J M Torrenti

F Benboudjema

F Barré

E Gallitre

AMS'

Keywords: Nuclear power plant, Prestressing, Creep, Concrete, Ageing

Several nuclear power plant containment buildings are made with biaxially prestressed concrete. In order to maintain enough prestressing in the structure, the delayed behaviour of concrete on the very long term should be correctly assessed. In this paper two questions are considered: is the very long term creep of concrete asymptotic or not? And how to deal with biaxial creep? The observation of the very long term deformations of containments, the deflections of concrete bridges and laboratory results of creep tests are used to give answers to these questions.

Introduction

The typical French reactor building consists of two concentric containments (fig. 1). The outer containment, designed to sustain external aggressions, is made with reinforced concrete. The inner containment is biaxially prestressed, from 80 to 120 cm thick, and designed to withstand an internal pressure of 0.5 MPa in case of an accident. In order to avoid tensile stresses in concrete, a prestress is applied, corresponding to compressive stresses in concrete of 8.5 MPa and 12 MPa along vertical (zz) and orthoradial (, respectively. In the previous generation of French nuclear power plant (NPP), there was no metallic liner inside the internal containment. So the tightness of the containment in case of an accident is only assured by concrete. To limit leakage, cracking should be avoided and, so, tensile stresses should remain below the tensile strength of concrete. That is why the evolution of prestressing forces during time is critical for the NPP operation and why a good prediction of the evolution of delayed strains of the containment is needed.

AMS'14

To predict the evolution of prestressing forces with time, the delayed behaviour of the constitutive materials (relaxation for steel and creep for concrete) has to be modelled precisely on the long term. In this paper only creep of concrete is considered and two problems are discussed: is the very long term creep asymptotic or not and how to deal with biaxial stresses?

2 Is the very long term creep asymptotic?

Laboratory tests

Classical creep tests in laboratory last generally less 2 or 3 years (and sometimes less). With such duration it is possible using these tests to fit as well an asymptotic or a logarithmic evolution of strains. There is nevertheless one set of creep tests performed by Brooks [START_REF] Brooks | 30-Year creep and shrinkage of concrete[END_REF] with a duration of 30 years. These tests performed on concretes with different water to cement ratio and with different aggregates types, show that creep is not asymptotic (figure 2).

Long term deformations of prestressed structures.

The observation of the deflections of several bridges built of prestressed box girders indicates rather clearly that long term deflections are not asymptotic [START_REF] Bažant | Excessive creep deflections: An awakening[END_REF][START_REF] Vítek | Deflections and Strains of Prestressed Concrete Bridges[END_REF][START_REF] Sellin | Delayed deformations of segmental prestressed concrete bridges: the case of the Savines Bridge[END_REF]. In the case of NPP, thanks to a large monitoring of the behaviour of the containment, global and local deformations are measured. Figure 3 presents the evolution of the deformations of the central part of the containment of a French NPP. Again, the long term behaviour is not asymptotic and seems in accordance with a logarithmic evolution with time. Note that delayed strains include here drying shrinkage and relaxation of the prestressing tendons.

Consequences

With the measurements of the deformations of a sample during a creep test, it is possible to adjust an asymptotic hyperbolic function (like in EC2-1 -equation 1) or a combination of an asymptotic and a logarithmic functions (like in MC2010 -equation 2) -see figure 4. In the latter case the logarithmic function corresponds to basic creep while drying creep is asymptotic.

( ) [eq. 1] [ ( ) ] [eq.2]
where a, b, c, d and e are fitted constants. f1, f2, g1 and g2 are functions of the notional size h given by EC2-1 and MC2010. The notional size is here equal to 80mm.

The creep models being optimized on a sample, it is possible to predict the creep deformations of a larger structure. In our case a NPP has a notional size h=800mm loaded under a constant uniaxial stress of 12.5MPa and a RH of 50%. Figure 5 shows that in this case at very long term large differences occur despite a good calibration of the model on the same test. This is of course due to the differences in the nature of the equations (asymptotic or not) and it shows the importance of a physical manner to model the very long term behaviour of concrete structures. A bad prediction during design will underestimate the loss of prestressing and could affect the service life of the NPP. Note also that, in this case the contribution of basic creep is the most important in the long term despite the large notional size (figure 6).

AMS'14

Figure 4 Comparison of experimental results [START_REF] Granger | Comportement différé du béton dans les enceintes de centrales nucléaires : analyse et modélisation[END_REF] and models corresponding to equations 1 and 2. The applied stress is equal to 12.5 MPa and the relative humidity to 50%.

Biaxial creep

Laboratory tests

Few experimental results are available in the literature. By extending isotropic elasticity theory to the visco-elasticity, it is possible to define a secant creep Poisson ratio c:

[ ] [eq.3]
where  is the creep strain tensor,  the stress tensor and J is the uniaxial specific creep. If it is assumed that the material remains isotropic, creep Poisson ratio can be calculated using either equation 4 or 5 (after developing eq. 3):

⁄ [ ] [ ] ⁄ [eq.4] ⁄ [ ] [ ] ⁄
[eq.5] An analysis of several experimental results leads to the fact, in most of 1D (with the measure of strain in the non-loaded direction), 2D and 3D creep tests, the creep Poisson ratio is closed to 0.15. However, in some tests unphysical values (less than -1 or greater than 0.5) has been found [START_REF] Benboudjema | Modélisation des déformations différées du béton sous sollicitations biaxiales. Application aux enceintes de confinement de bâtiments réacteurs des centrales nucléaires[END_REF], which may be due to uncertainty of sensors, tightness flaws in basic creep tests and friction between the concrete specimen and the loading frame. The analysis of available biaxial basic creep test is reported for biaxial creep tests in Figure 7.

For experiments where strains have been measured in the 3 principal directions [START_REF] Galenne | Prediction of Delayed Strain of Nuclear Containment Building: from laboratory tests to an industrial mock-up[END_REF][START_REF] Gopalakrishnan | Creep Poisson's Ratio of Concrete Under Multiaxial Compression[END_REF][START_REF] Jordaan | The Creep of Sealed Concrete under Multiaxial Compressive Stresses[END_REF], the use of eq. 4 and 5 has leaded to similar results. For biaxial creep tests for which compressive stresses are different, it has been found very close results using either eq. 4 or eq. 5. These results show a significant discrepancy between results. After 100 days, the basic creep Poisson ratio is comprised between about 0.14 and 0.35. Even after about 4 years, its value seems to be continuously evolving. It is difficult to conclude at this time.

Long term deformation of a NPP

In Figure 3, the evolution of the biaxial deformations during times in a NPP was presented. From these deformations one can express the evolutions of the sum and the difference of the vertical and tangential strains. These evolutions are presented figure 8 for three different NPPs [START_REF] Chauvel | Assessment of long term concrete deformations of nuclear structures based on in situ measurements[END_REF]. It can be seen that the delayed strains in the containment wall follow two distinct kinetics:

the difference between tangential and vertical strains, which exhibits a fast kinetics and becomes almost constant after 500 days, -the average of tangential and vertical strains, which corresponds to a slow kinetics. If vertical and tangential strains are equal despite different values for the prestressing, this means that in the long term biaxial creep deformations are only sensible to the mean stress and independent of deviatoric ones. If biaxial creep is expressed using a tangent Poisson ratio, this is equivalent to a Poisson ratio equal to -1. If we assume that the kinetics proposed by MC2010 are correct, the long term slow kinetics corresponds to basic creep (figure 6). So there is a contradiction between the basic creep Poisson ratio obtained with experimental results and the one obtained from the global behaviour of the NPP.

Conclusion

The observation of the global behaviour of containments of nuclear power plants confirms laboratory results and the observation of the long term behaviour of bridges: creep of concrete does not seem to have an asymptotic behaviour but an evolution with the logarithm of time. And it is very important to take into account this phenomenon in construction codes for the prediction of the very long term behaviour of structures. Concerning biaxial prestressed structures like the containment of a NPP, the global behaviour does not seem in accordance with laboratory tests. The analysis of these structures should be continued in order to understand the physical sources of this discrepancy. T-V NPP3

Figure 1

 1 Figure 1 Simplified diagram of the containment and its prestressing

Figure 2

 2 Figure 2Creep of concreteexample of the results of Brook's tests[START_REF] Brooks | 30-Year creep and shrinkage of concrete[END_REF]

14 Figure 3

 143 Figure 3 Example of the evolution of the deformations in the central part of a containment of a NPP (V=vertical strains, T=tangential strains)

Figure 5 1 MC2010 14 Figure 6

 51146 Figure 5 Comparison of the estimated creep strains using equations 1 and 2.

14 Figure 7

 147 Figure 7 Experimental evolution of basic creep Poisson ratio.

Figure 8

 8 Figure 8 evolutions of the tangential (T) and vertical (V) deformations in the central part of the containment of three NPPs

  Proceedings of the Intern. Conference on Ageing of Materials & Structures Delft 26 -28 May 2014, The Netherlands Paper No. 218885


Kelly Olivier
email: kelly.olivier@ens-cachan.fr

Aveline Darquennes
email: aveline.darquennes@ens-cachan.fr

Farid Benboudjema
email: farid.benboudjema@dgc.ens-cachan.fr

Richard Gagné
email: richard.gagne@usherbrooke.ca

Impact of material characteristics on the self-healing kinetic of cementitious materials

Keywords: Self-healing, blast-furnace slag, cement, SEM, EDS, X-ray tomography

Introduction

Concrete made with ground granulated blast-furnace slag (GGBFS) cement is often used in civilengineering structures. It offers many advantages such as reclamation of an industrial waste, decreased CO 2 emissions produced during the clinkerization process in cement manufacturing, and several enhanced material properties, such as workability and long-term compressive strength. It is also appropriate for structures in aggressive environments (farm silos, water-treatment plants, mining) because of its high resistance to sulfate attack and chloride diffusion [START_REF] Darquennes | Behaviour of slag cement concrete under restraint conditions[END_REF].

Nevertheless, some civil-engineering structures made with GGBFS cement present cracking at early age due to restrained shrinkage [START_REF] Darquennes | Long term deformations and cracking risk of concrete with high content of mineral additions[END_REF][START_REF] Briffaut | Analysis of cracking due to shrinkage restraint in a concrete tunnel[END_REF]. This cracking can lead to an increased risk of penetration by aggressive agents such as CO 2 and chloride.

Structures made with GGBFS cement, however, have evidenced some self-healing of these cracks. This phenomenon seems to be related to the hydration of anhydrous particles at early age (GGBFS, clinker) [START_REF] Hearn | Self-sealing, autogenous healing and continued hydration: What is the difference ?[END_REF][START_REF] Ter Heide | Crack healing in hydrating concrete[END_REF] and/or calcite formation due to portlandite carbonation on the long term [START_REF] Argouges | Mécanismes et cinétique de l'autocicatrisation dans des mortiers cimentaires : influence de la refissuration[END_REF]. New hydrates formed in the cracks can partially or completely fill the crack volume. It is not yet clear how GGBFS is involved in the self-healing phenomenon. Indeed, the few available experimental results [START_REF] Qian | Self-healing behavior of strain hardening cementitious composites incorporating local waste materials[END_REF][START_REF] Van Tittelboom | Influence of mix composition on the extent of autogenous crack healing by continued hydration or calcium carbonate formation[END_REF] are sometimes contradictory. For example, Van Tittelboom (2012) indicated that the addition of GGBFS positively influenced self-healing, whereas [START_REF] Qian | Self-healing behavior of strain hardening cementitious composites incorporating local waste materials[END_REF] found the opposite.

In this study, X-ray computed tomography (X-ray CT) was used to monitor self-healing evolution and to determine the effect of GGBFS addition on a macro-scale. X-ray CT offers several advantages over conventional analysis methods such as permeability tests, mechanical tests and scanning electronic microscopy (SEM) [START_REF] Jacobsen | SEM observations of the microstructure of frost deteriorated and self-healed concretes[END_REF][START_REF] Granger | Experimental characterization of the self-healing of cracks in an ultra high performance cementitious material: Mechanical tests and acoustic emission analysis[END_REF][START_REF] Van Tittelboom | Self-healing efficiency of cementitious materials containing tubular capsules filled with healing agent[END_REF]. For example, it is nondestructive and does not require pretreatment of samples. Moreover, X-ray CT can assess the spatial and temporal evolutions of self-healing on the same sample throughout an experimental campaign and yield a 3-D distribution of the healing products in the sample volume [START_REF] Poinard | Compression triaxial behavior of concrete : the role of the mesostructure by analysis of X-ray tomographic images[END_REF][START_REF] Landis | Emerging imaging techniques applied to concrete durability[END_REF].

