κ 1 , κ 2 such that, if V ′ ⊆ V is the set of nodes awake in all the interval

[τ 1 -κ 1 (D + log 2 n), τ 1 ], ν i is the input-value assigned to node i ∈ V ′ at time τ 1 , ν max = max i∈V ′ ν i , ν min = min i∈V ′ ν i , and ν = i∈V ′ ν i /|V ′ |
[(ν|V ′ | -f ν min )/(|V ′ | -f ), (ν|V ′ | -f ν max )/(|V ′ | -f )]
and the expected number of transmissions is in O(n((f + 1) log n + ∆/ log n + log ∆)) w.h.p.

Proof: As explained before, the failure of a slug node i only impacts the computation if i is running the COLLEC-TION phase and did not pass its input-value to its delegate yet. Due to the assumption of coverage, even under failures, delegate nodes that fail during the COLLECTION and COM-PUTATION phases are replaced in the tree-based algorithm by one or more of its slugs, introducing a O(log 2 n) overhead for each failure as shown in Lemma 3. Thus, the claimed range of the result follows. Regarding the running time, given that broadcast and convergecast is run in reserved slots the time taken by the last two phases is O(D + f log 2 n), using Lemmas 3, 4, and 5, given that D and ∆ can not be in o(log 3 n) simultaneously, and given that the number of failures is at most f the claim follows. The claimed number of transmissions is a direct consequence of Lemmas 3, 4, 5, Remark 2 and including the worst-case overhead of replacing the failed delegates. Mass-distribution Algorithm. For clarity, let us assume first that nodes do not fail. In the mass-distribution protocol used in this paper, after aggregating input-values in the COLLECTION phase, delegate nodes share a fraction with each delegate neighbor. More precisely, recall that [START_REF] Bawa | Estimating aggregates on a peer-to-peer network[END_REF]. Upon termination of the COLLECTION phase, each delegate node i ∈ M computes a local average ν i = σ i /ω i . From there on, the computation progresses as if all nodes in the network (slugs and delegates) were participating in it using this initial local-average value, although delegate nodes take on the task for the slug nodes. More precisely, in each round, each delegate node passes its weight and a fraction 1/2δ∆ of its weighted average to each neighboring delegate node, keeping the rest of the weighted average for itself. Then, delegate nodes update their average values appropriately using the shares and weights received, and repeat. After sufficient number of iterations, all average values converge to the average sought. We call this protocol Mass Distribution.

max i∈M {|N (i)|} ≤ 3⌈2β/α √ 3⌉(⌈2β/α √ 3⌉ + 1) as shown in Section III. Then, fix δ = 1 + 3⌈2β/α √ 3⌉(⌈2β/α √ 3⌉ +
Given that the shares are the same for all neighbors and δ and ∆ are known 4 , delegate nodes do not need to specify the destination and simply transmit the average and weight. After enough number of rounds of Mass Distribution, each delegate node i obtains the average with the accuracy desired. Furthermore, the DISSEMINATION phase is integrated in the COMPUTATION phase by default given that, although averages and weights are transmitted to neighboring delegate nodes, all neighboring nodes receive those transmissions because they are produced in reserved slots. Notice that Mass Distributiondoes not violate the memory restrictions since only a constant number of values are received in each round and the average and weight values are still polynomially upper bounded. Of course, precision limitations due to real number computations are still in order.

In presence of the slug and delegate failures described in Section III-B, slug nodes do nothing and delegate nodes adjust their delegate neighborhood appropriately. If a delegate node fails before broadcasting its values, the failure has the same impact on the computation as if it fails at the beginning of the round. If, on the other hand, the node fails after broadcasting its values, the failure has the same impact in the computation as if it fails at the end of the round. Therefore, without loss of generality, to analyze the convergence of Mass Distributionwe assume that delegatenode failures occur between rounds.

We analyze now Mass Distribution 5 . Assume first that nodes do not fail. Given that the fraction shared in Mass Distributionis round independent, the algorithm can be characterized by a matrix of shares as follows. Let ν (t) = (ν