The X-ray CT analysis was completed by SEM (scanning electron microscopy) and EDS (energy dispersion spectroscopy) to identify the microstructure of self-healing products. A thermogravimetric analysis was also used to quantify the evolution of boundary water and the portlandite content of the different formulations.

Materials and methods

Materials

The three mortar compositions were (1) a mortar with a binder containing 100% Portland cement (CEMI 52.5 N CP2 NF), (2) a mortar with 50% Portland cement (CEMI 52.5 N CP2 NF) and 50% GGBFS and (3) a mortar with 100% blended cement (CEMIII/A) composed with 62% GGBFS and 32% clinker (same clinker as CEMI 52.5 N CP2 NF used for others formulations) (Table 1. Mortar compositions.). Their water/binder (W/B) ratios were equal to 0.50, 0.52 and 0.50; their binder contents were 563, 539 and 563 kg/m 3 , respectively. The cement paste content, a parameter affecting strongly the shrinkage amplitude, was kept constant for both mixes and the same normalized siliceous sand was used . and 4.06% of MgO. The hydraulic factor ( (Eq. 1)) was up to 1 for the GGBFS (1.76), indicating a good reactivity. Table 3 presents the results of laser particle-size distribution (PSD) testing. They reveal that the slag is finer than the CEMI and CEMIII/A with 48.68% of its particles below 10 µm compared to 36.37% for the CEMI and 40.91% CEMIII/A.

X-ray tomography

Specimen dimensions play a significant role in X-ray CT picture resolution. To achieve a reasonable voxel size (20 µm), cylindrical specimens with a 4 cm diameter (height h = 10 cm) were tested. After mixing, the specimen were kept in a room at 23°C ± 1°C and 45% ± 5% of relative humidity (RH). The molds were removed after 2 days and the specimens were stored underwater at 23°C ± 1°C. At 6 days, they were confined with a two-component resin reinforced with glass fibers. At 7 days, the specimens were cracked along the diameter with a splitting-tensile test. Then, the samples were sawn into 2 cm thick disks and scanned using X-ray CT to determine their initial state. The samples were then stored in tap water at 23°C ± 1°C in individual containers. The specimens were regularly analyzed with tomography during a month to monitor self-healing.

During the tomography, a sample was placed on a rotating support between the X-ray tube and detector. Three scans were performed for each table position (600 positions). The test parameters of position number for one test, high-voltage, target power, and frames per second were 600, 160 kV, 5.5 W, and 0.4, respectively. Afterwards, a 3-D image of the specimen was reconstructed (using efX-CT software). For these 3-D images, the material volumes characterized by lower density such as air, show a darker grey level. Artefacts around the specimen can be seen after reconstruction. To limit the effect of these artefacts during the analysis, a sub-volume was isolated inside the specimen (Figure 1. Location of sub-volume).

Figure 1. Location of sub-volume

To assess the evolution of crack volume, the 3-D X-ray tomography images (1300 x 300 x 600 voxels, with 256 gray levels) underwent a post-processing. Firstly, the 3-D image was transformed into a set of pictures along the Z-axis containing 600 pictures with a size of 1300 by 300 pixels.

A segmentation algorithm and a connected-component analysis algorithm were applied in each picture to determine the gray-level threshold corresponding to the cracked volume [START_REF] Landis | Emerging imaging techniques applied to concrete durability[END_REF].

In this study, one specimen was tested for the CEMI and two for the CEMI+S and three for CEMIII/A to estimate the repeatability.

SEM-EDS

One year after the X-ray CT campaign, scanning electron microscopy (SEM) with energy dispersive X-ray spectroscopy (EDS) analysis was performed on specimens to determine new products formed in the crack. The specimens were stored in tap water.

Thermogravimetric analysis

Thermogravimetric analysis (TGA) was performed on mortar specimens to determine their hydration kinetics. The weight-loss evolution of a specimen was monitored between 50°C and 1,000°C with a heat rate of 10°C/min in an argon environment. The samples were tested at 2, 3, 7, 14, 28, and 90 days. Before the test, they were stored in tap water in a room controlled at 23°C. A few minutes before the test, 2 g of mortar was ground and about 60 mg placed in an alumina crucible. An empty crucible was used as reference and placed close to the sample. Three tests were performed for each measurement.

The boundary water was quantified by the mass-loss difference from 105°C to 1,000°C and from 600°C to 880°C (decarbonation of CaCO 3 ). The temperature of 105°C corresponds to free-water evaporation. The boundary water ( (Eq. 2) is expressed as cement mass [START_REF] Popic | Influence des ajouts minéraux, des granulats fins, de la température et du rapport eau/liant sur le développement et le contrôle du retrait endogène des matrices cimentaires[END_REF]):

(Eq. 2)

where w(t) is the boundary water (g/g binder ) at time t, Δm i→j is the mass loss between temperatures i to j (g), m sand is the sand mass in the sample (g), LOI sand is the sand's loss on ignition (%), m k is the cement or slag mass (g), LOI k is the loss on ignition of the cement or slag (%), and m b is the binder mass in the sample (g). Moreover, it was necessary to determine the ultimate boundary water w ∞ . To compute w ∞ , we used the experimental approach proposed by [START_REF] Bentz | A three-dimensional cement hydration and microstructure program. I. Hydration rate, heat of hydration and chemical shrinkage[END_REF]. This consists in mixing a cement paste (water/binder ratio equal to 3.0) with steel balls in a hermetic bottle during 56 days. Then, TGA was performed to determine the boundary water necessary to hydrate whole grains of cement and slag. Before the test, the cement paste was dried for 1 h at 80°C to produce a dry material for the TGA. Finally, the portlandite content was determined to assess the evolution of slag reaction. The portlandite content is computed with ( (Eq. 3):

(Eq. 3)
where Δm 400→600°C is the mass loss from 400°C to 600°C (g), m b is the binder mass (g), m sand is the sand mass in the sample (g), LOI sand is the sand's loss on ignition (%) and M Ca(OH)2 and M H2O are, respectively, the molar mass of portlandite and water (g.mol -1 ).

Results

Volume evolution of self-healing products Figure 2. Volume evolution of healing products

Figure 2. Volume evolution of healing products shows that the volume of healing products increased for all the mortar compositions with the duration of water storage. The change is faster during the first 15 days after cracking. Subsequently, healing-product formation slowed (the crack volume continued to decrease but at a slower rate). This is in agreement with [START_REF] Argouges | Étude des mécanismes et de la cinétique de l'autocicatrisation dans des mortiers cimentaires fissurés[END_REF], who showed that the kinetics of self-healing is characterized by two stages. Argouges followed selfhealing after one and three month. In that case, self-healing is faster during the first month after cracking. Two CEMI+S mortars and three CEMIII/A mortars were tested to check test repeatability. All specimens of CEMIII/A show a same self-healing behavior with standard deviations between 0.02 and 0.8 mm 3 . Both CEMI+S samples have the same tendency: the kinetic of the formation of selfhealing products decreases after about 21 days (age of sample). Indeed, it is equal to 1.22 mm 3 .d -1 for CEMI+S_1 and 1.32 mm 3 .d -1 for CEMI+S_2 before 21 days and to 0.38 mm 3 .d -1 for slag CEMI+S_1 and 0.48 mm 3 .d -1 for CEMI+S_2 after this time. In both cases, the CEMI+S mortar presents greater self-healing than the CEMI and the CEMIII/A mortars. At 7 and 14 days after cracking, the average volume of the self-healed products are, respectively, (1) 4.08 mm 3 and 7.77 mm 3 for the CEMI mortar, (2) 5.66 mm 3 ± 0.53 mm 3 , 17.78 mm 3 ± 0.71 mm 3 for the CEMI+S mortar and (3) 3.54 mm 3 ± 0.76 mm 3 , 6.10 mm 3 ± 0.04 mm 3 for the CEMIII/A mortar. Thus, the formation of new products is higher for the CEMI+S mortar and lower for the CEMIII/A than for the CEMI. Results in Figure 2. Volume evolution of healing products show that the CEMI+S mortar is characterized by a faster initial self-healing evolution rate than the CEMI mortar and CEMIII/A. The self-healing evolution rate of the CEMI+S is approximately 2 times faster for CEMI and 3 time faster for CEMIII/A. The healed volume is 9.88 mm 3 at 42 days for the CEMI mortar, 22.45 mm 3 and 25.20 mm 3 at 35 days respectively for the CEMI+S _1 and the CEM+S_2 and respectively 6.31 mm 3 , 4.74 mm 3 and 6.53 mm 3 at 35 days for CEMIII/A_1, CEMIII/A_2 and CEMIII/A_3. Two factors can account for this and are discussed below: initial crack size and degree of hydration.

Initial Crack Size

The self-healing kinetics depends on the crack width [START_REF] Gagné | A study of the natural self-healing of mortars using air-flow measurements[END_REF]. Small cracks tend to self-heal more completely at long term because of the smaller initial volume having be filled [START_REF] Gagné | A study of the natural self-healing of mortars using air-flow measurements[END_REF][START_REF] Edvardsen | Water permeability and autogenous healing of cracks in concrete[END_REF]). The initial crack opening in the CEMI mortar was 113 µm, while the crack opening in the CEMI+S_1 mortar was 71 µm and 115 µm for CEMI+S_2. For the CEMIII/A, the three samples (CEMIII/A_1, CEMIII/A_2, CEMIII/A_3) had respectively an initial crack opening: 159 µm, 289 µm and 214 µm. Even if these initial crack opening are different, the volume evolution of self-healing products are closed for the three CEMIII/A samples and for the two CEMI+S samples. Therefore, the cracking-size effect does not responsible for the different behaviour between slag and control mortar.

Evolution of Degree of Hydration

Binders with GGBFS have a slower evolution rate of hydration at early age according to [START_REF] Darquennes | How to assess the hydration of slag cement concretes ?[END_REF]. This leads to a greater amount of anhydrous products at the cracking age (7 days) and, therefore, a higher self-healing potential. From thermogravimetric analysis, the degree of hydration for the two formulations is calculated using the boundary water content (Eq. 4).

(Eq. 4)

Two parameters need to be determined in this relation:

-The final boundary water w ∞ . Based on the experimental approach developed by [START_REF] Bentz | A three-dimensional cement hydration and microstructure program. I. Hydration rate, heat of hydration and chemical shrinkage[END_REF], the values of the final boundary water w ∞ for the CEMI, CEMI+S and CEMIII/A mortars are 22.90 g/g binder , 20.45 g/g binder and 21.69 g/g binder , respectively and are in agreement with [START_REF] Popic | Influence des ajouts minéraux, des granulats fins, de la température et du rapport eau/liant sur le développement et le contrôle du retrait endogène des matrices cimentaires[END_REF].

-The ultimate hydration degree α u . In the case of cementitious materials with mineral additions, [START_REF] Schindler | Heat of hydration models for cementitious materials[END_REF] suggested a relation providing the ultimate hydration value (Eq. 5):

(Eq. 5)

where W/C is the water-cement ratio, p FA and p slag are the fly ash and the slag in percentages. The α u for CEMI, CEMI+S and CEMIIIA/ are, respectively, 0.74, 0.90 and 0.93. During 90 days, the portlandite evolution of CEMI, CEMI+S and CEMIII/A are respectively: 76%, 40%, 16% with final values equal to: 10.83.10 -2 g/g binder , 5.76.10 -2 g/g binder and 3.92.10 -2 g/g binder .

The slower evolution of GGBFS mortars is explained by (1) the smaller amount of clinker in the GGBFS mortars, and (2) the consumption of portlandite for reaction of GGBFS particles. Before the two first weeks, the lower hydration of CEMI+S (14% lower than CEMI and CEMIII/A at 7 days) and higher hydration of CEMIII/A can be explained respectively by the GGBFS's latent hydration properties and the material fineness. The GGBFS's latent hydration property is related to two phenomena [START_REF] Moranville-Regourd | Cements made from blastfurnace slag, Lea's chemistry of cement and concrete[END_REF][START_REF] Taylor | Cement Chemistery[END_REF]):

(1) initiating GGBFS dissolution requires high pH in the interstitial water, which is directly related to its alkali content (e.g., Ca 2+ ); and (2) the GGBFS reaction requires portlandite (see previously, Figure 4. Portlandite content) to create supplementary C-S-H. After this time, hydration degree of CEMI+S increases and is 20% upper than CEMI and 31% upper than CEMIII/A. The portlandite content of CEMIII/A is lower than the two others formulations during the test. However, before 7 days, its hydration degree is closed to that of the CEMI. Even if CEMIII/A mortar is composed by 62% of slag, its good reactivity at 2 and 3 days can be explained by its high fineness (Table 2). Subsequent hydration continued slowly, achieving a constant value near the ultimate degree of hydration for CEMI and CEMI+S (0.75 for the CEMI mortar, 0.90 for the CEMI+S mortar at 90 days). At the opposite, the hydration degree of CEMIII/A at 90 days (0.68) was not close to the ultimate hydration degree. The results reveal an average variability of 0.015 (degree-of-hydration value).

These results confirm the slower rate of hydration and, therefore, the greater number of anhydrous particles in the CEMI+S at early age. The different behaviour of CEMIII/A can be explained by its higher initial reactivity. The fast hydration at 2 and 3 days can form a high quantity of hydration products around anhydrous particles and slow down their hydration.

Microstructure of Self-Healing Products

Figure 5. Self-healing products in CEMI, Figure 6. Self-healing products in CEMI+S and Figure 7.

Self-healing products in CEMIII/A illustrate self-healed cracks in the CEMI, CEMI+S and CEMIII/A mortars. While sawing may have slightly damaged the new products, the bridging between the two crack faces are clearly visible. EDS analysis of specimens reveal that the new products are C-S-H. In the case of the CEMI+S and CEMIII/A mortars, the C/S ratio is lower than in the CEMI mortar. Moreover, EDS analysis of the new hydrated products in the slag mortars show the presence of alumina and iron ions from GGBFS. These differences between C-S-H in the slag and CEMI mortars have been reported in the literature [START_REF] Moranville-Regourd | Cements made from blastfurnace slag, Lea's chemistry of cement and concrete[END_REF][START_REF] Taylor | Cement Chemistery[END_REF]. The C-S-H formation confirms that the self-healing of early-age cracks corresponds to ongoing hydration.

Conclusions

This study examined the self-healing phenomenon in cementitious materials with high GGBFS content by conventional (SEM) and original (X-ray CT) means, yielding more information about the self-healing phenomenon and underlining the benefit of GGBFS addition in accelerating this phenomenon. The main results of this study are summarized below. -X-ray tomography is an effective technique for observing the self-healing phenomenon in 3-D as well as to assess crack evolution and the formation of healing products. The advantage of Xray CT is that the technique is nondestructive and it provides 3D information on the formation of the self-healing products in the crack. -For a low hydration degree at the cracking day, mixes with ground granulated blast-furnace slag have a significant capacity for self-healing due to its latent hydraulic and pouzzolanic properties, leading to a significant quantity of unhydrated GGBFS particles at early age. -Under favorable storage conditions (underwater), at early age and when crack size is limited, the self-healing phenomenon improves material durability for all the studied materials. -C-S-H was the main self-healing product for all the studied mortar compositions. The early age cracking and the storage underwater (without CO 2 ) justified these products formations.

FigureFigure 3 .

 3 Figure 3. Hydration degree Figure 3. Hydration degree and Figure 4. Portlandite content present the averages degree of hydration and portlandite content over time.During 90 days, the portlandite evolution of CEMI, CEMI+S and CEMIII/A are respectively: 76%, 40%, 16% with final values equal to: 10.83.10 -2 g/g binder , 5.76.10 -2 g/g binder and 3.92.10 -2 g/g binder . The slower evolution of GGBFS mortars is explained by (1) the smaller amount of clinker in the GGBFS mortars, and (2) the consumption of portlandite for reaction of GGBFS particles. Before the two first weeks, the lower hydration of CEMI+S (14% lower than CEMI and CEMIII/A at 7 days) and higher hydration of CEMIII/A can be explained respectively by the GGBFS's latent hydration properties and the material fineness. The GGBFS's latent hydration property is related to two phenomena[START_REF] Moranville-Regourd | Cements made from blastfurnace slag, Lea's chemistry of cement and concrete[END_REF][START_REF] Taylor | Cement Chemistery[END_REF]): (1) initiating GGBFS dissolution requires high pH in the interstitial water, which is directly related to its alkali content (e.g., Ca 2+ ); and (2) the GGBFS reaction requires portlandite (see previously, Figure4. Portlandite content) to create supplementary C-S-H. After this time, hydration degree of CEMI+S increases and is 20% upper than CEMI and 31% upper than CEMIII/A. The portlandite content of CEMIII/A is lower than the two others formulations during the test. However, before 7 days, its hydration degree is closed to that of the CEMI. Even if CEMIII/A mortar is composed by 62% of slag, its good reactivity at 2 and 3 days can be explained by its high fineness (Table2).

Figure

  Figure 4. Portlandite content

Figure

  Figure 5. Self-healing products in CEMI

Table 1 . Mortar compositions.

 1 Table2) were performed to characterize the cementitious materials. X-ray diffraction revealed that the Portland cement was principally composed of C 3 S, C 2 S, C 3 A, and C 4 AF. The GGBFS showed amorphous behavior with some crystalline peaks of akermanite, melilite, and merwinite. The CEMIII/A was principally composed of C 3 S, C 2 S, C 3 A, and C 4 AF with an incurved spectrum as GGBFS (bump between [20-40°]). The CEMI was mainly composed of CaO (64.85%). The slag contained a smaller proportion of CaO (43.21%) than the CEMI, but greater amounts of SiO 2 (33.08%), Al 2 O 3 (9.50%), and MgO (5.77%). The composition of CEMIII/A is between these two materials with 51.85% of CaO, 27.95% of SiO 2 , 7.76% of Al 2 O 3

		Cement [kg/m 3 ] Water [kg/m 3 ]	Sand [kg/m 3 ]	GGBFS [kg/m 3 ]
	CEMI	563	281	1409	/
	CEMI+S	281	281	1409	258
	CEMIII/A	563	281	1409	/
	X-ray diffraction and X-ray fluorescence (

Table 2 . Chemical compositions of the CEMI, CEMI+S and CEMIII/A.

 2

	Oxides [%]	CEMI	GGBFS	CEMIII/A
	CaO	64.85	43.21	51.85
	SiO 2	18.60	33.08	27.95
	Al 2 O 3	4.92	9.50	7.76
	MgO	1.15	5.77	4.06

Table 3 . Particle-size distribution of the CEMI, CEMI+S and CEMIII/A.

 3

	% <	CEMI	CEMI+S	CEMIII/A
	1 µm	4.31	4.36	3.97
	10 µm	36.37	48.68	40.91
	100 µm	99.94	100.00	100.00

Acknowledgment

This work has benefited from the support of the French "Agence Nationale de la Recherche", through the "Investissements d'avenir" program under the reference "ANR-10-EQPX-37 MATMECA.


F Benboudjema
email: farid.benboudjema@ens-cachan.fr

J M Torrenti

On the very long term delayed behavior of biaxially prestressed structures: the case of the containments of nuclear power plants

The typical French reactor building consists of two concentric containments. The outer containment is made with reinforced concrete. The inner containment is biaxially prestressed, from 80 to 120 cm thick, and designed to withstand an internal pressure of 0.5 MPa (absolute) in case of an accident. In order to avoid tensile stresses in concrete, a prestress is applied, corresponding to compressive stresses in concrete of about 8.5 MPa and 12 MPa along vertical and tangential directions, respectively. In order to maintain enough prestressing in the structure, the delayed behavior of concrete on the very long term should be correctly assessed.

The measurements of the very long term deformations of larges bridges and of nuclear containments seem to indicate that, on the very long term, creep strains are function of the logarithm of time. In particular, the measurements of delayed strains in the containment wall show two distinct kinetics:

-the difference between tangential and vertical strains, which exhibits a fast kinetics and becomes almost constant after 500 days, -the average of tangential and vertical strains undergoes a slow kinetics. If vertical and tangential strains rates are equal (despite different values of the prestressing), this means that the long term biaxial creep deformations are only sensible to the mean stress and independent of deviatoric ones.

In the recent Model Code 2010 creep is expressed as a combination of an asymptotic and a logarithmic function. The logarithmic function corresponds to basic creep while drying creep is asymptotic. Using this model for delayed strains, and adding the possibility to adjust the parameters of the relations proposed by MC2010 on experimental results, the evolution of the strains is calculated and compared to in situ measurements. We finally discuss the set of adjusted possible parameters.

INTRODUCTION -BASIC CREEP OF CONCRETE

The typical French reactor building consists of two concentric containments (fig. 1). The outer containment, designed to sustain external aggressions, is made with reinforced concrete. The inner containment is biaxially prestressed, from 80 to 120 cm thick, and designed to withstand an internal pressure of 0.5 MPa (absolute) in case of an accident. In order to avoid tensile stresses in concrete, a prestress is applied, Rev. 11/2014 corresponding to compressive stresses in concrete of about 8.5 MPa and 12 MPa along vertical (zz) and tangential ( directions, respectively. In the previous generation of French Nuclear Power Plant (NPP), there was no metallic liner inside the internal containment. So the tightness of the containment in case of an accident is only assured by concrete. To limit leakage, cracking should be avoided and, so, tensile stresses should remain below the tensile strength of concrete. That is why the evolution of prestressing forces during time is critical for the NPP operation and why a good prediction of the evolution of delayed strains of the containment is needed.

To predict the evolution of prestressing forces with time, the delayed behavior of the constitutive materials (relaxation for steel and creep and shrinkage for concrete) has to be modelled precisely on the long term. In this paper only creep of concrete is considered and two problems are discussed: is the very long term creep asymptotic or not and how to deal with biaxial stresses?

IS THE VERY LONG TERM CREEP ASYMPTOTIC?

Laboratory tests

Classical creep tests in laboratory last generally less 2 or 3 years (and sometimes less). With such duration it is possible using these tests to fit as well an asymptotic or a logarithmic evolution of strains. There are nevertheless some basic creep tests performed by Brooks [START_REF] Brooks | 30-Year creep and shrinkage of concrete[END_REF] with a duration of 30 years, Troxell [START_REF] Troxell | Long-time creep and shrinkage tests of plain and reinforced concrete[END_REF] and Russell [START_REF] Russel | Thirteen Years of Deformations in Water Tower Place[END_REF] with a duration of 20 years. These tests performed on concretes with different water to cement ratio and with different aggregates types, show that basic and total creep are not asymptotic (figure 2). It should be emphasized that, the specimens are either tested under water [START_REF] Brooks | 30-Year creep and shrinkage of concrete[END_REF] or tested under autogeneous conditions [START_REF] Troxell | Long-time creep and shrinkage tests of plain and reinforced concrete[END_REF][START_REF] Russel | Thirteen Years of Deformations in Water Tower Place[END_REF]. Thus, it means in the latter case, it is not sure that the specimens are not undergoing some parasite drying.

Long term deformations of prestressed structures

The observation of the deflections of several bridges built by means of prestressed box girders indicates rather clearly that long term deflections are not asymptotic (Bazant et al., 2011) (Vitek and[START_REF] Vítek | Deflections and Strains of Prestressed Concrete Bridges[END_REF] [START_REF] Sellin | Delayed deformations of concrete structures: the Savines bridge and the Cheviré bridge[END_REF]. In the case of NPP, thanks to a large monitoring of the behavior of the containment, global and local deformations are measured. Figure 3 presents the evolution of the deformations of the central part of the containment of three French NPP. The long term behavior in the two directions of the prestressing is not asymptotic and seems in accordance with a logarithmic evolution with time. Note that in the global behavior of the structure, the effects of drying shrinkage and relaxation of the prestressing tendons are included in the delayed strains.