(t) 1 . . . ν (t)
n ), i ∈ V be the vectors6 of averages held by nodes after round t. (Let the average held by a slug node be the average held by its chosen delegate.) Let P = (p ij ) be a matrix in R n×n such that p ij = 1/2δ∆ if i and j have chosen delegates r and s respectively such that r ∈ N (s), p ij = 1 -( s∈N (r) ω s )/2δ∆ if j = i and i has chosen delegate r, and p ij = 0 otherwise. Then, ν (t) = ν (0) P t is the vector of averages in round t. Given that P is stochastic, this characterization can be also seen as a Markov chain X = {X t } where the state space is V and the transition matrix is P . Mass-distribution algorithms only converge to the result. Hence, a metric of such an approximation has to be defined. In this paper, we use the relative point-wise distance, which is defined as max i |ν i -ν|/ν. The correctness of the average

  and the references therein. Lemma 3. (a) For any node i running the first phase of preprocessing, for any 0 < α ≤ 1, at least one node within distance αr of i becomes a delegate within O(log 2 n) time steps and no two delegate nodes are within distance αr of each other w.h.p. The expected number of transmissions of i during this phase is in O(log n) w.h.p. (b) For any node i running the second phase of preprocessing, if i is a delegate node, after O(log n) time steps i reserves a block of b ∈ O(1) steps every γ ∈ O(1) steps for local use, i.e., this block does not overlap with the block of any other delegate node separated by a distance at most r, w.h.p. During this phase, if i is a delegate node the expected number of transmissions of i is in O(log n) w.h.p., and if i is a slug node it does not transmit.

  , the following holds. If T ≥ κ 2 log 2 n, w.h.p., within O(∆+D+f log 2 n) time steps after τ 1 -κ 1 (D + log 2 n), all nodes running the algorithm receive (or hold) the same value, that value is in the range

A parameterized event Ep occurs w.h.p. if, for any constant γ > 0, there exists a valid value p such that P r{Ep} ≥ 1n -γ .

Throughout this paper, log means log 2 unless otherwise stated.

δ is a constant that can be stored before deployment. ∆ is broadcasted in the TRIGGER phase.

We assume familiarity with Markov chains and spectral graph theories. For an introduction refer to[START_REF] Chung | Spectral graph theory[END_REF],[START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF].

Throughout the paper, we use row vectors for clarity.

computation implemented with mass distribution algorithms is a well-known fact that can be proved using the fundamental theorem of Markov chains [START_REF] Motwani | Randomized Algorithms[END_REF]. We establish formally the correctness of the average computation in the following lemma. For the sake of brevity, the proof is left to the full version of this paper [START_REF] Fernández Anta | An early-stopping protocol for computing aggregate functions in sensor networks[END_REF].

Lemma 7. (Correctness) Let V be the set of n nodes in a Sensor Network, ν i be the input-value of node i ∈ V , and ν = i∈V ν i /n their average. Let ν (t) i be the average held by the delegate node chosen by node i ∈ V obtained t rounds after the COLLECTION phase of the Aggregate Computation Scheme. Then, if delegate nodes do not fail, implementing the COMPUTATION phase using Mass Distribution, there exists a τ 3 ≥ 0 such that, for all t ≥ τ 3 , |ν -ν (t) i |/ν ≤ ε, for all i ∈ V and for a given parameter ε > 0.

In presence of node failures, the fraction of average shared in Mass Distributionis not round independent. Therefore, instead, we characterize the computation carried out by Mass Distributionunder failures as a sequence of matrices P 0 , P 1 , P 2 , . . . such that P k , k ≥ 0, is the matrix of shares characterizing the algorithm between failures k and k + 1.

Given that these matrices are stochastic, each of these characterizations can be also seen as a Markov chain X k where the state space is V k , the set of nodes whose delegate is active between failures k and k + 1, and the transition matrix is P k .

To analyze the efficiency of Mass Distribution, we leverage the vast body of research work on bounding the mixing time of Markov chains. We bound the mixing time using the conductance as in [START_REF] Sinclair | Approximate counting, uniform generation and rapidly mixing markov chains[END_REF]. The conductance is a natural notion for Markov chains with underlying graphs with geometric properties. To show the overall efficiency of Mass Distribution, we bound the convergence time of each of the Markov chains and combine the effect of all of them in Lemma 8. For the sake of brevity, the details of the proof are left to the full version of this paper [START_REF] Fernández Anta | An early-stopping protocol for computing aggregate functions in sensor networks[END_REF].

Lemma 8. Given a Sensor Networkwith a set of nodes V , where ν max = max i∈V ν i and ν min = min i∈V ν i are the maximum and minimum input-values assigned to nodes respectively, nodes are running the Aggregate Computation Scheme as described, and the COMPUTATION phase is implemented using Mass Distributionas described, after O((f + log(1/ε) + log(ν max /ν min ))/Φ 2 min ) rounds, where f is the number of node failures and Φ min = min k∈{0,1,...,f } Φ k where Φ k is the conductance of the underlying graph after the kth failure, all nodes have converged to a value with relative error 0 < ε < 1.