T NPP1 V NPP1 T NPP2 V NPP2 T NPP3 V NPP3 Rev. 11/2014

Consequences in codes

In the actual Eurocode 2 (EN1992-1-1, 2005), creep is calculated by means of an asymptotic hyperbolic function (equation 1) while in the recent Model Code 2010 creep is expressed as a combination of an asymptotic and a logarithmic function (equation 2) [START_REF] Muller | Concrete: treatment of types and properties in MC 2010[END_REF]. In the latter case the logarithmic function corresponds to basic creep while drying creep is asymptotic.

𝜀 = 𝜎 𝐸𝑐 𝑎 𝑓 1 (ℎ) ( 𝑡-𝑡0 𝑏 𝑔 1 (ℎ)+(𝑡-𝑡0) ) 𝑐 [eq. 1] 𝜀 = 𝜎 𝐸𝑐 [𝑎 𝑓 2 (ℎ) ( 𝑡-𝑡0 𝑏 𝑔 2 +(𝑡-𝑡0) ) 𝑐 + 𝑑 ln (1 + 𝑒(𝑡 -𝑡0))] [eq. 2]
where a, b, c, d and e are fitted constants. f1, f2, g1 and g2 are functions of the notional size h and the relative humidity RH. These functions are given by Eurocode 2 and MC2010.

Using these equations, it is possible to adjust the parameters on experimental results obtained in the laboratory by Granger [START_REF] Granger | Comportement différé du béton dans les enceintes de centrales nucléaires : analyse et modélisation[END_REF]. Figure 4 shows that it is possible to find a set of parameters that could fit the experimental results (at least on the long term which is the most important), for a sample with a diameter of 16cm stored in a relative humidity RH=50%, despite the fact that the used equations are very different.

The creep models being optimized on a sample, it is possible to predict the creep deformations of a larger structure. In our case, a structure with a notional size h=800mm and loaded under a constant uniaxial stress of 12MPa and a RH of 50% is considered. Figure 5 shows that in this case at very long term large differences occur despite a good calibration of the model on the same test. This is of course due to the differences in the nature of the equations (asymptotic or not) and it shows the importance of the physical manner to model the very long term behaviour of concrete structures. It indicates also that the observation of the behaviour of a sample, even during 3 years, is not enough to predict the correct behaviour of a structure during several decades.

Analysis of the behavior of containment vessels

In Figure 3, the evolution of the biaxial deformations during times in a NPP was presented. From these deformations one can express the evolutions of the average and the difference of the vertical and tangential strains. These evolutions are presented figures 6 for the same three NPPs [START_REF] Barré | Enhancement of the assessment of the concrete creep and shrinkage strains[END_REF]. It can be seen that the delayed strains in the containment wall follow two distinct kinetics:

-the difference between tangential and vertical strains, which exhibits a fast kinetics and becomes almost constant after 500 days, -the average of tangential and vertical strains, which corresponds to a slow kinetics. If vertical and tangential strain rates are equal despite different values for the prestressing, this means that in the long term biaxial creep deformations are only sensible to the mean stress and independent of deviatoric ones. If biaxial creep is expressed using a tangent Poisson ratio, this is equivalent to a Poisson ratio equal to -1. This is what is proposed actually in Eurocode 2 part 1 (French national annex) for drying creep (NF EN 1992-2, 2005), as for basic creep it is proposed to use a Poisson ratio equal to the elastic one. It could be explained for instance if one assumes that drying creep is only an amplification of drying creep due to the loading (as proposed in [START_REF] Sellier | Vers une modélisation simple et unifiée du fluage propre, du retrait et du fluage en dessiccation du béton[END_REF]. In fact, the behavior of the containment vessel is also affected by other phenomena like drying shrinkage and relaxation of the prestressing. So a more complete modelling is needed.

Modelling

To model the biaxial behavior of the containment the following assumptions are made:

 Free boundary conditions are assumed (no structural effect),  Boltzmann superposition principle is used,  MC2010 relations are used for drying shrinkage, basic and drying creep of concrete and for steel relaxation,  Kinetics and amplitude of drying shrinkage, basic and drying shrinkage could be adjusted by means of 6 parameters (see eq.2) to experimental results (laboratory tests or in-situ measurements),  Uniaxial relations for creep are extended to 3D by means of two constant Poisson ratio for basic and desiccation creepsee [START_REF] Vandamme | Difference Between Creep And Relaxation Poisson's Ratios: Theoretical And Practical Significance For Concrete Creep Testing[END_REF] for a discussion about Poisson ratio for creep,  The vertical prestressing corresponds to a vertical stress of 8,5MPa and a tangential stress of 12MPa. To limit the number of parameters when adjustments are made Poisson ratio is taken equal to 0,35 for basic creep [START_REF] Galenne | Prediction of Delayed Strain of Nuclear Containment Building: from laboratory tests to an industrial mock-up[END_REF] and -1 for drying creep (as recommended in the French national annex of Eurocode 2 part 1). Then, three scenarii are tested:

 1st scenario: MC2010 relations are directly used  2nd scenario: MC2010 relations are adjusted to laboratory data (like in figure 4)  3rd scenario: MC2010 relations are adjusted to in-situ measurements Figure 7 shows the comparison between experimental results and the modellings following the three different scenarii. One can see that:

(V+T)/2 NPP1 T-V NPP1 (V+T)/2 NPP2 T-V NPP2 (V+T)/2 NPP3 T-V NPP3 Rev. 11/2014
 MC2010 relations underestimate the delayed deformations (scenario 1),  with adjustment of the MC2010 relations for creep on laboratory tests, delayed strains are closer of the measured ones but still underestimated (scenario 2),  it is possible to adjust MC2010 relations to obtain a good agreement with insitu measurements (scenario 4). Figure 8 shows the results of the modelling following this scenario and measurements corresponding to figure 6 (difference between tangential and vertical strains). These results indicate that it is possible to predict the behavior of the containment on a reasonable period of time. It should be emphasized that due to the large number of unknown factors when using only measurements on structures (at least 10 parameters: Young modulus; elastic, basic creep and drying creep Poisson ratio; amplitude and kinetic correction factor related to drying shrinkage, basic and drying creep), it is possible to find by inverse analysis a large number of material parameters allowing for reproducing rather correctly the experimental measurements. The analysis of the possible sets of parameters is actually conducted. Besides, the vertical and tangential prestress is supposed to be applied instantaneously. In reality, the duration of the prestress application last approximatively 3 and 7 months for the tangential and vertical prestress, respectively. It explains partially that it is difficult with here to capture the beginning of the vertical strain evolution.

CONCLUSION

The main conclusions of this work are the following ones:

-Very long term creep in laboratory or of real structures seems to be nonasymptotic -Using a long test (3 years) in a laboratory is not enough to assess the long term behavior of a massive structure for several decades. We need physical relations for creep in codes in order to predict correctly the delayed behavior of massive structures! -Biaxial creep of nuclear power plant could be modelled using data of the structure itself. This would allow to predict the delayed behavior of these structures. Further work is needed to improve the prediction in the design phase.

In short-term perspective, effect of temperature will be investigated (the service temperature in a NPP is around 40°C), since it will increase delayed strains and prestress relaxation. In mid-term perspective, numerical simulations will be undertaken (Benboudjema and Torrenti, 2014;[START_REF] Hilaire | Modeling basic creep in concrete at early-age under compressive and tensile loading[END_REF], and compared in order to investigate the structural effects such as gradients of shrinkage and restraint (dome and raft foundation).

Figure 1 .

 1 Figure 1. Simplified diagram of the containment and its prestressing.

  Figure 2. Basic creep of concrete; [1]=(Brooks, 2005), [2]=[START_REF] Troxell | Long-time creep and shrinkage tests of plain and reinforced concrete[END_REF] and [3]=[START_REF] Russel | Thirteen Years of Deformations in Water Tower Place[END_REF]

Figure 3 .

 3 Figure 3. Evolution of the deformations in the central part of the containment of 3 NPP (V=vertical strains, T=tangential strains)

Figure 4 .Figure 5 .

 45 Figure 4. Comparison of experimental results [8] and models corresponding to equations 1 and 2. 12MPa, RH=50%, h=80mm.

Figure 6 .

 6 Figure 6. Evolutions of the tangential (T) and vertical (V) deformations in the central part of the containment of the three NPPs -same data as in figure 3.

Figure 7 .

 7 Figure 7. Comparison between in-situ measurements of tangential and vertical strains and modelling following the three scenarii

Figure 8 .

 8 Figure 8. Comparison between in-situ measurements of tangential and vertical strains and of the difference between theses strains and modelling following the scenario 3.

Table 1 . set of parameters for the three scenarii of figure 7.

 1

			Scenario 1	Scenario 2 Scenario 3
	Drying	Amplitude	1	0.95	2.4
	shrinkage	kinetics	1	1.05	0.95
	Basic	Amplitude	1	2.71	3.2
	creep	kinetics	1	0.063	0.73
	Drying	Amplitude	1	1.28	2.3
	creep	kinetics	1	0.43	1.93


K Olivier
email: kelly.olivier@ens-cachan.fr

Assistant Professor Aveline Darquennes
email: aveline.darquennes@ens-cachan.fr

F Benboudjema
email: farid.benboudjema@ens-cachan.fr

R Gagné

Gagné Effect Of Self-Healing

EFFECT OF SELF-HEALING ON THE CHLORIDE DIFFUSIVITY AT EARLY AGE

Keywords: Blast-furnace slag, Cracking, Diffusion, Migration test, Self-healing

published or not. The documents may come

INTRODUCTION

A major pathology affecting reinforced concrete structures is the corrosion of steel reinforcement due to carbonation and chloride diffusion. This deterioration is amplified when concrete cover is cracked [START_REF] Djerbi | Influence of traversing crack on chloride diffusion into concrete[END_REF][START_REF] Gérard | Influence of cracking on the diffusion properties of cementbased materials -Part I : Influence of continuous cracks on the steady-state regime[END_REF]. In marine environment, concrete made with Ground Granulated Blast-Furnace-Slag (GGBFS) cement is often used for civil engineering structures as it offers many advantages such as reclamation of industrial wastes, decreased CO 2 emissions produced during the clinkerization process in cement manufacturing (by limiting the clinker content in the cementitious material) and enhanced material properties such as workability and long-term compressive strength [START_REF] Dubovoy | Effects of Ground Granulated Blast-Furnace Slags on Some Properties of Pastes, Mortars and Concrete, Blended Cements[END_REF][START_REF] Itim | Compressive strength and shrinkage of mortar containing various amounts of mineral additions[END_REF]. Given the low heat release, this type of cementitious material is recommended for massive structures such as dams and bridges. Nevertheless, several civil engineering structures made with GGBFS cement presented cracking at early age due to restrained shrinkage [START_REF] Darquennes | Behaviour of slag cement concrete under restraint conditions[END_REF][START_REF] Briffaut | Analysis of cracking due to shrinkage restraint in a concrete tunnel[END_REF]. This cracking reduces the mechanical properties and leads to an increased penetration risk by aggressive agents such as CO 2 and chloride. Structures made with GGBFS cement showed however evidence of self-healing of cracks that seems to be related to the hydration of anhydrous particles at early age (GGBFS, clinker) [START_REF] Hearn | Self-sealing, autogenous healing and continued hydration : What is the difference ?[END_REF][START_REF] Ter Heide | Crack healing in hydrating concrete[END_REF] and/or calcite formation due to portlandite carbonation on the long term [START_REF] Clear | The effects of autogenous healing upon the leakage of water through cracks in concrete[END_REF][START_REF] Gagné | A study of the natural self-healing of mortars using air-flow measurements[END_REF].

In this study, experimental investigations are performed to analyze the effect of a traversing crack obtained by splitting tests on the chloride diffusion in steady-state conditions and the effect of selfhealing. Three mortar compositions characterized by different blast-furnace slag contents are studied at early age by means of an accelerated migration test under an electrical field.

EXPERIMENTATION PROCEDURE

Materials

Three mortar compositions are studied in this experimental campaign: a Portland cement mortar (CEMI 52.5 N CP2), a slag cement mortar (CEM III/A 52,5 L PM-ES-CP1) and a blended mortar with 50% Portland cement (CEMI 52.5 N CP2) and 50% ground granulated blast-furnace slag (GGBFS) (volumic substitution). They are referred hereafter as CEMI, CEMIIIA and 50GGBS respectively. The content of GGBFS for the slag cement CEM III/A is equal to 60% (mass percentage). The clinker and ground granulated blast-furnace slag are of the same origin. The chemical composition of these components is given in Table 1. Details on the mortar compositions are given in Table 2. Their water/binder (W/B) ratios are equal to 0.50 for the first two compositions and 0.52 for the third one. The paste volume, a parameter affecting the shrinkage evolution, is kept constant for all mortar compositions. The same normalized siliceous sand is used.

Testing procedure

The effective chloride diffusion coefficient was determined by chloride migration under accelerated and steady-state conditions. Cylindrical specimens (diameter Ø = 110 mm, height h = 220 mm) were made for all compositions. After mixing, the specimens were stored in a room at 23°C ± 1°C and 45% ± 5% relative humidity (R.H.). The molds were removed after 2 days and the specimens stored under water for 4 days in a room at 23°C ± 1°C. At the 6 th day, 3 cylindrical disks (h = 30 mm) were sawn from the specimens. Each disk was confined in a two-component resin reinforced with glass fibers and then cracked by means of a splitting test at 7 days. The crack width was measured with an optical microscope. One specimen was tested immediately; the others were stored for 14 or 21 days under tap water in individual boxes. Before the diffusion test, the specimens were saturated by hydroxidesodium solution (0.3 mol/l) for 24 hours. An uncracked specimen was tested for each composition at 7 days.

For the chloride diffusion testing, a mortar specimen was placed between two cells and an electrical potential difference of 10 V was maintained across the specimen ends (Figure 1). The upstream cell was filled with a sodium-chloride solution (0.7 mol/l). The anodic solution corresponds to seawater concentration. The downstream cell was filled with a sodium-hydroxide solution (0.3 mol/l). The cathode solution is close to the interstitial solution concentration. The test lasted 7 days. Temperature was measured in one cell with a thermocouple. The solutions in the cells were removed every 12 hours and the chloride ions content of the downstream cell was quantified by titration according to the Charpentier-Volhard method described hereafter. An important amount of silver nitrate (AgNO 3 ) is first added to the solution. The Ag ions react with the available chloride ions and the surplus is determined by adding potassium thyocyanate (KSCN) at the end the chemical reaction (a). The end of this reaction is determined with iron oxide (Fe 2 O 3 ), leading to the formation of a red iron thyocyanate precipitate (Fe(SCN) 3 ) subsequent to the chemical reaction (b). Note that nitric acid was initially added to avoid the formation of metallic hydroxide.

𝐴𝑔 + + 𝑆𝐶𝑁 -↔ 𝐴𝑔𝑆𝐶𝑁 (a) 𝐹𝑒 3+ + 3𝑆𝐶𝑁 -↔ 𝐹𝑒(𝑆𝐶𝑁) 3 (b)
During the accelerated migration test, the ions migration is primarily due to the electrical field. The diffusion coefficient is determined with Equation (1):

𝐷 𝑒𝑓𝑓 = 𝑅 * 𝑇 * ℎ * 𝐽 𝑐 𝑧 * 𝐹 * 𝑈 * 𝐶 𝑎 (1)
where R is the gas constant (8.314 J/mol K), T the temperature (K), h the specimen height (m), z the chloride valence (1), F the Faraday constant (96500 J/V mol), U the differential potential (V), C a the anodic solution concentration (mol/m 3 ) and J c the effective flux (mol/m 2 s 1 ). This last parameter is computed with Equation [START_REF] Gérard | Influence of cracking on the diffusion properties of cementbased materials -Part I : Influence of continuous cracks on the steady-state regime[END_REF].

𝐽 𝑐 = 𝑉 * ∆𝐶 𝐴 * ∆𝑡 ( 2
)
where V is the cell volume (l), ΔC the differential concentration (mol/l), A the exposed specimen area (m²), and Δt the time increment (s).

EXPERIMENTAL RESULTS AND DISCUSSION

Chloride concentration evolution and diffusion coefficient

The chloride concentration evolution in the downstream cell is presented in Figure 2. For all the specimens, initially the concentration slowly increases. During this first stage, chloride ions migrate through saturated pores in the mortar specimen but they do not yet reach the downstream cell. After 3 days, the concentration rate significantly increases and becomes constant (steady-state condition) leading to a constant chloride ions flux. The beginning of this second stage depends on the mortar porosity and the chloride binding capacity of the cementitious matrix [START_REF] Truc | A new way for determining the chloride diffusion coefficient in concrete from steady state diffusion test[END_REF]. These results are in agreement with [START_REF] Djerbi | Influence of traversing crack on chloride diffusion into concrete[END_REF][START_REF] Jang | Effect of crack width on chloride diffusion coefficients of concrete by steady-state migration tests[END_REF]. The difference on the behavior of the uncracked specimens is directly related to the paste matrix properties. The lower final chloride concentration (Figure 2) and the effective diffusion coefficient (Figure 3) of CEMIIIA is linked to its lower connected porosity [START_REF] Darquennes | Effect of autogenous deformation on the cracking risk of slag cement concretes[END_REF] related to the formation of additional C-S-H. The higher value for 50GGBFS can be linked to its slow hydration kinetics leading to a higher connected porosity at early age [START_REF] Darquennes | Long term deformations and cracking risk of concrete with high content of mineral additions[END_REF]. After cracking, the final value and the evolution rate of the chloride concentration (Table 3) in the downstream cell under steady-state conditions are significantly increased for all the compositions. They are directly proportional to the average specimen crack width. Nevertheless, self-healing strongly decreases the final value and the evolution rate of the chloride ions concentration, particularly after the first 14 th curing days. Moreover, these parameters are lower for the specimens containing blast-furnace slag (CEMIIIA and 50GGBFS) despite an initial wider crack. It seems therefore that these compositions are characterized by a greater self-healing capacity. Indeed, these trends reveal the presence of new products inside the cracks that narrow the crack widths and improve the chloride migration resistance of the material. This is confirmed by the average crack width evolution (Table 3). At 28 days, the evolution of chloride concentration for 50GGBFS is very close to that of the respective uncracked specimen. This last observation suggests that the self-healed 50GGBFS matrix is characterized by similar transfer properties as the respective uncracked matrix. The slope of the curve for the final linear segment (Figure 2) is used to calculate the effective diffusion coefficient of chloride (Figure 3) using equations 1 and 2. Similar trends with the diffusion coefficient are observed. The uncracked specimens exhibit the lowest diffusion coefficients (2.99 10 -12 m²/s, 2.43 10 -12 m²/s and 2.84 10 -12 m²/s for CEMI, CEMIIIA and 50GGBFS respectively). Despite a lower diffusion coefficient for uncraked mortar, CEMIIIA is characterized by a larger chloride diffusion (7.4 10 -12 m²/s) after cracking. This is directly related to its larger crack width (153 µm). A significant increase of this parameter is also observed for the other compositions. Indeed, the value of the diffusion coefficient of the cracked specimens increased by 115%, 148% and 147% for CEMI, CEMIIIA and 50GGBFS respectively. After twenty-one days under water curing, this parameter decreases by 37%, 40% and 49% for CEMI, CEMIIIA and 50GGBFS respectively. These observations confirm the positive role of self-healing on the chloride migration intensity and kinetics, particularly for mortars containing blast-furnace slag. These results are related to the reaction potential of blast-furnace slag. Indeed, the hydration degree of mortars containing blast-furnace slag is lower at 7 days [START_REF] Olivier | Impact de l'ajout de laitier de haut-fourneau sur l'auto-cicatrisation des matériaux cimentaires. 14 ème Edition des journées scientifiques du Regroupement francophone pour la recherche et la formation sur le béton (RF)²B[END_REF]. It should be noted that the kinetics of self-healing is more important during the first 14 days under water for all the compositions. Finally, the results obtained with the chloride diffusion test confirm the potential of this method to monitor self-healing as suggested by [23,24], particularly for cementitious materials with mineral additions.

Relation between crack size and diffusion coefficient

Figure 3 presents the effective diffusion coefficients ratio for the cracked and the uncracked specimens as function of the crack width. A linear behavior is observed for all the compositions. Similar observations are mentioned for ordinary and high-performance concretes [START_REF] Djerbi | Influence of traversing crack on chloride diffusion into concrete[END_REF]. This ratio is slightly higher for mortars with blast-furnace slag (CEMIIIA and 50GGBFS). This can be explained by the lower migration/matrix interaction of chloride ions in the perpendicular direction to the crack due to their denser cementitious matrix. This is in accordance with the lower diffusion coefficient for their respective uncracked specimen (Figure 2).

Comparison between the theoretical and experimental diffusion coefficients

To determine the effect of crack width on the chloride penetration, Equation 3 is used to estimate a theoretical diffusion coefficient (D theo ). In this relation, the crack is considered delimited by two parallel planes perpendicular to the specimen extremities [START_REF] Gérard | Influence of cracking on the diffusion properties of cementbased materials -Part I : Influence of continuous cracks on the steady-state regime[END_REF].

𝐷 𝑡ℎ𝑒𝑜 = 𝐴 𝑐 𝐷 𝑓 +𝐴 𝑢𝑛 𝐷 𝑢𝑛 𝐴 𝑐 +𝐴 𝑢𝑛 (3)
where A c is the area of the crack (m²), A un the uncracked area of the specimen (m²), D f and D un are the chloride ions diffusion coefficient in the free solution (2.032 10 -9 m²/s) and of the uncracked material (m²/s) respectively.

In order to compare the theoretical and the experimental results, the diffusion coefficients are divided by the measured diffusion coefficient of the uncracked material in Table 4. It appears that the theoretical relation leads to an over-or underestimation of the diffusion coefficient of the chloride ions. The absolute error is sometimes significant (50GGGFS at 28 days, for example). Simplifying the crack by opting for a parallel model without considering roughness and tortuosity can explain these differences.

CONCLUSIONS

This study focuses on the self-healing phenomenon in cementitious materials with different contents of blast-furnace slag. Chloride diffusion tests were used to better understand the effect of mineral additions on the kinetics of self-healing and to underline the benefit of GGBFS additions in accelerating and amplifying this phenomenon. The main results are summarized below: -For uncracked specimens, the effective chloride diffusion coefficient is lower for mortar with blast-furnace slag. -The chloride diffusion coefficient of cracked samples increases with increasing crack width.

These two parameters are characterized by a linear relation for all the studied mortar compositions. -The ratio of the diffusion coefficients for cracked and uncracked specimens is lower for the Portland cement mortar and it is less sensitive to the crack width. This can be explained by the more important interaction between the flow through the crack and the cementitious matrix (lower density). -The chloride diffusion test is an efficient method to monitor the self-healing phenomenon for cementitious materials with mineral additions. It confirms that self-healing has a positive role in terms of durability. -The self-healing kinetics of mortars stored under water evolves quickly during the first two weeks after cracking. The kinetics is faster for mortars with blast-furnace slag at early age, and it is directly related to the slow hydration degree evolution for this type of composition.

Figure 1 .

 1 Figure 1. Accelerated chloride diffusion testing.

Figure 2 .

 2 Figure 2. Chloride cumulative concentration for 7 days old uncracked (7D) and 7 days old freshlycracked (C-7D) specimens, and 21 days old (21D) and 28 days old (28D) self-healing specimens.

Figure 2 .

 2 Figure 2. Diffusion coefficients and crack widths for all compositions

Figure 3 .

 3 Figure 3. Evolution of the D/D 0 ratio as function of the crack width (D 0 = the effective diffusion coefficient of the uncracked specimen).

Table 1 .

 1 Chemical composition

	Oxides (%) CEMI CEMIIIA GGBFS
	CaO	65.9	49.9	42.7
	SiO 2	20.8	29.1	36.7
	Al 2 O 3	5.4	8.5	11.3
	MgO	1.1	5.0	7.0
	SO 3	3.4	2.67	0.2

Table 2 .

 2 Mortar

			composition (kg/m 3 )
		CEMI CEMIIIA 50GGBS
	Cement	563	563	281
	GGBFS	/	/	258
	Water		281
	Sand		1409

Table 3 .