The following theorem shows the overall efficiency of the Aggregate Computation Scheme when the mass-distribution algorithm is used. Theorem 9. Given a Sensor Networkwith a set of nodes V running the Aggregate Computation Scheme as described, where ∆ is a tight upper bound on the maximum number of neighbors of any node, D is a tight upper bound of the diameter of the network during all the execution of the algorithm, and τ 1 is the time slot at which the inputvalues are assigned. Under the model described in Section I, if the number of node failures after τ 1 is bounded by f ≥ 0, and T is the minimum time between any pair of consecutive failures. There exist positive constants κ 1 , κ 2 such that, if V ′ ⊆ V is the set of nodes awake in all the interval

, where Φ min = min k∈{0,1,...,f } Φ k where Φ k is the conductance of the underlying graph after the kth failure, all nodes running the algorithm have converged to the same result in the range [ν max , ν min ], with relative error 0 < ε < 1 w.h.p., and the expected number of transmissions is in O(n(log n + ∆/ log n + log ∆ + (f + log(1/ε) + log(ν max /ν min )))/Φ 2 min log n) w.h.p. Proof: The running time is a direct consequence of Lemmas 3, 4, 5 and, given that each round takes O(1) time steps, Lemma 8, and using that D and ∆ can not be in o(log 3 n) simultaneously. The claimed number of transmissions follows from Lemmas 3, 4, 5, 8, and Remark 2.


Aliou Niang
email: aliou.niang@ensiacet.fr

Julitte Huez
email: bjulitte.huez@ensiacet.fr

Jacques Lacaze
email: cjacques.lacaze@ensiacet.fr

Bernard Viguier
email: bernard.viguier@ensiacet.fr

Characterizing precipitation defects in nickel based 718 alloy

Keywords: Nickel base superalloys, Inconel 718, Ni 3 Nb, Gamma double prime, HRTEM, Stacking faults

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Introduction

Polycrystalline nickel base 718 alloy is widely used in aerospace and energy industry, specially for turbine disks and other applications that require high strength materials for temperature up to 650°C. This alloy exhibits good mechanical properties in the intermediate temperature range and good ability for forming processes. The strength of this alloy comes from coherent precipitates, which are for a small part γ'-Ni 3 (Al,Ti) but mostly γ"-Ni 3 Nb [1][2][3][4]. Both of these precipitates are ordered forms of the γ Ni rich matrix (face centred cubic-A1). The ordered γ' phase presents the cubic L1 2 structure and the γ" is body centred tetragonal corresponding to the D0 22 structure. The γ" phase is a metastable form of Ni 3 Nb which tends to stabilize to the orthorhombic δ phase with the structure D0 a . The evolution of the precipitation state which can occur during thermal and mechanical ageing, is responsible for the evolution of mechanical properties of the alloy [5,6]. The softening of the alloy, through the transformation of fine J'' precipitates to the stable phase G is particularly important [START_REF] Dong | 706 and Various Derivatives[END_REF]. It has been shown [2,[START_REF] Dong | 706 and Various Derivatives[END_REF]8] that the growth of G at the expense of γ" precipitates is often associated with the formation of stacking faults (SF's) within these latter. The aim of the paper is focused on such SF's which may exist in γ" precipitates. We first present the crystallographic structure of γ, γ" and δ phases, showing how the transformations between those phases may occur. Then we present an experimental characterization of SF's observed in γ" precipitates and we discuss possible mechanism for their formation, in relation with the generation of stable G precipitates.

Crystallographic structure of phases

The crystallographic structure of the three phases involved in the present study, namely J, J" and G, are shown in Fig. 1. The unit cells are shown in Fig. 1(a, b andc). They respectively consist in a) a fcc cell with a = 0.3616 nm ; b) a bct cell with a = 0.3624 nm and c = 0.7046 nm (note that these parameters lead to c/a ≈ 1.94 so that the tetragonal cell can be viewed as two cubes superposed along the c axis) and c) an orthorhombic cell with a = 0.5141 nm, b = 0.4231 nm and c = 0.4534 nm. The J" and G precipitates are commonly observed with epitaxial relations with the fcc structure of the J matrix. Such relations, which are expressed as:

] 0 1 1 //[ ] 0 1 1 [ J J and G J 010 // ) 111 ( - G J ] 100 //[ ] 0 1 1 [ ,
actually meaning that there exists a correspondence between the compact planes and directions of the three structures. Those compact planes are shown as grey surfaces within the unit cells in Fig 1 (a, b andc) and the atomic arrangement of the atoms in the planes are depicted in Fig. 1(d, e andf) in which the surfaces greyed in the unit cells have been reported. These drawing show that the compact planes of each structure: (111) γ , (112) γ" and (010) δ , correspond to a close packing of atoms, with an additional chemical order in the case of J" and G phases. It is important to notice that these compact planes are identical for both ordered phases, with the same rectangular arrangement of Nb atoms within the plane. In this view it appears that the difference between J and J" phases is based on a chemical ordering of Nb atoms on the same cubic geometry. For the sake of simplicity the directions and plane indices will be given in the cubic crystallographic basis (with the relation [001] J // [001] J" ) so that the compact plane denoted (111) corresponds to (112) J" and (010) G . In this ordered plane, see Fig. The difference between the structures of the γ"-Ni 3 Nb and the δ-Ni 3 Nb phases lies in the difference in the stacking mode of these close-packed planes. In the D0 22 structure, this stacking sequence is similar to the cubic stacking fashion based on a sequence of three planes (ABCABC..). However, due to the chemical order within the compact plane in the case of D0 22 structure the stacking sequence extends over six planes depending on the location of the Nb atoms. This is better viewed in Fig. 2 12) their chemical nature. This stacking insures that each Nb atom is surrounded by twelve Ni atoms so that there never exists in the structure two Nb atoms as nearest neighbours. The stacking of compact planes in the D0 a structure is shown in Fig. 2(b), it corresponds to a sequence of two planes ABABAB…(similar to hexagonal compact packed structure, however as the compact plane shows a two fold symmetry not six fold due to the rectangular arrangement of Nb atoms, the structure possesses orthorhombic symmetry). Note that such stacking also insures that each Nb atom is surrounded by twelve Ni nearest neighbours and that there never exists any Nb atoms as nearest neighbours, just like in J" phase. We have seen that J" and G phases only differ in the way the ordered close-packed planes are stacked. Just like in disordered fcc structure, a localised alteration of the stacking sequence corresponds to a SF which is associated with the shearing of the crystal by one Shockley partial dislocation [START_REF] Hirth | Theory of Dislocations[END_REF]. Partial dislocation Burgers vectors are shown in Fig. 3, in which two overlapping ordered close-packed planes of the D0 22 structure are represented. As one can see in Fig. 3, the three <112> directions are not equivalent. Indeed, the displacement provoked by ] 2

(equivalent to

J c c ] 1 1 1 [ 6 1
) partial dislocation changes the type of plane without changing the neighbouring of the selected Nb atom. In the D0 22 stacking sequence, such Burgers vector would transform a B1 plane to C2, leading to the stacking A2B2C2A1C2A1B1C2 which includes two AC sequences of planes corresponding locally to the D0 a structure. Alternatively, the displacement by the Burgers vectors ] 1 1 2 [ J ) implies that the Nb atom which was sitting on a three Ni atoms triangle is now in contact with a Nb atom as nearest neighbour. In other words the resulting plane does not correspond to any of the A1 to C2 plane of the D0 22 staking sequence nor to the A or B planes of the D0 a sequence. Such displacement, which modifies the neighbouring of the Nb atoms, corresponds to a complex SF and is likely to be energetically quite unfavourable. The above crystallographic considerations show that for each compact plane in the J" phase there exists only one Burgers vector available to create a low energy stacking fault, corresponding to the formation of G like structure over four crystallographic planes. In this case the J" → G transformation can be viewed as a purely geometrical modification. Despite those remarks were already mentioned for quite a long time [2,8,[START_REF] Kusabiraki | [END_REF], no experimental characterization of the type of SF was reported previously. The consequences of this particularity on the formation of stacking faults in J" precipitates were neither discussed. These two points will be addressed below.

Experimental details

Direct experimental observations of SF's within γ" precipitates have been conducted on specimens extracted from an alloy 718 billet provided by Aubert & Duval. The observations reported here were realised on the as-received and heat treated 1000°C/1s samples taken from a study of the J" and G precipitation kinetics [11]. Thin foils were classically prepared from bulk samples by cutting and grinding 3 mm diameter disks to the final thickness of about 100 µm.

Results and discussions

In the present study the observations made on grains containing quite large J" precipitates are reported, since SFs are commonly observed in such large precipitates. It is worth noting that the sample observed did not undergo any plastic deformation after high temperature treatment, so that the SF's observed were formed during the growth of J" precipitates. In the different grains, that were objects of study, the presence of the three possible J" variants in the form of circular plates (about 100 nm in diameter and 20 nm thick) was noticed. Numerous SF's are observed in γ" precipitates as indicated by the white arrows in Fig. 4 (a). Similar observations of SF's in large J" precipitates have been often reported in the literature [START_REF] Kusabiraki | [END_REF]12,13]. However such observations do not allow characterizing precisely the exact morphology and nature of the SF's, the interpretation is almost limited to the statement that there exists some SF's. A more precise characterization of those defects can be achieved by HRTEM observations. Such an image is shown in Fig. 4 Low magnification observations show that the whole defect entirely lies within a J" precipitate.