 3 Width of the initial and healed crack -Chloride concentration rate in steady state

			CEMI		CEMIIIA		50GGBFS
		Initial	Crack	Concentration	Initial	Crack	Concentration	Initial	Crack	Concentration
		crack	width	rate	crack	width	rate	crack	width	rate
		width	after	(10 -3 mol/l*	width	after	(10 -3 mol/ l*	width	after	(10 -3 mol/ l*
		(µm)	healing	day)	(µm)	healing	day)	(µm)	healing	day)
			(µm)			(µm)			(µm)
	7D	/	/	75	/	/	61	/	/	71
	C7D	126	/	161	152	/	185	143	/	185
	21D	131	43	113	161	48	112	138	26	101
	28D	137	41	101	168	44	97	148	22	83

Table 4 .

 4 Comparison between theoretical (Dtheo) and experimental (Dexp) diffusion coefficients

			CEMI			CEMIIIA			50GGBFS
		D theo /D un D exp /D un Absolute	D theo /D un D exp /D un Absolute	D theo /D un D exp /D un Absolute
				error			error			error
	C7D	2.0	2.2	0.08	3.3	3.0	0.09	3.8	2.6	0.45
	21D	1.3	1.5	0.11	2.3	1.8	0.25	2.8	1.4	0.99
	28D	1.3	1.4	0.02	2.3	1.6	0.42	2.8	1.2	1.39

ACKNOWLEDGMENTS

The authors would like to thank Z. Djamai for his valuable contribution to the project.


Pierre Andreoletti

Sinai's walk : a statistical aspect

Keywords: random environment, random walk, Sinai's regime, non-parametric statistics

We consider Sinai's random walk in random environment. We prove that the logarithm of the local time is a good estimator of the random potential associated to the random environment. We give a constructive method allowing us to built the random environment from a single trajectory of the random walk.

Introduction and results

In this paper we are interested in Sinai's walk i.e., a one dimensional random walk in random environment with three conditions on the random environment: two necessaries hypothesis to get a recurrent process (see [Solomon(1975)]) which is not a simple random walk and an hypothesis of regularity which allows us to have a good control on the fluctuations of the random environment.

The asymptotic behavior of such walk has been understood by [Sinai(1982)] : this walk is subdiffusive and at an instant n it is localized in the neighborhood of a well defined point of the lattice. It is well known, see (Zeitouni [2001] for a survey) that this behavior is strongly dependent of the random environment or, equivalently, by the associated random potential defined Section 1.2.

The question we solve here is the following: given a single trajectory of a random walk (X l , 1 ≤ l ≤ n) where the time n is fixed, can we estimate the trajectory of the random potential where the walk lives ? Let us remark that the law of this potential is unknown as-well. In their paper, [Adelman and Enriquez(2004)] are interested in the question of the distribution of the random environment that could be deduced from a single trajectory of the walk, on the other hand, our purpose is to get an approximation of the trajectory of the random potential. In the paper [V. Baldazzi and Monasson(2006)] the authors are interested in a method to predict the sequence of DNA molecules. They model the unzipping of the molecule as a one-dimensional biased random walk for the fork position (number of open base pair) k in this landscape. The elementary opening (k → k + 1) and closing (k → k -1) transitions happen with a probability that depends on the unknown sequence. This probability of transition follows an Arrhénius law wich is closed to the one we discuss here. The question they answer is: given an unzipping signal can we predict the uniziping sequence ? Their approach is based on a Bayesian inference method which gives very good probabilities of prediction for a large amount of data. This means, in term of the walk, several trajectory on the same environment. Our approach is purely probailistic, it is based on good properties of the local time of the random walk which is the amount of time the walk spends on the points of the lattice. We treat a general case with a very few information on the random environment. We are able to reconstruct the random potential in a significant interval where the walk spends most of its time. Our proof is based on the results of [Andreoletti(2005)], in particular in a weak law of large number for the local time on the point of localization of the walk.

The largest part of this paper is devoted to the proof of a theoretical result (Theorem 1.7), we also present, at the end of the document, numerical simulations to illustrate our result. We give the main steps of the algorithm we use to rebuilt the random potential only by considering a trajectory of the walk. As an introduction we would like to comment one of these simulations: In blue we have represented the logarithm of the local time and in red the potential associated to the random environment. First, remark that we get a good approximation on a large neighborhood of the bottom of the valley around the coordinate -80. Outside this neighborhood and especially after the coordinate -20, the approximation is not precise at all. We will explain this phenomena by the fact that after the walk has reached the bottom of the valley, the walk will not return frequently to the points with coordinate larger than -20, so we lose information for this part of the latice.

Our method of estimation give us two crucial information: a confidence interval for the differencies of potential in sup-norm, on an observable set of sites "sufficiently" visited by the walk and a localization result for the bottom of the valley linked with the hitting time of the maximum of the local times. First we need to define the process:

1.1 Definition of Sinai's walk Let α = (α i , i ∈ Z) be a sequence of i.i.d. random variables taking values in (0, 1) defined on the probability space (Ω 1 , F 1 , Q), this sequence will be called random environment. A random walk in random environment (denoted R.W.R.E.) (X n , n ∈ N) is a sequence of random variable taking value in Z, defined on (Ω, F, P) such that • for every fixed environment α, (X n , n ∈ N) is a Markov chain with the following transition proba-bilities, for all n ≥ 1 and i ∈

Z P α [X n = i + 1|X n-1 = i] = α i , (1.1) P α [X n = i -1|X n-1 = i] = 1 -α i ≡ β i .
We denote (Ω 2 , F 2 , P α ) the probability space associated to this Markov chain.

• Ω = Ω 1 × Ω 2 , ∀A 1 ∈ F 1 and ∀A 2 ∈ F 2 , P [A 1 × A 2 ] = A 1 Q(dw 1 ) A 2 P α(w 1 ) (dw 2 ).
The probability measure P α [ .| X 0 = a] will be denoted P α a [.], the expectation associated to P α a : E α a , and the expectation associated to Q: E Q . Now we introduce the hypothesis we will use in all this work. The two following hypothesis are the necessaries hypothesis

E Q log 1 -α 0 α 0 = 0, (1.2) Var Q log 1 -α 0 α 0 ≡ σ 2 > 0.
(1.3) [Solomon(1975)] shows that under 1.2 the process (X n , n ∈ N) is P almost surely recurrent and 1.3 implies that the model is not reduced to the simple random walk. In addition to 1.2 and 1.3 we will consider the following hypothesis of regularity, there exists 0

< η 0 < 1/2 such that sup {x, Q [α 0 ≥ x] = 1} = sup {x, Q [α 0 ≤ 1 -x] = 1} ≥ η 0 . (1.4)
We call Sinai's random walk the random walk in random environment previously defined with the three hypothesis 1.2, 1.3 and 1.4. Let us define the local time L, at k (k ∈ Z) within the interval of time [1, T ] (T ∈ N * ) of (X n , n ∈ N)

L (k, T ) ≡ T i=1 I {X i =k} .
(1.5)

I is the indicator function (k and T can be deterministic or random variables). Let V ⊂ Z, we denote

L (V, T ) ≡ j∈V L (j, T ) = T i=1 j∈V I {X i =j} . (1.6)
To end, we define the following random variables

L * (n) = max k∈Z (L(k, n)) , F n = {k ∈ Z, L(k, n) = L * (n)} , (1.7) k * = inf{|k|, k ∈ F n } (1.8) L * (n)
is the maximum of the local times (for a given instant n), F n is the set of all the favourite sites and k * the smallest favorite site.

The random potential and the valleys

From the random environment we define what we will call random potential, Let

ǫ i ≡ log 1 -α i α i , i ∈ Z,
(1.9) define :

Definition 1.1. The random potential (S m , m ∈ Z) associated to the random environment α is defined in the following way:

S k = 1≤i≤k ǫ i , if k > 0, -k+1≤i≤0 ǫ i , if k < 0, S 0 = 0. Sj < 0 (S k , k) 0 (k ∈ Z) (S k -Sj) > 0 Sm > 0 k m j
Figure 2: Trajectory of the random potential Definition 1.2. We will say that the triplet {M ′ , m, M ′′ } is a valley if

S M ′ = max M ′ ≤t≤m S t ,
(1.10)

S M ′′ = max m≤t≤ M ′′ S t , (1.11) S m = min M ′ ≤t≤M ′′ S t .
(1.12)

If m is not unique we choose the one with the smallest absolute value.

Definition 1.3. We will call depth of the valley {M ′ , m, M ′′ } and we will denote it d([M ′ , M ′′ ]) the quantity min(S M ′ -S m , S M ′′ -S m ).

(1.13) Now we define the operation of refinement Definition 1.4. Let {M ′ , m, M ′′ } be a valley and let M 1 and m 1 be such that m ≤ M 1 < m 1 ≤ M ′′ and

S M 1 -S m 1 = max m≤t ′ ≤t ′′ ≤M ′′ (S t ′ -S t ′′ ).
(1.14)

We say that the couple (m 1 , M 1 ) is obtained by a right refinement of {M ′ , m, M ′′ }. If the couple (m 1 , M 1 ) is not unique, we will take the one such that m 1 and M 1 have the smallest absolute value.

In a similar way we define the left refinement operation. We denote log 2 = log log, in all this section we will suppose that n is large enough such that log 2 n is positive.

d([M ′ , m, M ′′ ]) = S M ′′ -Sm (k ∈ Z) 0 m M ′′ M ′ M1 m1 (S k , k)
Definition 1.5. Let n > 3, γ > 0, and Γ n ≡ log n + γ log 2 n, we say that a valley {M ′ , m, M ′′ } contains 0 and is of depth larger than Γ n if and only if

1. 0 ∈ [M ′ , M ′′ ], 2. d ([M ′ , M ′′ ]) ≥ Γ n , 3. if m < 0, S M ′′ -max m≤t≤0 (S t ) ≥ γ log 2 n , if m > 0, S M ′ -max 0≤t≤m (S t ) ≥ γ log 2 n .
The basic valley {M n ′ , m n , M n } We recall the notion of basic valley introduced by Sinai and denoted here {M n ′ , m n , M n }. The definition we give is inspired by the work of [Kesten(1986)]. First let {M ′ , m n , M ′′ } be the smallest valley that contains 0 and of depth larger than Γ n . Here smallest means that if we construct, with the operation of refinement, other valleys in {M ′ , m n , M ′′ } such valleys will not satisfy one of the properties of Definition 1.5. M n ′ and M n are defined from m n in the following way: if m n > 0

M n ′ = sup l ∈ Z -, l < m n , S l -S mn ≥ Γ n , S l -max 0≤k≤mn S k ≥ γ log 2 n , (1.15) M n = inf {l ∈ Z + , l > m n , S l -S mn ≥ Γ n } . (1.16) if m n < 0 M n ′ = sup {l ∈ Z -, l < m n , S l -S mn ≥ Γ n } , (1.17) M n = inf l ∈ Z + , l > m n , S l -S mn ≥ Γ n , S l -max mn≤k≤0 S k ≥ γ log 2 n . (1.18) if m n = 0 M n ′ = sup {l ∈ Z -, l < 0, S l -S mn ≥ Γ n } , (1.19) M n = inf {l ∈ Z + , l > 0, S l -S mn ≥ Γ n } . (1.20)
{M n ′ , m n , M n } exists with a Q probability as close to one as we need. In fact it is not difficult to prove the following lemma

Mn (k ∈ Z) (S k , k) mn 0 d([M ′ n , mn, Mn]) ≥ Γn ≥ γ log 2 n M ′ n Figure 4: Basic valley, case m n > 0
Lemma 1.6. There exists c > 0 such that if 1.2, 1.3 and 1.4 hold, for all γ > 0 and n we have

Q {M n ′ , m n , M n } = ∅ = 1 - cγ log 2 n log n .
(1.21)

Proof.

One can find the proof of this Lemma in Section 5.2 of [Andreoletti(2006)].

Main results

We start with some definitions that will be used all along this work. Let x ∈ Z, define

T x = inf{k ∈ N * , X k = x} +∞, if such k does not exist. (1.22)
Let n > 1, k ∈ Z, and c 0 > 0, define:

S n k,mn = 1 - 1 log n (S k -S mn ), (1.23) Ŝn k = log(L(k, n)) log n , (1.24) u n = c 0 log 3 n log n . (1.25)
S n k,mn is the function of the potential we want to estimate, Ŝn k is the estimator and u n is an error function. Now let us define the following random sub-set of Z:

L γ n =    l ∈ Z, n j=T k * I X j =l ≥ (log n) γ    , (1.26)
recall that γ > 0. This set L γ n is fundamental for our result, we notice that it depends only on the trajectory of the walk and more especially of its local time: L γ n is the set of points for which we are able to give an estimator of the the random potential. We will see that this set is large and contains a great amount of the points visited by the walk (see Proposition 1.9). We recall that T k * is the first time the walk hit a favorite site. In words, l ∈ L γ n , if and only if: The local time of the random walker in l after the instant T * k is large enough (larger than (log n) γ ). Our main result is the following: Theorem 1.7. Assume 1.2, 1.3 and 1.4 hold, there exists three constants c 0 , c 1 , c 2 and c ′ 2 such that for all γ > 6, there exists n 0 such that for all n > n 0 there exists

G n ⊂ Ω 1 with Q [G n ] ≥ 1 -φ 1 (n) and inf α∈Gn P α   k∈L γ n Ŝn k -S n k,mn < u n   ≥ 1 -φ 2 (n). (1.27)
where

φ 1 (n) = c 1 γ log 2 n log n , (1.28) φ 2 (n) = c 2 (log n) γ/2 + c ′ 2 (log n) γ-6 .
(1.29)

The fact that our result depends on m n seems to be restrictive, we would like to know where is the bottom of the valley only by considering the local time of the walk, we prove the following:

Proposition 1.8. Assume 1.2, 1.3 and 1.4 hold, there exists a constant c 3 > 0 such that for all γ > 6, there exists n 0 such that for all n > n 0 there exists

G n ⊂ Ω 1 with Q [G n ] ≥ 1 -φ 1 (n) and inf α∈Gn P α 0 max x∈Fn |m n -x| ≤ (log 2 n) 2 ≥ 1 -φ 3 (n), (1.30) inf α∈Gn P α 0 |T mn -T k * | ≤ (log n) 3 ≥ 1 -φ 3 (n), (1.31) where φ 3 (n) = c 3 /(log n) γ-6 .
Notice that the distance between m n (coordinate of the point visited by the walk where the minimum of the potential is reached) and a favorite site is negligible comparing to a typical fluctuation of the walk (of order (log n) 2 ). Thanks to Proposition 1.8 we can replace 1.27 in Theorem 1.7 by

inf α∈Gn P α   k∈L γ n Ŝn k -S n k,k * < u n   ≥ 1 -φ 2 (n). (1.32)
Now let us give a result giving the main properties of L γ n .

Proposition 1.9. Assume 1.2 and 1.3 and 1.4 hold, there exists a constant c 3 > 0 such that for all γ > 6, there exists n 0 such that for all n > n 0 there exists

G n ⊂ Ω 1 with Q [G n ] ≥ 1 -φ 1 (n), inf α∈Gn P α 0 [L(L γ n , n) = n(1 -o(1))] ≥ 1 -φ 2 (n), (1.33) inf α∈Gn P α 0 |L γ n | ≈ (log n) 2 ≥ 1 -φ 2 (n), (1.34) (1.35)
where lim n→+∞ o(1) = 0.

Remark 1.10. By definition we have

F n ⊆ L γ n .
Theorem 1.7 is known to be the quenched result that means for a fixed environment α, a simple consequence (see Remark 2.4) is the following annealed result:

Corollary 1.11. Assume 1.2, 1.3 and 1.4 hold, there exists three constants c 0 , c 1 and c 2 such that for all γ > 6, there exists n 0 such that for all n > n 0

P   k∈L γ n Ŝk k -S n k,k * < u n   ≥ 1 -φ(n), (1.36) where φ(n) = φ 1 (n) + φ 2 (n).
We would like to notice, that for our purpose the result above is not very interesting, because the aim is to reconstruct one environment whereas the result above give the mean of the probability for the walk over all the possible environments. This paper is organized as follows. In Section 2 we give the proof of Theorems 1.7 (we easily get the corrolary from Remark 2.4), we have split this proof into two parts, the first one deals with the random environment and the other one with the random walk itself. In section 3 we sketch the proofs of Propositions 1.8 and 1.9. In Section 4, as an application of our result, we present an algorithm and some numerical simulations. For completeness, we recall in the appendix, some basic facts on birth and death processes.

2 Proof of Theorem 1.7

The proof of a result with a random environment involves both arguments and properties for the random environment and arguments for the random walk itself. I will start to give the properties I need for the random environment. Then we will use it to get the result for the walk.

Properties needed for the random environment

2.1.1 Construction of (G n , n ∈ N)
Let k and l be in Z, define

E α k (l) = E α k [L(l, T k )] (2.1)
in the same way, let A ⊂ Z, define andω ∈ Ω 1 , we will say that α ≡ α(ω) is a good environment if there exists n 0 such that for all n ≥ n 0 the sequence (α i , i ∈ Z) = (α i (ω), i ∈ Z) satisfies the properties 2.3 to 2.5

E α k (A) = l∈A E α k [L(l, T k )] . (2.2) Definition 2.1. Let d 0 > 0, d 1 > 0,
• {M n ′ , m n , M n } = ∅, (2.3) • M n ′ ≥ -d 0 (σ -1 log 2 n log n) 2 , M n ≤ d 0 (σ -1 log 2 n log n) 2 , (2.4) • E α mn (W n ) ≤ d 1 (log 2 n) 2 , (2.5)
where

W n = {M ′ n , M ′ n + 1, • • • , m n , • • • , M n }.
Remark 2.2. We will see in Section 2 that we use some results of [Andreoletti(2006)]. Considering this, we need extra properties on the random environment in addition to the three mentioned above, but as we don't need them for our computations we do not make them appear.

Define the set of good environments

G n ≡ G n (d 0 , d 1 ) = {ω ∈ Ω 1 , α(ω) is a good environment} . (2.6)
G n depends on d 0 , d 1 and n, however we only make explicit the n dependence.

Proposition 2.3. There exists two constants d 0 > 0 and d 1 > 0 such that if 1.2, 1.3 and 1.4 hold, there exists n 0 such that for n > n 0

Q [G n ] ≥ 1 -φ 1 (n), (2.7)
where φ 1 (n) is given by 1.28.

Proof.

We can find the proof for the first three properties 2.3-2.5 in [Andreoletti(2006)], see Definition 4.1 and Proposition 4.2.

To end the section we would like to make the following elementary remark on the decomposition of P:

Remark 2.4. Let C n ∈ σ (X i , i ≤ n) and G n ⊂ Ω 1 , we have :

P [C n ] ≡ Ω 1 Q(dω) Cn dP α(ω) (2.8) ≥ Gn Q(dω) Cn dP α(ω) .
(2.9)

So assume that Q[G n ] ≡ e 1 (n) ≥ 1 -φ 1 (n) and assume that for all ω ∈ G n , Cn dP α(ω) ≡ e 2 (ω, n) ≥ 1 -φ 2 (n) we get that P [C n ] ≥ e 1 (n) × min w∈Gn (e 2 (w, n)) ≥ 1 -φ 1 (n) -φ 2 (n).
(2.10)

Arguments for the walk

Let (ρ 1 (n), n ∈ N) a strictly positive decreasing sequence such that lim n→∞ ρ 1 (n) = 0. First let us show that the Theorem 1.7 is a simple consequence of the following Proposition 2.5. Assume 1.2, 1.3 and 1.4 hold, there exists n 0 such that for all n > n 0 there exists

G n ⊂ Ω 1 with Q [G n ] ≥ 1 -φ 1 (n) and sup α∈Gn    P α 0   k∈L γ n L(k, n) n - E α mn (k) E α mn (W n ) ≤ w k,n      ≥ 1 -φ 2 (n) (2.11)
where w k,n = ρ 1 (n)

E α mn (k) E α mn (Wn) , φ 1 (n) and φ 2 (n) are given just after 1.27 Taking the logarithm and for n large enough, using Taylor series expansion, we remark that

E α mn (k) E α mn (W n ) (1 -ρ 1 (n)) ≤ L(k, n) n ≤ E α mn (k) E α mn (W n ) (1 + ρ 1 (n)) (2.12) implies -2ρ 1 (n) -log(E α mn (W n )) ≤ log L(k, n) -log n -log(E α mn (k)) ≤ -log(E α mn (W n )) + ρ 1 (n),
rearranging the terms and using A.1 (see the Appendix) we get

1 log n (R α n (k) -2ρ 1 (n)) ≤ Ŝn k -S n k,mn ≤ 1 log n (R α n (k) -ρ 1 (n)) (2.13)
where R α n (k) = log αm n β k a k,mn -log(E α mn (W n )) and a k,mn is given by A.2. Now using A.4 and Property 2.5 we get the Theorem. The proof of Proposition 2.5 is based on the following results (Lemma 2.6) of [Andreoletti(2005)],

Known facts

Let (ρ(n), n ∈ N) be a positive decreasing sequence such that lim n→∞ ρ(n) = 0, we define

A 1 = L(m n , n) n - 1 E α mn (W n ) > ρ(n) E α mn (W n ) , (2.14) A 2 = T mn ≤ n/(log n) 4 , L(W n , n) = 1 . (2.15)
Lemma 2.6. Assume 1.2, 1.3 and 1.4 hold, there exists a constant b 1 > 0 such that for all γ > 6, there exists n 0 such that for all n > n 0 there exists

G n ⊂ Ω 1 with Q [G n ] ≥ 1 -φ 1 (n) and sup α∈G ′ n {P α 0 [A 1 ]} ≤ r 1 (n), (2.16)
where r 1 (n) = b 1 /(log n) γ-6 .

Proof.

We do not give the details of the computations because the reader can find it in the referenced paper (Theorem 3.8 of [Andreoletti(2006)]), just notice that comparing to the Theorem 3.8 we have a better rate of convergence for the probability obtained just by using a weaker result for the concentration of the walk.

We will also need the following elementary fact :

Lemma 2.7. Assume 1.2, 1.3 and 1.4 hold, there exists a constant b 2 > 0 such that for all γ > 2, there exists n 0 such that for all n > n 0 there exists

G n ⊂ Ω 1 with Q [G n ] ≥ 1 -φ 1 (n) and sup α∈G ′ n {P α 0 [A 2 ]} ≤ r 2 (n), (2.17) where r 2 (n) = b 2 /(log n) γ-2 .
Proof.

Once again this can be find in [Andreoletti(2006)]: Proposition 4.7 and Lemma 4.8.

Using these results we can give the proof of Proposition 2.5 into two steps :

Step 1

Let us define the following subsets :

vn 1 ≡ {M ′ n ≤ k ≤ m n -1, ( max k≤j≤mn S j -S mn ) < log n - γ 2 log 2 n}, (2.18) vn 2 ≡ {m n + 1 ≤ k ≤ M n , ( max mn≤j≤k S j -S mn ) < log n - γ 2 log 2 n}, (2.19)
and

V γ n = vn 1 ∩ vn 2 .
(2.20)

In words V γ n is a subset of points included in W n , such that for all k ∈ V γ n the largest difference of potential between m n and k is smaller than log n -γ/2 log 2 n. For the walk we will see (Lemma below) that if k ∈ V γ n then the walk will hit k after it has reached m n and it will hit this point k a number of time large enough (see figure 5).

First let us prove the following Lemma :

Lemma 2.8. Assume 1.2, 1.3 and 1.4 hold, there exists a constant b 3 > 0 such that for all γ > 6, there exists n 0 such that for all n > n 0 there exists

G n ⊂ Ω 1 with Q [G n ] ≥ 1 -φ 1 (n) sup α∈G ′ n {P α 0 [L γ n ⊆ V γ n ]} ≥ 1 -r 3 (n), (2.21) where r 3 (n) = b 3 /(log n) γ/2 .
Notice that L γ n is a P random variable (with two levels of randomness) whereas V γ n is only a Q random variable (with one level of randomness), this Lemma makes the link between a trajectory of the walk and the random environment.

D γ n (k ∈ Z) mn 0 M ′ n Mn log n -γ/2 log 2 n (S k , k) V γ n Figure 5: V γ n with m n > 0, case 1: D γ n ≡ max k∈v 1 max k≤j≤mn (S j -S mn ) ≤ log n -γ log 2 n
Proof.