A Burgers circuit drawn around the whole defect evidences a closure vector within the . The identification of the Burgers vector of each partial dislocation and the structure of planar faults would necessitate the use of computer simulated images [14] and the comparison between simulated and experimental images as this was done in different intermetallic compounds [15]. The present observation also shows that a dislocation with a Burgers vector of the type 011 2 1 can dissociate into several partial dislocations and form complex configuration. It must be noted that the observed configuration is not mobile but would instead act as a lock. The mechanism by which during the plastic deformation of the alloy, the matrix dislocations shear the precipitates and create SF's [START_REF] Dong | 706 and Various Derivatives[END_REF]8] is clearly different from the dissociation mode observed in the present study. Such crossing mechanism still needs to be elucidated, particularly it is not clear how the same matrix dislocation would be able to cross J" precipitates of different variants.

In some circumstances, the direct interpretation of HRTEM images allows to determine the type of SF. This is illustrated in Fig. 5, corresponding to observations performed along ] 0 1 1 [ zone axis. In this case, (002) planes are projected and correspond to a succession of alternate Ni/Nb rows and pure Ni rows. This structure results in a image with modulated contrast over two (002) planes as can be seen in Fig. 5 (equivalent to

J c c ] 1 1 1 [ 6 1
), i.e. the vector that creates a purely geometric SF without chemical fault. Although it has been anticipated for quite a long time that this kind of fault is more likely to occur than a fault that would change the nearest neighbours of atoms [2], the present observation is to our knowledge its first experimental evidence.

Conclusions

1. The crystallographic structure of the J, J" and G phases appearing in alloy 718 are examined in details, showing that the J → J'' transformation consists of a chemical ordering of the Nb atoms without changing the geometry, whilst the J'' → G transformation is purely a geometrical modification of the compact planes stacking sequence. 2. Dissociation of ½<011> type dislocation in the form of V shaped lock has been evidenced. 3. HRTEM observations along [110] zone axis allow to identify the type of fault which is observed, we reported observation of purely geometric SF.

Fig. 1 .

 1 Fig. 1. Unit cell (a, b and c) and atomic arrangement in the corresponding compact planes (d, e and f) for the phases γ (A 1 ), γ" (D0 22 ) and δ(D0 a ) respectively.

  (a) which represents the projection of the structure along the ] 10 1 [ direction. The resulting sequence can be written as A1B1C1A2B2C2A1… where the letters (ABC) indicate the Ni Nb geometric position of the atoms and the numbers (

2 .

 2 Illustration of the stacking of the compact planes in γ" (a) and δ (b) phases projected along ] 10 1 [ as indicated in Fig.1. The large disks indicate the atoms in the plane, they correspond to projected rows of alternated Nb and Ni atoms. The small disks represent the atoms beyond the projection plane, they correspond to pure Ni rows. Note that for the J" phase (a) we have written directions in the cubic J basis (with the relation [001] J // [001] J" ).

Fig. 3 .

 3 Fig. 3. Stacking of two successive ordered compact (111) planes showing three possible partial dislocation Burgers vectors. Large disks represent atoms on the ground plane, and the smaller disk atom of the plane which lies on top of the earlier (crystallographic indices are given in the cubic J basis).

Fig. 4 .

 4 Fig. 4. TEM observations of SF's (white arrows) within γ" precipitates imaged using conventional diffraction contrast (a) and HRTEM (b).

Fig. 5 .

 5 (b), in which one can see two SF's lying on two secant {111} planes and that are connected to form a V shape. This image is taken along a [101] type direction, along which the ordered structure of J" is not detectable since all projected rows include alternatively Nb and Ni atoms, see Fig.1(e). (a) HRTEM image of a SF within a γ" precipitate viewed along ] compared with the schematic projection of atoms in the case of a SF created by ]

  corresponds to the projection of a Burgers vector of the type 011 2 1

  (a). Thanks to this special contrast, the shift in (002) planes across the SF can be directly quantified and compared with the schematics of Figs 5 (b) and (c). It is indeed clear that the insert in Fig.5(a) compares very well with Fig.5(b) and not with Fig.5 (c). The SF is thus associated with a displacement by the vector ]

' '