To prove this Lemma we use Proposition 1.8. First notice that

P α 0 [L γ n ⊆ V γ n ] = 1 -P α 0   k∈(v 1 n ∪v 2 n ) {k ∈ L γ n }   (2.22)
where

v n 1 ≡ {M ′ n ≤ k ≤ m n -1, ( max k≤j≤mn S j -S mn ) ≥ log n - γ 2 log 2 n}, (2.23) v n 2 ≡ {m n + 1 ≤ k ≤ M n , ( max mn≤j≤k S j -S mn ) ≥ log n - γ 2 log 2 n}.
(2.24)

Let k ∈ v n 1 let us give an upper bound for

P α 0 k ∈ L γ n , |T k * -T mn | ≤ (log n) 3 ≤ P α 0   n j=T k * I X j =k ≥ (log n) γ , |T k * -T mn | ≤ (log n) 3   ≤ P α 0   n j=Tm n I X j =k ≥ (log n) γ -(log n) 3   ≤ P α mn   Tm n,n j=1 I X j =k ≥ (log n) γ -(log n) 3   , (2.25)
for the third inequality we have used the strong Markov property, where

T mn,j ≡ inf{k > T mn,j-1 , X k = m n }, j ≥ 2 +∞, if such k does not exist. T mn,1 ≡ T mn (see 1.22).
Now using the Markov inequality and Lemma A.1 we get

P α 0 k ∈ L γ n , |T k * -T mn | ≤ (log n) 3 ≤ nE α mn [L(k, T mn )] (log n) γ -(log n) 3 (2.26) ≤ n η 0 exp(S k -S mn )((log n) γ -(log n) 3 ) (2.27) ≤ 1 η 0 (log n) γ/2 (1 -(log n) 3 /(log n) γ ) , (2.28)
notice that in the last inequality we have used the fact that k ∈ v n 1 . A similar computation give the same inequality when k ∈ v n 2 . Collecting what we did above, and using the Property 2.4 together with 1.31 yields the Lemma.

Step 2

This second step is devoted to the proof of the following Lemma.

Lemma 2.9. For all α and n we have

P α 0 L(k, n) n - E α mn (k) E α mn (W n ) > w k,n , A 1 , A 2 ≤ 2 exp(-n/2ψ α 2 (n)) (2.29) recall that w k,n = ρ 1 (n) E α mn (k) E α mn (Wn) and ψ α 2 (n) = 2 (ρ 1 (n)-ρ(n)) 2 1+ρ(n) (αm n ∧βm n ) |k-mn| exp(-(S M k -Sm n )) E α mn (Wn) . M k is such that S M k = max mn+1≤j≤k S j if k > m n and conversly if k < m n S M k = max k≤j≤mn-1 S j .

Proof.

We essentially use an inequality of concentration (see [Ledoux(2001)]), for simplicity we only give the proof for k > m n , the other case (k ≤ m n ) is very similar. Using the Markov property and the fact that L(k, T mn ) = 0, we get

P α 0 L(k, n) n - E α mn (k) E α mn (W n ) > w k,n , A 1 , A 2 ≤ P α mn L(k, n) n - E α mn (k) E α mn (W n ) > w k,n , A 1 . (2.30)
We have

P α mn L(k, n) n - E α mn (k) E α mn (W n ) > w k,n , A 1 ≤ P α mn L(k, n) n - E α mn (k) E α mn (W n ) > w k,n , L(m n , n) n - 1 E α mn (W n ) ≤ ρ(n) E α mn (W n ) (2.31) ≤ P α mn L(k, T mn,n 1 ) n - E α mn (k) E α mn (W n ) > w k,n (2.32)
≡ P α mn L(k, T mn,n 1 ) n - E α mn (k) E α mn (W n ) (1 + ρ(n)) > w ′ k,n (2.33)
where

n 1 = n E α mn (Wn) (1 + ρ(n))
, notice that n 1 is not necessarily an integer but for simplicity we disregard that, and

w ′ k,n = E α mn (k) E α mn (Wn) (ρ 1 (n) -ρ(n)).
The strong Markov property implies that L(k, T mn,n 1 ) is a sum of n 1 i.i.d. random variables, the inequality of concentration gives

P α mn L(k, T mn,n 1 ) n - E α mn (k) E α mn (W n ) > w ′ k,n , A 1 ≤ exp - n 2 E α mn (W n ) Var mn (L(k, T mn )) (w ′ k,n ) 2 1 + ρ(n) .(2.34)
With the same method we also get

P α mn L(k, T mn,n 1 ) n - E α mn (k) E α mn (W n ) < -w ′ k,n , A 1 ≤ exp - n 2 E α mn (W n ) Var mn (L(k, T mn )) (w ′ k,n ) 2 1 + ρ(n) .
(2.35)

Using A.3 we get Lemma 2.9.

End of the proof of the Theorem

Using Lemmata 2.6, 2.7 and 2.8 we have:

P α 0   k∈L a n L(k, n) n - E α mn (k) E α mn (W n ) > w k,n   ≤ |V γ n | sup k∈V γ n P α 0 L(k, n) n - E α mn (k) E α mn (W n ) > w k,n , A 1 , A 2 + 3 max 1≤i≤3 {r i (n)},
then using Lemma 2.9 we get sup

k∈V γ n P α 0 L(k, n) n - E α mn (k) E α mn (W n ) > w k,n , A 1 , A 2 ≤ 2 sup k∈V γ n exp(-n/2ψ α 2 (k, n)) ≤ 2 exp(-(log n) γ/2-2 /(ρ 1 (n) log 2 n)), (2.36)
where the last inequality comes from the definition of V γ n (see 2.20) and the Properties 2.4 and 2.5. To end we use again the Property 2.4 together with the definition of V γ n .

3 Proof of Proposition 1.8 and 1.9

Sketch of the proof of Proposition 1.8 1.30 is an improvement of the proof of Corollary 3.17 of [Andreoletti(2006)] in order to get a better rate of convergence for the probability. To get 1.31, we have used the same idea of the proof of Corollary 3.17 of [Andreoletti(2006)], so once again we will not repeat the computations here. We recall just the intuitive idea: once the walk has reached k * , we know from 1.30 that m n is at most at a distance (log 2 n) 2 , therefore the walk need at most the amount of time exp( ((log 2 n) 2 )) = (log n) to reach m n . We take (log n) 3 to get a better rate of convergence for the probability. Sketch of the proof of Proposition 1.9 The first two properties can be deduced from the following inequality, let ǫ > 1, for all n large enough and all α ∈ G n :

P α 0 V 2(γ+ǫ) n ⊆ L γ n ≥ φ 3 (n) + r 1 (n) + cte(log t) 2 exp -(log n) γ+ǫ-2 (1 -(log n) 1-ǫ ) . (3.1)
Indeed,thanks to 3.1 we have

P α [L(L γ n , n) ≥ n(1 -o(1))] ≥ P α L(V 2(γ+ǫ) n , n) ≥ n(1 -o(1)) (3.2)
we get 1.33 by using the same method [Andreoletti(2006)] uses to get Theorem 3.1. To get 1.34, we only need to show that |V γ+ǫ n | ≈ (log t) 2 , which is a basic fact for a simple random walk. Now, to get 3.1, first we notice that, by using a similar method of the proof of Theorem 1.7 we can get

P α 0 V 2(γ+ǫ) n L γ n ≤ |V γ+ǫ n | max k∈V 2(γ+ǫ) n P α mn   n 1 j=1 η k j < (log n) γ   + φ 3 (n) + r 1 (n). (3.3)
where (η k j , j) is a i.i.d. sequence with the law of L(k, T mn ). Then using an inequality of concentration, we get 3.1.

Algorithm and Numerical simulations 4.1 General and recall of the main definitions

First notice that we have no criteria to determine wether or not we can apply this method to an unknown series of data. All we know is that it works for Sinai's walk, however we can apply the following algorithm to every process. Let us recall the basic random variables that will be used for our simulations, let x ∈ Z, n ∈ N,

T x = inf{k ∈ N * , X k = x} +∞, if such k does not exist. , (4.1) L (x, n) ≡ n i=1 I {X i =x} , (4.2)
L * (n) = max k∈Z (L(k, n)) , F n = {k ∈ Z, L(k, n) = L * (n)} , (4.3)
k * = inf{|k|, k ∈ F n }. (4.4) (4.5)
We recall also the set L γ n , the function of the potential we want to estimate and its estimator:

L γ n =    k ∈ Z, n j=T k * I X j =k ≥ (log n) γ    , (4.6) S n k,mn = 1 - 1 log n (S k -S mn ), (4.7) Ŝn k = log(L(k, n)) log n , (4.8)
We also recall that thanks to Proposition 1.8, in probability we have

|m n -k * | ≤ cte(log 2 n) 2 .

Main steps of the algorithm

Step 1: We have to determine L γ n and to get it we have to compute T k * and therefore the local time of the process. First we compute L(k, n) for every k, notice that L(k, n) is not equal to zero only if k has been visited by the walk within the interval of time [1, n]. Then we can compute L * (n) and determine k * and T k * . Notice that T k * is not a stopping time, therefore we need two passes to compute what we need. We are now able to determine L γ n computing n j=T k * I X j =k .

Step 2: We can check that L γ n is connex, contains k * and that its size is of the order of a typical fluctuation of the walk. Now, keeping only the k that belongs to L γ n we compute for those k: Ŝn

k = log(L(k,n)) log n
the estimator of the potential. We localize the bottom of the valley m n using k * .

Simulations

For the first simulation (Figure 6) we show a case where L γ n is large i.e. L γ n contains most of the points visited by the walk. The trajectory of the random potential is in red the interval of confidence in x,mn , in blue Ŝn x -u n , in green Ŝn x + u n blue and green. We took n = 500000 and γ = 7, notice that the larger is γ, the smaller is L γ n but better is the rate of convergence of the probability. We get that L γ n = [10,94]. In Figure 7 we plot the difference S n

x,mn -Ŝn x and its the linear regression. We notice that the slope of the linear regression is -5 . We also notice that we have taken n = 500000, so the error function u n ≈ log 3 n log n ≈ 0, 7 this match with the max x (S n x,mn -Ŝn x ) ≈ 0.8 for this simulation. Now let us choose another example where L γ n is much more smaller. For the following simulation (Figure 8) we have only changed the sequence of random number. We get that L γ n = [-150, -85]. We x,mn , in blue Ŝn x -u n , in green Ŝn x + u n notice that for the coordinates larger than -85 and especially after -40, our estimator is not good at all. In fact once the walk has reached the minimum of the valley (coordinate -111) it will never reach again one of the points of coordinate larger than -40 before n = 500000, so our estimator can not say anything about the difference S n x,mn -Ŝn x . However if we look in the past of the walk and especially at a the time T k * which is the first time it has reached the coordinate -111, the favorite point for this time is localized around the point -2, so a good estimator between the coordinate -40 and 10 may be given by ( log(L(k,T * )) log T * , k). The difference S n

x,mn -Ŝn x and the linear regression in the interval L γ n = [-150, -85] is presented Figure 9. A Basic results for birth and death processes

For completeness we recall an explicit expression for the mean and an upper bound for the variance of the local times at a certain stopping time, we can be found a proof of these elementary facts in [Révész(1989)] (page 279)

Figure 1 :

 1 Figure 1: The logarithm of the local time (in blue) and the random potential (in red)

Figure 3 :

 3 Figure 3: Depth of a valley and refinement operation

  potential using the local time

Figure 6 :

 6 Figure 6: in red S n x,mn , in blue Ŝn x -u n , in green Ŝn x + u n

Figure 7 :

 7 Figure 7: in magenta S n x,mn -Ŝn x , in red the linear regression

  Figure 8: in red S n x,mn , in blue Ŝn x -u n , in green Ŝn x + u n

Figure 9 :

 9 Figure 9: in magenta S n x,mn -Ŝn x , in red the linear regression

Lemma A. 1 .

 1 For all α, Let k > m n E α mn [L(k, T mn )Var mn [L(k, T mn )] ≤ 2(E α mn [L(k, T mn )]) 2 e S M k -Sm n β k |k -m n |. (A.3) M k is such that S M k = max mn+1≤j≤k-1 S j .For Q-a.a. is true for k < m n and E α mn [L(m n , T mn )] = 1.

Acknowledgements I would like to thank Nicolas Klutchnikoff for stimulating discussions.